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Acta Physica Sinica

Properties of geometric phase under Galilean transformation

Zheng Ying-Hong, Tong Chen, Ping Wang, Zhe Chang, 中国科学院高能物理研究所,北京 100049

Published Jan 1, 2007• 0 citations
Publisher

Abstract

Properties of geometric phase under Galilean transformation are investigated within a general framework. We found that the geometric phase in the system commonly met with in the laboratory is Galilean invariant. However, this is not always true for general non-relativistic quantum systems. The physical origins of the frame dependence of geometric phase is discussed in detail. We emphasize that the change of geometric phase caused by Galilean boosts does not destroy the results of measurement for two-state interference. It is consistent with the general constraint of Galilean invariance on a non-relativistic quantum system.

DOI
10.7498/aps.56.6199

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Properties of geometric phase under Galilean transformation are investigated within a general framework. We found that the geometric phase in the system commonly met with in the laboratory is Galilean invariant. However, this is not always true for general non-relativistic quantum systems. The physical origins of the frame dependence of geometric phase is discussed in detail. We emphasize that the change of geometric phase caused by Galilean boosts does not destroy the results of measurement for two-state interference. It is consistent with the general constraint of Galilean invariance on a non-relativistic quantum system.
Neither geometric phases nor differences in geometric phases are generally invariant under time-dependent unitary transformations (unlike differences in total phases), in particular under local gauge transformations and Galilei transformations. (This was pointed out originally by Aharonov and Anandan, and in the case of Galilei transformations has recently been shown explicitly by Sjoeqvist, Brown and Carlsen.) In this paper, I introduce a phase, related to the standard geometric phase, for which phase differences are both gauge- and Galilei-invariant, and, indeed, invariant under transformations to linearly accelerated coordinate systems. I discuss in what sense this phase can also be viewed as geometric, what its relation is to earlier proposals for making geometric phases invariant under gauge or Galilei transformations, and what is its classical analogue. I finally apply this invariant phase to Berry's derivation of the Aharonov-Bohm effect.
Closed systems in Newtonian mechanics obey the principle of Galilean relativity. However, the usual Lagrangian for Newtonian mechanics, formed from the difference of kinetic and potential energies, is not invariant under the full group of Galilean transformations. In quantum mechanics, Galilean boosts require a non-trivial transformation rule for the wave function and a concomitant “projective representation” of the Galilean symmetry group. Using Feynman's path integral formalism, this latter result can be shown to be equivalent to the non-invariance of the Lagrangian. Thus, using path integral methods, the representation of certain symmetry groups in quantum mechanics can be simply understood in terms of the transformation properties of the classical Lagrangian and conversely. The main results reported here should be accessible to students and teachers of physics—particularly classical mechanics, quantum mechanics, and mathematical physics—at the advanced undergraduate and beginning graduate levels, providing a useful exposition for those wanting to explore topics such as the path integral formalism for quantum mechanics, relativity principles, Lagrangian mechanics, and representations of symmetries in classical and quantum mechanics.
We consider symmetries and reduction in non-relativistic many-body quantum mechanics, with the aim of identifying physically meaningful observables in systems such as molecules and crystalline solids. To this end, we propose a unified framework based on two additional postulates supplementing the standard quantum-mechanical formalism. For stable systems, the physically relevant states are normalizable stationary states, while physically meaningful observables are required to be invariant under a selected subgroup of the symmetry group and under Galilean boosts. In addition, we postulate the existence of a map from the set of all observables allowed by quantum mechanics to the corresponding invariant physically meaningful observables. The originality of the present work does not lie in specific reductions, but in the unified framework that connects symmetry reduction and relational many-body quantum theory. We interpret entities like superselection rules and quantum reference frames as important parts of the postulated process of obtaining the physically meaningful relational description. In particular, the requirement of Galilean-boost invariance added strengthens the criterion for physical observability by excluding quantities that depend on the choice of inertial frame. An important consequence of the postulates is that in the considered cases every physically meaningful observable necessarily depends on more than one non-invariant observable, the latter being typically associated with degrees of freedom assigned to a single particle. The postulates thus lead to theories that are well aligned with the literature on reduction and the description of molecules, while at the same time being consistent with the relational interpretation of quantum mechanics, according to which the complete physical description of a system is defined only relative to other systems.
The developments of special relativity and quantum mechanics marked beginning of modern physics age. The former has taught us that while space and time are frame dependent notions, there is a quantity---the space-time interval---whose value all inertial observers agree upon. This reveals, so to speak, a genuine fact of universe, a relativistic invariant. On other hand, since dawn of quantum mechanics, there is no consensus on what theory is all about. The situation is admittedly subtler: quantum theory is grounded on a complex vector space and very notions of and reference frame are controversial. Here we construct a theoretical framework within which a given combination of quantum resources is shown to be a Galilean invariant. To this end, we postulate a principle of relational symmetry between the observer and the observed and employ notion of quantum reference frame. Unitary transformations then follow that allow us to perceive physical resources seen from viewpoint of any quantum system. Interestingly, we find that one needs more than quantum coherence and quantum correlations to prove quantum resources covariance. Finally, we show that notion of physical reality implied by quantum mechanics is not absolute.