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arXiv:1204.5627 · Journal of Physics A Mathematical and Theoretical

Kinematics and dynamics in noninertial quantum frames of reference

R. M. Angelo, A D Ribeiro

Published Oct 31, 2012• 38 citations
arXivHTMLPDF

Abstract

From the principle that there is no absolute description of a physical state, we advance the approach according to which one should be able to describe the physics from the perspective of a quantum particle. The kinematics seen from this frame of reference is shown to be rather unconventional. In particular, we discuss several subtleties emerging in the relative formulation of central notions, such as vector states, the classical limit, entanglement, uncertainty relations and the complementary principle. A Hamiltonian formulation is also derived which correctly encapsulates effects of fictitious forces associated with the accelerated motion of the frame. Our approach shows, therefore, how to formulate nonrelativistic quantum mechanics within noninertial reference frames which can be consistently described by the theory, with no appeal to classical elements.

DOI
10.1088/1751-8113/45/46/465306

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From the principle that there is no absolute description of a physical state, we advance the approach according to which one should be able to describe the physics from the perspective of a quantum particle. The kinematics seen from this frame of reference is shown to be rather unconventional. In particular, we discuss several subtleties emerging in the relative formulation of central notions, such as vector states, the classical limit, entanglement, uncertainty relations and the complementary principle. A Hamiltonian formulation is also derived which correctly encapsulates effects of fictitious forces associated with the accelerated motion of the frame. Our approach shows, therefore, how to formulate nonrelativistic quantum mechanics within noninertial reference frames which can be consistently described by the theory, with no appeal to classical elements.
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.
In physics, every observation is made with respect to a frame of reference. Although reference frames are usually not considered as degrees of freedom, in all practical situations it is a physical system which constitutes a reference frame. Can a quantum system be considered as a reference frame and, if so, which description would it give of the world? Here, we introduce a general method to quantise reference frame transformations, which generalises the usual reference frame transformation to a "superposition of coordinate transformations". We describe states, measurement, and dynamical evolution in different quantum reference frames, without appealing to an external, absolute reference frame, and find that entanglement and superposition are frame-dependent features. The transformation also leads to a generalisation of the notion of covariance of dynamical physical laws, to an extension of the weak equivalence principle, and to the possibility of defining the rest frame of a quantum system.
Abstract The subject of quantum reference frame transformations gets popular lately with some interesting new theoretical development partly for the reason that the physics involved is becoming experimentally accessible. The position of a position eigenstate when observed from an object with ‘uncertainty’ in position would be seen with ‘uncertainty’. In fact, even the existence of entanglement is reference frame-dependent. We present an improved formulation of such a transformation and give a novel way to describe exactly by ‘how much’ the ‘value of the position’ has changed which fully encodes all information about the changes, including the ‘uncertainty’ and entanglement. That is an application of the notion of noncommutative values of physical quantities we introduced to understand the reality of quantum physics and beyond. Some implications on fundamental physics will also be discussed. In particular, we suggest thinking about quantum gravity as a theory of general quantum relativity, alleviating Penrose’s notion of incompatibility of qauntum mechanics with the relativity principle.
The observational basis of quantum theory in accelerated systems is studied. The extension of Lorentz invariance to accelerated systems via the hypothesis of locality is discussed and the limitations of this hypothesis are pointed out. The nonlocal theory of accelerated observers is briefly described. Moreover, the main observational aspects of Dirac's equation in noninertial frames of reference are presented. The Galilean invariance of nonrelativistic quantum mechanics and the mass superselection rule are examined in the light of the invariance of physical laws under inhomogeneous Lorentz transformations.
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.
Abstract Quantum reference frame as in the example of observing a quantum particle from another has been a topic of much recent interest. Quantum spatial translations, quantum rotations, and quantum Lorentz boosts in the sense have been studied to some extent. The article aims at using a consistent formulation of all that to give a full picture of what would be the symmetry of Special Quantum Relativity as special relativity as seen from quantum particles. We first look at a direct formulation of the mathematical quantum analog of any classical one-parameter Lie group of such transformations suggested by our Heisenberg picture thinking of quantum mechanics. A quantum rotation and a quantum Lorentz boost are presented. Analysis of the compositions of two such transformations shows unnatural results with various problems, especially with their commutators. From a physical point of view, we argue for a modified formulation. A quantum reference frame transformation, as the description of firstly the spacetime position of an object, idealized as a particle, as seen from another has to involve a full set of four-coordinate quantum quantities instead of one in a single coordinate only. Unlike the classical case, we illustrate that the analog of a change by the zero classical coordinate values has nontrivial effects in the transformation. A consistent full picture of the symmetry is then obtained. Results that could be interpreted essentially as the quantum analogs of the translation of a single coordinate observable, including angular ones for a rotation or a boost, can be retrieved from that but only as approximations. Under the traditional perspective of relativity symmetry being about spacetime reference frames, our task is then completed. We further look at the quantum version of Heisenberg–Weyl symmetry and perspectives on its relation with the subject matter and argue for the necessity of our background noncommutative geometric perspective for a conceptual consistent story.