Field Theoretic Approach to Hydrodynamics I. The Clebsch Transformation of the Hydrodynamic Equation

The object of this paper is to apply Takahashi's Galilei-invariant field theory to the derivation of the Clebsch transformation of hydrodynamic equations. In this paper, only normal fluids are considered, while in the second paper we will develop a field theory of superfluid hydrodynamics. Given view point adopted here, the fundamental objects of the theory are Galilei scalars which are invariant under the Galilei transformation. In terms of such scalars, the Lagrangian density is written explicitly. It is shown how the Clebsch transformation can be reformulated from a field theoretical viewpoint.     

Non-relativistic conformal symmetries in fluid mechanics

The symmetries of a free incompressible fluid span the Galilei group, augmented with independent dilations of space and time. When the fluid is compressible, the symmetry is enlarged to the expanded Schrödinger group, which also involves, in addition, Schrödinger expansions. While incompressible fluid dynamics can be derived as an appropriate non-relativistic limit of a conformally invariant relativistic theory, the recently discussed conformal Galilei group, obtained by contraction from the relativistic conformal group, is not a symmetry. This is explained by the subtleties of the non-relativistic limit.     

Galilei-Newton law by symplectic realizations

We classify all the coadjoint orbits of the central extension of the one spatial dimensional Galilei group G. Taking them in support of the generic symplectic realizations of the Galilei group G, we give their possible physical interpretations and also find that mass and force have their origins in the cohomological theory of the Galilei group.This has been presented as a seminar in Trieste (ICTP) during the workshop on Lie groups and their representations (10–20 November 1986).     

Axiomatic foundations of nonrelativistic quantum mechanics: A realistic approach

A realistic axiomatic formulation of nonrelativistic quantum mechanics for a single microsystem with spin is presented, from which the most important theorems of the theory can be deduced. In comparison with previous formulations, the formal aspect has been improved by the use of certain mathematical theories, such as the theory of rigged spaces, and group theory. The standard formalism is naturally obtained from the latter, starting from a central primitive concept: the Galilei group.     

On the theory of turbulence: A non eulerian renormalized expansion

A non Eulerian framework for a renormalized theory of isotropic homogeneous steady state turbulence at high Reynold's numbers is developed. By construction it is invariant under random Galilei transformations. A direct interaction factorization is free of infrared singularities and yields Kolmogorov scaling for the static as well as for the dynamic correlation and response functions.     

An extended Galilei group and its application

The Galilei group is combined with two one-dimensional groups, to form a twelve-dimensional extended Galilei group. Irreducible representations of this group depend upon two indicesm, s that can, respectively, be interpreted as the mass and spin of a non-relativistic particle. It turns out that the true irreducible representations of the ordinary Galilei group correspond tom=0, and this explains why these representations have no physical interpretation.     

Die Eindeutigkeit der Ortsobservablen in einer irreduziblen unitären Darstellung bis auf einen Faktor der Galilei-Gruppe

Uniqueness of the Position Observable in an Irreducible Unitary Representation up to a Factor of the Galilei Group. An elementarary quantum mechanical system with non vanishing mass is characterized by a continuous irreducible unitary representation up to a factor of the Galilei group in Hilbert space. An argument is given concerning the continuous iurreducible unitary representations of the universal covering group of the Euclidean group such that the position observable is uniquely determined by the transformation properties under the representation of the Galilei group.     

Spatial contraction of the Poincaré group and Maxwell’s equations in the electric limit

The contraction of the Poincaré group with respect to the space translations subgroup gives rise to a group that bears a certain duality relation to the Galilei group, that is, the contraction limit of the Poincaré group with respect to the time translations subgroup. In view of this duality, we call the former the dual Galilei group. A rather remarkable feature of the dual Galilei group is that the time translations constitute a central subgroup. Therewith, in unitary irreducible representations (UIRs) of the group, the Hamiltonian appears as a Casimir operator proportional to the identity H = EI, with E (and a spin value s) uniquely characterizing the representation. Hence, a physical system characterized by a UIR of the dual Galilei group displays no non-trivial time evolution. Moreover, the combined U(1) gauge group and the dual Galilei group underlie a non-relativistic limit of Maxwell’s equations known as the electric limit. The analysis presented here shows that only electrostatics is possible for the electric limit, wholly in harmony with the trivial nature of time evolution governed by the dual Galilei group.     

Minimal gravitational coupling in the Newtonian theory and the covariant Schrödinger equation

The role of the Bargmann group (11-dimensional extended Galilei group) in nonrelativistic gravitation theory is investigated. The generalized Newtonian gravitation theory (Newton-Cartan theory) achieves the status of a gauge theory about as much as general relativity and couples minimally to a complex scalar field leading to a four-dimensionally covariant Schrödinger equation. Matter current and stress-energy tensor follow correctly from the Lagrangian. This theory on curved Newtonian space-time is also shown to be a limit of the Einstein-Klein-Gordon theory.Partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A8059.     

Hierarchical order of Galilei and Lorentz invariance in the structure of matter

The structure of matter shows a hierarchical order: (1) from Lorentz invariance in high-energy physics; (2) to Galilei invariance in the low-energy nonrelativistic limit of high-energy physics; and (3) again to Lorentz invariance in condensed matter physics (where the velocity of sound takes the place of the velocity of light). The hierarchical order can be continued downward further to: (4) non-relativistic (velocity small compared to the velocity of sound) condensed matter excitons, obeying Galilei invariance; and (5) to excitonic matter obeying Lorentz invariance with an excitonic matter sound velocity. It was previously conjectured that Lorentz invariance of high-energy physics is preceded by Galilei invariance at the Planck scale. Still further, the conjectured Galilei invariance at the Planck scale may be the result of an underlying five-dimensional non-Euclidean conform invariant metric structure, with three spatial and two time dimensions, compactified onto three spatial and one time dimension.     

The Most General Set of Lagrange Functions Yielding Galilei Covariant Equations of Motion Having the Same Set of Solutions

The nonrelativistic case of two point particles in the (1 + 1)-dimensional space is considered. The existence of an autonomous Lagrange function is assumed, whose Euler-Lagrange Equations are forminvariant under the Galilei group. We show how to find all autonomous Lagrange functions, giving rise to Euler-Lagrange Eqzations, which again are Galilei covariant and whose set of solutions coincides with the set of solutions of the original equations, we started with. We are going to construct explicitly the most general expressions for the Lagrange functions as well as for the Equations of Motion. We supplement our considerations by some simple examples. We give also a short account on an extension of our formalism to the case of Equations of Motion, which are no longer Galilei covariant but whose solutions belong still to the same set as the previous one.     

Galilei Covariance Does Not Imply Minimal Electromagnetic Coupling

The postulate of Galilei covariance in one-particle classical and quantum mechanics is reinvestigated, with particular intent to correct some current misconceptions concerning the rǒle of minimal electromagnetic coupling in Galilei covariant theories.     

Exact periodic solution in coupled nonlinear Schodinger equations

The coupled nonlinear Schodinger equations （CNLSEs） of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Hamilton dynamics theory and Galilei transform. Making use of a transform for CNLSEs and using the ansatz with Jacobi elliptic function form, this paper obtains the exact optical pulse solutions.     

The Noncommutative Harmonic Oscillator Based on Symplectic Representation of Galilei Group

We study symplectic unitary representations for the Galilei group and derive the Schrödinger equation in phase space. Our formalism is based on the noncommutative structure of the star product. Guided by group theoretical concepts, we construct a physically consistent phase-space theory in which each state is described by a quasi-probability amplitude associated with the Wigner function. As applications, we derive the Wigner functions for the 3D harmonic oscillator and the noncommutative oscillator in phase space.     

Particle dynamics on Snyder space

We examine Hamiltonian formalism on Euclidean Snyder space. The latter corresponds to a lattice in the quantum theory. For any given dynamical system, it may not be possible to identify time with a real number parametrizing the evolution in the quantum theory. The alternative requires the introduction of a dynamical time operator. We obtain the dynamical time operator for the relativistic (nonrelativistic) particle, and use it to construct the generators of Poincaré (Galilei) group on Snyder space.     

Quantum mechanics in phase space: I. Unicity of the Wigner distribution function

The joint distribution function in phase space is related to the density matrix by an integral transformation which depends on the rule of correspondence used. All the requirements which can be restrictive for the kernel function defining the transformation are studied. It is shown that the conditions of Galilei invariance, unitarity, reality and normalization lead to the Wigner kernel function in a unique way. Galilei invariance, the requirement that the free particle behaves classically, and the conditions to obtain the correct mixed distributions also lead to the same result.     

Temperature, entropy, and kinetic theory

The relation between the concepts of temperature and entropy and the kinetic theory of gases is discussed, with particular attention to the aspects which are frequently treated as obvious or not even mentioned. In order to show that the usual thermodynamic relations are by no means obvious and may be contradictory, the model of a discrete velocity gas is used. It is also shown that the usual relation between the entropy rate and the heat supplied to a gas is not valid (even close to equilibrium) unless the theory is Galilei invariant (which is obviously not the case for a discrete velocity gas) and must be replaced by another one that eliminates all the paradoxical aspects of the matter.     

On the gauge variance of action functions under transformations on space-time

The problem of the gauge variance or invariance of action functions in classical mechanics is discussed from a group and path-theoretic viewpoint. By using the elementary theory of the cohomology of groups, criteria are introduced which enable one to decide when action functions gauge variant under a kinematical group are equivalent to action functions invariant under the transformations of the group. The criteria are applied to action functions gauge variant under Lorentz and Galilei transformations, where we deduce that any action function gauge variant under the Lorentz group is equivalent to an action function invariant under Lorentz transformations, whilst action functions gauge variant under the Galilei group are not necessarily equivalent to Galilei-invariant action functions. It is also shown that any action function gauge variant in a more restricted fashion which we define in the text, is necessarily equivalent to a kinetic-energy action.     

Lagrangian mechanics and the geometry of configuration spacetime

Holonomic rheonomic systems having a finite number of degrees of freedom are considered in classical nonrelativistic mechanics. It is shown that the configuration spacetime manifold $\text{M}$ of such a system can be furnished with a linear symmetric connection (called the “dynamical connection”) in such a way that the worldline of the system is a geodesic on $\text{M}$. The connection is based upon a degenerate metric structure (called a “generalized Galilei structure”) which in turn is uniquely determined by the system and the forces acting on it. The connection is compatible with the generalized Galilei structure in the sense that the covariant derivatives of the latter vanish. Systems which can be described in terms of a Lagrangian give rise to a particularly interesting class of dynamical connections, called “Lagrange connections,” whose geometry is studied in some detail. Within the class of generalized Galilei connections they are characterized by a geometrical condition imposed on the affine curvature tensor. Noether symmetries of the dynamical system turn out to be equivalent to “isometries” of the generalized Galilei structure together with collineations of the Lagrange connection. They form a Lie group. Spacelike generators of Noether symmetries are linked to the existence of “conservors” (i.e., covectors with vanishing symmetrized covariant derivatives). Timelike generators of Noether symmetries give rise to (second rank) Killing tensors.     

Most General Form of the Lagrange Function for Classical Two Particle Equation of Motion Covariant under Galilei Transformation in (1 + 1) and (3 + 1) Dimensional Spaces

In the present note we give the most general form of a non-relativistic Lagrange function for two point particles moving according to some classical equations of motion. We do not specify the form of these equations. We require only that these equations should be form invariant with respect to the Galilei transformations and should be derivable from a Lagrange function as its Euler-Lagrange Equations. We present also some of the generators of the Galilei group and their Lie-Cartan commutation relations.