Quantum mechanics in noninertial reference frames: Violations of the nonrelativistic equivalence principle

In previous work we have developed a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group that includes transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as is the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. A special feature of these previously constructed representations is that they all respect the non-relativistic equivalence principle, wherein the fictitious forces associated with linear acceleration can equivalently be described by gravitational forces. In this paper we exhibit a large class of cocycle representations of the Galilean line group that violate the equivalence principle. Nevertheless the classical mechanics analogue of these cocycle representations all respect the equivalence principle.     

Quantum mechanics in non-inertial reference frames: Time-dependent rotations and loop prolongations

This is the fourth in a series of papers on developing a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group to include transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. In previous work, we have shown that there exist representations of the Galilean line group that uphold the non-relativistic equivalence principle as well as representations that violate the equivalence principle. In these previous studies, the focus was on linear accelerations. In this paper, we undertake an extension of the formulation to include rotational accelerations. We show that the incorporation of rotational accelerations requires a class of loop prolongations of the Galilean line group and their unitary cocycle representations. We recover the centrifugal and Coriolis force effects from these loop representations. Loops are more general than groups in that their multiplication law need not be associative. Hence, our broad theoretical claim is that a Galilean quantum theory that holds in arbitrary non-inertial reference frames requires going beyond groups and group representations, the well-established framework for implementing symmetry transformations in quantum mechanics.     

Quantum mechanics in noninertial reference frames: Relativistic accelerations and fictitious forces

One-particle systems in relativistically accelerating reference frames can be associated with a class of unitary representations of the group of arbitrary coordinate transformations, an extension of the Wigner–Bargmann definition of particles as the physical realization of unitary irreducible representations of the Poincaré group. Representations of the group of arbitrary coordinate transformations become necessary to define unitary operators implementing relativistic acceleration transformations in quantum theory because, unlike in the Galilean case, the relativistic acceleration transformations do not themselves form a group. The momentum operators that follow from these representations show how the fictitious forces in noninertial reference frames are generated in quantum theory.     

Projective representations of the inhomogeneous Hamilton group: Noninertial symmetry in quantum mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries include the Weyl–Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl–Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves the Heisenberg commutation relations invariant is essentially a projective representation of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of the Hamilton equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.     

Die Eindeutigkeit der Ortsobservablen in einer irreduziblen unitären Darstellung bis auf einen Faktor der Galilei-Gruppe

Uniqueness of the Position Observable in an Irreducible Unitary Representation up to a Factor of the Galilei Group. An elementarary quantum mechanical system with non vanishing mass is characterized by a continuous irreducible unitary representation up to a factor of the Galilei group in Hilbert space. An argument is given concerning the continuous iurreducible unitary representations of the universal covering group of the Euclidean group such that the position observable is uniquely determined by the transformation properties under the representation of the Galilei group.     

Unitarity of Induced Representations from Coisotropic Quantum Subgroups

We study unitarity of the induced representations from coisotropic quantum subgroups which were introduced by Ciccoli in math.QA/9804138. We define a real structure on coisotropic subgroups which determines an involution on the homogeneous space. We give general invariance properties for functionals on the homogeneous space which are sufficient to build a unitary representation starting from the induced one. We present the case of the one-dimensional quantum Galilei group where we have to use in all generality our definition of quasi-invariant functional.     

Non-linear Liouville and Shrödinger equations in phase space

Unitary representations of the Galilei group are studied in phase space, in order to describe classical and quantum systems. Conditions to write in general form the generator of time translation and Lagrangians in phase space are then established. In the classical case, Galilean invariance provides conditions for writing the Liouville operator and Lagrangian for non-linear systems. We analyze, as an example, a generalized kinetic equation where the collision term is local and non-linear. The quantum counter-part of such unitary representations are developed by using the Moyal (or star) product. Then a non-linear Schrödinger equation in phase space is derived and analyzed. In this case, an association with the Wigner formalism is established, which provides a physical interpretation for the formalism.     

Physical theories in Galilean space-time and the origin of Schrödinger-like equations

A method to develop physical theories of free particles in space-time with the Galilean metric is presented. The method is based on a Principle of Analyticity and a Principle of Relativity, and uses the Galilei group of the metric. The first principle requires that state functions describing the particles are analytic and the second principle demands that dynamical equations for these functions are Galilean invariant. It is shown that the method can be used to formally derive Schrödinger-like equations and to determine modifications of the Galilei group of the metric that are necessary to fullfil the requirements of analyticity and Galilean invariance. The obtained results shed a new light on the origin of Schrödinger’s equation of non-relativistic quantum mechanics.     

Perturbations of unitary representations of the Galilei group: Mass operators with continuous spectra

We present a systematic procedure for constructing mass operators with continuous spectra for a system of particles in a manner consistent with Galilean relativity. These mass operators can be used to construct what may be called point-form Galilean dynamics. As in the relativistic case introduced by Dirac, the point-form dynamics for the Galilean case is characterized by both the Hamiltonian and momenta being altered by interactions. An interesting property of such perturbative terms to the Hamiltonian and momentum operators is that, while having well-defined transformation properties under the Galilei group, they also satisfy Maxwell’s equations. This result is an alternative to the well-known Feynman-Dyson derivation of Maxwell’s equations from non-relativistic quantum physics.     

Conformally covariant vector–spinor field in de Sitter space

Unlike the Lorentz transformation which replaces the Galilean transformation among inertial frames at high relative velocities, there seems to be no such a consensus in the case of coordinate transformation between inertial frames and uniformly rotating ones. There have been some attempts to generalize the Galilean rotational transformation to high rotational velocities. Here we introduce a modified version of one of these transformations proposed by Philip Franklin in 1922. The modified version is shown to resolve some of the drawbacks of the Franklin transformation, specially with respect to the corresponding spacetime metric in the rotating frame. This new transformation introduces non-inertial eccentric observers on a uniformly rotating disk and the corresponding metric in the rotating frame is shown to be consistent with the one obtained through Galilean rotational transformation for points close to the rotation axis. Employing the threading formulation of spacetime decomposition, spatial distances and time intervals in the spacetime metric of a rotating observer’s frame are also discussed.     

Constitution of objects in classical mechanics and in quantum mechanics

The constitution of objects is discussed in classical mechanics and in quantum mechanics. The requirement of objectivity and the Galilei invariance of classical and quantum mechanics leads to the postulate of covariance which must be fulfilled by observable quantities. Objects are then considered as carriers of these covariant observables and turn out to be representations of the Galilei group. Individual systems can be defined in classical mechanics by their trajectories in phase space. However, in quantum mechanics the characterization of individuals can only be achieved approximately by means of unsharp observables.     

Induced Representations of the One-Dimensional Quantum Galilei Group

We study the representations of the quantum Galilei group by a suitable generalization of the Kirillov method on spaces of noncommutative functions. On these spaces, we determine a quasi-invariant measure with respect to the action of the quantum group by which we discuss unitary and irreducible representations. The latter are equivalent to representations on ², i.e. on the space of square summable functions on a one-dimensional lattice.     

Perturbations of unitary representations of finite dimensional Lie groups

In quantum physical theories, interactions in a system of particles are commonly understood as perturbations to certain observables, including the Hamiltonian, of the corresponding interaction-free system. The manner in which observables undergo perturbations is subject to constraints imposed by the overall symmetries that the interacting system is expected to obey. Primary among these are the spacetime symmetries encoded by the unitary representations of the Galilei group and Poincaré group for the non-relativistic and relativistic systems, respectively. In this light, interactions can be more generally viewed as perturbations to unitary representations of connected Lie groups, including the non-compact groups of spacetime symmetry transformations. In this paper, we present a simple systematic procedure for introducing perturbations to (infinite dimensional) unitary representations of finite dimensional connected Lie groups. We discuss applications to relativistic and non-relativistic particle systems.     

Realisations of the representations of para-Fermi algebras—Part IV: Fock spaces of para-Bose and para-Fermi operators

The representations of the para-Fermi algebra in the Fock spaces of para-Bose and para-Fermi operators are constructed. The unitary equivalence of different representations is proved. The Bardeen-Cooper-Schrieffer pair creation and annihilation operators and the four fermion interaction appear as particular realisations of the para-Fermi algebra. The para-Fermi algebra representations in quantum mechanics are discussed.     

Unitary equivalence of different bipartite entangled states under unitary transformations

The unitary equivalence of different bipartite entangled states with continuous variables under unitary transformations are investigated. With the help of the technique of integration within an ordered product of operators, the corresponding unitary operators are also derived. These results may deepen people's understanding to the various bipartite entangled states, and enrich the representations and transformations theory in quantum mechanics.     

Invariance Properties of the Entropy Production,and the Entropic Pairing of Inertial Frames of Reference by Shear-Flow Systems

This study examines the invariance properties of the thermodynamic entropy production in its global (integral), local (differential), bilinear, and macroscopic formulations, including dimensional scaling, invariance to fixed displacements, rotations or reflections of the coordinates, time antisymmetry, Galilean invariance, and Lie point symmetry. The Lie invariance is shown to be the most general, encompassing the other invariances. In a shear-flow system involving fluid flow relative to a solid boundary at steady state, the Galilean invariance property is then shown to preference a unique pair of inertial frames of reference—here termed an entropic pair—respectively moving with the solid or the mean fluid flow. This challenges the Newtonian viewpoint that all inertial frames of reference are equivalent. Furthermore, the existence of a shear flow subsystem with an entropic pair different to that of the surrounding system, or a subsystem with one or more changing entropic pair(s), requires a source of negentropy—a power source scaled by an absolute temperature—to drive the subsystem. Through the analysis of different shear flow subsystems, we present a series of governing principles to describe their entropic pairing properties and sources of negentropy. These are unaffected by Galilean transformations, and so can be understood to “lie above” the Galilean inertial framework of Newtonian mechanics. The analyses provide a new perspective into the field of entropic mechanics, the study of the relative motions of objects with friction.     

Theory of symmetry in the quantum mechanics of infinite systems: II. Isotropic representations of the Galilei group and its central extensions

The isotropic bundle representations of the Galilei group and its central extension are classified, and the natural cross-section, action of the group on the base manifold and the canonical cocyle are determined for all cases. Projective bundle representations of the Galilei group are defined and the extension of Bargmann's superselection rule is established. Coordinate transformations on the base space are discussed in all cases, and the notion of generalized coordinate transformations is introduced. It is then shown that the bundle representations being considered do not violate the principle of Galilean relativity as it is commonly understood. The physical interpretation of the irreducible and some reducible representations is discussed. It is found that some bundle representations might correspond to objects which can act as sources or sinks of linear and/or angular momentum.     

Relativistic Few-Body Physics

I discuss different formulations of the relativistic few-body problem with an emphasis on how they are related. I first discuss the implications of some of the differences with non-relativistic quantum mechanics. Then I point out that the principle of special relativity in quantum mechanics implies that the quantum theory has a Poincaré symmetry, which is realized by a unitary representation of the Poincaré group. This representation can always be decomposed into direct integrals of irreducible representations and the different formulations differ only in how these irreducible representations are realized. I discuss how these representations appear in different formulations of relativistic quantum mechanics and discuss some applications in each of these frameworks.     

Gauge fixing,BRS invariance and Ward identities for randomly stirred flows

The Galilean invariance of the Navier–Stokes equation is shown to be akin to a global gauge symmetry familiar from quantum field theory. This symmetry leads to a multiple counting of infinitely many inertial reference frames in the path integral approach to randomly stirred fluids. This problem is solved by fixing the gauge, i.e., singling out one reference frame. The gauge fixed theory has an underlying Becchi–Rouet–Stora (BRS) symmetry which leads to the Ward identity relating the exact inverse response and vertex functions. This identification of Galilean invariance as a gauge symmetry is explored in detail, for different gauge choices and by performing a rigorous examination of a discretized version of the theory. The Navier–Stokes equation is also invariant under arbitrary rectilinear frame accelerations, known as extended Galilean invariance (EGI). We gauge fix this extended symmetry and derive the generalized Ward identity that follows from the BRS invariance of the gauge-fixed theory. This new Ward identity reduces to the standard one in the limit of zero acceleration. This gauge-fixing approach unambiguously shows that Galilean invariance and EGI constrain only the zero mode of the vertex but none of the higher wavenumber modes.