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arXiv:0802.2401

A Bicycle Built for Two: The Galilean and U(1) Gauge Invariance of the Schrödinger Field

V. E. Colussi, S. Wickramasekara

Published Feb 17, 2008• 2 citations
arXivHTMLPDF

Abstract

This paper undertakes a study of the nature of the force associated with the local U (1) gauge symmetry of a non-relativistic quantum particle. To ensure invariance under local U (1) symmetry, a matter field must couple to a gauge field. We show that such a gauge field necessarily satisfies the Maxwell equations, whether the matter field coupled to it is relativistic or non-relativistic. This result suggests that the structure of the Maxwell equations is determined by gauge symmetry rather than the symmetry transformation properties of space-time. In order to assess the validity of this notion, we examine the transformation properties of the coupled matter and gauge fields under Galilean transformations. Our main technical result is the Galilean invariance of the full equations of motion of the U (1) gauge field.

DOI
10.48550/arxiv.0802.2401

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This paper undertakes a study of the nature of the force associated with the local U (1) gauge symmetry of a non-relativistic quantum particle. To ensure invariance under local U (1) symmetry, a matter field must couple to a gauge field. We show that such a gauge field necessarily satisfies the Maxwell equations, whether the matter field coupled to it is relativistic or non-relativistic. This result suggests that the structure of the Maxwell equations is determined by gauge symmetry rather than the symmetry transformation properties of space-time. In order to assess the validity of this notion, we examine the transformation properties of the coupled matter and gauge fields under Galilean transformations. Our main technical result is the Galilean invariance of the full equations of motion of the U (1) gauge field.
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.
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 $\ensuremath{\chi}(\mathbf{r},t)$, 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=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.
Gauge symmetry respectively gauge invariance plays a crucial role in modern quantum field theory. Historically, the first locally gauge-invariant equations of motion are Maxwell’s original equations for the electrical charges and currents and the electromagnetic fields created by them. In order to keep the relation to classical mechanics as close as possible, this chapter starts with a critical examination of global gauge invariance, basing on Helmholtz’s interpretation of the potential energy as “disposable work storage”. Then, Helmholtz’s explorations of the relationships between forces and energies are sketched and extended. Their generalization to quantum mechanics leads to the gauge invariance of the stationary and timedependent Schrödinger equations. An important application is the Ehrenberg-Siday- Aharonov-Bohm effect and its gravito-electromagnetic analog. For the sake of the unity of physics, it is shown, how the (gravito-)electromagnetic equations can be deduced. Here, Newton’s imaginations about the field of gravity are also exploited.
The Schrödinger equation of a charged particle in a uniform electric field can be specified in either a time-independent or a time-dependent gauge. The wave functions in these two gauges are related by a phase factor reflecting the gauge symmetry of the problem. We show that the effect of such a gauge transformation connecting the two wave functions can be mimicked by the effect of two successive extended Galilean transformations. An extended Galilean transformation connects two reference frames such that one is accelerating with respect to the other.
The Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.
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.