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Consequences of a Two-Time Relativistic Bohmian Model

Effects of a Bohmian quantum-relativistic theory are explored. The model is obtained by introducing a new and independent time parameter whose relative motions are not directly observable and cause the quantum uncertainties of the observables. The quantum potential does not affect directly the observable motion, as for a normal external potential, but it determines that one relative to the new time variable. It turns out that the Zitterbewegung of a free particle is just an example of these hidden motions and a more general law of it is obtained. A relativistic revision of the uncertainty principle, for particles with non-zero rest mass, also derives from the theory.


I. INTRODUCTION
At today, the de Broglie-Bohm (or pilot-waves) theory [1][2][3][4][5][6] is considered, with good consensus, a valid determinist approach to non-relativistic quantum mechanics. The picture is not so favorable as regards the attempts at relativistic generalization. For some time, the explicit non-locality of the theory -i.e. the presence of hidden variables that allow instantaneous actions at a distancehas been considered a manifest violation of the Lorentz invariance [7,8]. A more radical criticism has pointed out that for a relativistic pilot-waves theory it is not reasonable to expect Lorentz invariance since in the nonrelativistic context it is not Galilean invariant [9]. More recently, impossibility of covariance has been concluded on the basis of fact that nonlocal hidden variables can't predict quantum events in a relativistic time-order invariant way [10]. These conclusions are today refused by the majority of the scientific community for the following reasons. Firstly, if existence of superluminal particles is admitted, the action at a distance does not necessarily imply a violation of the Lorentz invariance [11][12][13][14]. Secondly, the Galilean invariance is replaced in the modern approach by the Lorentz transformations, which are interpreted as a generalization of space rotations; then, the non-relativistic pilot-waves theory results Lorentz (although not Galilean) invariant [15]. Finally, the timeordering invariance it is a property that does not belong to the genuine Lorentz invariance [15].
Currently, the most authoritative Lorentz-covariant pilot waves theories can be grouped in this way, in terms of the proposed strategy [16]: 1. Introducing a temporal parameterization that synchronizes the trajectories of all particles [11-13, 17, 18]. This approach has the drawback of not making concrete statistical predictions, although subsequent developments suggest this possibility under certain circumstances [15,[19][20][21].
2. Defining a future light-cone as surface of simultaneity [22]. This theory is unable to predict any probability. 3. Rejecting the decomposition of the wavefunction into norm and phase, and interpreting it as an real quantum field exerting force on the particle [23].
Here, tachyon particles are avoided but it is not clear if and how the model can be verified.
4. Defining simultaneity through "slicings" of the space-time (foliations) by spacelike hypersurfaces. The majority of researchers favor a preferred foliation, unique for all particles [24][25][26][27][28]30] but there are also those who suggest a degenerate foliation [29] or an union of all possible foliations [31]. These hidden geometrical structures are considered by some researchers to be the most feasible way to obtain a Bohmian Lorentz-covariant mechanics capable of making predictions [32,33].
Despite the conceptual stability of some of these models, this panorama is not very satisfactory. The few theories capable of making predictions, do this at the cost of introducing additional spacetime structures and admitting superluminal velocities (and hence travels backwards in time in some Lorentz frames) for massive particles, in violation of Special Relativity (although not of Lorentz covariance). Furthermore, none of these theories make predictions beyond the standard interpretation, indicating, for example, a discriminating experiment. But if the advantage of adopting a deterministic model were only conceptual, then it would be superfluous from a strictly physical point of view. On the other hand, it should be highlighted that important physical matter such as spin, Zitterbewegung [34,35] and non-local entanglement are not satisfactorily explained by the standard approach.
It would certainly be desirable a relativistic Bohmian theory which, possibly under weaker assumptions, is able to propose, together with a determinist ontology, elucidation of known quantum properties and prediction of new effects to be tested.
In this work, after observing that the more spontaneous relativistic generalization of the original de Broglie-Bohm theory is unsatisfactory as regards the continuity of the quantum potential with the non-relativistic case, we try to overcome this issue by introducing a new independent time parameter as the only hidden variable, whose relative motions are recognized to be instantaneous with respect to the traditional time. In this deterministic and explicitly non-local approach, a more general Zitterbewegung, as motion in the new time variable caused by the quantum potential, arises; and it would then be not directly observable, in accordance with recent theoretical advances [36][37][38]. A further consequence is a modification of the uncertainty principle for particles with non-zero rest mass, which, since it depends on the direction of motion, becomes anisotropic. This property could be observed in high energy accelerators.

II. RELATIVISTIC GENERALIZATION ISSUE FOR THE QUANTUM POTENTIAL IN CASE OF CONSTANT MOMENTUM
In the de Broglie-Bohm non-relativistic theory [1][2][3][4][5][6], the wavefunction: with R > 0 and S real functions, is inserted into the Schrödinger equation, yielding the following equations: Identifying S with the Hamilton's principal function, we have ∇S = p and ∂S ∂t = −H, so eq. (2) means the Hamilton-Jacobi equation with Hamiltonian: where the last addend represents an energy, called quantum potential, that takes into account the quantum effects: According to the Bohm's original proposal, the motion equation is: In the case of a free particle (V =0), both a stationary solution with positive constant V Q and a solution with variable V Q are admitted [39,40].
The direct relativistic generalization of this approach, still for a free particle, pushes to write: ∇S = p = m γ v and ∂S ∂t = −H = −m γ c 2 − V Q , with V Q to be determined. We will start by assuming that p is constant, while for V Q , and therefore H, a possible dependence on time.
From eq. (1), taking the gradient, then the divergence and finally dividing by ψ, we obtain: being ∇ · v = 0. Comparison with the Klein-Gordon equation: provides: having applied the identity On the other hand, deriving twice eq. (1) over time, one gets: Substituting eq. (10) into (9) and matching real and imaginary parts, we have: Now we assume that the continuity equation (3), with mγ instead of m, still holds. By employing ∇ · v = 0, we get just the expansion of dR 2 dt . Of course, we can refer it simply to R: as well as to any function of R. Indeed, since v is constant, eq. (13) also holds for any function of any-order derivative of R over time or space, and thus for V Q and H. We are therefore assuming dH dt = 0 but with the possibility that ∂H ∂t = 0, and the same for V Q . This is possible if V Q is not a generalized potential. Substituting eq. (13) into eq. (12) one finds: where, looking for non-trivial solutions, one assumes m = 0, and ∂R ∂t = 0. Now we have the cases: 1. H (and therefore V Q ) with non-zero partial derivative. The motion equation reads: then, considerating eq. (13) applied to V Q , one obtains ∂VQ ∂t = 0 : impossible. 2. H (and therefore V Q ) with null partial derivative.
Eq. (14) implies H = m γ c 2 and then V Q = 0; so in general we have not continuity for the quantum potential with the non-relativistic case.

III. THE TWO-TIME BOHMIAN MODEL
Apparently, the spontaneous Bohmian relativistic generalization for the motion of a free particle requires p not be constant. Inside the Heisenberg picture of the standard interpretation, Schrödinger and Dirac recognized that: where v o is the velocity actually observed in mean value and v i an omnidirectional velocity due to a sinusoidal movement of frequency 2mγc 2 and amplitude about 2mγc , known as Zitterbewegung [34,35]. This motion has been much commented on, sometimes related to spin [41][42][43][44] and quantum potential [45], but the debate on its exact interpretation, and even observability, is not yet resolved [36][37][38][46][47][48][49][50][51][52][53][54][55]. Naturally, to find this peculiar movement in a deterministic approach could clarify its nature. A smooth idea is to consider it as due to the quantum potential; however it has been shown that putting d pi dt = − ∇V Q leads to inconsistency in [1 + 1] dimensions [56].
Here we explore the effect of introducing a new independent time variable for the spatial coordinates of the particle. Two time physics has been considered in various contexts. An additional temporal parameter is considered e.g. in [57][58][59][60][61] and, with a different approach, in [62][63][64][65] in order to more accurately explain different quantum properties or obtain supersymmetries [66]. But in these theories the new time is not independent as we wish.
Let's admit, then, that the coordinates depend not only on commonly experienced time t, but also on another, totally independent and hidden, time parameter τ . Referring for simplicity only to x(t, τ ), we get two velocities, one classic and one intrinsic: having so: If we suppose that v c and v i depend, respectively, only on t and τ , we have: having set t 0 = τ 0 = 0. The intrinsic motion is not directly observable and, for the independence of τ , is perceived as instantaneous with respect to time t. Of course, this has dramatic effects on the measurements of the observable quantities. As for the position, eq. (18) allows to the particle, observed in its trajectory as a function of t, to jump instantly from one point of space to another, in principle arbitrarily distant, through the time dimension τ . Actually, quantum mechanics has accustomed us to similar situations, not only in the case of an electron spreaded in an atomic orbital or in a quantum diffraction and entaglement, but more in general in any motion. According to Feynman's sum of histories interpretation, in fact, the physical properties resulting from a displacement between two points A and B, for a particle or photon, can be correctly explained only by admitting that it has traveled simultaneously all the possible paths from A to B [67].
For a particle in presence of a potential V the wavefunction is written: The motion is decomposed into a classical and a purely quantum part. The law of classical motion is well known; in particular, if V is conservative, the classic total energy is constant: and deriving over time one can get: So, the Bohm's motion equation (6) should be incorrect. Indeed, if we take the total derivative over t of eq. (4), we re-obtain the eq. (21), because For t-motion we also require the continuity equation: As regards the intrinsic motion, it is spontaneous to consider that it is the direct cause of the quantum uncertainties. More, we assume that the Restricted Relativity does not hold for it. This condition is consistent with the known Zitterbewegung formula, where the mass of the particle does not undergo any relativistic increaseother than that one due to the mean velocity v o -despite it reaches the speed c. Being mγ c the mass involved in the intrinsic motion, we have p i = mγ c v i .
For the τ -motion we have to admit the validity of the following equation: To try to justify it, we can observe that it derives from a continuity equation in which the addends subtract instead of adding up. This would remind the behavior of an antiparticle, although in the following we will not make any reference to antimatter and only use positive energy values. Since τ -motion is not directly influenced by V , we can consider that v i only depends on τ ; consequently, eq. (23) holds for any function of any-order derivative of R over t or space. So we admit that it can be applied to V Q , at least in case of conservative V . Conservation of energy for motion in τ reads: representing the purely quantum total energy, independent explicitally on τ .
Taking the ∂ ∂τ of eq. (24) and using eq. (23) for V Q , one gets: The motion equations (21) and (25) have so to replace the Bohm's original one (6). The sum E q + E c represents the full total energy of the particle.
The new Bohmian model can thus be so summarized: 1. The spatial coordinates of the particle have to vary as a function of two independent temporal parameters, t and τ . Motion in τ is not directly observable and is responsible for quantum uncertainties. 4. The Bohmian law of motion (6) must be replaced by eq. (21) for motion in t and by eq. (25) for motion in τ , for which the Restricted Relativity is not valid. The total momentum of the particle is: We also observe that taking the gradient of ψ and applying ∇S = p c one can easily get the guidance equation:

IV. ZITTERBEWEGUNG IN THE NEW MODEL
Studying a free particle in the new theory reveals a Zitterbewegung motion in the τ variable.
Here, basically, we will follow the deduction made in [68]; but, thanks to the more general form of eq. (25), obtained using eq. (23), we will get a simpler and more rigorous derivation.
The classical motion equation (21) no longer leads to an absurdity but informs us that p c is constant. All equations found in section II -except (6) -are valid. Again, eq. (22) can be rewritten: being f any function of any-order derivative of R over t or space. In particular it holds for V Q and H. We will begin by studying only the rectilinear trajectory, called x = xv c . Here, eq. (26) becomes the equation of a progressive wave: with a dot denoting the derivative with respect to x. Deriving again we obtain the d'Alembert equation: So, V Q and H are waves that follow the particle in tmotion. Reasoning in the same way on eq. (23), we find instead that they are regressive waves in τ -motion. Using the above equations for f=R, eqs. (11) and (14) can be rewritten as: In the following we assume m = 0, v c = 0,Ṙ = 0 and limit to consider only non-negative values for the Hamiltonian. Substituting eq. (29) into eq. (30), we get: Using eq. (27) for f = β, one finds: Multiplying by dx and integrating member by member we can obtain: with c 1 arbitrary constant. So: and the quantum potential is V Q = mγ c c 2 c1 R 2 . Compatibility with the stationary non-relativistic case requires c 1 > 0. By squaring eq. (33) one finds: Recalling that R > 0, a first integration provides: with c 2 arbitrary constant. We look for a solution where R has a maximum, called R M , at some point x M , having in eq. (36) sign -for x > x M and + for x < x M . Froṁ R(x M ) = 0 we determine c 2 , getting: Placing f ≡ c1 R 2 M > 0, the quantum potential is rewritten: and its minimum value is in (39) which cannot be simplified by means of standard functions.
In a small neighborhood of the maximum, i.e. for R → R M , eq. (39) gives for R a quadratic dependence on x − x M . Indeed, here the rooting at first member is approximated by 2(f + 2)(1 − R RM ), so by integrating we have: where c 4 = 0 by imposing R(x M ) = R M . Hence: Then, from eq. (38) the quantum potential around x M can be approximated by the potential of a harmonic oscillator: By replacing it into the τ -motion equation (25) and imposing x(t, 0) = x t0 + x τ0 + v c t = x M , we ultimately get the following solutions: The angular frequency is more general than that one of the known Zitterbewegung. We get this latter, 2mγcc 2 , for γ c f 2(f + 2) = 2; and by imposing also v iMAX = c, we find A = 2mγcc , the relativistic Compton length. But in this model dependence of f on γ c appears undue. By only imposing v iMAX = c, we get A = , where we will assume that f is of the order of unity. According to point 1 of the theory, the uncertainty principle is obtained by multiplying: with an unexpected Lorentz factor to divide. Looking at eq. (29) this seems at all correct, because there is not only a γ c which increases the relativistic mass but also another γ c which decreases the influence of . At speeds close to c, quantum uncertainty in direction of motion should be negligible for particles with non-zero rest mass.
Eq. (43) shows that V Q , and hence also H, oscillates with an amplitude 1 2 mγ c c 2 : this value represents so the quantum uncertainty in a measure of energy, for a free particle with a minimum location given by Let's find again the uncertainty principle by multiplying by half-period of oscillation: The purely quantum total energy E q ≡ V Q + 1 2 mγ c v 2 i is: represents the full total-energy of the particle.
For V Q = E q , R has minimum value: R m = RM Finally, a normalization of R 2 (not exact, for the nonintegrability of eq. (39)), can determine R M as a function of f.

A. Zitterbewegung in arbitrary direction
In an arbitrary directionŝ, which forms an angle ϑ withv c , eq. (26), for f=R, is written: where v cs = v c cos ϑ and the dot denots derivative over s. So eqs. (29) and (30) are generalized by: Placing: λ 2 ≡ 2 m 2 c 2 γ 2 c γ 2 cs , we get back the same equations we have considered. Expressions of the Hamiltonian and quantum potential, given by (34) and (38), do not change, not even the amplitude of their oscillations. By imposing the motion equation: where v is is the intrinsic velocity in directionŝ, we obtain an intrinsic harmonic motion described again by (44) and (45), but with the new value of λ. The consequence on the uncertainty principle -eqs. (46,47) -is the presence of γ cs instead of γ c to divide. This gives rise to a spatial anisotropy of the uncertainty principle: In particular, forŝ which tends to be perpendicular tô v c , γ cs → 1 and the standard uncertainty principle, independent of speed, is re-established. However, the case of exact perpendicularity is singular: for v cs = 0 eq. (51) implies H = const ⇒ V Q = const ⇒ v is = const, instead of being oscillatory.
B. Non-relativistic limit. Singularity in vc = 0 For the non-relativistic (v c ≪ c) and v c → 0 cases, it is sufficient to approximate γ c in the previous equations (respectively, with 1 + v 2 c 2c 2 and 1), obtaining still high frequency oscillatory motions in τ . These results cannot be obtained starting from the standard Schrödinger equation because from it the non-relativistic approximation of eq. (30) cannot be deduced. This happens since the explicit dependence of R on t causes the Klein-Gordon equation to no longer tend to the Schrödinger equation. However, the non-relativistic potential of Bohm (5) can be achieved by approximating eq. (50) [69].
In v c = 0 there is another singularity; in fact, by eq. (30) we get again v i = const, contrary to the case v c → 0.

C. Standard Zitterbewegung
Interpretating the oscillatory motion in τ as occurring in t, the standard Zitterbewegung is re-obtained. In the standard view, R is considered time independent, so the second side of eq. (10) is just − H 2 2 , with H = const. In the new theory this means demanding:

V. DISCUSSION
A model in which the wavefunction of a particle is a function on r, t and a new temporal parameter τ , has been explored. The interpretation of S does not change, so all equations referring to time t, as they use partial derivatives, remain the same. By assuming for the τmotion an apt continuity equation, new laws of motion, which must replace that one of Bohm's original theory, derive. The peculiarity of the intrinsic motion is due to the retroactive nature of the quantum potential, which originates from the same wavefunction of the particle. This latter, interacting with its own wave, spreads out in a region of space that depends on the conditions of the physical surrounding. For a free particle, a more general Zitterbewegung motion has been found, within a region of the Compton length order radius, assuming f ∼ 1. In presence of a V = 0, the spreading will affect the classic motion that obeys d pc dt = − ∇V : we would find eg an orbital, for an atomic electron.
The intrinsic movements are the cause of the uncertainty relations and do not obey to the Special Relativity. Nonetheless, the spreading is judged as instantaneous with respect to time t, measured in any reference system, making it impossible to directly detect violations of the Lorentz covariance.
The imposition of eq. (23), necessary to deduce the general equation (25), might seem a weakness of the theory. Anyway, it is legitimate that even the movement in τ possesses own equation of continuity, like that one in t. Its peculiarity (regressive wave) reminds the motion of an antiparticle.
A question concerns the determination of the parameter f. By assuming f = 1/2, one gets a full total energy 2mγ c c 2 , half classical and half quantum: this latter could represent the energy of the antiparticle. But, naturally, the determination of f should derive by rigorous considerations and it is not excluded that it may depend on the type of particle and/or its properties here not considered.
The evanescence of the uncertainty principle in direction of motion, for non-zero rest mass and v c → c, is recognizable just by observing the general eq. (50): there, a Lorentz factor -equal to 1 only in direction tending to perpendicular to the rectilinear classic motion -directly reduces the quantum potential, which causes the motion in τ and thus the quantum uncertainties. Then, the spreading region of the particle should be contracted in direction of motion.
Believing that the high energy available in today's accelerators should have already detected deviations based on eq. (53), would be quite hasty. It is necessary to consider not only that high speeds make localization measurements hard but above all that simultaneous measurements of complementary observables are prohibited by the same principle of causality in the Special Relativity. These and several other problems make it very difficult to experimentally verify the uncertainty principle [70,71]. An angular test could aid the verification.
Compared to spacetime structures planned by the current most accredited theories, the introduction of a second time variable is a more radical but also simpler criterion. Plus there is the exciting possibility that, through the new hidden time parameter, the Feynman's paths are redefined as real physical trajectories (although not directly observable) and the sum of the histories represented as a properly mathematical integral.
By traveling in a hidden time dimension, you can deterministically explain why a particle is able to interact instantly with more slits in a diffraction, to run across all the possible paths between two points, to admit a non-local relationship with another particle... Actually all observed facts.