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Time and energy operators in the canonical quantization of special relativity

Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) is shown to provide a unified origin for: i) the complex vector space formulation of Quantum Mechanics (QM); ii) the momentum and space commutation relations and the corresponding representations; iii) the Dirac Hamiltonian in the formulation of Relativistic Quantum Mechanics (RQM); iv) the existence of a self adjoint Time Operator that circumvents Pauli's objection.


Introduction
Quantum mechanics (QM) fails to treat time and space coordinates on the almost equal footing accorded by Special Relativity (SR) as it does with momentum and energy. In QM time appears as a parameter not as a dynamical variable. It is a c-number, following Dirac's designation [1]. This is the Problem of Time (PoT) in QM, that results in the extensive discussion of the existence and meaning of a time operator [2,3], and of a time energy uncertainty relation [4,5] in view of Pauli's objection [6].
The procedure usually termed "canonical quantization". arises from applying to the Hamiltonian formulation of classical physics the rule of substituting dynamical variables by self adjoint operators acting on normalized vectors representing the physical system. In addition to considering the existance of Lorentz invariants, the Born reciprocity principle is brought into play [7]. This proposed principle arises from noting that the Hamiltonian formulation of classical mechanics is invariant under the transformations x i → p i , p i → −x i and from the equivalence of the configuration and momentum representations in QM. Although Born acknowledges to be unsuccessful in his intended applications 1 , the reciprocity principle is currently receiving a renewed interest [8,9,10].In the present paper it is shown that it complements the required Lorentz invariance in the canonical quantization of Special Relativity (SR) to provide a unified origin for: i) the complex vector space formulation of QM; ii) the momentum and position operators' commutation relations and their corresponding representations; iii) the Hamiltonian in Dirac's formulation of Relativistic Quantum Mechanics (RQM) [1,11,12,13]; iv) the existence of a self adjoint Time Operator that circumvents Pauli's objection [14,15,16].
2 Lorentz and reciprocity invariants in the canonical quantization of special relativty In SR the invariants under Lorentz transformations for a free particle are the scalar products of the fourvectors in a Minkowski space with metric η µν = diag(1, −1, −1, −1) , namely: where c is the constant light velocity and the constants m 0 (the rest mass) and s 0 (to be interpreted) are internal properties of the physical system (Einstein's summation convention is assumed). To be included in addition are the constant products: where the symetrization is introduced as these dynamical variables will be transformed to operators where order matters.
i) The Dirac free particle Hamiltonian From the momentum invariant, first relation in Eq.1, one obtains upon quantization the QM constraint This can be factorized as provided that, to cancel the cross terms, the momentum operators satisfy the commutation relation [p µ ,p ν ] = 0 and the coefficients ρ µ the anticommutation relation {ρ µ ρ ν + ρ ν ρ µ } = 2η µν I 4 where I 4 is the 4 × 4 identity matrix. Thus the coefficients ρ µ obey a Clifford algebra and are represented by matrices. Then the constraint is satisfied with the linear equation: Multiplying by cρ 0 and defining ρ 0 := β, ρ 0 ρ i := α i one obtains : that exhibits the Dirac Hamiltonian H D = cα.p + βm 0 c 2 . One recognizes here the procedure followed by Dirac to obtain a first order linear equation in energy and momentum that agrees with the second order one resulting from the energy momentum relation in Eq.1 [1,11,12,13].
ii) The free particle time operator In exactly the same way, from the second relation in Eq.1, the displacement invariant yields upon quantization the QM constraint This can be factorized as The constraint is then satisfied with the linear equation: or, denoting s 0 = cτ 0 where τ 0 would be an internal time property of the system: Here T = α.r/c + βτ 0 is the time operator introduced earlier by analogy to the Dirac Hamiltonian [15].
iii) The Born reciprocity invariant The invariants: (11) give the constraint: [{x 0p0 ±p 0x0 } − {r.p ±p.r}] |Ψ = 0 (12) or equivalently: The operator O + is self adjoint and represents a real invariant. On the other hand O − is a pure imaginary invariant as (O − ) † = −O − , but is the one to satisfy reciprocity invariance. An obvious choice for O − to satisfy Eqs.12 and 13 is: where is the reduced Planck constant.
Thus reciprocity invariance complements Lorentz invariance to yield the commutation relations of the operatorsx µ andp ν , namely: as an alternative to the postulate of transforming Poisson brackets to quantum commutators. In Appendix A it is shown that these commutation relations are sufficient to derive: a) the continuity from −∞ to +∞ of the spectra ofx µ andp µ ; b) the representations ofx µ andp µ in the corresponding orthogonal eigenvector basis: c) the Fourier transformation between the configuration and momentum representation of the system vector and d) the Heisenberg uncertainty relations, including Bohr's interpretation of the time-energy uncertainty relation (Appendix B). Such unified relationship is unfortunately not present in QM textbooks, where some of these elements are usually introduced as independent antzats.

Configuration and momentum representations
Considering Eq.6.in the configuration representation wherep ν → −i ∂ ∂xν this equation reads: Substituting from SR x 0 = ct , the result is Dirac's relativistic equation as usually formulated, namely: On the other hand, in the momentum representation wherex µ → i ∂ ∂pµ , Eq.10 yields: Substituting cp 0 = e, one obtains for the time operator the equation: 4 The energy and time spectra, and Pauli's objection As is well known, the energy spectrum of the Dirac Hamiltonian has both positive and negative real values, namely e(p) = ± (cp) 2 + m 0 c 2 ,.separated by a 2m 0 c 2 gap. In the same way, the spectrum of the time operator contains positive and negative real values separated by a gap 2τ 0 , as τ (r) = ± (r/c) 2 + τ 0 . Now, the actual interpretation of the effect of the Dirac Hamiltonian H D and the time operator T is seen from the fact that they are self adjoint. By Stone-vonNewmann's theorem [17] they are generators of unitary transformations of the state vectors. Then it can be shown that for infinitesimal changes [16]: a) U T = exp{i(δe)T / } ≈ exp{i(δe)α.r/c } exp{i(δe)βτ 0 } generates continous displacements in momentum δp = (α/c)δe = cα(δe/c 2 ) and changes in phase δφ = βτ 0 δe/ . A 2π finite phase change for positive energy waves ( β = 1) is obtained from setting: Then τ 0 is seen to be the deBroglie period T B , in agreement with deBroglie's daring assumptions [18,19]. It is an intrinsic time property associated with the rest mass [20]. b) U HD = exp{i(δt)H D / } ≈ exp{i(δt)cα.p/ } exp{i(δt)βm 0 c 2 } generates displacements in space δr = cαδt and changes in phase δφ = βm 0 c 2 δt/ . For β = 1, a 2π finite change of phase requires a time lapse: For wave packets the expectation value cα is the group velocity v gp and the space displacement in a time lapse ∆t generated by H D corresponds to the classical v gp ∆t. On the other hand T acts on the continous momentum space, generating a change of momentum ∆p = mv gp = m 0 γv gp , where γ = [1 − (v/c) 2 ] −1 is the Lorentz factor, and consequently an energy change from E(p) to E(p + mv gp ) in both branches of the relativistic energy spectrum. This circumvents Pauli's correct objection that a commutation relation [T, H] = i where T acts on the energy spectrum, necessarily implies a continuum energy spectrum from −∞ to +∞ , contradicting the fact that the energy expectation value is expected to be positive and that there also may be discrete eigenvalues [6].
To be remarked finally is that, from Eq.20, the energy gap 2m 0 c 2 and the time gap 2h/m 0 c 2 are complementary of each other. The mass dependence of the Zitterbewegung period (twice the deBroglie period) has been correctly exhibited in the experimental simulation of the Dirac equation with trapped ions [21,22].

Conclusion
It has been shown that the canonical quantization of SR that preserves the Lorentz and reciprocity invariants, is at the origin of the (usually postulated or inferred separately) commutation relations of the configuration and momentum dynamical operators, as well as of the Dirac relativistic Hamiltonian together with a self adjoint relativistic "time operator". Furthermore, it brings about the derived properties -infinite continuous space and momentum spectra, ensuing representations, uncertainty relation as shown in Appendix A and B, that unfortunately in most QM textbooks are introduced as independent antzats.
To be stressed also, it is the reciprocity invariance which introduces an imaginary constant, opening the formulation to complex functions which are necessary to allow for "a non-negative probability function that is constant in time when integrated over the whole space " [6], the basis for a probabilistic interpretation of QM.
The problem of time is very much present in the canonical quantization of General Relativity (GR), with many facets: indefinition of the spacetime foliation ("many fingered time"), disappearence of time ("frozen formalism"), and so on [24,25,26,27]. However one condition to be satisfied is local concordance with SR, i.e., any acceptable theory of Quantum Gravity (QG) must allow to recover the classical spacetime in the appropiate limit [28]. It follows that a venue to be explored is whether this bottom up completion of Dirac's RQM with a time operator as derived above helps to resolve some of the issues noted [22,23]. To represent observables the operatorsx µ andp µ are selfadjoint (x µ =x † µ , p µ =p † µ ) , which insures real eigenvalues. Then, for each componentx µ and p µ it follows:

1) Spectrum[13]
Consider the eigenvalue equation : By Stone-von Newmann's theorem the operator U (α) = exp(−iαp/ℏ) with a real is unitary [17]. Then it can be shown that: Therefore {U (α) |x } = C |x + α . As α is arbitrary, it follows that the eigenvalues ofx are continous from −∞ to +∞ , and that the eigenvectors satisfy: where δ(x ′ − x) is the Dirac delta function and I is the identity operator.
In the same way one can prove that the eigenvalues inp |p = p |p span a continuum from −∞ to +∞ and that the eigenvectors satisfy: and It follows: Introducing the vectors: their representations in configuration space are: and using Eq.A.5: i.e., the representation in configuration space of the vector |Φ =p |Ψ is obtained by taking the derivative of the representation of the vector |Ψ , while the representation in configuration space of the vector |Θ =x |Ψ is obtained multiplying by x the representation of |Ψ . To conclude, in configuration space one has: In the same way in momentum space:

3) Transformation between representations
Consider: Developing: one obtains: which is satisfied if: Finally: and: i.e., the representations of the state vector in the configuration and momentum spaces are Fourier transforms of each other. To preserve normalization one requires C = C * = 1/ √ 2πℏ. where K = β(2s.l/ 2 + 1) is a constant of motion [11]. In the usual manner an uncertainty relation follows, namely: This correspponds to Bohr's interpretation that the uncertainty in the time of passage at a certain point is given by the width of the wave packet, which is complementary to the momentum uncertainty, and thus to the energy uncertainty [29].