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Journal of Physics A Mathematical and General

Pointed weak energy and quantum state evolution in Pancharatnam–Fubini-Study configuration space

Allen Parks

Published Dec 21, 2005• 7 citations
Publisher

Abstract

The weak energy for a quantum state evolving relative to its fixed initial state (pointed weak energy) in a Hilbert space of arbitrary dimension is defined and aspects of the classical mechanical, transformational and geometric relationships between this quantity and the motion of the associated quantum state are studied in a two-dimensional configuration space with Pancharatnam (P) phase and Fubini-Study (FS) metric distance as generalized coordinates (PFS configuration space). It is shown that when expressed in these coordinates: (1) the evolution of the quantum state is described by an equation of motion for an associated correlation amplitude (pointed correlation amplitude) in which the dynamics are induced by the interaction of the pointed weak energy with this correlation amplitude; (2) the pointed weak energy defines a 1-form with the transformational properties of a U (1) gauge potential (pointed weak energy gauge potential) which produces infinitesimal changes in the pointed correlation amplitude via its interaction with the amplitude; (3) the pointed correlation amplitude is defined by an exponential function of a line integral of the pointed weak energy gauge potential; (4) pointed weak energy gauge transformations exist that are equivalent to local U (1) gauge transformations of quantum states and—when examined within the context of the pointed probability current and pointed weak energy stationary action principles—offer an interpretation of gauge invariance analogous to that found in classical mechanics; and (5) integral invariants exist which relate line integrals of the weak energy gauge potential to geometric phase and the Aharonov–Anandan connection 1-form in the associated Hilbert space. States which irreversibly decay and Grover's quantum search algorithm serve as simple illustrations of the theory.

DOI
10.1088/0305-4470/39/3/012

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Mathematics•Jun 1, 2015•1 citation
The weak energy for a quantum state evolving relative to its fixed initial state (pointed weak energy) in a Hilbert space of arbitrary dimension is defined and aspects of the classical mechanical, transformational and geometric relationships between this quantity and the motion of the associated quantum state are studied in a two-dimensional configuration space with Pancharatnam (P) phase and Fubini-Study (FS) metric distance as generalized coordinates (PFS configuration space). It is shown that when expressed in these coordinates: (1) the evolution of the quantum state is described by an equation of motion for an associated correlation amplitude (pointed correlation amplitude) in which the dynamics are induced by the interaction of the pointed weak energy with this correlation amplitude; (2) the pointed weak energy defines a 1-form with the transformational properties of a U (1) gauge potential (pointed weak energy gauge potential) which produces infinitesimal changes in the pointed correlation amplitude via its interaction with the amplitude; (3) the pointed correlation amplitude is defined by an exponential function of a line integral of the pointed weak energy gauge potential; (4) pointed weak energy gauge transformations exist that are equivalent to local U (1) gauge transformations of quantum states and—when examined within the context of the pointed probability current and pointed weak energy stationary action principles—offer an interpretation of gauge invariance analogous to that found in classical mechanics; and (5) integral invariants exist which relate line integrals of the weak energy gauge potential to geometric phase and the Aharonov–Anandan connection 1-form in the associated Hilbert space. States which irreversibly decay and Grover's quantum search algorithm serve as simple illustrations of the theory.
It is shown that the actual paths in Hilbert space followed by a finite set of n ⩾ 2 quantum states evolving between initial and final end point configurations are such that an associated weak energy functional defined by Pancharatnam phases and state separation distances in projective Hilbert space determined by the generalized Fubini-Study metric is stationary for all variations of these phases, separations and time which vanish at the end points. Noether's theorem is used to identify two weak energy conservation laws which are shown to be the analogues of the momentum and energy conservation laws of Langrangian mechanics.
The pointed weak energy for an arbitrarily evolving quantum state defines a complex valued phase which is the sum of a dynamic phase and a purely geometric phase—the pointed geometric phase. The pointed geometric phase is concisely expressed as a time integral which depends upon the energy uncertainty, the associated evolving state, and its orthogonal companion state. The real part of the pointed geometric phase is to within a sign the geometric phase for arbitrary evolutions defined by Mukunda and Simon and that of Aharonov and Anandan for cyclic evolutions. The imaginary part of the pointed geometric phase governs the survival probability of the initial state. Several general rate of change relationships associated with the real and imaginary parts of the pointed geometric phase are deduced from this concise expression, and it is used to calculate the pointed geometric phase acquired as a spin- particle precesses under the influence of a uniform magnetic field.
The Aharonov–Vaidman gauge—which additively transforms the mean value of a quantum mechanical observable into an associated weak value—is introduced. It is shown that the unusual eccentric properties of weak values are inherited from this gauge and that a weak value of an observable can be considered as its mean value measured in the Aharonov–Vaidman gauge. The total time derivative of this sum also transforms an observable's mean value equation of motion into the associated weak value equation of motion and it is shown that the weak energy of evolution which influences the evolution of a weak value is intrinsic to the rate of change of the Aharonov–Vaidman gauge. Both of these equations of motion can be expressed in terms of time varying generalized coordinates and their rates of change. These equations satisfy the Euler–Lagrange equations which—in turn—define conjugate momenta and provide for their coordinate/momentum/time Poincaré representations. The underlying mathematical forms of these two representations are identical except for three symbols which distinguish them physically and identify three simple replacement operations that are required to transform the mean value Poincaré representation into the weak value Poincaré representation. This transformational relationship between Poincaré representations defines the notion of quasi-form invariance and the replacement operations encode the peculiar physical properties induced by the Aharonov–Vaidman gauge, i.e. the complexification and increased dimension of phase space, and the absorption of the weak energy of evolution by the conjugate momenta. Simple examples are used to illustrate the theory.
The equation of motion for a time-dependent weak value of a quantum-mechanical observable is known to contain a complex valued energy factor (the weak energy of evolution) that is defined by the dynamics of the pre-selected and post-selected states which specify the observable's weak value. In this paper, the mechanism responsible for the creation of this energy is identified and it is shown that the cumulative effect over time of this energy is manifested as dynamical phases and pure geometric phases (the intrinsic phases of evolution) which govern the evolution of the weak value during its measurement process. These phases are simply related to a Pancharatnam phase and Fubini–Study metric distance defined by the Hilbert space evolution of the associated pre-selected and post-selected states. A characterization of time-dependent weak value evolution as Pancharatnam phase angle rotations and Fubini–Study distance scalings of a vector in the Argand plane is discussed as an application of this relationship. The theory of weak values is also reviewed and simple ‘gedanken experiments’ are used to illustrate both the time-independent and the time-dependent versions of the theory. It is noted that the direct experimental observation of the weak energy of evolution would strongly support the time-symmetric paradigm of quantum mechanics and it is suggested that weak value equations of motion represent a new category of nonlocal equations of motion.
The theory of weak values for quantum mechanical observables has come to serve as a useful basis for contemporary discussions concerning such varied topics as the tunnelling-time controversy and quantum stochastic processes. An intrinsic complex-valued weak energy has recently been observed experimentally and reported in the literature. In this paper it is shown that: (a) the real and imaginary valued parts of this weak energy have geometric interpretations related to a phase acquired from parallel transport in Hilbert space and the variational dynamics occurring in the associated projective Hilbert space, respectively; (b) the weak energy defines functions which translate correlation amplitudes and probabilities in time; (c) correlation probabilities can be controlled by manipulating the weak energy and there exists a condition of weak stationarity that guarantees their time invariance; and (d) a time-weak energy uncertainty relation of the usual form prevails when a suitable set of dynamical constraints are imposed.
The Aharonov-Vaidman gauge additively transforms the mean energy of a quantum mechanical system into a weak valued system energy. In this paper, the equation of motion of this weak valued energy is used to provide a mathematical statement of an extended 1st Law of Thermodynamics that is applicable to the mean energy of a closed quantum system when the mean energy is expressed in the Aharonov-Vaidman gauge, i.e., when the system’s energy is weak valued. This is achieved by identifying the generalized heat and work exchange terms that appear in the equation of motion for weak valued energy. The complex valued contributions of the additive gauge term to these generalized exchange terms are discussed and this extended 1st Law is shown to subsume the usual 1st Law that is applicable for the mean energy of a closed quantum system. It is found that the gauge transformation introduces an additional energy uncertainty exchange term that—while it is neither a heat nor a work exchange term—is necessary for the conservation of weak valued energy. A spin-1/2 particle in a uniform magnetic field is used to illustrate aspects of the theory. It is demonstrated for this case that the extended 1st Law implies the existence of a gauge potential ω and that it generates a non-vanishing gauge field F. It is also shown for this case that the energy uncertainty exchange accumulated during the evolution of the system along a closed evolutionary cycle C in an associated parameter space is a geometric phase. This phase is equal to both the path integral of ω along C and the integral of the flux of F through the area enclosed by C.