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Journal of Physics A Mathematical and Theoretical

Weak values and the Aharonov–Vaidman gauge

Allen Parks

Published Dec 17, 2009• 6 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/43/3/035305

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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 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.
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.
The time derivative of a physical property often gives rise to another meaningful property. Since weak values provide empirical insights that cannot be derived from expectation values, this paper explores what physical properties can be obtained from the time derivative of weak values. It demonstrates that, in general, the time derivative of a weak value that is invariant under an electromagnetic gauge transformation is neither a weak value nor a gauge-invariant quantity. Left- and right-hand time derivatives of a weak value are defined, and two necessary and sufficient conditions are presented to ensure that they are also gauge-invariant weak values. Finite-difference approximations of the left- and right-hand time derivatives of weak values are also presented, yielding results that match the theoretical expressions when these are gauge invariant. With these definitions, a local Ehrenfest-like theorem can be derived, giving a natural interpretation for the time derivative of weak values. Notably, a single measured weak value of the system's position provides information about two additional unmeasured weak values: the system's local velocity and acceleration, through the first- and second-order time derivatives of the initial weak value, respectively. These findings also offer guidelines for experimentalists to translate the weak value theory into practical laboratory setups, paving the way for innovative quantum technologies. An example illustrates how the electromagnetic field can be determined at specific positions and times from the first- and second-order time derivatives of a weak value of position.
The conservation of physical quantities under coordinate transformations, known as gauge invariance, has been the foundation of theoretical frameworks in both quantum and classical theory. The finding of gauge-invariant quantities has enabled the geometric and topological interpretations of quantum phenomena with the Berry phase, or the separation of quark and gluon contributions in quantum chromodynamics. Here, with an example of quantum geometric quantities-Berry connection, phase, and curvature-we extract a new gauge-invariant quantity by applying a "weak value picture". By employing different pre- and post-selections in the derivation of the Berry phase in the context of weak values, we derive the gauge-invariant vector potential from the Berry connection that is originally gauge-dependent, and show that the obtained vector potential corresponds to the weak value of the projected momentum operator. The local nature of this quantity is demonstrated with an example of the Aharonov-Bohm effect, proving that this gauge-invariant vector potential can be interpreted as the only source of the Berry curvature in the magnetic field. This weak value decomposition approach will lead to the extraction of new measurable quantities from traditionally unobservable quantities.
The distinction between gauge independence and gauge covariance in electrodynamics is considered for quantities of physical interest. The gauge independence of directly observable quantities, such as the spectra of operators and energy conserving transition rates, is discussed and contrasted with the gauge covariance of other quantities such as scalar products. The quasigauge-invariance of the classical Lagrangian and the invariance of the Euler-Lagrange equations under the addition of a total time derivative to the Lagrangian are discussed. The consequent inherent nonuniqueness of the Lagrangian and Hamiltonian formulations of classical mechanics is pointed out. A physical interpretation of the explicit gauge dependence of classical canonical momenta and of the expectation values of the corresponding quantum mechanical operators is presented. The gauge independence of energy conserving transition rates calculated in the conventional finite-order time dependent perturbation theory is discussed and illustrated. The gauge dependence of the conventional time dependent transition amplitudes in the presence of electromagnetic fields is then discussed, and gauge independent transition amplitudes are constructed. An alternative formulation of the quantum mechanics of charged particles is obtained in terms of a new gauge independent Hamiltonian, which is, as might be expected, unique only within an arbitrary canonical transformation.