{'fraction_non_alphanumeric': 0.03284491060042162, 'fraction_numerical': 0.008282799364963693, 'mean_word_length': 4.307628552681562, 'pattern_counts': {'":': 1, '<': 7, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A New Look at the Position Operator in Quantum Theory

The postulate that coordinate and momentum representations are related to each other by the Fourier transform has been accepted from the beginning of quantum theory by analogy with classical electrodynamics. As a consequence, an inevitable effect in standard theory is the wave packet spreading (WPS) of the photon coordinate wave function in directions perpendicular to the photon momentum. This leads to several paradoxes. The most striking of them is that coordinate wave functions of photons emitted by stars have cosmic sizes and strong arguments indicate that this contradicts observational data. We argue that the above postulate is based neither on strong theoretical arguments nor on experimental data and propose a new consistent definition of the position operator. Then WPS in directions perpendicular to the particle momentum is absent and the paradoxes are resolved. Different components of the new position operator do not commute with each other and, as a consequence, there is no wave function in coordinate representation. Implications of the results for entanglement, quantum locality and the problem of time in quantum theory are discussed.

1 Why do we need position operator in quantum theory?
It has been postulated from the beginning of quantum theory that the coordinate and momentum representations of wave functions are related to each other by the Fourier transform. The historical reason was that in classical electrodynamics the coordinate and wave vector k representations are related analogously and we postulate that p =hk where p is the particle momentum. Then, although the interpretations of classical fields on one hand and wave functions on the other are fully different, from mathematical point of view classical electrodynamics and quantum mechanics have much in common (and such a situation does not seem to be natural). Similarity of classical electrodynamics and quantum theory is reflected even in the terminology of the latter. The terms "wave function", "particle-wave duality" and "de Broglie wave length" have arisen at the beginning of quantum era in efforts to explain quantum behavior in terms of classical waves but now it is clear that no such explanation exists. The notion of wave is purely classical; it has a physical meaning only as a way of describing systems of many particles by their mean characteristics. In particular, such notions as frequency and wave length can be applied only to classical waves, i.e. to systems consisting of many particles. If a particle state vector contains exp[i(pr − Et)/h], where E is the energy, then by analogy with the theory of classical waves one might say that the particle is a wave with the frequency ω = E/h and the (de Broglie) wave length λ = 2πh/|p|. However, such defined quantities ω and λ are not real frequencies and wave lengths measured on macroscopic level. A striking example showing that on quantum level λ does not have a usual meaning is that from the point of view of classical theory an electron having the size of the order of the Bohr radius cannot emit a wave with λ = 21cm (this observation has been pointed out to me by Volodya Netchitailo).
In quantum theory the photon and other particles are characterized by their energies, momenta and other quantities for which there exist well defined operators while the notion of coordinates on quantum level is a problem which is investigated in the present paper. The term "wave function" might be misleading since in quantum theory it defines not amplitudes of waves but only amplitudes of probabilities. So, although in our opinion the term "state vector" is more pertinent than "wave function" we will use the latter in accordance with the usual terminology, and the phrase that a photon has a frequency ω and the wave length λ will be understood only such that ω = E/h and λ = 2πh/|p|.
One of the examples of the above similarity follows. As explained in textbooks on quantum mechanics (see e.g. Ref. [1]), if the coordinate wave function ψ(r, t) contains a rapidly oscillating factor exp[iS(r, t)/h], where S(r, t) is the classical action as a function of coordinates and time, then in the formal limith → 0 the Schrödinger equation becomes the Hamilton-Jacoby equation which shows that quantum mechanical wave packets are moving along classical trajectories. This situation is called semiclassical approximation and it is analogous to the approximation of geometrical optics in classical electrodynamics (see e.g. Ref. [2]) when the fields contain a rapidly oscillating factor exp[iϕ(r, t)] where the function ϕ(r, t) is called eikonal. It satisfies the eikonal equation which coincides with the relativistic Hamilton-Jacobi equation for a particle with zero mass. This shows that classical electromagnetic wave packets are moving along classical trajectories for particles with zero mass what is reasonable since it is assumed that such packets consist of photons.
Another example follows. In classical electrodynamics a wave packet mov-ing even in empty space inevitably spreads out and this fact has been known for a long time. For example, as pointed out by Schrödinger (see pp. 41-44 in Ref. [3]), in standard quantum mechanics a packet does not spread out if a particle is moving in a harmonic oscillator potential in contrast to "a wave packet in classical optics, which is dissipated in the course of time". However, as a consequence of the similarity, a free quantum mechanical wave packet inevitably spreads out too. This effect is called wave packet spreading (WPS) and it is described in textbooks and many papers (see e.g. Ref. [4] and references therein). Moreover, as shown in Sec. 7, in quantum theory this effect is pronounced even in a much greater extent than in classical electrodynamics. In particular, the WPS effect has been investigated by de Broglie, Darwin and Schrödinger. The fact that WPS is inevitable has been treated by several authors as unacceptable and as an indication that standard quantum theory should be modified. For example, de Broglie has proposed to describe a free particle not by the Schrödinger equation but by a wavelet which satisfies a nonlinear equation and does not spread out (a detailed description of de Broglie's wavelets can be found e.g. in Ref. [5]). Sapogin writes (see Ref. [6] and references therein) that "Darwin showed that such packet quickly and steadily dissipates and disappears" and proposes an alternative to standard theory which he calls unitary unified quantum field theory.
At the same time, it has not been explicitly shown that numerical results on WPS are incompatible with experimental data. For example, it is known (see Sec. 3) that for macroscopic bodies the effect of WPS is extremely small. Probably it is also believed that in experiments on the Earth with atoms and elementary particles spreading does not have enough time to manifest itself although we have not found an explicit statement on this problem in the literature. Probably for these reasons the majority of physicists do not treat WPS as a drawback of the theory.
However, a natural problem arises what happens to photons which can travel from distant objects to Earth even for billions of years. For example, as shown in Sec. 9, in the case when the major part of photons emitted by stars are in wave packet states (what is the most probable scenario) the effect of WPS for photons emitted even by close stars is so strong that we should see not separate stars but rather an almost continuous background from all stars. In addition, data on relic radiation and gamma-ray bursts, signals from radio antennas to planets and space probes, signals from space probes and signals from pulsars show no signs of spreading of photon wave functions. We call those facts the WPS paradoxes. The consideration given in the present paper shows that the reason of the paradoxes is that standard position operator is not consistently defined. Hence the inconsistent definition of the position operator is not only an academic problem but leads to the above paradoxes.
Usual arguments in favor of choosing the standard position and momentum operators are that these operators have correct properties in semiclassical approximation. However, the requirement that an operator should have correct properties in semiclassical approximation does not define the operator unambiguously.
One of the arguments in favor of choosing standard position and momentum operators is that the nonrelativistic Schrödinger equation correctly describes the hydrogen energy levels, the Dirac equation correctly describes fine structure corrections to these levels etc. Historically these equations have been first written in coordinate space and in textbooks they are still discussed in this form. However, from the point of view of the present knowledge those equations should be treated as follows.
A fundamental theory describing electromagnetic interactions on quantum level is quantum electrodynamics (QED). This theory proceeds from quantizing classical Lagrangian which is only an auxiliary tool for constructing S-matrix. The argument x in the Lagrangian density L(t, x) cannot be treated as a position operator because L(t, x) is constructed from field functions which do not have a probabilistic interpretation. When the quantization is accomplished, the results of QED are formulated exclusively in momentum space and the theory does not contain space-time at all.
In particular, as follows from Feynman diagrams for the one-photon exchange, in the approximation (v/c) 2 the electron in the hydrogen atom can be described in the potential formalism where the potential acts on the wave function in momentum space. So for calculating energy levels one should solve the eigenvalue problem for the Hamiltonian with this potential. This is an integral equation which can be solved by different methods. One of the convenient methods is to apply the Fourier transform and get standard Schrödinger or Dirac equation in coordinate representation with the Coulomb potential. Hence the fact that the results for energy levels are in good agreement with experiment shows only that QED defines the potential correctly and standard coordinate Schrödinger and Dirac equations are only convenient mathematical ways of solving the eigenvalue problem. For this problem the physical meaning of the position operator is not important at all. One can consider other transformations of the original integral equation and define other position operators. The fact that for non-standard choices one might obtain something different from the Coulomb potential is not important on quantum level. On classical level the interaction between two charges can be described by the Coulomb potential but this does not imply that on quantum level the potential in coordinate representation should be necessarily Coulomb.
Another example follows. It is said that the spatial distribution of the electric charge inside a system can be extracted from measurements of form-factors in the electron scattering on this system. However, the information about the experiment is again given only in terms of momenta and conclusions about the spatial distribution can be drawn only if we assume additionally how the position operator is expressed in terms of momentum variables. On quantum level the physical meaning of such a spatial distribution is not fundamental.
In view of the fact that the coordinate and momentum representations are related to each other by the Fourier transform, one might think that the position and momentum operators are on equal footing. However, they are not on equal footing for the following reasons. In quantum theory each elementary particle is described by an irreducible representation (IR) of the symmetry algebra. For example, in Poincare invariant theory the set of momentum operators represents three of ten linearly independent representation operators of the Poincare algebra and hence those operators are consistently defined. On the other hand, among the representation operators there is no position operator. In view of the above discussion, since the results of existing fundamental quantum theories describing interactions on quantum level (QED, electroweak theory and QCD) are formulated exclusively in terms of the S-matrix in momentum space without any mentioning of space-time, for investigating such stationary quantum problems as calculating energy levels, form-factors etc., the notion of the position operator is not needed.
However, the choice of the position operator is important in nonstationary problems when evolution is described by the time dependent Schrödinger equation (with the nonrelativistic or relativistic Hamiltonian). For any new theory there should exist a correspondence principle that at some conditions the new theory should reproduce results of the old well tested theory with a good accuracy. In particular, quantum theory should reproduce the motion of a particle along the classical trajectory defined by classical equations of motion. Hence the position operator is needed only in semiclassical approximation and it should be defined from additional considerations.
In standard approaches to quantum theory the existence of space-time background is assumed from the beginning. Then the position operator for a particle in this background is the operator of multiplication by the particle radius-vector r. As explained in textbooks on quantum mechanics (see e.g. Ref. [1]), the result −ih∂/∂r for the momentum operator can be justified from the requirement that quantum theory should correctly reproduce classical results in semiclassical approximation. However, as noted above, this requirement does not define the operator unambiguously.
A standard approach to Poincare symmetry on quantum level follows. Since Poincare group is the group of motions of Minkowski space, quantum states should be described by representations of the Poincare group. In turn, this implies that the representation generators should commute according to the commutation relations of the Poincare group Lie algebra: where P µ are the operators of the four-momentum, M µν are the operators of Lorentz angular momenta, the diagonal metric tensor η µν has the nonzero components η 00 = −η 11 = −η 22 = −η 33 = 1 and µ, ν = 0, 1, 2, 3. It is usually said that the above relations are written in the system of units c =h = 1. However, as we argue in Ref. [7], quantum theory should not contain c andh at all; those quantities arise only because we wish to measure velocities in m/s and angular momenta in kg × m 2 /s. The above approach is in the spirit of Klein's Erlangen program in mathematics. However, as we argue in Refs. [7,8], quantum theory should not be based on classical space-time background. The notion of space-time background contradicts the basic principle of physics that a definition of a physical quantity is a description of how this quantity should be measured. Indeed one cannot measure coordinates of a manifold which exists only in our imagination.
As we argue in Refs. [7,8] and other publications, the approach should be opposite. Each system is described by a set of independent operators. By definition, the rules how these operators commute with each other define the symmetry algebra. In particular, by definition, Poincare symmetry on quantum level means that the operators commute according to Eq. (1). This definition does not involve Minkowski space at all. Such a definition of symmetry on quantum level is in the spirit of Dirac's paper [9].
The fact that an elementary particle in quantum theory is described by an IR of the symmetry algebra can be treated as a definition of the elementary particle. In Poincare invariant theory the IRs can be implemented in a space of functions χ(p) such that |χ(p)| 2 d 3 p < ∞ (see Sec. 4). In this representation the momentum operator P is defined unambiguously and is simply the operator of multiplication by p. A standard assumption is that the position operator in this representation is ih∂/∂p.
As explained in textbooks on quantum mechanics (see e.g. Ref. [1] and Sec. 2), semiclassical approximation cannot be valid in situations when the momentum is rather small. Consider first a one-dimensional case. If the value of the x component of the momentum p x is rather large, the definition of the coordinate operator x = ih∂/∂p x can be justified but this definition does not have a physical meaning in situations when p x is small.
Consider now the three-dimensional case. If all the components p j (j = 1, 2, 3) are rather large then there are situations when all the operators ih∂/∂p j are semiclassical. A semiclassical wave function χ(p) in momentum space should describe a narrow distribution around the mean value p 0 . Suppose now that the coordinate axes are chosen such p 0 is directed along the z axis. Then in view of the above remarks the operators ih∂/∂p j cannot be physical for j = 1, 2, i.e. in directions perpendicular to the particle momentum. Hence the standard definition of all the components of the position operator can be physical only for special choices of the coordinate axes and there exist choices when the definition is not physical. The situation when a definition of an operator is physical or not depending on the choice of the coordinate axes is not acceptable and hence standard definition of the position operator is not physical.
In the present paper we propose a consistent definition of the position operator. As a consequence, in our approach WPS in directions perpendicular to the particle momentum is absent regardless of whether the particle is nonrelativistic or relativistic. Hence the above paradoxes are resolved. Moreover, for an ultrarelativistic particle the effect of WPS is absent at all. In our approach different components of the position operator do not commute with each other and, as a consequence, there is no wave function in coordinate representation.
Our presentation is self-contained and for reproducing the results of the calculations no special knowledge is needed. The paper is organized as follows. In Secs. 2 and 4 we discuss the approach to the position operator in standard nonrelativistic and relativistic quantum theory, respectively. An inevitable consequence of this approach is the effect of WPS of the coordinate wave function which is discussed in Secs. 3 and 5 for the nonrelativistic and relativistic cases, respectively. In Sec. 7 we discuss a relation between the WPS effects for a classical wave packet and for photons comprising this packet. In Sec. 8 the problem of WPS in coherent states is discussed. In Sec. 9 we show that the WPS effect leads to several paradoxes mentioned above. As discussed in Sec. 10, in standard theory it is not possible to avoid those paradoxes. Our approach to a consistent definition of the position operator and its application to WPS are discussed in Secs. 11-13. Finally, in Sec. 14 we discuss implications of the results for entanglement, quantum locality and the problem of time in quantum theory.

Position operator in nonrelativistic quantum mechanics
In quantum theory, states of a system are represented by elements of a projective Hilbert space. The fact that a Hilbert space H is projective means that if ψ ∈ H is a state then const ψ is the same state. The matter is that not the probability itself but only relative probabilities of different measurement outcomes have a physical meaning. In this paper we will work with states ψ normalized to one, i.e. such that ||ψ|| = 1 where ||...|| is a norm. It is defined such that if (..., ...) is a scalar product in H then ||ψ|| = (ψ, ψ) 1/2 .
In quantum theory every physical quantity is described by a selfadjoint operator. Each selfadjoint operator is Hermitian i.e. satisfies the property (ψ 2 , Aψ 1 ) = (Aψ 2 , ψ 1 ) for any states belonging to the domain of A. If A is an operator of some quantity then the mean value of the quantity and its uncertainty in state ψ are given byĀ = (ψ, Aψ) and ∆A = ||(A −Ā)ψ||, respectively. The condition that a quantity corresponding to the operator A is semiclassical in state ψ can be defined such that ∆A ≪ |Ā|. This implies that the quantity can be semiclassical only if |Ā| is rather large. In particular, ifĀ = 0 then the quantity cannot be semiclassical.
Let B be an operator corresponding to another physical quantity andB and ∆B be the mean value and the uncertainty of this quantity, respectively. We A question arises whether two physical quantities corresponding to the operators A and B can be simultaneously semiclassical in state ψ. Since ||ψ 1 ||||ψ 2 || ≥ |(ψ 1 , ψ 2 )|, we have that This condition is known as a general uncertainty relation between two quantities. A well-known special case is that if P is the x component of the momentum operator and X is the operator of multiplication by x then [P, X] = −ih and ∆p∆x ≥h/2. The states where ∆p∆x =h/2 are called coherent ones. They are treated such that the momentum and the coordinate are simultaneously semiclassical in a maximal possible extent. A well-known example is that if thenX = x 0 ,P = p 0 , ∆x = a/ √ 2 and ∆p =h/(a √ 2). Consider first a one dimensional motion. In standard textbooks on quantum mechanics, the presentation starts with a wave function ψ(x) in coordinate space since it is implicitly assumed that the meaning of space coordinates is known. Then a question arises why P = −ihd/dx should be treated as the momentum operator. The explanation follows.
Consider wave functions having the form ψ(x) = exp(ip 0 x/h)a(x) where the amplitude a(x) has a sharp maximum near x = x 0 ∈ [x 1 , x 2 ] such that a(x) is not small only when x ∈ [x 1 , x 2 ]. Then ∆x is of the order x 2 − x 1 and the condition that the coordinate is semiclassical is ∆x will be approximately the eigenfunction of −ihd/dx with the eigenvalue p 0 if |p 0 a(x)| ≫h|da(x)/dx|. Since |da(x)/dx| is of the order of |a(x)/∆x|, we have a condition |p 0 ∆x| ≫h. Therefore if the momentum operator is −ihd/dx, the uncertainty of momentum ∆p is of the order ofh/∆x, |p 0 | ≫ ∆p and this implies that the momentum is also semiclassical. At the same time, |p 0 ∆x|/2πh is approximately the number of oscillations which the exponent makes on the segment [x 1 , x 2 ]. Therefore the number of oscillations should be much greater than unity. In particular, semiclassical approximation cannot be valid if ∆x is very small, but on the other hand, ∆x cannot be very large since it should be much less than x 0 . Another justification of the fact that −ihd/dx is the momentum operator is that in the formal limith → 0 the Schrödinger equation becomes the Hamilton-Jacobi equation.
We conclude that the choice of −ihd/dx as the momentum operator is justified from the requirement that in semiclassical approximation this operator becomes the classical momentum. However, it is obvious that this requirement does not define the operator uniquely: any operatorP such thatP − P disappears in semiclassical limit, also can be called the momentum operator.
One might say that the choice P = −ihd/dx can also be justified from the following considerations. In nonrelativistic quantum mechanics we assume that the theory should be invariant under the action of the Galilei group, which is a group of transformations of Galilei space-time. The x component of the momentum operator should be the generator corresponding to spatial translations along the x axis and −ihd/dx is precisely the required operator. In this consideration one assumes that the space-time background has a physical meaning while, as discussed in Refs. [7,8] and references therein, this is not the case.
As noted in Refs. [7,8] and references therein, one should start not from space-time but from a symmetry algebra. Therefore in nonrelativistic quantum mechanics we should start from the Galilei algebra and consider its IRs. For simplicity we again consider a one dimensional case. Let P x = P be one of representation operators in an IR of the Galilei algebra. We can implement this IR in a Hilbert space of functions χ(p) such that ∞ −∞ |χ(p)| 2 dp < ∞ and P is the operator of multiplication by p, i.e. P χ(p) = pχ(p). Then a question arises how the operator of the x coordinate should be defined. In contrast to the momentum operator, the coordinate one is not defined by the representation and so it should be defined from additional assumptions. Probably a future quantum theory of measurements will make it possible to construct operators of physical quantities from the rules how these quantities should be measured. However, at present we can construct necessary operators only from rather intuitive considerations.
By analogy with the above discussion, one can say that semiclassical wave functions should be of the form χ(p) = exp(−ix 0 p/h)a(p) where the amplitude a(p) has a sharp maximum near p = p 0 ∈ [p 1 , p 2 ] such that a(p) is not small only when p ∈ [p 1 , p 2 ]. Then ∆p is of the order of p 2 −p 1 and the condition that the momentum is semiclassical is ∆p ≪ |p 0 |. Since ihdχ(p)/dp = x 0 χ(p) + ihexp(−ix 0 p/h)da(p)/dp, we see that χ(p) will be approximately the eigenfunction of ihd/dp with the eigenvalue x 0 if |x 0 a(p)| ≫h|da(p)/dp|. Since |da(p)/dp| is of the order of |a(p)/∆p|, we have a condition |x 0 ∆p| ≫h. Therefore if the coordinate operator is X = ihd/dp, the uncertainty of coordinate ∆x is of the order ofh/∆p, |x 0 | ≫ ∆x and this implies that the coordinate defined in such a way is also semiclassical. We can also note that |x 0 ∆p|/2πh is approximately the number of oscillations which the exponent makes on the segment [p 1 , p 2 ] and therefore the number of oscillations should be much greater than unity. It is also clear that semiclassical approximation cannot be valid if ∆p is very small, but on the other hand, ∆p cannot be very large since it should be much less than p 0 . By analogy with the above discussion, the requirement that the operator ihd/dp becomes the coordinate in classical limit does not define the operator uniquely. In nonrelativistic quantum mechanics it is assumed that the coordinate is a well defined physical quantity even on quantum level and that ihd/dp is the most pertinent choice.
The above results can be directly generalized to the three-dimensional case. For example, if the coordinate wave function is chosen in the form then the momentum wave function is It is easy to verify that the uncertainty of each component of the coordinate operator is a/ √ 2 and the uncertainty of each component of the momentum operator ish/(a √ 2). Hence one might think that Eqs. (4) and (5) describe a state which is semiclassical in a maximal possible extent.
Let us make the following remark about semiclassical vector quantities. We defined a quantity as semiclassical if its uncertainty is much less than its mean value. In particular, as noted above, a quantity cannot be semiclassical if its mean value is small. In the case of vector quantities we have sets of three physical quantities. Some of them can be small and for them it is meaningless to discuss whether they are semiclassical or not. We say that a vector quantity is semiclassical if all its components which are not small are semiclassical and there should be at least one semiclassical component.
For example, if the mean value of the momentum p 0 is directed along the z axes then the xy components of the momentum are not semiclassical but the threedimensional vector quantity p can be semiclassical if p 0 is rather large. However, in that case the definitions of the x and y components of the position operator as x = ih∂/∂p x and y = ih∂/∂p y become inconsistent. The situation when the validity of an operator depends on the choice of directions of the coordinate axes is not acceptable and hence the above definition of the position operator is at least problematic. Moreover, as already mentioned, it will be shown in Sec. 9 that the standard choice of the position operator leads to the WPS paradoxes.
Let us note that semiclassical states can be constructed not only in momentum or coordinate representations. For example, instead of momentum wave functions χ(p) one can work in the representation where the quantum numbers (p, l, µ) in wave functions χ(p, l, µ) mean the magnitude of the momentum p, the orbital quantum number l (such that a state is the eigenstate of the orbital momentum squared L 2 with the eigenvalue l(l + 1)) and the magnetic quantum number µ (such that a state is the eigenvector or L z with the eigenvalue µ). A state described by a χ(p, l, µ) will be semiclassical with respect to those quantum numbers if χ(p, l, µ) has a sharp maximum at p = p 0 , l = l 0 , µ = µ 0 and the widths of the maxima in p, l and µ are much less than p 0 , l 0 and µ 0 , respectively. However, by analogy with the above discussion, those widths cannot be arbitrarily small if one wishes to have other semiclassical variables (e.g. the coordinates). Examples of such situations will be discussed in Sec. 12.
3 Wave packet spreading in nonrelativistic quantum mechanics As noted by Pauli (see p. 63 of Ref. [10]), at early stages of quantum theory some authors treated time t as the operator commuting with the Hamiltonian as [H, t] = ih but such a treatment is not correct (for example, one cannot construct the eigenstate of the time operator with the eigenvalue 5000 BC or 2014 AD). Hence the quantity t can be only a classical parameter (see also Ref. [11]). We see that the principle of quantum theory that every physical quantity is defined by an operator does not apply to time. The problem of time in quantum theory is discussed in a wide literature and remarks on this problem are made in Sec. 14. However, for now we assume that standard treatment of time is valid, i.e. that time is a classical parameter such that the dependence of the wave function on time is defined by the Hamiltonian according to the Schrödinger equation.
In nonrelativistic quantum mechanics the Hamiltonian of a free particle with the mass m is H = p 2 /2m and hence, as follows from Eq. (5), in the model discussed above the dependence of the momentum wave function on t is given by It is easy to verify that for this state the mean value of the operator p and the uncertainty of each momentum component are the same as for the state χ(p), i.e. those quantities do not change with time.
Consider now the dependence of the coordinate wave function on t. This dependence can be calculated by using Eq. (7) and the fact that The result of a direct calculation is ψ(r, t) = 1 π 3/4 a 3/2 (1 + iht where v 0 = p 0 /m is the classical velocity. This result shows that the semiclassical wave packet is moving along the classical trajectory r(t) = r 0 + v 0 t. At the same time, it is now obvious that the uncertainty of each coordinate depends on time as where ∆x j (0) = a/ √ 2, i.e. the width of the wave packet in coordinate representation is increasing. This fact, known as the wave-packet spreading (WPS), is described in many textbooks and papers (see e.g. the textbooks [4] and references therein). It shows that if a state was semiclassical in the maximal extent at t = 0, it will not have this property at t > 0 and the accuracy of semiclassical approximation will decrease with the increase of t. The characteristic time of spreading can be defined as t * = ma 2 /h. For macroscopic bodies this is an extremely large quantity and hence in macroscopic physics the WPS effect can be neglected. In the formal limith → 0, t * becomes infinite, i.e. spreading does not take place. This shows that WPS is a pure quantum phenomenon. For the first time the result (9) has been obtained by Darwin in Ref. [12].
One might pose a problem whether the WPS effect is specific only for Gaussian wave functions. One might expect that this effect will take place in general situations since each component of the standard position operator ih∂/∂p does not commute with the Hamiltonian and so the distribution of the corresponding physical quantity will be time dependent. A good example showing inevitability of WPS follows. If at t = 0 the coordinate wave function is ψ 0 (r) then, as follows from Eqs. (5) and (8), As follows from this expression, if ψ 0 (r) = 0 only if r belongs to a finite vicinity of some vector r 0 then at any t > 0 the support of ψ(r, t) belongs to the whole threedimensional space, i.e. the wave function spreads out with an infinite speed. One might think that in nonrelativistic theory this is not unacceptable since this theory can be treated as a formal limit c → ∞ of relativistic theory. In the next section we will discuss an analogous situation in relativistic theory. As shown in Ref. [13] titled "Nonspreading wave packets", for a onedimensional wave function in the form of an Airy function, spreading does not take place and the maximum of the quantity |ψ(x)| 2 propagates with constant acceleration even in the absence of external forces. Those properties of Airy packets have been observed in optical experiments [14]. However, since such a wave function is not normalizable, we believe that the term "wave packet" in the given situation might be misleading since the mean values and uncertainties of the coordinate and momentum cannot be calculated in a standard way. Such a wave function can be constructed only in a limited region of space. As explained in Ref. [13], this wave function describes not a particle but rather families of particle orbits. As shown in Ref. [13], one can construct a normalized state which is a superposition of Airy functions with Gaussian coefficients and "eventually the spreading due to the Gaussian cutoff takes over". This is an additional argument that the effect of WPS is an inevitable consequence of standard quantum theory.
Since quantum theory is invariant under time reversal, one might ask the following question: is it possible that the width of the wave packet in coordinate representation is decreasing with time? From the formal point of view, the answer is "yes". Indeed, the solution given by Eq. (9) is valid not only when t ≥ 0 but when t < 0 as well. Then, as follows from Eq. (10), the uncertainty of each coordinate is decreasing when t changes from some negative value to zero. However, eventually the value of t will become positive and the quantities ∆x j (t) will grow to infinity. In the present paper we consider situations when a photon is created on atomic level and hence one might expect that its initial coordinate uncertainties are not large. However, when the photon travels a long distance to Earth, those uncertainties become much greater, i.e. the term WPS reflects the physics adequately.

Position operator in relativistic quantum mechanics
The problem of the position operator in relativistic quantum theory has been discussed in a wide literature and different authors have different opinions on this problem. In particular, some authors state that in relativistic quantum theory no position operator exists. As already noted, the results of fundamental quantum theories are formulated only in terms of the S-matrix in momentum space without any mentioning of space-time. This is in the spirit of the Heisenberg S-matrix program that in relativistic quantum theory it is possible to describe only transitions of states from the infinite past when t → −∞ to the distant future when t → +∞. On the other hand, since quantum theory is treated as a theory more general than classical one, it is not possible to fully avoid space and time in quantum theory. For example, quantum theory should explain how photons from distant objects travel to Earth and even how macroscopic bodies are moving along classical trajectories. Hence we can conclude that: a) in quantum theory (nonrelativistic and relativistic) we must have a position operator and b) this operator has a physical meaning only in semiclassical approximation.
Let us first consider the definition of elementary particle. Although theory of elementary particles exists for a rather long period of time, there is no commonly accepted definition of elementary particle in this theory. In Refs. [7,8] and references therein we argue that, in the spirit of Wigner's approach to Poincare symmetry [15], a general definition, not depending on the choice of the classical background and on whether we consider a local or nonlocal theory, is that a particle is elementary if the set of its wave functions is the space of an IR of the symmetry algebra in the given theory.
There exists a wide literature describing how IRs of the Poincare algebra can be constructed. In particular, an IR for a spinless particle can be implemented in a space of functions ξ(p) satisfying the condition where ǫ(p) = (m 2 + p 2 ) 1/2 is the energy of the particle with the mass m. The convenience of the above requirement is that the volume element dρ(p) is Lorentz invariant. In that case it can be easily shown by direct calculations (see e.g. Ref. [16]) that the representation operators have the form where L is the orbital angular momentum operator, N is the Lorentz boost operator, P is the momentum operator, E is the energy operator and these operators are For particles with spin these results are modified as follows. For a massive particle with spin s the functions ξ(p) also depend on spin projections which can take 2s + 1 values −s, −s + 1, ...s. If s is the spin operator then the total angular momentum has an additional term s and the Lorentz boost operator has an additional term (s×p)/(ǫ(p) + m) (see e.g. Eq. (2.5) in Ref. [16]). Hence corrections of the spin terms to the quantum numbers describing the angular momentum and the Lorentz boost do not exceed s. We assume as usual that in semiclassical approximation the quantum numbers characterizing the angular momentum and the Lorentz boost are much greater than unity and hence in this approximation spin effects can be neglected. For a massless particle with the spin s the spin projections can take only values −s and s and those quantum numbers have the meaning of helicity. In this case the results for the representation operators can be obtained by taking the limit m → 0 if the operators are written in the light front variables (see e.g. Eq. (25) in Ref. [7]). As a consequence, in semiclassical approximation the spin corrections in the massless case can be neglected as well. Hence for investigating the position operator we will neglect spin effects and will not explicitly write the dependence of wave functions on spin projections.
In the above IRs the representation operators are Hermitian as it should be for operators corresponding to physical quantities. In standard theory (over complex numbers) such IRs of the Lie algebra can be extended to unitary IRs of the Poincare group. In the literature elementary particles are described not only by such IRs but also by nonunitary representations induced from the Lorentz group [17]. Since the factor space of the Poincare group over the Lorentz group is Minkowski space, the elements of such representations are fields Ψ(x) depending on four-vectors x in Minkowski space and possibly on spin variables. Since those functions describe nonunitary representations, their probabilistic interpretation is problematic. The Pauli theorem [18] states that for fields with an integer spin it is impossible to define a positive definite charge density and for fields with a half-integer spin it is impossible to define a positive definite energy density. Hence a problem arises whether such fields have a physical meaning. The answer is that after quantizing they become quantum fields defining the Hermitian operators (M µν , P µ ) and the S-matrix for systems of quantum fields in quantum field theory (QFT). All those operators are given only in terms of integrals of local quantum fields and so the quantity x is only an integration parameter (i.e. a problem of whether there are quantum operators corresponding to x does not arise). So local quantum fields (in this situation even the term "local" is not clear) are only auxiliary tools for constructing physical operators and the S-matrix in QFT.
Let us note that although QFT has achieved very impressive successes in explaining many experimental data, a problem of its mathematical substantiation has not been solved yet. The main mathematical inconsistency of QFT is that since interacting local quantum fields can be treated only as operatorial distributions, their products at the same space-time points are not well defined (see e.g. Ref. [19]). One of ideas of the string theory is that if products of fields at the same points (zerodimensional objects) are replaced by products where the arguments belong to strings (one-dimensional objects) then there is hope that infinities will be less singular. In view of such controversial properties of local quantum fields, many authors posed a question whether local fields will survive in the future quantum theory. Nevertheless, in the literature the problem of position operator is mainly discussed in the approach when elementary particles are described by local fields rather than unitary IRs. Below we discuss the both approaches but first we consider the case of unitaty IRs.
As follows from Eq. (1), the operator I 2 = E 2 −P 2 is the Casimir operator of the second order, i.e. it is a bilinear combination of representation operators commuting with all the operators of the algebra. As follows from the well-known Schur lemma, all states belonging to an IR are the eigenvectors of I 2 with the same eigenvalue m 2 . Note that Eq. (13) contains only m 2 but not m. The choice of the energy sign is only a matter of convention but not a matter of principle. Indeed, the energy can be measured only if the momentum p is measured and then it is only a matter of convention what sign of the square root should be chosen. However, it is important that the sign should be the same for all particles. For example, if we consider a system of two particles with the same values of m 2 and the opposite momenta p 1 and p 2 such that p 1 + p 2 = 0, we cannot define the energies of the particles as ǫ(p 1 ) and −ǫ(p 2 ), respectively, since in that case the total four-momentum of the two-particle system will be zero what contradicts experiment.
The notation I 2 = m 2 is justified by the fact that for all known particles I 2 is greater or equal than zero. Then the mass m is defined as the square root of m 2 and the sign of m is only a matter of convention. The usual convention is that m ≥ 0. However, from mathematical point of view, IRs with I 2 < 0 are not prohibited. If the velocity operator v is defined as v = P/E then for known particles |v| ≤ 1, i.e. |v| ≤ c in standard units. However, for IRs with I 2 < 0, |v| > c and, at least from the point of view of mathematical construction of IRs, this case is not prohibited. The hypothetical particles with such properties are called tachyons and their possible existence is widely discussed in the literature. If the tachyon mass m is also defined as the square root of m 2 then this quantity will be imaginary. However, this does not mean than the corresponding IRs are unphysical since all the operators of the Poincare group Lie algebra depend only on m 2 .
As follows from Eqs. (12) and (13), in the nonrelativistic approximation dρ(p) = d 3 p/m and N = −im∂/∂p. Therefore in this approximation N is proportional to standard position operator and one can say that the position operator is in fact present in the description of the IR.
In relativistic case the operator i∂/∂p is not selfadjoint since dρ(p) is not proportional to d 3 p. However, one can perform a unitary transformation ξ(p) → χ(p) = ξ(p)/ǫ(p) 1/2 such that the Hilbert space becomes the space of functions χ(p) satisfying the condition |χ(p)| 2 d 3 p < ∞. It is easy to verify that in this implementation of the IR the operators (L, P, E) will have the same form as in Eq. (13) but the expression for N will be In this case one can define ih∂/∂p as a position operator but now we do not have a situation when the position operator is present among the other representation operators.
A problem of the definition of the position operator in relativistic quantum theory has been discussed since the beginning of the 1930s and it has been noted that when quantum theory is combined with relativity the existence of the position operator with correct physical properties becomes a problem. The above definition has been proposed by Newton and Wigner in Ref. [20]. They worked in the approach when elementary particles are described by local fields rather than unitary IRs. The Fourier transform of such fields describes states where the energy can be positive and negative and this is interpreted such that local quantum fields describe a particle and its antiparticle simultaneously. Newton and Wigner first discuss the spinless case and consider only states on the upper Lorentz hyperboloid where the energy is positive. For such states the representation operators act in the same way as in the case of spinless unitary IRs. With this definition the coordinate wave function ψ(r) can be again defined by Eq. (4) and a question arises whether such a position operator has all the required properties.
For example, in the introductory section of the well-known textbook [21] the following arguments are given in favor of the statement that in relativistic quantum theory it is not possible to define a physical position operator. Suppose that we measure coordinates of an electron with the mass m. When the uncertainty of coordinates is of the order ofh/mc, the uncertainty of momenta is of the order of mc, the uncertainty of energy is of the order of mc 2 and hence creation of electronpositron pairs is allowed. As a consequence, it is not possible to localize the electron with the accuracy better than its Compton wave lengthh/mc. Hence, for a particle with a nonzero mass exact measurement is possible only either in the nonrelativistic limit (when c → ∞) or classical limit (whenh → 0). In the case of the photon, as noted by Pauli (see p. 191 of Ref. [10]), the coordinate cannot be measured with the accuracy better thanh/p where p is the magnitude of the photon momentum. The quantity λ = 2πh/p is called the photon wave length although, as noted in Sec. 1, the meaning of this quantity in quantum case might be fully different than in classical one. Since λ → 0 in the formal limith → 0, Pauli concludes that "Only within the confines of the classical ray concept does the position of the photon have a physical significance".
Another argument that the Newton-Wigner position operator does not have all the required properties follows. A relativistic analog of Eq. (11) is As a consequence, the Newton-Wigner position operator has the "tail property": if ψ 0 (r) = 0 only if r belongs to a finite vicinity of some vector r 0 then at any t > 0 the function ψ(r, t) has a tail belonging to the whole three-dimensional space, i.e. the wave function spreads out with an infinite speed. Hence at any t > 0 the particle can be detected at any point of the space and this contradicts the requirement that no information should be transmitted with the speed greater than c. The tail property of the Newton-Wigner position operator has been known for a long time (see e.g. Ref. [22] and references therein). It is characterized as nonlocality leading to the action at a distance. Hegerfeldt argues [22] that this property is rather general because it can be proved assuming that energy is positive and without assuming a specific choice of the position operator. The Hegerfeldt theorem [22] is based on the assumption that there exists an operator N(V ) whose expectation defines the probability to find a particle inside the volume V . However, the meaning of time on quantum level is not clear and for the position operator proposed in the present paper such a probability does not exist because there is no wave function in coordinate representation (see Sec. 11 and the discussion in Sec. 14).
One might say that the requirement that no signal can be transmitted with the speed greater than c has been obtained in Special Relativity which is a classical (i.e. nonquantum) theory operating only with classical space-time coordinates. For example, in classical theory the velocity of a particle is defined as v = dr/dt but, as noted above, the velocity should be defined as v = p/E (i.e. without mentioning space-time) and then on classical level it can be shown that v = dr/dt. In QFT local quantum fields separated by space-like intervals commute or anticommute (depending on whether the spin is integer or half-integer) and this is treated as a requirement of causality and that no signal can be transmitted with the speed greater than c. However, as noted above, the physical meaning of space-time coordinates on quantum level is not clear. Hence from the point of view of quantum theory the existence of tachyons is not prohibited. Note also that when two electrically charged particles exchange by a virtual photon, a typical situation is that the four-momentum of the photon is space-like, i.e. the photon is the tachyon. We conclude that although in relativistic theory such a behavior might seem undesirable, there is no proof that it must be excluded. Also, as argued by Griffiths (see Ref. [23] and references therein), with a consistent interpretation of quantum theory there are no nonlocality and superluminal interactions. In Sec. 14 we argue that the position operator proposed in the present paper sheds a new light on this problem.
Another striking example is a photon emitted in the famous 21cm transition line between the hyperfine energy levels of the hydrogen atom. The phrase that the lifetime of this transition is of the order of τ = 10 7 years implies that the width of the level is of the order ofh/τ , i.e. experimentally the uncertainty of the photon energy ish/τ . Hence the uncertainty of the photon momentum ish/(cτ ) and with the above definition of the coordinate operators the uncertainty of the longitudinal coordinate is cτ , i.e. of the order of 10 7 light years. Then there is a nonzero probability that immediately after its creation at point A the photon can be detected at point B such that the distance between A and B is 10 7 light years.
A problem arises how this phenomenon should be interpreted. On one hand, one might say that in view of the above discussion it is not clear whether or not the requirement that no information should be transmitted with the speed greater than c should be a must in relativistic quantum theory. On the other hand (as pointed out to me by Alik Makarov), we can know about the photon creation only if the photon is detected and when it was detected at point B at the moment of time t = t 0 , this does not mean that the photon travelled from A to B with the speed greater than c since the time of creation has an uncertainty of the order of 10 7 years. Note also that in this situation a description of the system (atom + electric field) by the wave function (e.g. in the Fock space) depending on a continuous parameter t has no physical meaning (since roughly speaking the quantum of time in this process is of the order of 10 7 years). If we accept this explanation then we should acknowledge that in some situations a description of evolution by a continuous classical parameter t is not physical and this is in the spirit of the Heisenberg S-matrix program. However, this example describes a pure quantum phenomenon while, as noted above, a position operator is needed only in semiclassical approximation.
For particles with nonzero spin, the number of states in local fields is typically by a factor of two greater than in the case of unitary IRs (since local fields describe a particle and its antiparticle simultaneously) but those components are not independent since local fields satisfy a covariant equation (Klein-Gordon, Dirac etc.). In Ref. [20] Newton and Wigner construct a position operator in the massive case but say that in the massless one they have succeeded in constructing such an operator only for Klein-Gordon and Dirac particles while in the case of the photon the position operator does not exist. On the other hand, as noted above, in the case of unitary IRs different spin components are independent and in semiclassical approximation spin effects are not important. So in this approach one might adopt the Newton-Wigner position operator for particles with any spin and any mass.
In view of the WPS paradoxes, we consider the photon case in greater details. In classical theory the notion of field, as well as that of wave, is used for describing systems of many particles by their mean characteristics. For example, the electromagnetic field consists of many photons. In classical theory each photon is not described individually but the field as a whole is described by the field strengths E(r, t) and B(r, t) which can be measured (in principle) by using macroscopic test bodies. In quantum theory one can formally define corresponding quantized field operators but the meaning of (r, t) on quantum level is not clear. In particular, the physical meaning of electric and magnetic fields of a single photon is problematic.
In standard textbooks on QED (see e.g. Ref. [24]) it is stated that in this theory there is no way to define a coordinate photon wave function and the arguments are as follows. The electric and magnetic fields of the photon in coordinate representation are proportional to the Fourier transforms of |p| 1/2 χ(p), rather than χ(p). As a consequence, the quantities E(r) and B(r) are defined not by ψ(r) but by integrals of ψ(r) over a region of the order of the wave length. However, this argument also does not exclude the possibility that ψ(r) can have a physical meaning in semiclassical approximation since, as noted above, the notions of the electric and magnetic fields of a single photon are problematic. In addition, since λ → 0 in the formal limith → 0, one should not expect that any position operator in semiclassical approximation can describe coordinates with the accuracy better than the wave length.
A detailed discussion of the photon position operator can be found in papers by Margaret Hawton and references therein (see e.g. Ref. [25]). In this approach the photon is described by a local field and the momentum and coordinate representations are related to each other by standard Fourier transform. The author of Ref. [25] discusses generalizations of the photon position operator proposed by Pryce [26]. However, the Pryce operator and its generalizations discussed in Ref. [25] differ from the Newton-Wigner operator only by terms of the order of the wave length. Hence in semiclassical approximation all those operators are equivalent.
The above discussion shows that on quantum level the physical meaning of the coordinate is not clear but in view of a) and b) (see the beginning of this section) one can conclude that in semiclassical approximation all the existing proposals for the position operator are equivalent to the Newton-Wigner operator ih∂/∂p. An additional argument in favor of this operator is that the relativistic nature of the photon might be somehow manifested in the longitudinal direction while in transverse directions the behavior of the wave function should be similar to that in standard nonrelativistic quantum mechanics. Another argument is that the photon wave func-tion in coordinate representation constructed by using this operator satisfies the wave equation in agreement with classical electrodynamics (see Sec. 6).
In addition, if we consider a motion of a free particle, it is not important in what interactions this particle participates and, as explained above, if the particle is described by its IR in semiclassical approximation then the particle spin is not important. Hence the effect of WPS for an ultrarelativistic particle does not depend on the nature of the particle, i.e. on whether the particle is the photon, the proton, the electron etc. For all the reasons described above and in view of a) and b), in the next section we consider what happens if the space-time evolution of relativistic wave packets is described by using the Newton-Wigner position operator.

Wave packet spreading in relativistic quantum mechanics
Consider first a construction of the wave packet for a particle with nonzero mass. A possible way of the construction follows. We first consider the particle in its rest system, i.e. in the reference frame where the mean value of the particle momentum is zero. The wave function χ 0 (p) in this case can be taken as in Eq. (5) with p 0 = 0. As noted in Sec. 2, such a state cannot be semiclassical. However, it is possible to obtain a semiclassical state by applying a Lorentz transformation to χ 0 (p). One can show (see e.g. Eq. (2.4) in Ref. [16]) that when the IR for a spinless particle is extended to the unitary representation of the Poincare group then the operator U(g) corresponding to a Lorentz transformation g is where p ′ is the momentum obtained from p by the Lorentz transformation g −1 . If g is the Lorentz boost along the z axis with the velocity v then where we use the subscript ⊥ to denote projections of vectors onto the xy plane. As follows from this expression, exp(−p ′ 2 a 2 /2h 2 ) as a function of p has the maximum at what represents the effect of the Lorentz contraction. If mv ≫h/a (in units where c = 1) then m ≫ |p ⊥ | and p z0 ≈ mv/(1 − v 2 ) 1/2 . In this case the transformed state is semiclassical and the mean value of the momentum is exactly the classical (i.e. nonquantum) value of the momentum of a particle with mass m moving along the z axis with the velocity v. However, in the opposite case when m ≪h/a the transformed state is not semiclassical since the uncertainty of p z is of the same order as the mean value of p z . If the photon mass is exactly zero then the photon cannot have the rest state. However, even if the photon mass is not exactly zero, it is so small that the relation m ≪h/a is certainly satisfied for any realistic value of a. Hence a semiclassical state for the photon or a particle with a very small mass cannot be obtained by applying the Lorentz transformation to χ 0 (p) and considering the case when v is very close to unity. In this case we will describe a semiclassical state by a wave function which is a generalization of the function (5): Here we assume that the vector p 0 is directed along the z axis and its z component is p 0 . In the general case the parameters a and b defining the momentum distributions in the transverse and longitudinal directions, respectively, can be different. In that case the uncertainty of each transverse component of momentum ish/(a √ 2) while the uncertainty of the z component of momentum ish/(b √ 2). In view of the above discussion one might think that, as a consequence of the Lorentz contraction, the parameter b should be very small. However, the above discussion shows that the notion of the Lorentz contraction has a physical meaning only if m ≫h/a while for the photon the opposite relation takes place. We will see below that in typical situations the quantity b is large and much greater than a.
In relativistic quantum theory the situation with time is analogous to that in the nonrelativistic case (see Sec. 3) and time can be treated only as a good approximate parameter describing the evolution according to the Schrödinger equation with the relativistic Hamiltonian. Then the dependence of the momentum wave function (18) on t is given by where p = |p| and we assume that the particle is ultrarelativistic, i.e. p ≫ m. Since at different moments of time the wave functions in momentum space differ each other only by a phase factor, the mean value and uncertainty of each momentum component do not depend on time. In other words, there is no WPS for the wave function in momentum space. As noted in Sec. 3, the same is true in the nonrelativistic case.
In view of the above discussion, the function ψ(r, t) can be again defined by Eq. (8) where now χ(p, t) is defined by Eq. (19). If the variable p z in the integrand is replaced by p 0 + p z then as follows from Eqs. (8,18,19) ψ(r, t) = ab 1/2 exp(ip 0 r/h) We now take into account the fact that in semiclassical approximation the quantity p 0 should be much greater than the uncertainties of the momentum in the longitudinal and transversal directions, i.e. p 0 ≫ p z and p 0 ≫ |p ⊥ |. Hence with a good accuracy we can expand the square root in the integrand in powers of |p|/p 0 . Taking into account the linear and quadratic terms in the square root we get Then the integral over d 3 p can be calculated as the product of integrals over d 2 p ⊥ and dp z and the calculation is analogous to that in Eq. (9). The result of the calculation is This result shows that the wave packet describing an ultrarelativistic particle (including a photon) is moving along the classical trajectory z(t) = z 0 +ct, in the longitudinal direction there is no spreading while in transversal directions spreading is characterized by the function The characteristic time of spreading can be defined as t * = p 0 a 2 /hc. The fact that t * → ∞ in the formal limith → 0 shows that in relativistic case WPS also is a pure quantum phenomenon (see the end of Sec. 3). From the formal point of view the result for t * is the same as in nonrelativistic theory but m should be replaced by E/c 2 where E is the energy of the ultrarelativistic particle. This fact could be expected since, as noted above, it is reasonable to think that spreading in directions perpendicular to the particle momentum is similar to that in standard nonrelativistic quantum mechanics. However, in the ultrarelativistic case spreading takes place only in this direction. If t ≫ t * the transversal width of the packet is a(t) =hct/p 0 a.
Hence the speed of spreading in the perpendicular direction is v * =hc/p 0 a.

Geometrical optics
The relation between quantum and classical electrodynamics is well-known and is described in textbooks (see e.g. Ref. [24]). As already noted, classical electromagnetic field consists of many photons and in classical electrodynamics the photons are not described individually. Instead, classical electromagnetic field is described by field strengths which represent mean characteristics of a large set of photons. For constructing the field strengths one can use the photon wave functions χ(p, t) or ψ(r, t) where E is replaced byhω and p is replaced byhk. In this connection it is interesting to note that since ω is a classical quantity used for describing a classical electromagnetic field, the photon is a pure quantum particle since its energy disappears in the formal limith → 0. Even this fact shows that the photon cannot be treated as a classical particle and the effect of WPS for the photon cannot be neglected.
With the above replacements the functions χ and ψ do not contain any dependence onh (note that the normalization factorh −3/2 in χ(k, t) will disappear since the normalization integral for χ(k, t) is now over d 3 k, not d 3 p). The quantities ω and k are now treated, respectively, as the frequency and the wave vector of the classical electromagnetic field and the functions χ(k, t) and ψ(r, t) are interpreted not such that they describe probabilities for a single photon but such that they describe classical electromagnetic field and E(r, t) and B(r, t) can be constructed from these functions as described in textbooks on QED (see e.g. Ref. [24]).
An additional argument in favor of the choice of ψ(r, t) as the coordinate photon wave function is that in classical electrodynamics the quantities E(r, t) and B(r, t) for the free field should satisfy the wave equation ∂ 2 E/c 2 ∂t 2 = ∆E and analogously for B(r, t). Hence if E(r, t) and B(r, t) are constructed from ψ(r, t) as described in textbooks (see e.g. Ref. [24]), they will satisfy the wave equation since, as follows from Eqs. (8,18,19), ψ(r, t) also satisfies this equation.
The geometrical optics approximation implies that if k 0 and r 0 are the mean values of the wave vector and the spatial radius vector for a wave packet describing the electromagnetic wave then the uncertainties ∆k and ∆r, which are the mean values of |k − k 0 | and |r − r 0 |, respectively, should satisfy the requirements ∆k ≪ |k 0 | and ∆r ≪ |r 0 |. Analogously, in full analogy with the derivation of Eq.
(3), one can show that for each j = 1, 2, 3 the uncertainties of the corresponding projections of the vectors k and r satisfy the requirement ∆k j ∆r j ≥ 1/2 (see e.g. Ref. [2]). In particular, an electromagnetic wave satisfies the approximation of geometrical optics in the greatest possible extent if ∆k∆r is of the order of unity.
The above discussion confirms what has been mentioned in Sec. 1 that the effect of WPS in transverse directions takes place not only in quantum theory but even in classical electrodynamics. Indeed, since the function ψ(r, t) satisfies the classical wave equation, the above consideration can be also treated as an example showing that even for a free wave packet in classical electrodynamics the WPS effect is inevitable. In the language of classical waves the parameters of spreading can be characterized by the function a(t) (see Eq. (23)) and the quantities t * and v * (see the end of the preceding section) such that in terms of the wave length λ = 2πc/ω 0 The last expression can be treated such that if λ ≪ a then the momentum has the angular uncertainty of the order of α = λ/(2πa). This result is natural from the following consideration. Let the mean value of the momentum be directed along the z-axis and the uncertainty of the transverse component of the momentum be ∆p ⊥ . Then ∆p ⊥ is of the order ofh/a, λ = 2πh/p 0 and hence α is of the order of ∆p ⊥ /p 0 ≈ λ/(2πa). This is analogous to the well known result in classical optics that the best angular resolution of a telescope with the dimension d is of the order of λ/d. Another well known result of classical optics is that if a wave encounters an obstacle having the dimension d then the direction of the wave diverges by the angle of the order of λ/d. The inevitability of WPS for a free wave packet in classical electrodynamics is obvious from the following consideration. Suppose that a classical wave packet does not have a definite value of the momentum. Then if a is the initial width of the packet in directions perpendicular to the mean momentum, one might expect that the width will grow as a(t) = a + αct and for large values of t, a(t) ≈ αct. As follows from Eq. (24), if t ≫ t * then indeed a(t) ≈ αct. In standard quantum theory we have the same result because the coordinate and momentum wave functions are related to each other by the same Fourier transform as the coordinate and k distributions in classical electrodynamics.
The quantity N || = b/λ shows how many oscillations the oscillating exponent in Eq. (22) makes in the region where the wave function or the amplitude of the classical wave is significantly different from zero. As noted in Sec. 2, for the validity of semiclassical approximation this quantity should be very large. In nonrelativistic quantum mechanics a and b are of the same order and hence the same can be said about the quantity N ⊥ = a/λ. As noted above, in the case of the photon we don't know the relation between a and b. In terms of the quantity N ⊥ we can rewrite the expressions for t * and v * in Eq. (24) as where T is the period of the classical wave. Hence the accuracy of semiclassical approximation (or the geometrical optics approximation in classical electrodynamics) increases with the increase of N ⊥ . In Ref. [27] the problem of WPS for classical electromagnetic waves has been discussed in the Fresnel approximation (i.e. in the approximation of geometrical optics) for a two-dimensional wave packet. Equation (25) of Ref. [27] is a special case of Eq. (21) and the author of Ref. [27] shows that, in his model the wave packet spreads out in the direction perpendicular to the group velocity of the packet. As noted at the end of the preceding section, in the ultrarelativistic case the function a(t) is given by the same expression as in the nonrelativistic case but m is replaced by E/c 2 . Hence if the results of the preceding section are reformulated in terms of classical waves then m should be replaced byhω 0 /c 2 and this fact has been pointed out in Ref. [27].

Wave packet width paradox
We now consider the following important question. We assume that a classical wave packet is a collection of photons. Let a cl be the quantity a for the classical packet and a ph be a typical value of a for the photons. What is the relation between a cl and a ph ?
My observation is that physicists answer this question in different ways. Quantum physicists usually say that in typical situations a ph ≪ a cl because a cl is of macroscopic size while in semiclassical approximation the quantity a ph for each photon can be treated as the size of the region where the photon has been created. On the other hand, classical physicists usually say that a ph ≫ a cl and the motivation follows.
Consider a decomposition of some component of classical electromagnetic field into the Fourier series: where σ is the polarization, x and p are the four-vectors such that x = (ct, x) and p = (|p|c, p), the functions a(p, σ) are the same for all the components, the functions u(p, σ) depend on the component and * is used to denote the complex conjugation. Then photons arise as a result of quantization when a(p, σ) and a(p, σ) * are understood not as usual function but as operators of annihilation and creation of the photon with the quantum numbers (p, σ) and * is now understood as Hermitian conjugation. Hence the photon is described by a plane wave which has the same magnitude in all points of the space. In other words, a ph is infinitely large and a finite width of the classical wave packet arises as a result of interference of different plane waves. The above definition of the photon has at least the following inconsistency. If the photon is treated as a particle then its wave function should be normalizable while the plane wave is not normalizable. In textbooks this problem is often circumvented by saying that we consider our system in a finite box. Then the spectrum of momenta becomes finite and instead of Eq. (26) one can write where j enumerates the points of the momentum spectrum.
One can now describe quantum electromagnetic field by states in the Fock space where the vacuum vector Φ 0 satisfies the condition a(p j , σ)Φ 0 = 0, ||Φ 0 || = 1 and the operators commute as Classical states are understood such that although the number of photons is large, it is much less than the number of possible momenta and in Eq. (29) all the photons have different momenta (this is analogous to the situation in classical statistics where mean numbers of particles in each state are much less than unity). Then it is not important whether the operators (a, a * ) commute or anticommute. However, according to the Pauli theorem on spin-statistics connection [18], they should commute and this allows the existence of coherent states where many photons have the same quantum numbers. Such states can be created in lasers and they are not described by classical electrodynamics. In the next section we consider position operator for coherent states while in this section we consider only quantum description of states close to classical.
The next problem is that one should take into account that in standard theory the photon momentum spectrum is continuous. Then the above construction can be generalized as follows. The vacuum state Φ 0 satisfies the same conditions ||Φ 0 || = 1 and a(p, σ)Φ 0 = 0 while the operators (a, a * ) satisfy the following commutation relations Then a general quantum state can be written as In the approximation when a classical wave packet is understood as a collection of independent photons (see the discussion in Sec. 10), the state of this packet has the form where χ j is the wave function of the jth photon and intersections of supports of wave functions of different photons can be neglected. This is an analog of the above situation with the discrete case where it is assumed that different photons in a classical wave packet have different momenta. In other words, while the wave function of each photon can be treated as an interference of plane waves, different photons can interfere only in coherent states but not in classical wave packets. We now describe a well known generalization of the results on IRs of the Poincare algebra to the description in the Fock space (see e.g. Ref. [28] for details). If A is an operator in the space of the photon IR then a generalization of this operator to the case of the Fock space can be constructed as follows. Any operator in the space of IR can be represented as an integral operator acting on the wave function as For example, if Aχ(p, σ) = ∂χ(p, σ)/∂p then A is the integral operator with the kernel We now require that if the action of the operator A in the space of IR is defined by Eq. (33) then in the case of the Fock space this action is defined as Then it is easy to verify that if A, B and C are operators in the space of IR satisfying the commutation relation [A, B] = C then the generalizations of these operators in the Fock space satisfy the same commutation relation. It is also easy to verify that the operators generalized to the action in the Fock space in such a way are additive, i.e. for a system of n photons they are sums of the corresponding single-particle operators. In particular, the energy of the n-photon system is a sum of the energies of the photons in the system and analogously for the other representation operators of the Poincare algebra -momenta, angular momenta and Lorentz boosts. We are interested in calculating mean values of different combinations of the momentum operator. Since this operator does not act over spin variables, we will drop such variables in the (a, a * ) operators and in the functions χ j . Then the explicit form of the momentum operator is P = pa(p) * a(p)d 3 p. Since this operator does not change the number of photons, the mean values can be independently calculated in each subspace where the number of photons is N.
Suppose that the momentum of each photon is approximately directed along the z-axis and the quantity p 0 for each photon approximately equals 2πh/λ. If ∆p ⊥ is a typical uncertainty of the transversal component of the momentum for the photons then a typical value of the angular uncertainty for the photons is α ph = ∆p ⊥ /p 0 ≈ λ/(2πa ph ). The total momentum of the classical wave packet consisting of N photons is a sum of the photon momenta: P = N i=1 p (i) . Suppose that the mean value of P is directed along the z-axis and its magnitude P 0 is such that P 0 ≈ Np 0 . The uncertainty of the x component of P is ∆P x = P 2 Then in the approximation of independent photons (see the remarks after Eq. (32)) where we have taken into account that P ⊥ have the the same order of magnitude equal to ∆p ⊥ then ∆P ⊥ ≈ N 1/2 ∆p ⊥ and the angular divergence of the classical vave packet is Since the classical wave packet is described by the same wave equation as the photon wave function, its angular divergence can be expressed in terms of the parameters λ and a cl such that α cl = λ/(2πa cl ). Hence a cl ≈ N 1/2 a ph and we conclude that a ph ≪ a cl .
Note that in this derivation no position operator has been used. Although the quantities λ and a ph have the dimension of length, they are defined only from considering the photon in momentum space because, as noted in Sec. 4, for individual photons λ is understood only as 2πh/p 0 , a ph defines the width of the photon momentum wave function (see Eq. (18)) and is of the order ofh/∆p ⊥ . As noted in Secs. 3 and 5, the momentum distribution does not depend on time and hence the result a ph ≪ a cl does not depend on time too. If photons in a classical wave packet could be treated as (almost) pointlike particles then photons do not experience WPS while the WPS effect for a classical wave packet is a consequence of the fact that different photons in the packet have different momenta.
However, in standard quantum theory this scenario does not take place for the following reason. Let a cl (t) be the quantity a(t) for the classical wave packet and a ph (t) be a typical value of the quantity a(t) for individual photons. With standard position operator the quantity a ph (t) is interpreted as the spatial width of the photon coordinate wave function in directions perpendicular to the photon momentum and this quantity is time dependent. As shown in Secs. 5 and 6, a(0) = a but if t ≫ t * then a(t) is inversely proportional to a and the coefficient of proportionality is the same for the classical wave packet and individual photons (see Eq. (24)). Hence in standard quantum theory we have a paradox that after some period of time a ph (t) ≫ a cl (t) i.e. individual photons in a classical wave packet spread out in a much greater extent than the wave packet as a whole. We call this situation the wave packet width (WPW) paradox (as noted above, different photons in a classical wave packet do not interfere with each other). The reason of the paradox is obvious: if the law that the angular divergence of a wave packet is of the order of λ/a is applied to both, a classical wave packet and photons comprising it then the paradox follows from the fact that the quantities a for the photons are much less than the quantity a for the classical wave packet. Note that in classical case the quantity a cl does not have the meaning of h/∆P ⊥ and λ is not equal to 2πh/P 0 .

Wave packet spreading in coherent states
In textbooks on quantum optics the laser emission is described by the following model (see e.g. Ref. [29]). Consider a set of photons having the same momentum p and polarization σ and, by analogy with the discussion in the preceding section, suppose that the momentum spectrum is discrete. Consider a quantum superposition Ψ = ∞ n=0 c n [a(p, σ) * ] n Φ 0 where the coefficients c n satisfy the condition that Ψ is an eigenstate of the annihilation operator a(p, σ). Then the product of the coordinate and momentum uncertainties has the minimum possible valueh/2 and, as noted in Sec. 2, such a state is called coherent. However, the term coherent is sometimes used meaning that the state is a quantum superposition of many-photon states [a(p, σ) * ] n Φ 0 .
In the above model it is not taken into account that (in standard theory) photons emitted by a laser can have only a continuous spectrum of momenta. Meanwhile for the WPS effect the width of the momentum distribution is important. In this section we consider a generalization of the above model where the fact that photons have a continuous spectrum of momenta is taken into account. This will make it possible to consider the WPS effect in coherent states.
In the above formalism coherent states can be defined as follows. We assume that all the photons in the state Eq. (31) have the same polarization. Hence for describing such states we can drop the quantum number σ in wave functions and a-operators. We also assume that all photons in coherent states have the same momentum distribution. These conditions can be satisfied by requiring that coherent states have the form where c n are some coefficients. Finally, by analogy with the description of coherent states in standard textbooks on quantum optics one can require that they are eigenstates of the operator a(p)d 3 p.
The dependence of the state Ψ in Eq. (35) on t is Ψ(t) = exp(−iEt/h)Ψ where, as follows from Eqs. (13) and (34), the action of the energy operator in the Fock space is E = pca(p) * a(p)d 3 p. Since exp(iEt/h)Φ 0 = Φ 0 , it readily follows from Eq. (30) that where the relation between χ(p, t) and χ(p) = χ(p, 0) is given by Eq. (19). A problem arises how to define the position operator in the Fock space. If this operator is defined by analogy with the above construction then we get an unphysical result that each coordinate of the n-photon system as a whole is a sum of the corresponding coordinates of the photons in the system. This is an additional argument that the position operator is less fundamental than the representation operators of the Poincare algebra and its action should be defined from additional considerations. In textbooks on quantum optics the position operator for coherent states is usually defined by analogy with the position operator in nonrelativistic quantum mechanics for the harmonic oscillator problem. The motivation follows. If the energy levelshω(n + 1/2) of the harmonic oscillator are treated as states of n quanta with the energieshω then the harmonic oscillator problem can be described by the operators a and a * which are expressed in terms of the one-dimensional position and momentum operators q and p as a = (ωq + ip)/(2hω) 1/2 and a * = (ωq − ip)/(2hω) 1/2 . However, as noted above, the model description of coherent states in those textbooks is one-dimensional because the continuous nature of the momentum spectrum is not taken into account. In addition, the above results on WPS give indications that the position operator in standard theory is not consistently defined. For all these reasons a problem arises whether the requirement that the state Ψ in Eq. (35) is an eigenvector of the operator a(p)d 3 p has a physical meaning. In what follows this requirement will not be used.
In nonrelativistic classical mechanics the radius vector of a system of n particles as a whole (the radius vector of the center of mass) is defined as R = (m 1 r 1 + ... + m n r n )/(m 1 + ... + m n ) and in works on relativistic classical mechanics it is usually defined as R = (ǫ 1 (p 1 )r 1 + ... + ǫ n (p n )r n )/(ǫ 1 (p 1 ) + ... + ǫ n (p n )) where ǫ i (p i ) = (m 2 i + p 2 i ) 1/2 . Hence if all the particles have the same masses and momenta, R = (r 1 + ... + r n )/n. These remarks make it reasonable to define the position operator for coherent states as follows. Let x j be the jth component of the position operator in the space of IR and A j (p, p ′ ) be the kernel of this operator. Then in view of Eq. (34) the action of the operator X j on the state Ψ(t) in Eq. (35) can be defined as If x j (t) and x 2 j (t) are the mean values of the operators x j and x 2 j , respec-tively then as follows from the definition of the kernel of the operator x j and in the case of IR the uncertainty of the quantity x j is ∆x j (t) = [x 2 j (t) − x j (t) 2 ] 1/2 . At the same time, if X j (t) and X 2 j (t) are the mean values of the operators X j and X 2 j , respectively then and the uncertainty of the quantity X j is ∆X j (t) = [X 2 j (t) − X j (t) 2 ] 1/2 . Our goal is to express ∆X j (t) in terms of x j (t), x 2 j (t) and ∆x j (t). If the function χ(p, t) is normalized to one (see Eq. (6)) then, as follows from Eq. (30), A direct calculation using Eqs. (30), (37), (38) and (39) gives It now follows from Eq. (40) and the definitions of the quantities ∆x j (t) and ∆X j (t) that Equation (42) is the key result of this section. It has been derived without using a specific choice of the single photon position operator. The consequence of this result follows. If the main contribution to the state Ψ(t) in Eq. (36) is given by very large values of n then |c 0 | is very small and the first term in this expression can be neglected. Suppose that the main contribution is given by terms where n is of the order ofn. Then, as follows from Eqs. (40) and (42), ∆X j (t) is of the order of ∆x j (t)/n 1/2 . This means that for coherent states where the main contribution is given by very large numbers of photons the effect of WPS is pronounced in a much less extent than for single photons.

Experimental consequences of WPS in standard theory
The problem of explaining the redshift phenomenon has a long history. Different competing approaches can be divided into two big sets which we call Theory A and Theory B. In Theory A the redshift has been originally explained as a manifestation of the Doppler effect but in recent years the cosmological and gravitational redshifts have been added to the consideration. In this theory the interaction of photons with the interstellar medium is treated as practically not important, i.e. it is assumed that with a good accuracy we can treat photons as propagating in empty space. On the contrary, in Theory B, which is often called the tired-light theory, the interaction of photons with the interstellar medium is treated as a main reason for the redshift. At present the majority of physicists believe that Theory A explains the astronomical data better than Theory B. Even some physicists working on Theory B acknowledged that any sort of scattering of light would predict more blurring than is seen (see e.g. the article "Tired Light" in Wikipedia). A problem arises whether or not WPS of the photon wave function is important for explaining the redshift. One might think that this effect is not important since a considerable WPS would also blur the images more than what is seen. However, as shown in the previous discussion, WPS is an inevitable consequence of standard quantum theory and moreover this effect also exists in classical electrodynamics. Hence it is not sufficient to just say that a considerable WPS is excluded by observations. One should try to estimate the importance of WPS and to understand whether our intuition is correct or not.
As follows from these remarks, in Theory A it is assumed that with a good accuracy we can treat photons as propagating in empty space. It is also reasonable to expect (see the discussion in the next section) that photons from distant stars practically do not interact with each other. Hence the effect of WPS can be considered for each photon independently and the results of the preceding sections make it possible to understand what experimental consequences of WPS are.
A question arises what can be said about characteristics of photons coming to Earth from distance objects. Since wave lengths of such photons are typically much less than characteristic dimensions of obstacles one might think that the radiation of stars can be described in the geometrical optics approximation. As discussed in Sec. 6, this approximation is similar to semiclassical approximation in quantum theory. This poses a question whether this radiation can be approximately treated as a collection of photons moving along classical trajectories.
Consider, for example, the Lyman transition 2P → 1S in the hydrogen atom on the Sun. In this case the mean energy of the photon is E 0 = 10.2eV , its wave length is λ = 121.6nm and the lifetime is τ = 1.6 · 10 −9 s. The phrase that the lifetime is τ is interpreted such that the uncertainty of the energy ish/τ , the uncertainty of the longitudinal momentum ish/cτ and b is of the order of cτ ≈ 0.48m. In this case the photon has a very narrow energy distribution since the mean value of the momentum p 0 = E 0 /c satisfies the condition p 0 b ≫h. At the same time, since the orbital angular momentum of the photon is a small quantity, the direction of the photon momentum cannot be semiclassical. Qualitative features of such situations can be described by the following model.
Suppose that the photon momentum wave function is spherically symmetric and has the form where C is a constant, and p is the magnitude of the momentum. Then the main contribution to the normalization integral is given by the region of p where |p − p 0 | is of the order ofh/b and in this approximation the integration over p can be taken from −∞ to ∞. As a result, the function normalized to one has the form The dependence of this function on t is χ( where r 0 (t) = r 0 + ct. The coordinate wave function is ψ(r, t) = 1 (2πh) 3/2 χ(p, t)e ipr/h d 3 p Since χ(p, t) is spherically symmetric it is convenient to decompose e ipr/h as a sum of spherical harmonics and take into account that only the term corresponding to l = 0 contributes to the integral. This term is j 0 (pr/h) = sin(pr/h)/(pr/h). Then the integral can be again taken from −∞ to ∞ and the result is We assume that r 0 (t) ≫ b and hence the term with exp[−(r + r 0 (t)) 2 /2b 2 ] can be neglected and r in the denominator can be replaced by r 0 (t). As follows from the above results, the mean value of r is r 0 (t). If λ is defined as λ = 2πh/p 0 then the requirement that p 0 b ≫h implies that b ≫ λ. The conditions p 0 b/h ≫ 1 and r 0 (t) ≫ b imply that the radial part of the photon state is semiclassical while the angular part is obviously strongly nonclassical. Suppose that we want to detect the photon inside the volume V where the coordinates are be the characteristic function of V , i.e. g(r) = 1 when r ∈ V and g(r) = 0 otherwise. Let P be the projector acting on wave functions as Pψ(r) = g(r)ψ(r). Then Assume that r 0 (t) ≫ d x , d y . Then r − r 0 (t) ≈ z − r 0 (t) + (x 2 + y 2 )/2r 0 (t). We also assume that r 0 (t) is so large then r 0 (t)λ ≫ (d 2 x + d 2 y ). Then We also assume that d z ≫ b. Then a simple calculation shows that where S = 4d x d y is the area of the cross section of V by the plane z = r 0 (t). The meaning of Eq. (50) is obvious: ||Pψ(r, t)|| 2 is the ratio of the cross section to the area of the sphere with the radius r 0 (t).
Let us now calculate the momentum distribution in the function Pψ(r, t). This distribution is defined as As follows from Eq. (48) where A(t) is a function of t. This result is similar to the well known result in optics that the best angular resolution is of the order of λ/d where d is the dimension of the optical device (see e.g. textbooks [29]). As noted in Sec. 1, the reason of the similarity is that in quantum theory the coordinate and momentum representations are related to each other by the Fourier transform by analogy with classical electrodynamics. Note also that since the fall off of the function j 0 (x) = sinx/x is not rapid enough when x increases, in the case when many photons are detected, a considerable part of them might be detected in the angular range much greater than λ/d. Let L be the distance to a pointlike source of spherically symmetric photons. From geometrical consideration one might expect that photons from this source will be detected in the angular range of the order of d/L. This quantity does not depend on λ while the quantity λ/d does not depend on L. Therefore the result given by Eq. (52) is counterintuitive. This problem is discussed in Sec. 13.
If R is the radius of a star then one might expect that the star will be visible in the angular range (R + d)/L ≈ R/L. Hence the standard result predicts that if λ/d ≥ R/L then the image of the star will be blurred. The experimental verification of this prediction is problematic since the quantities R/L are very small and at present star radii cannot be measured directly. Conclusions about them are made from the data on luminosity and temperature assuming that the major part of the radiation from stars comes not from transitions between atomic levels but from processes which can be approximately described as a blackbody radiation.
A theoretical model describing blackbody radiation (see e.g. Ref. [30]) is such that photons are treated as an ideal Bose gas weakly interacting with matter and such that typical photon energies are not close to energies of absorption lines for that matter (hence the energy spectrum of photons is almost continuous). It is also assumed that the photons are distributed over states with definite values of momenta. With these assumptions one can derive the famous Planck formula for the spectral distribution of the blackbody radiation (this formula is treated as marking the beginning of quantum theory). As explained in Ref. [30], when the photons leave the black body, their distribution in the phase space can be described by the Liouville theorem; in particular it implies that the photons are moving along classical trajectories.
Although the blackbody model is not ideal, numerous experimental data indicate that it works with a good accuracy. One of the arguments that the major part of the radiation consists of semiclassical photons is that the data on deflection of light by the Sun are described in the eikonal approximation which shows that the light from stars consists mainly of photons approximately moving along classical trajectories.
If we accept those arguments then the main part of photons emitted by stars can be described in the formalism considered in Sec. 5. In that case we cannot estimate the quantity b as above and it is not clear what criteria can be used for estimating the quantity a. The estimation a ≈ b ≈ 0.48m seems to be very favorable since one might expect that the value of a is of atomic size, i.e. much less than 0.48m. With this estimation for yellow light (with λ = 580nm) N ⊥ = a/λ ≈ 8 · 10 5 . So the value of N ⊥ is rather large and in view of Eq. (25) one might think that the effect of spreading is not important. However, this is not the case because, as follows from Eq. (25), t * ≈ 0.008s. Since the distance between the Sun and the Earth is approximately t = 8 light minutes and this time is much greater than t * , the value of a(t) (which can be called the half-width of the wave packet) when the packet arrives to the Earth is v * t ≈ 28km. In this case standard geometrical interpretation does not apply. In addition, if we assume that the initial value of a is of the order of several wave lengths then the value of N ⊥ is much less and the width of the wave packet coming to the Earth is much greater. An analogous estimation shows that even in the favorable scenario the half-width of the wave packet coming to the Earth from Sirius will be approximately equal to 15 · 10 6 km but in less favorable situations the half-width will be much greater. Hence we come to the conclusion that even in favorable scenarios the assumption that photons are moving along classical trajectories does not apply and a problem arises whether or not this situation is in agreement with experiment.
For illustration we consider the following example. Let the Earth be at point A and the center of Sirius be at point B. Suppose for simplicity that the Earth is a pointlike particle. Suppose that Sirius emitted a photon such that its wave function in momentum space has a narrow distribution around the mean value directed not along BA but along BC such that the angle between BA and BC is α. As noted in Sec. 5, there is no WPS in momentum space but, as follows from Eq. (24), the function a(t) defining the mean value of the radius of the coordinate photon wave function in perpendicular directions is a rapidly growing function of t. Let us assume for simplicity that α ≪ 1. Then if L is the length of AB, the distance from A to BC is approximately d = Lα. So if this photon is treated as a point moving along the classical trajectory then the observer on the Earth will not see the photon. Let us now take into account the effect of WPS in directions perpendicular to the photon momentum. The front of the photon wave function passes the Earth when t ≈ t 1 = L/c. As follows from Eq. (24) and the definition of the quantity N ⊥ , if t 1 ≫ t * then a(t 1 ) = L/(2πN ⊥ ). If a(t 1 ) is of the order of d or greater and we look in the direction AD such that AD is antiparallel to BC then there is a nonzero probability that we will detect this photon. So we can see photons coming from Sirius in the angular range which is of the order of a(t 1 )/L. If R is the radius of Sirius and a(t 1 ) is of the order of R or greater, the image of Sirius will be blurred. As noted above, a very optimistic estimation of a(t 1 ) is 15 · 10 6 km. However, one can expect that a more realistic value of N ⊥ is not so large and then the estimation of a(t 1 ) gives a much greater value. Since R = 1.1 · 10 6 km this means that the image of Sirius will be extremely blurred. Moreover, in the above angular range we can detect photons emitted not only by Sirius but also by other objects. Since the distance to Sirius is "only" 8.6 light years, for the majority of stars the effect of WPS will be pronounced even to a much greater extent. So if WPS is considerable then we will see not separate stars but an almost continuous background from many objects.
In the case of planets it is believed that we see a light reflected according to the laws of geometrical optics. Therefore photons of this light are in wave packet states and WPS for them can be estimated by using Eq. (24). The effect of blurring depends on the relation between the radii of planets and the corresponding quantities a(t 1 ) = L/(2πN ⊥ ). Then it is obvious that if N ⊥ is not very large then even the images of planets will be blurred.
In the infrared and radio astronomy wave lengths are much greater than in the optical region but typical values of a ph are expected to be much greater. As a consequence, predictions of standard quantum theory on blurring of astronomical images are expected to be qualitatively the same as in the optical region.
In the case of gamma-ray bursts (GRBs) wave lengths are much less than in the optical region but this is outweighed by the facts that, according to the present understanding of the GRB phenomenon (see e.g. Ref. [31]), gamma quanta created in GRBs typically travel to Earth for billions of years and typical values of a ph are expected to be much less than in the optical region. The location of sources of GBRs are determined with a good accuracy and the data can be explained only assuming that the gamma quanta are focused into narrow jets (i.e. GRBs are not spherically symmetric) which are observable when Earth lies along the path of those jets. However, in view of the above discussion, the results on WPS predicted by standard quantum theory are fully incompatible with the data on GRBs.
A striking example illustrating the problem with the WPS effect follows. The phenomenon of the relic (CMB) radiation is treated as a case where the approximation of the blackbody radiation works with a very high accuracy. As noted above, photons emitted in this radiation are treated as moving along classical trajectories i.e. that they are in wave packet states. Since their wave lengths are much greater than wave lengths in the optical region and the time of their travel to Earth is several billions of years, the quantity a(t 1 ) should be so large that no anisotropy of CMB should be observable. Meanwhile the anisotropy is observable and in the literature different mechanisms of the anisotropy are discussed (see e.g. Ref. [32]). However, the effect of WPS is not discussed.
On the other hand, the effect of WPS is important only if a particle travels a rather long distance. Hence one might expect that in experiments on the Earth this effect is negligible. Indeed, one might expect that in typical experiments on the Earth the quantity t 1 is so small that a(t 1 ) is much less than the size of any macroscopic source of light. However, a conclusion that the effect of WPS is negligible for any experiment on the Earth might be premature.
As an example, consider the case of protons in the LHC accelerator. According to Ref. [33], protons in the LHC ring injected at the energy E = 450GeV should be accelerated to the energy E = 7T eV within one minute during which the protons will turn around the 27km ring approximately 674729 times. Hence the length of the proton path is of the order of 18 · 10 6 km. The protons cannot be treated as free particles since they are accelerated by strong magnets. A problem of how the width of the proton wave function behaves in the presence of strong electromagnetic field is very complicated and the solution of the problem is not known yet. It is always assumed that the WPS effect for the protons can be neglected. We will consider a model problem of the WPS for a free proton which moves for t 1 = 1min with the energy in the range [0. 45,7]T eV .
In nuclear physics the size of the proton is usually assumed to be a quantity of the order of 10 −13 cm. Therefore for estimations we take a = 10 −13 cm. Then the quantity t * defined after Eq. (23) is not greater than 10 −19 s, i.e. t * ≪ t 1 . Hence, as follows from Eq. (23), the quantity a(t 1 ) is of the order of 500km if E = 7T eV and in the case when E = 450GeV this quantity is by a factor of 7/0.45 ≈ 15.6 greater. This fully unrealistic result cannot be treated as a paradox since, as noted above, the protons in the LHC ring are not free. Nevertheless it shows that a problem of what standard theory predicts on the width of proton wave functions in the LHC ring is far from being obvious.
Consider now WPS effects for radio wave photons. In radiolocation it is important that a beam from a directional antenna has a narrow angular distribution and a narrow distribution of wave lengths. Hence photons from the beam can be treated as (approximately) moving along classical trajectories. This makes it possible to communicate even with very distant space probes. For this purpose a set of radio telescopes can be used but for simplicity we consider a model where signals from a space probe are received by one radio telescope having the diameter D of the dish.
The Cassini spacecraft can transmit to Earth at three radio wavelengths: 14cm, 4cm and 1cm [34]. A radio telescope on Earth can determine the position of Cassini with a good accuracy if it detects photons having momenta in the angular range of the order of D/L where L is the distance to Cassini. The main idea of using a system of radiotelescopes is to increase the effective value of D. As a consequence of the fact that the radio signal sent from Cassini has an angular divergence which is much greater than D/L, only a small part of photons in the signal can be detected.
Consider first the problem on classical level. For the quantity a = a cl we take the value of 1m which is of the order of the radius of the Cassini antenna. If α = λ/(2πa) and L(t) is the length of the classical path then, as follows from Eq. (24), a cl (t) ≈ L(t)α. As a result, even for λ = 1cm we have a cl (t) ≈ 1.6 · 10 6 km. Hence if the photons in the beam are treated as (approximately) pointlike particles, one might expect that only a [D/a cl (t)] 2 part of the photons can be detected.
Consider now the problem on quantum level. The condition t ≫ t * is satisfied for both, the classical and quantum problems. Then, as follows from Eq. (24), a ph (t) = a cl (t)a cl /a ph , i.e. the quantity a ph (t) is typically greater than a cl (t) and in Sec. 7 this effect is called the WPW paradox. The fact that only photons in the angular range D/L can be detected can be described by projecting the states χ = χ(p, t) (see Eqs. (18), and (19)) onto the states χ 1 = Pχ where χ 1 (p, t) = ρ(p)χ(p, t) and the form factor ρ(p) is significant only if p is in the needed angular range. We choose ρ(p) = exp(−p 2 ⊥ a 2 1 /2h 2 ) where a 1 is of the order ofhL/(p 0 D). Since a 1 ≫ a ph , it follows from Eqs. (18), and (19) that ||Pχ|| 2 = (a ph /a 1 ) 2 . Then, as follows from Eq. (24), (a ph /a 1 ) 2 is of the order of [D/a ph (t)] 2 as expected and this quantity is typically much less than [D/a cl (t)] 2 . Hence the WPW paradox would make communications with space probes much more difficult.
Consider now the effect called Shapiro time delay. The meaning of the effect follows. An antenna on Earth sends a signal to Mercury, Venus or an interplanetary space probe and receives the reflected signal. If the path of the signal nearly grazes the Sun then the gravitational influence of the Sun deflects the path from a straight line. As a result, the path becomes longer by S ≈ 75km and the signals arrive with a delay S/c ≈ 250µs. This effect is treated as the fourth test of General Relativity and its theoretical consideration is based only on classical geometry. In particular, it is assumed that the radio signal is moving along the classical trajectory.
However, in standard quantum theory the length of the path has an uncertainty which can be defined as follows. As a consequence of WPS, the uncertainty of the path is In contrast to the previous example, this quantity is quadratic in α and one might think that it can be neglected. However, this is not the case. For example, in the first experiment on measuring the Shapiro delay [35] signals with the frequency 8GHz were sent by the MIT Haystack radar antenna [36] having the diameter 37m. If we take for a ph a very favorable value which equals the radius of the antenna then α 2 ≈ 10 −7 . As a result, when the signal is sent to Venus, ∆L(t) ≈ 25km but since a ph is typically much less than a cl then in view of the WPW paradox the value of ∆L(t) will be much greater. However, even the result 25km is incompatible with the fact that the accuracy of the experiment was of the order of 5%.
In classical consideration the Shapiro delay is defined by the parameter γ which depends on the theory and in General Relativity γ = 1. At present the available experimental data are treated such that the best test of γ has been performed in measuring the Shapiro delay when signals from the DSS-25 antenna [37] with the frequencies 7.175GHz and 34.136GHz were sent to the Cassini spacecraft when it was 7AU away from the Earth. The results of the experiment are treated such that γ −1 = (2.1 ±2.3) · 10 −5 [38]. For estimating the quantity ∆L(t) in this case we take a favorable scenario when the frequency is 34.136GHz and a ph equals the radius of the DSS-25 antenna which is 17m. Then α ≈ 8 · 10 −5 and ∆L(t) ≈ 6.7km but in view of the WPW effect this quantity will be much greater. This is obviously incompatible with the fact that the accuracy of computing γ is of the order of 10 −5 .
The last example follows. The astronomical objects called pulsars are treated such that they are neutron stars with radii much less than radii of ordinary stars. Therefore if mechanisms of pulsar electromagnetic radiation were the same as for ordinary stars then the pulsars would not be visible. The fact that pulsars are visible is explained as a consequence of the fact that they emit beams of light which can only be seen when the light is pointed in the direction of the observer with some periods which are treated as periods of rotation of the neutron stars. In popular literature this is compared with the light of a lighthouse. However, by analogy with the case of a signal sent from Cassini, only a small part of photons in the beam can reach the Earth. At present the pulsars have been observed in different regions of the electromagnetic spectrum but the first pulsar called PSR B1919+21 was discovered in 1967 as a radio wave radiation with λ ≈ 3.7m [39]. This pulsar is treated as the neutron star with the radius R = 0.97km and the distance from the pulsar to the Earth is 2283 light years. If for estimating a cl (t) we assume that a cl = R then we get α ≈ 6 · 10 −4 and a cl (t) ≈ 1.3ly ≈ 12 · 10 12 km. Such an extremely large value of spreading poses a problem whether even predictions of classical electrodynamics are compatible with the fact that pulsars are observable. However, in view of the WPW paradox, the value of a ph (t) will be even much greater and no observation of pulsars would be possible.
Our conclusion is that we have several fundamental paradoxes posing a problem whether predictions of standard quantum theory for the WPS effect are correct.
10 Discussion: is it possible to avoid the WPS paradoxes in standard theory?
As shown in the preceding section, if one assumes that photons coming to Earth do not interact with the interstellar or interplanetary medium and with each other then a standard treatment of the WPS effect contradicts the facts that there is no blurring of astronomical images, anisotropy of the CMB radiation, GRBs and pulsars are observable, communication with space probes is possible and the Shapiro delay can be explained in classical theory. Hence a question arises whether this assumption is legitimate. As shown in textbooks on quantum optics (see e.g. Ref. [29]), quantum states describing the laser emission are strongly coherent and the approximation of independent photons is not legitimate. As shown in Sec. 8, the WPS effect in coherent states is pronounced in a much less extent than for individual photons. However, laser emission can be created only at very special conditions when energy levels are inverted, the emission is amplified in the laser cavity etc. At the same time, the main part of the radiation emitted by stars is understood such that it can be approximately described as the blackbody radiation and in addition a part of the radiation consists of photons emitted from different atomic energy levels. In that case the emission of photons is spontaneous rather than induced and one might think that the photons can be treated independently. Several authors (see e.g. Ref. [40] and references therein) discussed a possibility that at some conditions the inverted population and amplification of radiation in stellar atmospheres might occur and so a part of the radiation can be induced. This problem is now under investigation. Hence we adopt a standard assumption that a main part of the radiation from stars is spontaneous. In addition, there is no reason to think that radiation of GRBs, radio antennas, space probes or pulsars is laser like.
The next question is whether the interaction of photons in the above phenomena is important or not. As explained in standard textbooks on QED (see e.g. Ref. [24]), the photon-photon interaction can go only via intermediate creation of virtual electron-positron or quark-antiquark pairs. If ω is the photon frequency, m is the mass of the charged particle in the intermediate state and e is the electric charge of this particle then in the case whenhω ≪ mc 2 the total cross section of the photon-photon interaction is [24] σ = 56 5πm 2 139 90 2 ( For photons of visible light the quantitieshω/(mc 2 ) and σ are very small and for radio waves they are even smaller by several orders of magnitude. At present the effect of the direct photon-photon interaction has not been detected, and experiments with strong laser fields were only able to determine the upper limit of the cross section [41]. The problem of WPS in the ultrarelativistic case has been discussed in a wide literature. As already noted, in Ref. [27] the effect of WPS has been discussed in the Fresnel approximation for a two-dimensional model and the author shows that in the direction perpendicular to the group velocity of the wave spreading is important. He considers WPS in the framework of classical electrodynamics. We believe that considering this effect from quantum point of view is even simpler since the photon wave function satisfies the relativistic Schrödinger equation which is linear in ∂/∂t. As noted in Sec. 6, this function also satisfies the wave equation but it is simpler to consider an equation linear in ∂/∂t than that quadratic in ∂/∂t. However, in classical theory there is no such an object as the photon wave function and hence one has to solve either a system of Maxwell equations or the wave equation. There is also a number of works where the authors consider WPS in view of propagation of classical waves in a medium such that dissipation is important (see e.g. Ref. [42]). In Ref. [43] the effect of WPS has been discussed in view of a possible existence of superluminal neutrinos. The authors consider only the dynamics of the wave packet in the longitudinal direction in the framework of the Dirac equation. They conclude that wave packets describing ultrarelativistic fermions do not experience WPS in this direction. However, the authors do not consider WPS in perpendicular directions.
In view of the above discussion, standard treatment of WPS leads to several fundamental paradoxes of modern theory. To the best of our knowledge, those paradoxes have never been discussed in the literature. For resolving the paradoxes one could discuss several possibilities. One of them might be such that the interaction of light with the interstellar or interplanetary medium cannot be neglected. On quantum level a process of propagation of photons in the medium is rather complicated because several mechanisms of propagation should be taken into account. For example, a possible process is such that a photon can be absorbed by an atom and reemitted. This process makes it clear why the speed of light in the medium is less than c: because the atom which absorbed the photon is in an excited state for some time before reemitting the photon. However, this process is also important from the following point of view: even if the coordinate photon wave function had a large width before absorption, as a consequence of the collapse of the wave function, the wave function of the emitted photon will have in general much smaller dimensions since after detection the width is defined only by parameters of the corresponding detector. If the photon encounters many atoms on its way, this process does not allow the photon wave function to spread out significantly. Analogous remarks can be made about other processes, for example about rescattering of photons on large groups of atoms, rescattering on elementary particles if they are present in the medium etc.
However, such processes have been discussed in Theory B and, as noted in Sec. 9, they probably result in more blurring than is seen.
The interaction of photons with the interstellar or interplanetary medium might also be important in view of hypotheses that the density of the medium is much greater than usually believed. Among the most popular scenarios are dark energy, dark matter etc. As shown in our papers (see e.g. Refs. [7,8,44] and references therein), the phenomenon of the cosmological acceleration can be easily and naturally explained from first principles of quantum theory without involving dark energy, empty space-background and other artificial notions. However, the other scenarios seem to be more realistic and one might expect that they will be intensively investigated. A rather hypothetical possibility is that the propagation of photons in the medium has something in common with the induced emission when a photon induces emission of other photons in practically the same direction. In other words, the interstellar medium amplifies the emission as a laser. This possibility seems to be not realistic since it is not clear why the energy levels in the medium might be inverted.
We conclude that at present in standard theory there are no realistic scenarios which can explain the WPS paradoxes. In the remaining part of the paper we propose a solution of the problem proceeding from a consistent definition of the position operator.

Consistent construction of position operator
The above results give grounds to think that the reason of the paradoxes which follow from the behavior of the coordinate photon wave function in perpendicular directions is that standard definition of the position operator in those directions does not correspond to realistic measurements of coordinates. Before discussing a consistent construction, let us make the following remark. On elementary level students treat the mass m and the velocity v as primary quantities such that the momentum is mv and the kinetic energy is mv 2 /2. However, from the point of view of Special Relativity, the primary quantities are the momentum p and the total energy E and then the mass and velocity are defined as m 2 c 4 = E 2 − p 2 c 2 and v = pc 2 /E, respectively. This example has the following analogy. In standard quantum theory the primary operators are the position and momentum operators and the orbital angular momentum operator is defined as their vector product. However, the operators P and L are consistently defined as representation operators of the Poincare algebra while the definition of the position operator is a problem. Hence a question arises whether the position operator can be defined in terms of P and L.
One might seek the position operator such that on classical level the relation r × p = L will take place. Note that on quantum level this relation is not necessary. Indeed, the very fact that some elementary particles have a half-integer spin shows that the total angular momentum for those particles does not have the or-bital nature but on classical level the angular momentum can be always represented as a cross product of the radius-vector and standard momentum. However, if the values of p and L are known and p = 0 then the requirement that r × p = L does not define r uniquely. One can define parallel and perpendicular components of r as r = r || p/p + r ⊥ where p = |p|. Then the relation r × p = L defines uniquely only r ⊥ . Namely, as follows from this relation, r ⊥ = (p × L)/p 2 . On quantum level r ⊥ should be replaced by a selfadjoint operator R ⊥ defined as where e jkl is the absolutely antisymmetric tensor, e 123 = 1, a sum over repeated indices is assumed and we assume that if L is given by Eq. (13) then the orbital momentum ishL. We define the operators F and G such that R ⊥ =hF/p and G is the operator of multiplication by the unit vector p/p. A direct calculation shows that these operators satisfy the following relations: The first two relations show that F and G are the vector operators as expected. The result for the anticommutator shows that on classical level F × G = L and the last two relations show that on classical level the operators in the triplet (F, G, L) are mutually orthogonal. Note that if the momentum distribution is narrow and such that the mean value of the momentum is directed along the z axis then it does not mean that on the operator level the z component of the operator R ⊥ should be zero. The matter is that the direction of the momentum does not have a definite value. One might expect that only the mean value of the operator R ⊥ will be zero or very small.
In addition, an immediate consequence of the definition (54) follows: Since the momentum and angular momentum operators commute with the Hamiltonian, the distribution of all the components of r ⊥ does not depend on time. In particular, there is no WPS in directions defined by R ⊥ . This is also clear from the fact that R ⊥ = hF/p where the operator F acts only over angular variables and the Hamiltonian depends only on p. On classical level the conservation of R ⊥ is obvious since it is defined by the conserving quantities p and L. It is also obvious that since a free particle is moving along a straight line, a vector from the origin perpendicular to this line does not change with time.
The above definition of the perpendicular component of the position operator is well substantiated since on classical level the relation r × p = L has been verified in numerous experiments. However, this relation does not make it possible to define the parallel component of the position operator and a problem arises what physical arguments should be used for that purpose.
A direct calculation shows that if ∂/∂p is written in terms of p and angular variables then where the operator R || acts only over the variable p: The correction 1/p is related to the fact that the operator R || is Hermitian since in variables (p, n) the scalar product is given by where do is the element of the solid angle. While the components of standard position operator commute with each other, the operators R || and R ⊥ satisfy the following commutation relation: An immediate consequence of these relation follows: Since the operator R || and different components of R ⊥ do not commute with each other, the corresponding quantities cannot be simultaneously measured and hence there is no wave function ψ(r || , r ⊥ ) in coordinate representation.
In standard theory −h 2 (∂/∂p) 2 is the operator of the quantity r 2 . As follows from Eq. (55), the two terms in Eq. (56) are not strictly orthogonal and on the operator level −h 2 (∂/∂p) 2 = R 2 || + R 2 ⊥ . A direct calculation using Eqs. (55) and (56) gives in agreement with the expression for the Laplacian in spherical coordinates. In semiclassical approximation, (h 2 /p 2 ) ≪ R 2 ⊥ since the eigenvalues of L 2 are l(l + 1), in semiclassical states l ≫ 1 and, as follows from Eq. (55), R 2 ⊥ = [h 2 (l 2 + l + 1)/p 2 ]. As follows from Eq. (59), [R || , p] = −ih, i.e. in the longitudinal direction the commutation relation between the coordinate and momentum is the same as in standard theory. One can also calculate the commutators between the different components of R ⊥ and p. Those commutators are not given by such simple expressions as in standard theory but it is easy to see that all of them are of the order ofh as it should be. Equation (56) can be treated as an implementation of the relation r = r || p/|p| + r ⊥ on quantum level. As argued in Secs. 1 and 2, the standard position operator ih∂/∂p j in the direction j is not consistently defined if p j is not sufficiently large. One might think however that since the operator R || contains ih∂/∂p, it is defined consistently if the magnitude of the momentum is sufficiently large.
In summary, we propose to define the position operator not by the set (ih∂/∂p x , ih∂/∂p y , ih∂/∂p z ) but by the operators R || and R ⊥ . Those operators are defined from different considerations. As noted above, the definition of R ⊥ is based on solid physical facts while the definition of R || is expected to be more consistent than the definition of standard position operator. However, this does not guarantee that the operator R || is consistently defined in all situations. As argued in Ref. [45], in a quantum theory over a Galois field an analogous definition is not consistent for macroscopic bodies (even if p is large) since in that case semiclassical approximation is not valid. In the remaining part of the paper we assume that for elementary particles the above definition of R || is consistent in situations when semiclassical approximation applies.
One might pose the following question. What is the reason to work with the parallel and perpendicular components of the position operator separately if, according to Eq. (56), their sum is the standard position operator? The explanation follows.
In quantum theory every physical quantity corresponds to a selfadjoint operator but the theory does not define explicitly how a quantity corresponding to a specific operator should be measured. There is no guaranty that for each selfadjoint operator there exists a physical quantity which can be measured in real experiments.
Suppose that there are three physical quantities corresponding to the selfadjoint operators A, B and C such that A + B = C. Then in each state the mean values of the operators are related asĀ +B =C but in situations when the operators A and B do not commute with each other there is no direct relation between the distributions of the physical quantities corresponding to the operators A, B and C. For example, in situations when the physical quantities corresponding to the operators A and B are semiclassical and can be measured with a good accuracy, there is no guaranty that the physical quantity corresponding to the operator C can be measured in real measurements. As an example, the physical meaning of the quantity corresponding to the operator L x + L y is problematic. Another example is the situation with WPS in directions perpendicular to the particle momentum. Indeed, as noted above, the physical quantity corresponding to the operator R ⊥ does not experience WPS and, as shown in Sec. 13, in the case of ultrarelativistic particles there is no WPS in the parallel direction as well. However, standard position operator is a sum of noncommuting operators corresponding to well defined physical quantities and, as a consequence, there are situations when standard position operator defines a quantity which cannot be measured in real experiments.

New position operator and semiclassical states
As noted in Sec. 2, in standard theory states are treated as semiclassical in greatest possible extent if ∆r j ∆p j =h/2 for each j and such states are called coherent. The existence of coherent states in standard theory is a consequence of commutation relations [p j , r k ] = −ihδ jk . Since in our approach there are no such relations, a problem arises how to construct states in which all physical quantities p, r || , n and r ⊥ are semiclassical.
One of the ways to prove this is to calculate the mean values and uncertainties of the operator R || and all the components of the operator R ⊥ in the state defined by Eq. (18). The calculation is not simple since it involves three-dimensional integrals with Gaussian functions divided by p 2 . The result is that these operators are semiclassical in the state (18) if p 0 ≫h/b, p 0 ≫h/a and r 0z has the same order of magnitude as r 0x and r 0y .
However, a more natural approach follows. Since R ⊥ =hF/p, the operator F acts only over the angular variable n and R || acts only over the variable p, it is convenient to work in the representation where the Hilbert space is the space of functions χ(p, l, µ) such that the scalar product is and l and µ are the orbital and magnetic quantum numbers, respectively, i.e.
In such an approximation one can define wave functions ψ(r) in the r || representation. By analogy with the consideration in Secs. 2 and 3 we define where the integral is formally taken from −∞ to ∞. Then Note that here the quantities r and r 0 have the meaning of coordinates in the direction parallel to the particle momentum, i.e. they can be positive or negative. Consider now states where the quantities F and G are semiclassical. One might expect that in semiclassical states the quantities l and µ are very large. In this approximation, as follows from Eqs. (64) and (65), the action of the operators F and G can be written as In view of the remark in Sec. 2 about semiclassical vector quantities, consider a state χ(l, µ) such that χ(l, µ) = 0 only if l ∈ [l 1 , l 2 ], µ ∈ [µ 1 , µ 2 ] where l 1 , µ 1 > 0, δ 1 = l 2 +1−l 1 , δ 2 = µ 2 +1−µ 1 , δ 1 ≪ l 1 , δ 2 ≪ µ 1 µ 2 < l 1 and µ 1 ≫ (l 1 −µ 1 ). This is the state where the quantity µ is close to its maximum value l. As follows from Eqs. (62) and (63), in this state the quantity L z is much greater than L x and L y and, as follows from Eq. (69), the quantities F z and G z are small. So on classical level this state describes a motion of the particle in the xy plane. The quantity L z in this state is obviously semiclassical since χ(l, µ) is the eigenvector of the operator L z with the eigenvalue µ. As follows from Eq. (69), the action of the operators (F + , F − , G + , G − ) on this state can be described by the following approximate formulas: where l 0 is a value from the interval [l 1 , l 2 ]. Consider a simple model when χ(l, µ) = exp[i(lα − µβ)]/(δ 1 δ 2 ) 1/2 when l ∈ [l 1 , l 2 ] and µ ∈ [µ 1 , µ 2 ]. Then a simple direct calculation using Eq. (70) gives where γ = α − β. Hence the vector quantities F and G are semiclassical since either |cosγ| or |sinγ| or both are much greater than (δ 1 + δ 2 )/(δ 1 δ 2 ).

New position operator and wave packet spreading
If the space of states is implemented according to the scalar product (61) then the dependence of the wave function on t is As noted in Secs. 3 and 5, there is no WPS in momentum space and this is natural in view of momentum conservation. Then, as already noted, the distribution of the quantity r ⊥ does not depend on time and this is natural from the considerations described in Sec. 11. At the same time, the dependence of the r || distribution on time can be calculated in full analogy with Sec. 3. Indeed, consider, for example a function χ(p, l, µ, t = 0) having the form χ(p, l, µ, t = 0) = χ(p, t = 0)χ(l, µ) Then, as follows from Eqs. (67) and (72), Suppose that the function χ(p, t = 0) is given by Eq. (66). Then in full analogy with the calculations in Sec. 3 we get that in the nonrelativistic case the r || distribution is defined by the wave function ψ(r, t) = 1 where v 0 = p 0 /m is the classical speed of the particle in the direction of the particle momentum. Hence the WPS effect in this direction is similar to that given by Eq. (9) in standard theory.
In the opposite case when the particle is ultrarelativistic, Eq. (74) can be written as ψ(r, t) = exp[ ī h p(r − ct)]χ(p, t = 0) dp (2πh) 1/2 (76) Hence, as follows from Eq. (68): In particular, for an ultrarelativistic particle there is no WPS in the direction of particle momentum and this is in agreement with the results of Sec. 5. We conclude that in our approach an ultrarelativistic particle (e.g. the photon) experiences WPS neither in the direction of its momentum nor in perpendicular directions, i.e. the WPS effect for an ultrarelativistic particle is absent at all.
Let us note that the absence of WPS in perpendicular directions is simply a consequence of the fact that a consistently defined operator R ⊥ commutes with the Hamiltonian, i.e. r ⊥ is a conserving physical quantity. On the other hand, the longitudinal coordinate is not a conserved physical quantity since a particle is moving along the direction of its momentum. However, in a special case of ultrarelativistic particle the absence of WPS is simply a consequence of the fact that the wave function given by Eq. (76) depends on r and t only via a combination of r − ct.
Consider now the model discussed in Sec. 9 when the momentum wave function is described by Eq. (43). As noted in Sec. 9, the standard choice leads to the result given by Eq. (52) which is counterintuitive. In view of the discussion at the end of Sec. 11, one might think that this result is a consequence of the fact that standard position operator is a sum of the operators corresponding to different noncommuting physical quantities the contributions of which should be considered separately.
The wave function given by Eq. (43) is spherically symmetric and is the eigenstate of the momentum operator L such that all the eigenvalues equal zero. Hence the physical quantity defined by the operator R ⊥ is not semiclassical and the problem arises whether in this situation the operator R ⊥ should be modified. It follows from Eq. (54) that ||R ⊥ χ|| ≤ λ||χ||. As noted in Sec. 4, one can expect that the coordinate wave function cannot define coordinates with the accuracy better than the wave length. Hence a reasonable approximation in this case is that the position operator contains only the parallel part GR || . In this approximation different components of the position operator commute with each other. Therefore one can define the coordinate wave function which in the given case again has the form (47).
Since p = Gp, G acts only on angular variables and R || acts only on the variable p we conclude that in the given case the angular parts of the position and momentum operators are the same in contrast to the situation in standard theory where those parts are related to each other by the Fourier transform.
As noted in Sec. 9, in standard theory the angular resolution corresponding to Eq. (52) is a quantity of the order of λ/d while from obvious geometrical considerations this quantity should be of the order of d/L. As noted in Sec. 9, for calculating the angular resolution one should project the coordinate wave function on the state having the support inside the volume V where the photon will be measured. Suppose that the volume V is inside the element do of the solid angle. Then in view of the fact that angular variables in the coordinate and momentum wave functions are the same, any measurement of the photon momentum inside V can give only the results where the direction of the photon momentum is inside do.
Therefore, as noted in Sec. 9, for a pointlike source of light the angular resolution is of the order of d/L and for a star with the radius R the resolution is of the order of R/L. Hence, in contrast to the situation discussed in Sec. 9, there is no blurring of astronomical images because the angular resolution is always ideal and does not depend on d. However, details of astronomical objects will be distinguishable only if d is rather large because, as follows from Eq. (50), the norm of the function Pψ(r, t) is of the order of d/L.

Discussion and conclusion
In the present paper we consider a problem of constructing position operator in quantum theory. As noted in Sec. 1, this operator is needed in situations where semiclassical approximation works with a high accuracy and the example with the spherically symmetric case discussed at the end of the preceding section indicates that this operator can be useful in other problems.
A standard choice of the position operator in momentum space is ih∂/∂p. A motivation for this choice is discussed in Sec. 2. We note that this choice is not consistent since ih∂/∂p j cannot be a physical position operator in directions where the momentum is small. Physicists did not pay attention to the inconsistency probably for the following reason: as explained in textbooks, transition from quantum to classical theory can be performed such that if the coordinate wave function contains a rapidly oscillating exponent exp(iS/h) where S is the classical action then in the formal limit h → 0 the Schrödinger equation becomes the Hamilton-Jacobi equation.
However, an inevitable consequence of standard quantum theory is the effect of wave packet spreading (WPS). This fact has not been considered as a drawback of the theory. Probably the reasons are that for macroscopic bodies this effect is extremely small while in experiments on the Earth with atoms and elementary particles spreading probably does not have enough time to manifest itself. However, for photons travelling to the Earth from distant objects this effect is considerable, and it seems that this fact has been overlooked by physicists.
As shown in Sec. 9, if the WPS effect for photons travelling to Earth from distant objects is as given by standard theory then we have several fundamental paradoxes: a) if the major part of photons emitted by stars are in wave packet states (what is the most probable scenario) then we should see not stars but only an almost continuous background from all stars; b) no anisotropy of the relic radiation could be observable; c) the effect of WPS is incompatible with the data on gamma-ray bursts; d) communication with distant space probes could not be possible; e) the Shapiro delay could not be explained only in the framework of classical theory; f) the fact that we can observe pulsars could not be explained. In addition, the consideration in Secs. 9 and 13 poses the following questions: g) how is it possible to verify that the angular resolution of a star in the part of the spectrum corresponding to transitions between atomic levels is of the order of λ/d rather than R/L?; h) are predictions of standard theory on the WPS effect for protons in the LHC ring compatible with experimental data? We have also noted that in the scenario when the quantities N ⊥ are not very large, even images of planets will be blurred.
In Sec. 7 it is shown that, from the point of view of standard quantum theory, there exists the WPW paradox that after some period of time the transversal widths of the coordinate wave functions for photons comprising a classical wave packet will be typically much greater than the transversal width of the classical packet as a whole. This situation seems to be fully unphysical since, as noted in Sec. 7, different photons in a classical wave packet do not interfere with each other. The calculations in Sec. 5 show that the reason of the WPW paradox is that in directions perpendicular to the particle momentum the standard position operator is defined inconsistently. At the same time, as shown in Sec. 8, for coherent states the WPS effect is pronounced in a much less extent than for individual photons.
We propose a new definition of the position operator which we treat as consistent for the following reasons. Our position operator is defined by two components -in the direction along the momentum and in perpendicular directions. The first part has a familiar form ih∂/∂p and is treated as the operator of the longitudinal coordinate if the magnitude of p is rather large. At the same condition the position operator in the perpendicular directions is defined as a quantum generalization of the relation r ⊥ ×p = L. So in contrast to the standard definition of the position operator, the new operator is expected to be physical only if the magnitude of the momentum is rather large.
As a consequence of our construction, WPS in directions perpendicular to the particle momentum is absent regardless of whether the particle is nonrelativistic or relativistic. Moreover, for an ultrarelativistic particle the effect of WPS is absent at all.
As noted in Sec. 7, in standard quantum theory photons comprising a classical electromagnetic wave packet cannot be (approximately) treated as pointlike particles in view of the WPW paradox. However, in our approach, in view of the absence of WPS for massless particles, the usual intuition is restored and photons comprising a divergent classical wave packet can be (approximately) treated as pointlike particles. Moreover, the phenomenon of divergence of a classical wave packet can now be naturally explained simply as a consequence of the fact that different photons in the packet have different momenta.
Our result resolves the above paradoxes and, in view of the above discussion, also poses a problem whether the results of classical electrodynamics can be applied for wave packets moving for a long period of time. For example, as noted in Sec. 9, even classical theory predicts that when a wave packet emitted in a gammaray burst or by a pulsar reaches the Earth, the width of the packet is extremely large (while the value predicted by standard quantum theory is even much greater) and this poses a problem whether such a packet can be detected. We believe that a natural explanation of why classical theory does not apply in this case follows. As noted in Sec. 4, classical electromagnetic fields should be understood as a result of taking mean characteristics for many photons. Then the fields will be (approximately) continuous if the density of the photons is high. However, for a divergent beam of photons their density decreases with time. Hence after a long period of time the mean characteristics of the photons in the beam cannot represent continuous fields. In other words, in this situation the set of photons cannot be effectively described by classical electromagnetic fields.
A picture that a classical wave packet can be treated as a collection of (almost) pointlike photons also sheds new light on the explanation of known phenomena. Suppose that a wide beam of visible light falls on a screen which is perpendicular to the direction of light. Suppose that the total area of the screen is S but the surface contains slits with the total area S 1 . We are interested in the question of what part of the light will pass the screen. The answer that the part equals S 1 /S follows from the picture that the light consists of many almost pointlike photons moving along geometrical trajectories and hence only the S 1 /S part of the photons will pass the surface. Numerous experiments show that deviations from the above answer begin to manifest in interference experiments where dimensions of slits and distances between them have the order of tens or hundreds of microns or even less. In classical theory interference is explained as a phenomenon arising when the wave length of the classical electromagnetic wave becomes comparable to dimensions of slits and distances between them. However, as noted in Sec. 1, the notion of wave length does not have the usual meaning on quantum level. From the point of view of particle theory, the phenomenon of interference has a natural explanation that it occurs when dimensions of slits and distances between them become comparable to the typical width of the photon wave function.
Our results on the position operator also pose a problem how the interference phenomenon should be explained on the level of single photons. The usual qualitative explanation follows. Suppose that the mean momentum of a photon is directed along the z axis perpendicular to a screen. If the (x, y) dependence of the photon wave function is highly homogeneous then the quantities ∆p x and ∆p y are very small. When the photon passes the screen with holes, its wave function is not homogeneous in the xy plane anymore. As a result, the quantities ∆p x and ∆p y become much greater and the photon can be detected in points belonging to the geometrical shadow. However, such an explanation is problematic for the following reason. Since the mean values of the x and y components of the photon momentum are zero, as noted in Secs. 2 and 4, the (p x , p y ) dependence of the wave function cannot be semiclassical and, as it has been noted throughout the paper, in that case standard position operator in the xy plane is not consistently defined.
Different components of the new position operator do not commute with each other and, as a consequence, there is no wave function in coordinate representation. In particular, there is no quantum analog of the coordinate Coulomb potential (see the discussion in Sec. 1). A possibility that coordinates can be noncommutative has been first discussed by Snyder [46] and it is implemented in several modern theories. In those theories the measure of noncommutativity is defined by a parameter l called the fundamental length (the role of which can be played e.g. by the Planck length or the Schwarzschild radius). In the formal limit l → 0 the coordinates become standard ones related to momenta by a Fourier transform. As shown in the present paper, this is unacceptable in view of the WPS paradoxes. One of ideas of those theories is that with a nonzero l it might be possible to resolve difficulties of standard theory where l = 0 (see e.g. Ref. [47] and references therein). At the same time, in our approach there can be no notion of fundamental length since commutativity of coordinates takes place only in the formal limith → 0.
The position operator proposed in the present paper is also important in view of the following. There exists a wide literature discussing the Einstein-Podolsky-Rosen paradox, locality in quantum theory, quantum entanglement, Bell's theorem and similar problems (see e.g. Ref. [23] and references therein). Consider, for example, the following problem in standard theory. Let at t = 0 particles 1 and 2 be localized inside finite volumes V 1 and V 2 , respectively, such that the volumes are very far from each other. Hence the particles don't interact with each other. However, as follows from Eq. (15), their wave functions will overlap at any t > 0 and hence the interaction can be transmitted even with an infinite speed. This is often characterized as quantum nonlocality, entanglement and/or action at a distance.
Consider now this problem in the framework of our approach. Since in this approach there is no wave function in coordinate representation, there is no notion of a particle localized inside a finite volume. Hence a problem arises whether on quantum level the notions of locality or nonlocality have a physical meaning. In addition, spreading does not take place in directions perpendicular to the particle momenta and for ultrarelativistic particles spreading does not occur at all. Hence, at least in the case of ultrarelativistic particles, this kind of interaction does not occur in agreement with classical intuition that no interaction can be transmitted with the speed greater than c. This example poses a problem whether the position operator should be modified not only in directions perpendicular to particle momenta but also in longitudinal directions such that the effect of WPS should be excluded at all.
A problem discussed in a wide literature is whether evolution of a quantum system can be always described by the time dependent Schrödinger equation. We will discuss this problem in view of the statements (see e.g. Refs. [48,49]) that t cannot be treated as a fundamental physical quantity. The reason is that all fundamental physical laws do not require time and the quantity t is obsolete on fundamental level. A hypothesis that time is an independently flowing fundamental continuous quantity has been first proposed by Newton. However, a problem arises whether this hypothesis is compatible with the principle that the definition of a physical quantity is a description of how this quantity can be measured.
Consider first the problem of time in classical mechanics. A standard treatment of this theory is that its goal is to solve equations of motion and get classical trajectories where coordinates and momenta are functions of t. In Hamiltonian mechanics the action can be written as S = S 0 − Hdt where S 0 does not depend on t and is called the abbreviated action. Then, as explained in textbooks, the dependence of the coordinates and momenta on t can be obtained from a variational principle with the action S. Suppose now that one wishes to consider a problem which is usually treated as less general: to find not the dependence of the coordinates and momenta on t but only possible forms of trajectories in the phase space without mentioning time at all. If the energy is a conserved physical quantity then, as described in textbooks, this problem can be solved by using the Maupertuis principle involving only S 0 .
However, the latter problem is not less general than the former one. For illustration we first consider the one-body case. Here the phase space can be described by the quantities (r || , r ⊥ , G, p) discussed in Sec. 11. Suppose that by using the Maupertuis principle one has solved the problem with some initial values of coordinates and momenta. One can choose r || such that it is zero at the initial point and increases along the trajectory. Then r || = s where s is the length along the spacial trajectory and a natural parametrization for the trajectory in the phase space is such that (r ⊥ , G, p) are functions of r || = s. This is an additional indication that our choice of the position operator is more natural than standard one. At this stage the problem does not contain t yet. We can note that in standard case ds/dt = |v(s)| = |p(s)|/E(s). Hence in the problem under consideration one can define t such that dt = E(s)ds/|p(s)| and hence the value of t at any point of the trajectory can be obtained by integration. In the case of many bodies one can define t by using the spatial trajectory of any body and the result does not depend on the choice of the body. Hence the general problem of classical mechanics can be formulated without mentioning t while if one wishes to work with t then, by definition, this value can flow only in positive direction.
Consider now the problem of time in quantum theory. In the case of one strongly quantum system (i.e. the system which cannot be described in classical theory) a problem arises whether there exists a quantum analog of the Maupertuis principle and whether time can be defined by using this analog. This is a difficult unsolved problem. A possible approach for solving this problem has been proposed in Ref. [48]. However, one can consider a situation when a quantum system under consideration is a small subsystem of a big system where the other subsystem -the environment, is strongly classical. Then one can define t for the environment as described above. The author of Ref. [49] considers a scenario when the system as a whole is described by the stationary Schrödinger equation HΨ = EΨ but the small quantum subsystem is described by the time dependent Schrödinger equation where t is defined for the environment as t = ∂S 0 /∂E.
One might think that this scenario gives a natural solution of the problem of time in quantum theory. Indeed, in this scenario it is clear why a quantum system is described by the Schrödinger equation depending on the classical parameter t which is not an operator: because t is the physical quantity characterizing not the quantum system but the environment. This scenario seems also natural because it is in the spirit of the Copenhagen interpretation of quantum theory: the evolution of a quantum system can be characterized only in terms of measurements which in the Copenhagen interpretation are treated as interactions with classical objects. However, this scenario encounters the following problems. As noted in Ref. [49], it does not solve the problem of quantum jumps. For example, as noted in Sec. 4, the 21cm transition in the hydrogen atom cannot be described by the evolution operator depending on the continuous parameter t. Another problem is that the environment can be a classical object only in some approximation and hence t can be only an approximately continuous parameter. Finally, the Copenhagen interpretation cannot be universal in all situations. For example, if the Big Bang hypothesis is correct then at the early stage of the Universe there were no classical objects but nevertheless physics should somehow describe evolution even in this situation.
Our result for ultrarelativistic particles can be treated as ideal: quantum theory reproduces the motion along a classical trajectory without any spreading. However, this is only a special case of one free elementary particle. If quantum theory is treated as more general than the classical one then it should describe not only elementary particles and atoms but even the motion of macroscopic bodies in the Solar System and in the Universe. We believe that the assumption that the evolution of macroscopic bodies can be described by the Schrödinger equation is unphysical. For example, if the motion of the Earth is described by the evolution operator exp[−iH(t 2 − t 1 )/h] where H is the Hamiltonian of the Earth then the quantity H(t 2 − t 1 )/h becomes of the order of unity when t 2 − t 1 is a quantity of the order of 10 −68 s if the Hamiltonian is written in nonrelativistic form and 10 −76 s if it is written in relativistic form. Such time intervals seem to be unphysical and so in the given case the approximation when t is a continuous parameter seems to be unphysical too. In modern theories (e.g. in the Big Bang hypothesis) it is often stated that the Planck time t P ≈ 10 −43 s is a physical minimum time interval. However, at present there are no experiments confirming that time intervals of the order of 10 −43 s can be measured.
The time dependent Schrödinger equation has not been experimentally verified and the major theoretical arguments in favor of this equation are as follows: a) the Hamiltonian is the generator of the time translation in the Minkowski space; b) this equation becomes the Hamilton-Jacobi one in the formal limith → 0. However, as noted in Sec. 1, quantum theory should not be based on the space-time background and the conclusion b) is made without taking into account the WPS effect. Hence the problem of describing evolution in quantum theory remains open.
Let us now return to the problem of the position operator. As noted above, in directions perpendicular to the particle momentum the choice of the position operation is based only on the requirement that semiclassical approximation should reproduce the standard relation r ⊥ × p = L. This requirement seems to be beyond any doubts since on classical level this relation is confirmed in numerous experiments. At the same time, the choice ih∂/∂p of the coordinate operator in the longitudinal direction is analogous to that in standard theory and hence one might expect that this operator is physical if the magnitude of p is rather large (see, however, the above remark about the entanglement caused by WPS).
It will be shown in a separate publication that the construction of the position operator described in this paper for the case of Poincare invariant theory can be generalized to the case of de Sitter (dS) invariant theory. In this case the interpretation of the position operator is even more important than in Poincare invariant theory. The reason is that even the free two-body mass operator in the dS theory depends not only on the relative two-body momentum but also on the distance between the particles.
As argued in Ref. [45], in dS theory over a Galois field the assumption that the dS analog of the operator ih∂/∂p is the operator of the longitudinal coordinate is not valid for macroscopic bodies (even if p is large) since in that case semiclassical approximation is not valid. We have proposed a modification of the position operator such that quantum theory reproduces for the two-body mass operator the mean value compatible with the Newton law of gravity and precession of Mercury's perihelion. Then a problem arises how quantum theory can reproduce classical evolution for macroscopic bodies.
The above examples show that at macroscopic level a consistent definition of the transition from quantum to classical theory is the fundamental open problem.