aa r X i v : . [ qu a n t - ph ] O c t Relativistic almost local hidden-variable theory
Hrvoje Nikoli´c
Theoretical Physics Division, Rudjer Boˇskovi´c Institute, P.O.B. 180, HR-10002 Zagreb, Croatia. ∗ (Dated: June 21, 2018)A simple relativistic quantum hidden-variable theory of particle trajectories, similar to the Bohmtheory but without nonlocal forces between the particles, is proposed. To provide compatibility withstatistical predictions of quantum mechanics one needs to assume the initial probability density | ψ | of particle positions in spacetime, which is the only source of nonlocality in the theory. Thisdemonstrates that the usual Bohm hidden-variable theory contains much more nonlocality thanrequired by the Bell theorem. PACS numbers: 03.65.Ta, 03.65.Ud
The Bell theorem [1] (as well as some other theorems[2, 3]) shows that quantum mechanics (QM) is, to a cer-tain extent, a nonlocal theory. In particular, the theoremimplies than any hypothetic hidden-variable completionof QM must necessarily be nonlocal. This is particularlymanifest in the Bohm [1, 4, 5] nonlocal hidden-variableformulation of nonrelativistic QM, where acceleration orvelocity of one particle depends on instantaneous posi-tions of all other particles. Yet, the existence of otherinterpretations of QM, in which such instantaneous in-fluences do not play any role, suggests that the Bohminterpretation might contain more nonlocality than nec-essary. For example, if the wave function itself satisfyingthe Schr¨odinger equation is the only objectively existingentity [6], then the only source of nonlocality is the non-separability of the wave function, while all equations ofmotion are local. The similar is true for the standardinstrumental view of QM, which is agnostic on the issueof objective reality. The fact that these “non-Bohmian”views of QM are also compatible with the Bell theo-rem suggests that the Bohm interpretation might containmuch more nonlocality than required by the Bell theo-rem.Such a view is also supported by a recent demonstra-tion that the Bohm theory can be reformulated in anapparently local form [7], but in a very complicated wayin terms of an infinite tower of auxiliary pilot waves inthe physical (rather than configuration) space, satisfyingan infinite coupled set of local equations of motion. How-ever, the fact that the set of equations is infinite leavesa space for suspicions that it could still be a nonlocaltheory in disguise.To provide a more compelling argument that the usualBohm formulation of QM contains much more nonlocal-ity than required by the Bell theorem, in this paper wepropose a simple hidden-variable theory of particle tra-jectories very similar to the Bohm theory, but withoutnonlocal forces between the particles. Instead, with agiven wave function in the configuration space, the ve-locity of a particle depends only on the position of that ∗ Electronic address: [email protected] particle. The compatibility with statistical predictionsof QM is encoded in the initial correlations between theparticles, which are nonlocal in accordance with the Belltheorem. These initial correlations turn out to be theonly source of nonlocality in the theory.Another distinguished feature of our theory is that it isexplicitly relativistic covariant. Time and space coordi-nates are treated on an equal footing. In particular, theusual space probability density given by | ψ | is general-ized to the spacetime probability density (see, e.g., [8, 9]for old forms of that idea). It seems that without such arelativistic probabilistic interpretation, the compatibilitybetween the local equations for particle trajectories andprobabilistic predictions of QM could not be achieved.Let x = { x µ } , µ = 0 , , ,
3, denotes the coordinatesof a position in spacetime. The state of n free (butpossibly entangled!) relativistic spin-0 particles can bedescribed by a many-time wave function ψ ( x , . . . , x n ).This wave function satisfies n Klein-Gordon equations(with the units ¯ h = c = 1 and the Minkowski metricsignature (+ , − , − , − ))( ∂ µa ∂ aµ + m a ) ψ ( x , . . . , x n ) = 0 , (1)one for each x a , a = 1 , . . . , n . (The Einstein conventionof summation over repeated indices refers only to vec-tor indices µ , not to particle labels a .) From this wavefunction one can construct the quantity j µ ...µ n ( x , . . . , x n ) ≡ (cid:18) i (cid:19) n ψ ∗ ↔ ∂ µ · · · ↔ ∂ µ n ψ, (2)where χ ↔ ∂ µ ϕ ≡ χ∂ µ ϕ − ( ∂ µ χ ) ϕ and ∂ µ a ≡ ∂/∂x µ a a . Thequantity (2) transforms as an n -vector [10]. Eq. (1) im-plies that this quantity satisfies the conservation equation ∂ µ j µ ...µ n = 0 and similar conservation equations withother ∂ µ a . Thus we have n conservation equations ∂ µ a j µ ...µ a ...µ n = 0 , (3)one for each x a . Assuming that ψ is a superposition ofpositive-frequency solutions to (1), ψ can be normalizedsuch that the n -particle Klein-Gordon norm is equal to1. Explicitly, this means that Z Σ dS µ · · · Z Σ n dS µ n n j µ ...µ n = 1 , (4)where Σ a are arbitrary 3-dimensional spacelike hypersur-faces and dS µ a a = d x a | g (3) a | / n µ a (5)is the covariant measure of the 3-volume on Σ a . Here n µ a is the unit future-oriented vector normal to Σ a , while g (3) a is the determinant of the induced metric on Σ a . The con-servation equations (3) imply that the left-hand side of(4) does not depend on the choice of timelike hypersur-faces Σ , . . . , Σ n .Now we introduce n j aµ ( x a ) byomitting the integration over dS µ a a in (4). For example,for a = 1, j µ ( x ) = Z Σ dS µ · · · Z Σ n dS µ n n j µµ ...µ n ( x , . . . , x n ) , (6)which does not depend on the choice of timelike hyper-surfaces Σ , . . . , Σ n and satisfies ∂ µ j µ = 0. This impliesthe conservation equation n X a =1 ∂ aµ j µa ( x a ) = 0 . (7)Next we study the integral curves of the vector fields j µa ( x a ). These integral curves can be represented by func-tions ˜ X µa (˜ s ) satisfying local differential equations d ˜ X µa (˜ s ) d ˜ s = j µa ( ˜ X a (˜ s )) , (8)where ˜ s is an auxiliary scalar parameter (a generalizedproper time [11]) that parameterizes the curves. How-ever, the curves in spacetime do not depend on their pa-rameterization. In particular, even though the equations(8) are local, nonlocal parameterizations can also be in-troduced. For example, along the integral curves the fol-lowing parameterization-independent equalities are valid dx µa dx νb = j µa ( x a ) j νb ( x b ) = v µa ( x , . . . , x n ) v νb ( x , . . . , x n ) , (9)where v µa ( x , . . . , x n ) ≡ j µa ( x a ) | ψ ( x , . . . , x n ) | . (10)Thus we see that the integral curves of j µa ( x a ) can alsobe parameterized by different functions X µa ( s ) satisfyingnonlocal equations of motion dX µa ( s ) ds = v µa ( X ( s ) , . . . , X n ( s )) . (11)The conservation equation (7) now can be written as n X a =1 ∂ aµ ( | Ψ | v µa ) = 0 , (12) where Ψ( x , . . . , x n ) = ψ ( x , . . . , x n ) N / , (13)and N is a normalization constant to be fixed later. SinceΨ( x , . . . , x n ) does not have an explicit dependence on s ,(12) can also be written as ∂ | Ψ | ∂s + n X a =1 ∂ aµ ( | Ψ | v µa ) = 0 . (14)In [12], the integral curves of j µa ( x a ) have been used asan auxiliary mathematical tool. Here, using the new re-sult above that these curves can also be viewed as integralcurves of v µa ( x , . . . , x n ), we propose a different interpre-tation of these curves. We propose that these integralcurves are the actual particle trajectories. The com-patibility with statistical predictions of the “standard”purely probabilistic interpretation of QM is provided byEq. (14), now interpreted as the relativistic equivarianceequation [5, 13–18]. Namely, if a statistical ensemble ofparticles has the probability distribution (on the rela-tivistic 4 n -dimensional configuration space) equal to ρ ( x , . . . , x n ) = | Ψ( x , . . . , x n ) | (15)for some initial s , then the equivariance equation (14)provides that the ensemble will have the distribution (15)for any s . (The scalar parameter s itself can be inter-preted as a relativistic analogue of the Newton absolutetime [11, 18].) In this sense, the particle trajectories (11)are compatible with the quantum-mechanical probabilitydistribution (15).A few additional remarks are in order. First, the prob-abilistic interpretation (15) implies that N in (13) shouldbe fixed to N = Z d x · · · d x n | ψ ( x , . . . , x n ) | . (16)To avoid dealing with an infinite N , one can confinethe whole physical system into a large but finite 4-dimensional spacetime box. Mathematically more rig-orous ways of dealing with wave functions that do notvanish at infinity also exist, such as the rigged Hilbertspace [19].Second, the spacetime probabilistic interpretation(15), generalizing the usual space probabilistic interpre-tation of nonrelativistic QM, has also been studied inolder literature, such as [8, 9]. A detailed discussion ofcompatibility of such a generalized probabilistic interpre-tation with the usual probabilistic interpretation is pre-sented in [17]. In particular, in [17] it is explained howparticle trajectories obeying (14) are compatible with all statistical predictions of QM, not only with statisticalpredictions on particle positions. (For example, eventhough (8) may lead to superluminal velocities, a mea-sured velocity cannot be superluminal [17].) The keyinsight is that all observations can be reduced to observa-tions of spacetime positions of some macroscopic pointerobservables.Third, in the classical limit, the Klein-Gordon equa-tions (1) reduce to the classical relativistic Hamilton-Jacobi equation [11]. Consequently, by evaluating (6)in that limit, it can be shown that (8) reduces to theclassical relativistic equation of motion.Fourth, it is straightforward to generalize the theoryabove to particles interacting with a classical gravita-tional or electromagnetic background, by replacing thederivatives ∂ µ with the appropriate covariant derivatives.Again, it can be shown that the resulting theory has thecorrect classical limit.Fifth, the theory can be generalized to particles withspin and even to quantum field theory, by appropriateadaptation of the formal developments presented in [16,18].To summarize, our main results can be reexpressedin the following way. Eqs. (11) are nonlocal Bohmian-like equations of motion, compatible with statistical pre-dictions of QM due to the equivariance equation (14).However, owing to the specific form of (10) in which allnonlocality is carried by a common scalar factor univer- sal to all components of v µa , the nonlocality of (11) isonly an apparent nonlocality. This apparent nonlocal-ity can be explicitly eliminated by choosing a differentparameterization of the particle trajectories, which leadsto the manifestly local equations of motion (8). In thissense, our theory of particle trajectories is much morelocal than the usual Bohmian formulation of QM. Yet,a certain nonlocal feature is still present. To provideconsistency with statistical predictions of QM, one mustassume that the a priori probabilities of initial particlepositions X µa (0) are given by (15). Thus, all nonlocalitycan be ascribed to initial nonlocal correlations betweenthe particle spacetime positions. The nonlocal forces be-tween the particles turn out to be superfluous.Even if the theory described above does not describethe true reality behind QM, at the very least it providesan example which explicitly demonstrates that quantumreality may be much less nonlocal then suggested by theusual form of the Bohm interpretation. We believe thatit significantly enriches the general understanding of non-locality and relativity in QM.This work was supported by the Ministry of Science ofthe Republic of Croatia under Contract No. 098-0982930-2864. [1] J. S. Bell, Speakable and Unspeakable in Quantum Me-chanics (Cambridge University Press, Cambridge, 1987).[2] D. M. Greenberger, M. Horne and A. Zeilinger, in
Bell’sTheorem, Quantum Theory, and Conceptions of the Uni-verse , edited by M. Kafatos (Kluwer Academic, Dor-drecht, 1989).[3] L. Hardy, Phys. Rev. Lett. , 2981 (1992).[4] D. Bohm, Phys. Rev. , 166 (1952); D. Bohm, Phys.Rev. , 180 (1952).[5] D. Bohm and B. J. Hiley, The Undivided Universe (Rout-ledge, London, 1993).[6] B. S. DeWitt and N. Graham (eds.),