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Journal of Mathematical Physics

Galilean Relativity, Locality, and Quantum Hydrodynamics

J. C. Garrison, Harry L. Morrison, Jack Wong

Published Feb 1, 1970• 9 citations
Publisher

Abstract

In this work we study the consequences of locality and Galilean covariance for the operators that occur in Landau's quantum hydrodynamics. We specifically consider the following requirements: (1) Galilean covariance of the velocity field, (2) locality of the velocity field, and (3) Landau's assumption that the momentum density is a symmetrized product of the velocity and density operators. It is demonstrated that the density-velocity commutation relation of the Landau theory is essentially a direct consequence of (1) and (2). The addition of (3) is sufficient to determine the velocity-velocity commutation relation, also in agreement with Laudau. We further show that the density-velocity commutation relation, independent of (3) or any specific form for the velocity field, is inconsistent with the nonnegative character of the local density.

DOI
10.1063/1.1665177

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In this work we study the consequences of locality and Galilean covariance for the operators that occur in Landau's quantum hydrodynamics. We specifically consider the following requirements: (1) Galilean covariance of the velocity field, (2) locality of the velocity field, and (3) Landau's assumption that the momentum density is a symmetrized product of the velocity and density operators. It is demonstrated that the density-velocity commutation relation of the Landau theory is essentially a direct consequence of (1) and (2). The addition of (3) is sufficient to determine the velocity-velocity commutation relation, also in agreement with Laudau. We further show that the density-velocity commutation relation, independent of (3) or any specific form for the velocity field, is inconsistent with the nonnegative character of the local density.
Landau's quantized hydrodynamics is derived from the microscopic Hamiltonian, which is written in terms of the density and velocity operators. In Euler's equation for the velocity operator, the force per unit mass has a form different from Landau's and is given explicitly in terms of the two-body potential. A quantum pressure term, which does not occur in Landau's theory, also appears. New realizations of the density and velocity operators are given which satisfy the commutation relations, but must act in a Hilbert space larger than Fock space. It is shown that the formalism is without meaning in Fock space owing to the nonexistence of certain operators.
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.
The nonequilibrium statistical operator method is used to construct a two-velocity relativistic hydrodynamics of a quantum-field system with broken symmetry and complex scalar order parameter. The approach is based on model expansions of the operators (the energy-momentum tensor, the current density, etc.) used for statistical description of the system in the hydrodynamic stage. In the first part of the paper, these expansions are used to construct the statistical operator and the thermodynamics of a local-equilibriu m state. Below, we construct a quantum relativistic two-velocity hydrodynamics on the basis of first principles using Zubarev's nonequilibrium statistical operator method (11}. In Sec.2, we establish the connection between the 4-velocity of the motion of the ordered component with the gradient of the phase of the order parameter. In Sec.3 we note that the methods of (2-7) based on the use of unitary transformations in the Galileo group to obtain the equations of two-velocity hydrodynamics do not admit direct generalization to the relativistic domain. In this connection, we propose an approach based on model expansions of the energy-momentum tensor and the current density (and, possibly, other operators used for statistical des- cription of systems in the hydrodynamic stage). These expansions are then used in Sec.4 to construct the statistical operator and the thermodynamics of the local-equilibrium state. The transport relations and the hydrodynamic equations will be discussed in Part If. Below, we use a system of units in which c = l~ = k = 1. The signature of the metric tensor is
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.