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

arXiv:2012.13836 · Physical Review A

Gauge transformations and Galilean covariance in nonlinear gauge-coupled quantum fluids

Yvan Buggy, Patrik Öhberg

Published Sep 30, 2020• 2 citations
arXivHTMLPDF

Abstract

We investigate certain invariance properties of quantum fluids subject to a nonlinear gauge potential. In particular, we derive the covariant transformation laws for the nonlinear potentials under a space-time Galilean boost and consider $U(1)$ gauge transformations. We find that the hydrodynamic canonical field equations are form-invariant in the case of external gauge functions $\ensuremath{\chi}(\mathbf{r},t)$, but not for nonlinear gauge functionals $\ensuremath{\chi}[\ensuremath{\rho}]$. Hence, nonlinear gauge potentials are nontrivial potentials which may not be ``gauged-away.'' Notably, for a superfluid in dimension $d=1$, attempting to do so generates the gauge pressure of the fluid in the Hamiltonian density. Furthermore, we investigate how the field equations transform under arbitrary Galilean transformations. We find that the immediate lack of Galilean covariance is restored under a suitably chosen transformation rule set for the potentials, which is identical in form to that of a Schr\"odinger particle coupled to external scalar and vector potentials.

DOI
10.1103/physreva.102.033342

Related papers

On the hydrodynamics of Bose-condensed fluids subject to density-dependent gauge potentials

Yvan Buggy, Patrik Öhberg

When the energy functional of a Bose-condensed state of matter features an effective gauge potential which depends on the density $ρ$ of the condensate, the kinetic energy density of the matter field becomes nonlinear in $ρ$ and additional flow-dependent terms enter the wave equation for the phase of the condensate wavefunction. To begin with, we consider a certain class of density-dependent `single-component' gauge potentials, and later extend this class to encompass more general `multi-component' potentials. The nonlinear flow terms are cast into the general form of an inner-product between the velocity field of the fluid and the gauge potential. This is achieved by introducing a coupling matrix of dimensionless functions $γ_{ij}\left( ρ\right)$, which characterises the particular functional form of the gauge potential and regulates the strengths of the nonlinear terms accordingly. In the momentum-transport equation of the fluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the fluid stress tensor features a flow-dependent pressure contribution given by the inner-product of the gauge potential and the current density of the fluid. This explicit dependence of the fluid pressure on the flow highlights the lack of Galilean invariance of the nonlinear fluid.

arXiv:1708.07712•Aug 25, 2017•1 citation

On the hydrodynamics of nonlinear gauge-coupled quantum fluids

Yvan Buggy, Lawrence Phillips, Patrik Öhberg

Abstract By constructing a hydrodynamic canonical formalism, we show that the occurrence of an arbitrary density-dependent gauge potential in the meanfield Hamiltonian of a Bose-condensed fluid invariably leads to nonlinear flow-dependent terms in the wave equation for the phase, where such terms arise due to the explicit dependence of the mechanical flow on the fluid density. In addition, we derive a canonical momentum transport equation for this class of nonlinear fluid and obtain an expression for the stress tensor. Further, we study the hydrodynamic equations in a particular nonlinear fluid, where the effective gauge potential results from the introduction of weak contact interactions in an ultracold dilute Bose gas of optically-addressed two-level atoms. In the Cauchy equation of mechanical momentum transport of the superfluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the stress tensor features a canonical flow pressure term given by the inner-product of the gauge potential and the canonical current density. By numerical simulation, we illustrate an interesting effect of the nonlinear gauge potential on the groundstate wavefunction of a superfluid in the presence of a foreign impurity. We find that the groundstate adopts a non-trivial local phase, which is antisymmetric under reversal of the gauge potential. The phase profile leads to a canonical-flow or phase-flow dipole about the impurity, resulting in a skirting mechanical flow. As a result, the pressure becomes asymmetric about the object and the condensate undergoes a deformation. Graphical abstract

arXiv:2012.13834•The European Physical Journal D•May 1, 2020•4 citations

Operator Gauge Transformations in Nonrelativistic Quantum Electrodynamics

Raymond Dale Gray

A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.

•Dec 1, 1982•0 citations

The Galilean covariance of quantum mechanics in the case of external fields

Harvey R. Brown, Peter Holland

Textbook treatments of the Galilean covariance of the time-dependent Schrödinger equation for a spinless particle seem invariably to cover the case of a free particle or one in the presence of a scalar potential. The principal objective of this paper is to examine the situation in the case of arbitrary forces, including the velocity-dependent variety resulting from a vector potential. To this end, we revisit the 1964 theorem of Jauch which purports to determine the most general form of the Hamiltonian consistent with “Galilean-invariance,” and argue that the proof is less than compelling. We then show systematically that the Schrödinger equation in the case of a Jauch-type Hamiltonian is Galilean covariant, so long as the vector and scalar potentials transform in a certain way. These transformations, which to our knowledge have appeared very rarely in the literature on quantum mechanics, correspond in the case of electrodynamical forces to the “magnetic” nonrelativistic limit of Maxwell’s equations in the sense of Le Bellac and Lévy-Leblond (1973). Finally, this Galilean covariant theory sheds light on Feynman’s “proof” of Maxwell’s equations, as reported by Dyson in 1990.

American Journal of Physics•Mar 1, 1999•70 citations

Gauge transformations in extended phase space

W P Healy

Gauge transformations of the potentials for a prescribed electromagnetic field acting on a non-relativistic charged particle are shown to induce canonical transformations in the extended phase space, in which the time and its conjugate momentum are treated as dynamical variables. The extended Hamiltonian is invariant and the extended canonical equations are covariant under a gauge transformation. The usual time-dependent wave equations in different gauges are obtained from the formalism by using Schrödinger representations.

Physica Scripta•Sep 1, 1993•0 citations

Quadratic Gauge Transformation

Sunita Singh, Akshit Sharma

Symmetries plays a significant role in understanding the conservation laws in Quantum field theories. Here, we attempted a quadratic type dimensionless gauge transformation to achieve the invariance in QFTs. We have shown the extensive study of invariance of complex scalar, Abelian and Non- Abelian theories and established the conservation laws. We included an explicit graphical analysis to invoke the invariance. This is studied in a physical context, where different field configurations correspond to the same physical state. The necessity of the covariant derivative is studied in detail, highlighting how it ensures consistent transformation under local symmetry operations. The meaning of covariance is clarified as the preservation of the form of physical laws under transformations.

hep-thCross-listedarXiv:2606.30177•Jun 30, 2026
We investigate certain invariance properties of quantum fluids subject to a nonlinear gauge potential. In particular, we derive the covariant transformation laws for the nonlinear potentials under a space-time Galilean boost and consider U(1)U(1)U(1) gauge transformations. We find that the hydrodynamic canonical field equations are form-invariant in the case of external gauge functions \ensuremathχ(r,t)\ensuremath{\chi}(\mathbf{r},t)\ensuremathχ(r,t), but not for nonlinear gauge functionals \ensuremathχ[\ensuremathρ]\ensuremath{\chi}[\ensuremath{\rho}]\ensuremathχ[\ensuremathρ]. Hence, nonlinear gauge potentials are nontrivial potentials which may not be ``gauged-away.'' Notably, for a superfluid in dimension d=1d=1d=1, attempting to do so generates the gauge pressure of the fluid in the Hamiltonian density. Furthermore, we investigate how the field equations transform under arbitrary Galilean transformations. We find that the immediate lack of Galilean covariance is restored under a suitably chosen transformation rule set for the potentials, which is identical in form to that of a Schr"odinger particle coupled to external scalar and vector potentials.
When the energy functional of a Bose-condensed state of matter features an effective gauge potential which depends on the density ρρρ of the condensate, the kinetic energy density of the matter field becomes nonlinear in ρρρ and additional flow-dependent terms enter the wave equation for the phase of the condensate wavefunction. To begin with, we consider a certain class of density-dependent `single-component' gauge potentials, and later extend this class to encompass more general `multi-component' potentials. The nonlinear flow terms are cast into the general form of an inner-product between the velocity field of the fluid and the gauge potential. This is achieved by introducing a coupling matrix of dimensionless functions γij(ρ)γ_{ij}\left( ρ\right)γij​(ρ), which characterises the particular functional form of the gauge potential and regulates the strengths of the nonlinear terms accordingly. In the momentum-transport equation of the fluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the fluid stress tensor features a flow-dependent pressure contribution given by the inner-product of the gauge potential and the current density of the fluid. This explicit dependence of the fluid pressure on the flow highlights the lack of Galilean invariance of the nonlinear fluid.
Abstract By constructing a hydrodynamic canonical formalism, we show that the occurrence of an arbitrary density-dependent gauge potential in the meanfield Hamiltonian of a Bose-condensed fluid invariably leads to nonlinear flow-dependent terms in the wave equation for the phase, where such terms arise due to the explicit dependence of the mechanical flow on the fluid density. In addition, we derive a canonical momentum transport equation for this class of nonlinear fluid and obtain an expression for the stress tensor. Further, we study the hydrodynamic equations in a particular nonlinear fluid, where the effective gauge potential results from the introduction of weak contact interactions in an ultracold dilute Bose gas of optically-addressed two-level atoms. In the Cauchy equation of mechanical momentum transport of the superfluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the stress tensor features a canonical flow pressure term given by the inner-product of the gauge potential and the canonical current density. By numerical simulation, we illustrate an interesting effect of the nonlinear gauge potential on the groundstate wavefunction of a superfluid in the presence of a foreign impurity. We find that the groundstate adopts a non-trivial local phase, which is antisymmetric under reversal of the gauge potential. The phase profile leads to a canonical-flow or phase-flow dipole about the impurity, resulting in a skirting mechanical flow. As a result, the pressure becomes asymmetric about the object and the condensate undergoes a deformation. Graphical abstract
A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.
Textbook treatments of the Galilean covariance of the time-dependent Schrödinger equation for a spinless particle seem invariably to cover the case of a free particle or one in the presence of a scalar potential. The principal objective of this paper is to examine the situation in the case of arbitrary forces, including the velocity-dependent variety resulting from a vector potential. To this end, we revisit the 1964 theorem of Jauch which purports to determine the most general form of the Hamiltonian consistent with “Galilean-invariance,” and argue that the proof is less than compelling. We then show systematically that the Schrödinger equation in the case of a Jauch-type Hamiltonian is Galilean covariant, so long as the vector and scalar potentials transform in a certain way. These transformations, which to our knowledge have appeared very rarely in the literature on quantum mechanics, correspond in the case of electrodynamical forces to the “magnetic” nonrelativistic limit of Maxwell’s equations in the sense of Le Bellac and Lévy-Leblond (1973). Finally, this Galilean covariant theory sheds light on Feynman’s “proof” of Maxwell’s equations, as reported by Dyson in 1990.
Gauge transformations of the potentials for a prescribed electromagnetic field acting on a non-relativistic charged particle are shown to induce canonical transformations in the extended phase space, in which the time and its conjugate momentum are treated as dynamical variables. The extended Hamiltonian is invariant and the extended canonical equations are covariant under a gauge transformation. The usual time-dependent wave equations in different gauges are obtained from the formalism by using Schrödinger representations.
Symmetries plays a significant role in understanding the conservation laws in Quantum field theories. Here, we attempted a quadratic type dimensionless gauge transformation to achieve the invariance in QFTs. We have shown the extensive study of invariance of complex scalar, Abelian and Non- Abelian theories and established the conservation laws. We included an explicit graphical analysis to invoke the invariance. This is studied in a physical context, where different field configurations correspond to the same physical state. The necessity of the covariant derivative is studied in detail, highlighting how it ensures consistent transformation under local symmetry operations. The meaning of covariance is clarified as the preservation of the form of physical laws under transformations.