Galilean-invariant Theory Research Articles

Hall viscosity, topological states and effective theories

Hall viscosity is a dissipationless transport coefficient whose value is quantized in units of the density in some topological phases and may be used as a measure of topological order. I give an overview of the Hall viscosity, its relation to Hall conductivity in Galilean invariant theories and its realization in effective theories.

Behavior of perturbations on spherically symmetric backgrounds in multi-Galileon theory

We consider multi-Galileon theory, the most general Galilean invariant theory with $N$ scalar fields linearly coupled to the trace of the stress-energy tensor. We study the behavior of perturbations on a static spherically symmetric background with a massive point source, and show that, under the assumptions of stability and successful Vainshtein screening, solutions cannot be found that are free of both superluminal propagation and slowly moving, strongly coupled fluctuations. The latter imply that the theoretical and phenomenological issues related to a very low strong-interaction scale cannot be avoided in this model.

Relativistic wave and particle mechanics formulated without classical mass

The fact that the concept of classical mass plays an important role in formulating relativistic theories of waves and particles is well-known. However, recent studies show that Galilean invariant theories of waves and particles can be formulated with the so-called ‘wave mass’, which replaces the classical mass and allows attaining higher accuracy of performing calculations [J.L. Fry and Z.E. Musielak, Ann. Phys. 325 (2010) 1194]. The main purpose of this paper is to generalize these results and formulate fundamental (Poincaré invariant) relativistic theories of waves and particles without the classical mass. In the presented approach, the classical mass is replaced by an invariant frequency that only involves units of time. The invariant frequencies for various elementary particles are deduced from experiments and their relationship to the corresponding classical and wave mass for each particle is described. It is shown that relativistic wave mechanics with the invariant frequency is independent of the Planck constant, and that such theory can attain higher accuracy of performing calculations. The choice of natural units resulting from the developed theories of waves and particles is also discussed.

Nonrelativistic closed string theory

We construct a Galilean invariant nongravitational closed string theory whose excitations satisfy a nonrelativistic dispersion relation. This theory can be obtained by taking a consistent low energy limit of any of the conventional string theories, including the heterotic string. We give a finite first order worldsheet Hamiltonian for this theory and show that this string theory has a sensible perturbative expansion, interesting high energy behavior of scattering amplitudes and a Hagedorn transition of the thermal ensemble. The strong coupling duals of the Galilean superstring theories are considered and are shown to be described by an eleven-dimensional Galilean invariant theory of light membrane fluctuations. A new class of Galilean invariant nongravitational theories of light-brane excitations are obtained. We exhibit dual formulations of the strong coupling limits of these Galilean invariant theories and show that they exhibit many of the conventional dualities of M theory in a nonrelativistic setting.

Core structure and oscillations of spinor vortices

We describe the topological vortices of spinor fields. The topological group of the invariant manifold is (U(1)⊛SU(2)) Z 2 , and the circulation of the vortex is half the one of a complex scalar field. In the context of a Galilean invariant theory, the vortex core structure is non-analytical at its axis. Since the typical speed becomes very large in the inner vortex core, one must include relativistic effects. We discuss the stability of non-relativistic vortices under the first relativistic correction. Furthermore, spinorial fields satisfying the Dirac equation possess vortices with an analytic behavior in the core that becomes locally magnetized. The relativistic inner core structure matches the non-relativistic solution at a distance of the order of the Compton length.

Theory of ionization in ion-atom collisions: Spectra of ejected electrons

A Galilean invariant theory of atomic collisions is constructed for straight-line trajectories of nuclei with arbitrary impact parameters. The theory is based on Sturmian expansions in Fourier space, and provides an exact description of excitation and ionization processes for a wide range of collision velocities. Advantages of Sturmian sets over conventional eigenstates are emphasized. Detailed spectra of electrons ejected in head-on ion-atom collisions are calculated.

Relativistic aspects of nonrelativistic quantum mechanics

It is argued that nonrelativistic quantum mechanics does not in all respects behave as a fully Galilean invariant theory. The difference is empirically significant, as is illustrated by the Sagnac effect. The conclusion of this article sheds some new light on a recent discussion concerning the status of de Broglie’s theory of matter waves.

An Eulerian Galilean-Invariant Theory for Homogeneous Turbulence, with Application to Inhomogeneous Turbulence with Arbitrary Mean Flows

A Galilean-invariant homogeneous-turbulence theory within Eulerian framework is proposed for the study of the small-scale turbulence. Two Galilean-invariant quantities, the one-time velocity covariance and the convection-free response, are chosen as fundamental ones instead of the usual two-time velocity covariance and response. The Direct-Interaction formalism is applied to find governing equations for these two quantities. Using them, Kolmogorov's law may be derived with the reasonable estimate of Kolmogorov's constant. This theory is also applied to the evaluation of the Reynolds stresses in the inhomogeneous turbulence with arbitrary mean flows. Results obtained are shown to coincide with those from the previous work (A. Yoshizawa: J. Phys. Soc. Jpn. 46 (1979) 669 and 47 (1979) 1665), apart from a slight change of numerical factors.

An Eulerian Galilean-Invariant Approach to Diffusion of a Passive Scalar in Homogeneous Turbulence

A statistical analysis is made of the diffusion of a passive scalar in homogeneous turbulence. An Eulerian Galilean-invariant theory is constructed with the convection-free response and the one-time scalar covariance chosen as fundamental physical quantities and with the aid of the Direct-Interaction formalism. The -5/3 power law for the spectrum of scalar covariance is derived in the intertial range, with the reasonable estimate of its proportional constant.

Lagrangian formulation for arbitrary spin. II. The fermion case

The Rarita-Schwinger formalism for fermion fields is brought to a Lagrangian form in the case of arbitrary spin. The requirement that all differential equations of the field should follow from the variation of an action integral necessitates the introduction of additional fields in the theory. By demanding that these auxiliary variables vanish in the case of no interaction, an explicit form is obtained for the Lagrangian. The resulting theory is found to reproduce the usual formalism in the case of spin $\frac{3}{2}$, and turns out to be in agreement with results obtained by Chang for spin values $\frac{5}{2}$ and $\frac{7}{2}$. The Galilean limit of the minimally coupled equations yields the minimal Galilean-invariant theory of Hagen and Hurley. The $g$ factor turns out to be $\frac{1}{s}$, in accordance with a long-standing conjecture.

Lagrangian formulation for arbitrary spin. I. The boson case

An explicit form is obtained for the Lagrangian of an arbitrary-spin boson field. This is achieved by introducing auxiliary field variables which are required to vanish in the free-field limit. For $s\ensuremath{\le}4$ the results are found to be in agreement with those obtained by Chang. Canonical commutation rules are derived and the equations of motion are brought to first-order form, thereby facilitating the introduction of minimal electromagnetic coupling. It is found that, upon taking the Galilean limit, the ($6s+1$)-component Galilean-invariant theory of Hagen and Hurley results. The $g$ factor is found to be $\frac{1}{s}$, thereby confirming a long-standing conjecture.

Particles Interacting through a Medium. I. Classical Galilean Theory

An analysis is presented of any Galilean-invariant classical Hamiltonian theory in which the particles do not interact directly with each other, but interact directly with a number of other dynamical variables. These other dynamical variables are collectively called "the medium." It is shown that only 12 of the variables which describe the medium are crucial for the Galilean invariance of the theory. The generators of Galilean transformations are constructed in their most general form. An example is given. Galilean-invariant approximations to Galilean-invariant theories are discussed.

Gauge invariance and Galilean invariance

The usual method of introducing electromagnetic interaction via a gauge principle is shown to fail for a Galilean-invariant theory. A clarification of the results is obtained by considering the non-relativistic limit of the Lorentz-invariant theory of the electromagnetic interaction of a complex scalar field.

Theories at Infinite Momentum

We construct Galilean-invariant theories (with Schr\odinger equations) at infinite momentum that describe interacting relativistic systems. Classes of both first- and second-quantized theories are presented. The formalism provides a general approach to the saturation of current algebra; positivity of the mass spectrum is guaranteed, and as much inelasticity as necessary may be introduced. More generally, however, such theories offer the hope of potential-theoretic intuition for relativistic physics.