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Canadian Journal of Physics

Current conservation as a geometric property of space–time

S. Twareque Ali, James A. Brooke, Paul Busch, R. Gagnon, Franklin E. Schroeck

Published Mar 1, 1988• 16 citations
Publisher

Abstract

Working within the framework of stochastic quantum mechanics, we prove that a mild reality constraint on the radial part of the resolution generator renders the (probability) current, for a single particle with only positive (or only negative) energies and having arbitrary integer spin, conserved. The result is proven both nonrelativistically and relativistically. This extends some earlier results that had been obtained for spinless systems. We argue that the origin of such a conservation law is a consequence of the particular geometry of space–time predicated by the formalism of stochastic quantum mechanics.

DOI
10.1139/p88-036

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Working within the framework of stochastic quantum mechanics, we prove that a mild reality constraint on the radial part of the resolution generator renders the (probability) current, for a single particle with only positive (or only negative) energies and having arbitrary integer spin, conserved. The result is proven both nonrelativistically and relativistically. This extends some earlier results that had been obtained for spinless systems. We argue that the origin of such a conservation law is a consequence of the particular geometry of space–time predicated by the formalism of stochastic quantum mechanics.
It is shown that to every Galilei-covariant nonrelativistic stochastic phase space representation of a system of massive particles, whose generator is rotationally invariant, corresponds a Poincaré-covariant relativistic representation sharing the same generator. The stochastic phase space probability densities of the two representations overlap in the limit of nonrelativistic velocities in the laboratory frame. The relativistic representations give rise to covariant and conserved probability currents at stochastic space–time points, in complete analogy with their nonrelativistic counterparts. This parallelism extends to the existence of a global representation of the proper Poincaré group in L2(Γ), which is reduced by each subspace of L2(Γ) spanned by the set of phase-space wavefunctions generated by some stochastic phase space representation of all pure states of the system.
By taking into account the quantum aspects intrinsic to any operational definition of spatio-temporal relationships, a stochastic concept of spacetime emerges. In relation to its classical counterpart is realized as a stochastic mean around which quantum fluctuations become negligible only in the limit of macroscopic spacetime intervals. The test-particle propagators used in the proposed quantum concept of spacetime are derived by solving in a consistent manner the localizability problem for relativistic particles. This is achieved in the framework of the stochastic phase space formulation of quantum mechanics, which in the nonrelativistic context is shown to result from systems of imprimitivity related to phase space conserved probability currents derivable from bona fide convariant probability densities in stochastic phase spaces of one particle systems, which can be interpreted as due to measurements performed with extended rather than pointlike test particles. The associated particle propagators can be therefore consistently related to coordinate probability densities measurable by the exchange of photons in between test particles from a chosen standard. Quantum spacetime is defined as the family of propagators corresponding to all conceivable coherent flows of test particles. This family of free-fall propagators has to satisfy certain self-consistency conditions as well as consistent laws ofmore » motion which inplicitly determine the stochastic geometro-dynamics of quantum space-time. Field theory on quantum spacetime retains many of the formal features of conventional quantum field theory. On a fundamental epistemological level stochastic geometries emerge as essential prerequisites in the construction of spacetime models that would be operationally based and yet consistent with the relativity principle as well as with the uncertinty principle.« less
In this paper, a conserved current for Klein--Gordon equation is derived. It is shown, for (1+11+11+1) dimensions, the first component of this current is non-negative and reduces to ${|\ensuremath{\phi}|}^{2}$ in nonrelativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from ${|\ensuremath{\phi}|}^{2}$ only when the uncertainty in momentum is greater than m0c{m}_{0}cm0​c.
The positivity of the energy in relativistic quantum mechanics implies that wave functions can be continued analytically to the forward tube T in complex spacetime. For Klein-Gordon particles, we interpret T as an extended (8D) classical phase space containing all 6D classical phase spaces as symplectic submanifolds. The evaluation maps ez:f→f(z)e_z: f\to f(z)ez​:f→f(z) of wave functions on T are relativistic coherent states reducing to the Gaussian coherent states in the nonrelativistic limit. It is known that no covariant probability interpretation exists for Klein-Gordon particles in real spacetime because the time component of the conserved "probability current" can attain negative values even for positive-energy solutions. We show that this problem is solved very naturally in complex spacetime, where ∣f(x−iy)∣2|f(x-iy)|^2∣f(x−iy)∣2 is interpreted as a probability density on all 6D phase spaces in T which, when integrated over the "momentum" variables y, gives a conserved spacetime probability current whose time component is a positive regularization of the usual one. Similar results are obtained for Dirac particles, where the evaluation maps eze_zez​ are spinor-valued relativistic coherent states. For free quantized Klein-Gordon and Dirac fields, the above formalism extends to n-particle/antiparticle coherent states whose scalar products are Wightman functions. The 2-point function plays the role of a reproducing kernel for the one-particle and antiparticle subspaces.
We unify Brownian motion and quantum mechanics in a single mathematical framework. In particular, we show that non-relativistic quantum mechanics of a single spinless particle on a flat space can be described by a Wiener process that is rotated in the complex plane. We then extend this theory to relativistic stochastic theories on manifolds using the framework of second order geometry. As a byproduct, our results suggest that a consistent path integral based formulation of a quantum theory on a Lorentzian (Riemannian) manifold requires an Ito deformation of the Poincare (Galilean) symmetry, arising due to the coupling of the quadratic variation to the affine connection.