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 B Atomic and Molecular Physics

On gauge invariance in quantum transition amplitudes

C. K. Au

Published Feb 14, 1984• 8 citations
Publisher

Abstract

The wavefunction of a quantum mechanical system can be evaluated in any gauge and can be expressed in terms of a complete set of eigenstates of a physical observable or of a mathematical operator. If the physical observable is expressed in the same gauge as the Hamiltonian from which the wavefunction is obtained, the expansion coefficients are gauge invariant and have definite experimental implications. This physical observable need not be the energy. If the basis set of vectors are eigenstates of a mathematical operator, the expansion coefficients are gauge dependent. This mathematical operator can be translated to a physical observable once the gauge under which the wavefunction is calculated is specified. The expansion coefficients can then be interpreted as probability amplitudes.

DOI
10.1088/0022-3700/17/3/001

Related papers

Questions of gauge and basis sets for the representation of electromagnetic interactions in quantum mechanics

Silviu Olariu, Iovitzu Popescu, C. B. Collins

Recent concerns about the gauge dependence of transition amplitudes describing the changes of state of quantized systems interacting with electromagnetic radiations have led to the development of invariant schemes for the expression of the time-dependent quantities involved. It is shown in this work that such schemes are not uniquely specified by the requirement for gauge invariance. Moreover, they simply exchange a dependence upon the gauge of the electromagnetic potentials for a previously unrecognized dependence upon the representation of basis states describing initial and final states of the process that is equivalent to a dependence upon gauge. However, general expressions are developed in this work which show that resonant transitions which correspond to physically observable processes involving the pure or mixed absorption and emission of $N$ photons are gauge invariant regardless of the multipolarity of the transition moments and independent of the representation of the basis states of the system. A preferred representation is introduced which is shown to accommodate the physical effects generally encountered.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Dec 15, 1979•14 citations

The physical significance of gauge independence and gauge covariance in quantum mechanics

T. E. Feuchtwang, Emil Kazes, Paul H. Cutler +1 more

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.

Journal of Physics A Mathematical and General•Jan 11, 1984•15 citations

Gauge transformations, Foldy-Wouthuysen transformations and conservation of energy

Kuo‐Ho Yang

The conventional interpretation of the quantum mechanical probability amplitudes and probabilities is examined in detail with respect to gauge transformations and the Foldy-Wouthuysen transformations. It is shown that the conventional interpretation is gauge dependent and, when applied to the relativistic (Dirac) and the non-relativistic quantum mechanics, cannot be consistent with itself through the Foldy-Wouthuysen transformations. It is also shown that these two difficulties are not present in a gauge invariant formulation that explicitly incorporates Poynting's theorem and the conservation law of energy into the definition of probability amplitudes.

Journal of Physics A Mathematical and General•Feb 1, 1982•25 citations

Formulation of quantum mechanics in terms of gauge transformations

Shiv R. Vatsya

Formulations of quantum mechanics incorporating the Weyl gauge transformations are studied in this article and developed further. In the process, impact of the method of observation on its outcome is interpreted in terms of the assigned gauges by incorporating properties of the corresponding experimental arrangement in defining them. Further, the assigned gauge is explicitly incorporated in the Feynman path integral formulation of quantum mechanics. The resulting wavefunction, which is not uniquely defined, represents a gauge equivalence class. The representative wavefunction is still obtained by the original path integral method. Methods to obtain the pertinent information about the assigned gauges supplementing the representative wavefunction are discussed. The probability density is shown to be a uniquely defined gauge invariant quantity but at the expense of some information describing the observable effects contained in gauge factors. In the standard quantum mechanics, a wavefunction is assumed to be defined within a phase factor while the probability density is phase-independent, paralleling these results. Also, the path integral method is used to deduce the Klein-Gordon equation for the representative wavefunction in the Riemannian spaces in a more streamlined manner than the previous derivations.

•May 29, 2014•0 citations

The Gauge Picture of Quantum Dynamics

Kevin Slagle

Although local Hamiltonians exhibit local time dynamics, this locality is not explicit in the Schrödinger picture in the sense that the wavefunction amplitudes do not obey a local equation of motion. We show that geometric locality can be achieved explicitly in the equations of motion by "gauging" the global unitary invariance of quantum mechanics into a local gauge invariance. That is, expectation values ⟨ψ|A|ψ⟩ are invariant under a global unitary transformation acting on the wavefunction |ψ⟩→U|ψ⟩ and operators A→UAU† , and we show that it is possible to gauge this global invariance into a local gauge invariance. To do this, we replace the wavefunction with a collection of local wavefunctions |ψJ⟩ , one for each patch of space J . The collection of spatial patches is chosen to cover the space; e.g. we could choose the patches to be single qubits or nearest-neighbor sites on a lattice. Local wavefunctions associated with neighboring pairs of spatial patches I and J are related to each other by dynamical unitary transformations UIJ . The local wavefunctions are local in the sense that their dynamics are local. That is, the equations of motion for the local wavefunctions |ψJ⟩ and connections UIJ are explicitly local in space and only depend on nearby Hamiltonian terms. (The local wavefunctions are many-body wavefunctions and have the same Hilbert space dimension as the usual wavefunction.) We call this picture of quantum dynamics the gauge picture since it exhibits a local gauge invariance. The local dynamics of a single spatial patch is related to the interaction picture, where the interaction Hamiltonian consists of only nearby Hamiltonian terms. We can also generalize the explicit locality to include locality in local charge and energy densities.

arXiv:2210.09314•Quantum•Mar 21, 2024•2 citations

Gauge invariance in quantum mechanics: zero electromagnetic field

Donald H. Kobe, Kuo‐Ho Yang

The case of a single charged particle with potential energy is considered when the classical electromagnetic field is zero. The vector and scalar potentials describing this situation are not necessarily zero. When the conventional approach to radiation problems using the interaction A⋅p plus p⋅A plus A2 plus A0 (constants omitted) is used, time-dependent probabilities are predicted if the vector potential A=−∇Λ≠0 even though the electromagnetic field is zero. The reason for this unphysical behavior is that eigenfunctions of the unperturbed Hamiltonian, which is not gauge invariant, are used to calculate the probability amplitudes. When a manifestly gauge-invariant procedure is applied to this case the probabilities are constant in all gauges. The reason that physical results are obtained is that eigenstates of the energy operator are used to calculate the probability amplitudes.

American Journal of Physics•Feb 1, 1983•13 citations
The wavefunction of a quantum mechanical system can be evaluated in any gauge and can be expressed in terms of a complete set of eigenstates of a physical observable or of a mathematical operator. If the physical observable is expressed in the same gauge as the Hamiltonian from which the wavefunction is obtained, the expansion coefficients are gauge invariant and have definite experimental implications. This physical observable need not be the energy. If the basis set of vectors are eigenstates of a mathematical operator, the expansion coefficients are gauge dependent. This mathematical operator can be translated to a physical observable once the gauge under which the wavefunction is calculated is specified. The expansion coefficients can then be interpreted as probability amplitudes.
Recent concerns about the gauge dependence of transition amplitudes describing the changes of state of quantized systems interacting with electromagnetic radiations have led to the development of invariant schemes for the expression of the time-dependent quantities involved. It is shown in this work that such schemes are not uniquely specified by the requirement for gauge invariance. Moreover, they simply exchange a dependence upon the gauge of the electromagnetic potentials for a previously unrecognized dependence upon the representation of basis states describing initial and final states of the process that is equivalent to a dependence upon gauge. However, general expressions are developed in this work which show that resonant transitions which correspond to physically observable processes involving the pure or mixed absorption and emission of NNN photons are gauge invariant regardless of the multipolarity of the transition moments and independent of the representation of the basis states of the system. A preferred representation is introduced which is shown to accommodate the physical effects generally encountered.
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.
The conventional interpretation of the quantum mechanical probability amplitudes and probabilities is examined in detail with respect to gauge transformations and the Foldy-Wouthuysen transformations. It is shown that the conventional interpretation is gauge dependent and, when applied to the relativistic (Dirac) and the non-relativistic quantum mechanics, cannot be consistent with itself through the Foldy-Wouthuysen transformations. It is also shown that these two difficulties are not present in a gauge invariant formulation that explicitly incorporates Poynting's theorem and the conservation law of energy into the definition of probability amplitudes.
Formulations of quantum mechanics incorporating the Weyl gauge transformations are studied in this article and developed further. In the process, impact of the method of observation on its outcome is interpreted in terms of the assigned gauges by incorporating properties of the corresponding experimental arrangement in defining them. Further, the assigned gauge is explicitly incorporated in the Feynman path integral formulation of quantum mechanics. The resulting wavefunction, which is not uniquely defined, represents a gauge equivalence class. The representative wavefunction is still obtained by the original path integral method. Methods to obtain the pertinent information about the assigned gauges supplementing the representative wavefunction are discussed. The probability density is shown to be a uniquely defined gauge invariant quantity but at the expense of some information describing the observable effects contained in gauge factors. In the standard quantum mechanics, a wavefunction is assumed to be defined within a phase factor while the probability density is phase-independent, paralleling these results. Also, the path integral method is used to deduce the Klein-Gordon equation for the representative wavefunction in the Riemannian spaces in a more streamlined manner than the previous derivations.
Although local Hamiltonians exhibit local time dynamics, this locality is not explicit in the Schrödinger picture in the sense that the wavefunction amplitudes do not obey a local equation of motion. We show that geometric locality can be achieved explicitly in the equations of motion by "gauging" the global unitary invariance of quantum mechanics into a local gauge invariance. That is, expectation values ⟨ψ|A|ψ⟩ are invariant under a global unitary transformation acting on the wavefunction |ψ⟩→U|ψ⟩ and operators A→UAU† , and we show that it is possible to gauge this global invariance into a local gauge invariance. To do this, we replace the wavefunction with a collection of local wavefunctions |ψJ⟩ , one for each patch of space J . The collection of spatial patches is chosen to cover the space; e.g. we could choose the patches to be single qubits or nearest-neighbor sites on a lattice. Local wavefunctions associated with neighboring pairs of spatial patches I and J are related to each other by dynamical unitary transformations UIJ . The local wavefunctions are local in the sense that their dynamics are local. That is, the equations of motion for the local wavefunctions |ψJ⟩ and connections UIJ are explicitly local in space and only depend on nearby Hamiltonian terms. (The local wavefunctions are many-body wavefunctions and have the same Hilbert space dimension as the usual wavefunction.) We call this picture of quantum dynamics the gauge picture since it exhibits a local gauge invariance. The local dynamics of a single spatial patch is related to the interaction picture, where the interaction Hamiltonian consists of only nearby Hamiltonian terms. We can also generalize the explicit locality to include locality in local charge and energy densities.
The case of a single charged particle with potential energy is considered when the classical electromagnetic field is zero. The vector and scalar potentials describing this situation are not necessarily zero. When the conventional approach to radiation problems using the interaction A⋅p plus p⋅A plus A2 plus A0 (constants omitted) is used, time-dependent probabilities are predicted if the vector potential A=−∇Λ≠0 even though the electromagnetic field is zero. The reason for this unphysical behavior is that eigenfunctions of the unperturbed Hamiltonian, which is not gauge invariant, are used to calculate the probability amplitudes. When a manifestly gauge-invariant procedure is applied to this case the probabilities are constant in all gauges. The reason that physical results are obtained is that eigenstates of the energy operator are used to calculate the probability amplitudes.