Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

The geometry and significance of weak energy

Allen Parks

Published Mar 23, 2000• 12 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/33/13/308

Related papers

Time-dependent weak values and their intrinsic phases of evolution

Allen Parks

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.

Journal of Physics A Mathematical and Theoretical•Jul 17, 2008•11 citations

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

Allen Parks

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.

Journal of Physics A Mathematical and General•Dec 21, 2005•7 citations

Pointed weak energy and quantum geometric phase

Allen Parks

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.

Journal of Physics A Mathematical and Theoretical•Feb 14, 2007•5 citations

Weak valued statistics as fundamental explanation of quantum physics

Holger Friedrich Hofmann

Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.

arXiv:1305.0053•Apr 30, 2013•1 citation

Quantum trajectories based on the weak value

Takuya Mori, Izumi Tsutsui

The notion of the trajectory of an individual particle is strictly inhibited in quantum mechanics because of the uncertainty principle. Nonetheless, the weak value, which has been proposed as a novel and measurable quantity definable to any quantum observable, can offer a possible description of trajectory on account of its statistical nature. In this paper, we explore the physical significance provided by this “weak trajectory” by considering various situations where interference takes place simultaneously with the observation of particles, that is, in prototypical quantum situations for which no classical treatment is available. These include the double slit experiment and Lloyd's mirror, where in the former case it is argued that the real part of the weak trajectory describes an average over the possible classical trajectories involved in the process, and that the imaginary part is related to the variation of interference. It is shown that this average interpretation of the weak trajectory holds universally under the complex probability defined from the given transition process. These features remain essentially unaltered in the case of Lloyd's mirror where interference occurs with a single slit.

arXiv:1412.0916•Progress of Theoretical and Experimental Physics•Apr 1, 2015•17 citations

CLASSICAL ANALOGUE OF WEAK VALUE IN STOCHASTIC PROCESS

H. TOMITA

One of the remarkable notions in the recent development of quantum physics is the weak value related to weak measurements. We emulate it as a two-time conditional expectation in a classical stochastic model. We use the well known symmetrized form of the master equation, which is formally equivalent to the wave equation in quantum mechanics apart from the fact that wave functions are always real. The origin of the unusual behaviors of the weak value such as the negative probability and the abnormal enhancement of some expectations becomes clearer in the present case, where the two-time conditional probability has no ambiguity of imaginary/complex values.

arXiv:1201.1329•Sep 27, 2012•0 citations
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 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 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.
Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.
The notion of the trajectory of an individual particle is strictly inhibited in quantum mechanics because of the uncertainty principle. Nonetheless, the weak value, which has been proposed as a novel and measurable quantity definable to any quantum observable, can offer a possible description of trajectory on account of its statistical nature. In this paper, we explore the physical significance provided by this “weak trajectory” by considering various situations where interference takes place simultaneously with the observation of particles, that is, in prototypical quantum situations for which no classical treatment is available. These include the double slit experiment and Lloyd's mirror, where in the former case it is argued that the real part of the weak trajectory describes an average over the possible classical trajectories involved in the process, and that the imaginary part is related to the variation of interference. It is shown that this average interpretation of the weak trajectory holds universally under the complex probability defined from the given transition process. These features remain essentially unaltered in the case of Lloyd's mirror where interference occurs with a single slit.
One of the remarkable notions in the recent development of quantum physics is the weak value related to weak measurements. We emulate it as a two-time conditional expectation in a classical stochastic model. We use the well known symmetrized form of the master equation, which is formally equivalent to the wave equation in quantum mechanics apart from the fact that wave functions are always real. The origin of the unusual behaviors of the weak value such as the negative probability and the abnormal enhancement of some expectations becomes clearer in the present case, where the two-time conditional probability has no ambiguity of imaginary/complex values.