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arXiv:hep-th/0504060 · International Journal of Modern Physics A

QUANTUM MECHANICS IN NONINERTIAL FRAMES WITH A MULTITEMPORAL QUANTIZATION SCHEME II: NONRELATIVISTIC PARTICLES

DAVID ALBA

Published Aug 10, 2006• 16 citations
arXivHTMLPDF

Abstract

The nonrelativistic version of the multitemporal quantization scheme of relativistic particles in a family of noninertial frames (see Ref. 1) is defined. At the classical level the description of a family of nonrigid noninertial frames, containing the standard rigidly linear accelerated and rotating ones, is given in the framework of parametrized Galilei theories. Then the multitemporal quantization, in which the gauge variables, describing the noninertial effects, are not quantized but considered as c-number generalized times, is applied to nonrelativistic particles. It is shown that with a suitable ordering there is unitary evolution in all times and that, after the separation of the center-of-mass, it is still possible to identify the inertial bound states. The few existing results of quantization in rigid noninertial frames are recovered as special cases.

DOI
10.1142/s0217751x0603254x

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The nonrelativistic version of the multitemporal quantization scheme of relativistic particles in a family of noninertial frames (see Ref. 1) is defined. At the classical level the description of a family of nonrigid noninertial frames, containing the standard rigidly linear accelerated and rotating ones, is given in the framework of parametrized Galilei theories. Then the multitemporal quantization, in which the gauge variables, describing the noninertial effects, are not quantized but considered as c-number generalized times, is applied to nonrelativistic particles. It is shown that with a suitable ordering there is unitary evolution in all times and that, after the separation of the center-of-mass, it is still possible to identify the inertial bound states. The few existing results of quantization in rigid noninertial frames are recovered as special cases.
The non-relativistic version of the multi-temporal quantization scheme of relativistic particles in a family of non-inertial frames (see hep-th/0502194) is defined. At the classical level the description of a family of non-rigid non-inertial frames, containing the standard rigidly linear accelereted and rotating ones, is given in the framework of parametrized Galilei theories. Then the multi-temporal quantization, in which the gauge variables, describing the non-inertial effects, are not quantized but considered as c-number generalized times, is applied to non relativistic particles. It is shown that with a suitable ordering there is unitary evolution in all times and that, after the separation of center of mass, it is still possible to identify the inertial bound states. The few existing results of quantization in rigid non-inertial frames are recovered as special cases.
After a review of the few attempts to define quantum mechanics in noninertial frames, we introduce a family of relativistic nonrigid noninertial frames (equal-time parallel hyper-planes with differentially rotating 3-coordinates) as a gauge fixing of the description of N positive energy particles in the framework of parametrized Minkowski theories. Then we define a multitemporal quantization scheme in which the particles are quantized, but not the gauge variables describing the noninertial frames: they are considered as c-number generalized times. We study the coupled Schrödinger-like equations produced by the first class constraints and we show that there is a physical scalar product independent both from time and generalized times and a unitary evolution. Since a path in the space of the generalized times defines a nonrigid noninertial frame, we can find the associated self-adjoint effective Hamiltonian [Formula: see text] for the noninertial evolution: it differs from the inertial energy operator for the presence of inertial potentials and turns out to be frame-dependent like the energy density in general relativity. After a separation of the relativistic center of mass from the relative variables by means of a recently developed relativistic kinematics, inside [Formula: see text] we can identify the self-adjoint relative energy operator (the invariant mass) [Formula: see text] corresponding to the inertial energy and producing the same levels for the spectra of atoms as in inertial frames. Instead the (in general time-dependent) effective Hamiltonian is responsible for the interferometric effects signaling the noninertiality of the frame. It cannot be interpreted as an energy (there is no relativity principle and no kinematic group in noninertial frames) and generically, like in the case of time-dependent c-number external electromagnetic fields, it has no associated eigenvalue equation defining a noninertial spectrum. This formulation should help to find relativistic Bel inequalities and to define a quantization scheme for canonical gravity after having found a ultraviolet regularization of the Tomonaga–Schwinger formalism in special relativity as required by the Torre–Varadarajan no-go theorem.
A review is given of a new relativistic classical mechanics for N-particle systems compatible with relativistic bound states. It is formulated in global non-inertial frames in Minkowski space-time by means of parametrized Minkowski theories and then restricted to the intrinsic inertial rest frame of the isolated system. There is a complete control on the relativistic collective variables (Newton-Wigner center of mass, Fokker-Pryce center of inertia, M/oller center of energy) and on the realization of the Poincare' algebra (with the explicit form of the interaction-dependent Lorentz boosts). The particle world-lines are found to correspond to the ones of predictive mechanics and localization problems are clarified. The model can be consistently quantized avoiding the instantaneous spreading of the center-of-mass wave packets (Hegerfeldt theorem), because the non-local non-covariant center of mass is a non-measurable quantity. This implies that the Hilbert space must be the tensor product of the center-of-mass Hilbert space with the one of relative motions. This spatial non-separability (due to the Lorentz signature of space-time) makes relativistic entanglement much more involved than the non-relativistic one.
A new formulation of relativistic quantum mechanics is proposed in the framework of the rest-frame instant form of dynamics, where the world-lines of the particles are parametrized in terms of the Fokker-Pryce center of inertia and of Wigner-covariant relative 3-coordinates inside the instantaneous Wigner 3-spaces, and where there is a decoupled (non-covariant and non-local) canonical relativistic center of mass. This approach: (a) allows us to make a consistent quantization in every inertial frame; (b) leads to a description of both bound and scattering states; (c) offers new insights on the relativistic localization problem; (d) leads to a non-relativistic limit with a Hamilton-Jacobi treatment of the Newton center of mass; (e) clarifies non-local aspects (spatial non-separability) of relativistic entanglement connected with Lorentz signature and not present in its non-relativistic treatment.
Noninertial transformations on time–position–momentum–energy space { t , q , p , e } with invariant Born–Green metric and the symplectic metric −d e ∧ d t + d p ∧ d q are studied. This group of transformations contains the Lorentz group as the inertial special case and, in the limit of small forces and velocities, reduces to the expected Hamilton transformations leaving invariant the symplectic metric and the nonrelativistic line element d s 2 = −d t 2 . The transformations bound relative velocities by c and relative forces by b . Spacetime is no longer an invariant subspace but is relative to noninertial observer frames. In the limit of b → ∞, spacetime is invariant. Born was lead to the metric by a concept of reciprocity between position and momentum degrees of freedom and for this reason we call this reciprocal relativity. For large b , such effects will almost certainly only manifest in a quantum regime. Wigner showed that special relativistic quantum mechanics follows from the projective representations of the inhomogeneous Lorentz group. Projective representations of a Lie group are equivalent to the unitary representations of its central extension. The same method of projective representations for the inhomogeneous group is used to define the quantum theory in the noninertial case. The central extension of the inhomogeneous group is the cover of the quaplectic group . is the Weyl–Heisenberg group. The group, and the associated Heisenberg commutation relations central to quantum mechanics, results directly from requiring projective representations. A set of second-order wave equations result from the representations of the Casimir operators.
Mathjax On | Off Abstract In the preceding chapter, we showed how one-particle quantum states can be obtained from unitary irreducible representations of the Galilei group, the transformation group of Newton relativity. Since the Galilei group defines transformations amongst inertial reference frames, this description of one-particle quantum states is limited to inertial observers. In particular, observers who are accelerating are excluded in this description. This motivates the question of whether it is possible to provide a description of one-particle quantum states that holds for non-inertial observers by expanding the notion of Newton relativity to include non-inertial reference frames.