Abstract Some introductory results concerning symmetries and conservation laws within the Projection Evolution (PEv) formalism are presented. It is shown that the PEv formalism conserves the quantum probability distributions after unitary transformations of the evolution operators represented by the orthogonal resolution of unity. Some conditions for transformations selecting the states with equal transition probabilities are found. A relation between a given symmetry represented by a Lie group and PEv conservation of the eigenvalues of the Casimir operator is derived.
Abstract Some introductory results concerning symmetries and conservation laws within the Projection Evolution (PEv) formalism are presented. It is shown that the PEv formalism conserves the quantum probability distributions after unitary transformations of the evolution operators represented by the orthogonal resolution of unity. Some conditions for transformations selecting the states with equal transition probabilities are found. A relation between a given symmetry represented by a Lie group and PEv conservation of the eigenvalues of the Casimir operator is derived.
Abstract In the previous paper we examined, for a quantum system, the relation between its n‐dimensional state space and the su (n) Lie algebra. The present paper is devoted to relations between unitary transformations in the state space and orthogonal transformations in Lie's algebra. Two cases can happen. First, the transformations are independently chosen in the two spaces; this amounts to changing the former relation. On the other hand, the relation is maintained and the unitary operators are then related to some of the orthogonal operators. This second case is used to study the evolution operators.
Quantum hydrodynamics represents quantum mechanics through two complementary models: the Eulerian picture, a direct transcription of wave mechanics, and the Lagrangian picture, in which the quantum state is represented by the collective motion of a continuum of fluid particles, the latter being obtained by continuously varying the particle label. The Lagrangian picture thus adds variables to the quantum formalism and exhibits a corresponding new quantum symmetry viz. a continuous particle-relabelling covariance group. Using Noether's theorem, the relabelling symmetry is derived as a component of a general investigation of symmetries and conservation laws in the quantum Lagrangian picture. Relations with symmetries and conservation laws in the Eulerian picture are explored in detail. Alongside the infinite relabelling group, the 12-parameter kinematical covariance group of the Schrödinger equation is derived. It is stressed that the role of label transformations extends beyond the class corresponding to Eulerian invariance. As an example, it is shown that the linear superposition of waves, a fundamental symmetry of the quantum Eulerian picture, can be generated by a deformation-dependent label transformation in the Lagrangian picture.
Using the projection evolution (PEv) approach, time can be included in the quantum mechanics as an observable. Having the time operator, it is possible to explore the temporal structure of various quantum events. In the present paper we discuss the possibility of constructing a quantum clock, which advances in time during its quantum evolution, in each step having some probability to localize itself on the time axis in the new position. We propose a working two-state model as the simplest example of such a clock.
A cornerstone of quantum mechanics is the characterization of symmetries provided by Wigner's theorem. Wigner's theorem establishes that every symmetry of the quantum state space must be either a unitary transformation or an antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated with the dynamics of closed systems and the stochastic evolutions associated with the outcomes of quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time-reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time-symmetric formulation of quantum theory must either restrict the set of the allowed evolutions or modify the operational interpretation of quantum states and processes. Here we propose a time-symmetric formulation of quantum theory where the allowed quantum evolutions are restricted to a suitable set, which includes both unitary evolution and projective measurements but excludes the deterministic preparation of pure states. The standard operational formulation of quantum theory can be retrieved from this time-symmetric version by introducing an operation of conditioning on the outcomes of past experiments.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0, and a conservation law, namely the invariance of expectation values of A. For non-Hermitian Hamiltonians, H† comes into play as a distinct operator that complements H in generalized unitarity relations. The above description of symmetries has to be extended to include also A-pseudohermiticity relations of the form AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of A operators forming Klein's 4-group: parity, time-reversal, parity\&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be more subtle for non-Hermitian than for Hermitian Hamiltonians.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.