Galilean Field Theories Research Articles

Quantization of interacting Galilean field theories

We present the quantum field description of Galilean electrodynamics minimally coupled to massless Galilean fermion in (3 + 1)-dimensions. At classical level, the Lagrangian is obtained as a null reduction of a relativistic theory in one higher dimension. We use functional techniques to develop the quantum field description of the theory. Quantum corrections to the propagators and vertex are obtained upto first order and the theory is found to be renormalizable to this order. The beta function of the theory is found to grow linearly; the theory is not asymptotically free.

Effective field theory for non-relativistic hydrodynamics

We write down a Schwinger-Keldysh effective field theory for non-relativistic (Galilean) hydrodynamics. We use the null background construction to covariantly couple Galilean field theories to a set of background sources. In this language, Galilean hydrodynamics gets recast as relativistic hydrodynamics formulated on a one dimension higher spacetime admitting a null Killing vector. This allows us to import the existing field theoretic techniques for relativistic hydrodynamics into the Galilean setting, with minor modifications to include the additional background vector field. We use this formulation to work out an interacting field theory describing stochastic fluctuations of energy, momentum, and density modes around thermal equilibrium. We also present a translation of our results to the more conventional Newton-Cartan language, and discuss how the same can be derived via a non-relativistic limit of the effective field theory for relativistic hydrodynamics.

2D Galilean field theories with anisotropic scaling

In this work, we study two-dimensional Galilean field theories with global translations and anisotropic scaling symmetries. We show that such theories have enhanced local symmetries, generated by the infinite dimensional spin-l Galilean algebra with possible central extensions, under the assumption that the dilation operator is diagonalizable and has a discrete and non-negative spectrum. We study the Newton-Cartan geometry with anisotropic scaling, on which the field theories could be defined in a covariant way. With the well-defined Newton-Cartan geometry we establish the state-operator correspondence in anisotropic GCFT, determine the two-point functions of primary operators, and discuss the modular properties of the torus partition function which allows us to derive Cardy-like formulae.

Anomalies for Galilean fields

We initiate a systematic study of ‘t Hooft anomalies in Galilean field theories, focusing on two questions therein. In the first, we consider the non-relativistic theories obtained from a discrete light-cone quantization (DLCQ) of a relativistic theory with flavor or gravitational anomalies. We find that these anomalies survive the DLCQ, becoming mixed flavor/boost or gravitational/boost anomalies. We also classify the pure Weyl anomalies of Schrödinger theories, which are Galilean conformal field theories (CFTs) with z=2z=2. There are no pure Weyl anomalies in even spacetime dimension, and the lowest-derivative anomalies in odd dimension are in one-to-one correspondence with those of a relativistic CFT in one dimension higher. These results classify many of the anomalies that arise in the field theories dual to string theory on Schrödinger spacetimes.

Internal supersymmetry and small-field Goldstini

The dynamics of the Goldstino mode of spontaneously broken supersymmetry is universal, being fully determined by the non-linearly realized symmetry. We investigate the small-field limit of this theory. This model non-linearly realizes an alternative supersymmetry algebra with vanishing anti-commutators between the fermionic generators, much like an internal supersymmetry. This Goldstino theory is akin to the Galilean scalar field theory that arises as the small-field limit of Dirac-Born-Infeld theory and non-linearly realizes the Galilean symmetry. Indeed, the small-field Goldstino is the partner of a complex Galilean scalar field under conventional supersymmetry. We close with the generalization to extended internal supersymmetry and a discussion of its higher-dimensional origin.

Galilean field theories and conformal structure

We perform a detailed analysis of Galilean field theories, starting with free theories and then interacting theories. We consider non-relativistic versions of massless scalar and Dirac field theories before we go on to review our previous construction of Galilean Electrodynamics and Galilean Yang-Mills theory. We show that in all these cases, the field theories exhibit non-relativistic conformal structure (in appropriate dimensions). The surprising aspect of the analysis is that the non-relativistic conformal structure exhibited by these theories, unlike relativistic conformal invariance, becomes infinite dimensional even in spacetime dimensions greater than two. We then couple matter with Galilean gauge theories and show that there is a myriad of different sectors that arise in the non-relativistic limit from the parent relativistic theories. In every case, if the parent relativistic theory exhibited conformal invariance, we find an infinitely enhanced Galilean conformal invariance in the non-relativistic case. This leads us to suggest that infinite enhancement of symmetries in the non-relativistic limit is a generic feature of conformal field theories in any dimension.

Aspects of hot Galilean field theory

We reconsider general aspects of Galilean-invariant thermal field theory. Using the proposal of our companion paper, we recast non-relativistic hydrodynamics in a manifestly covariant way and couple it to a background spacetime. We examine the concomitant consequences for the thermal partition functions of Galilean theories on a time-independent, but weakly curved background. We work out both the hydrodynamics and partition functions in detail for the example of parity-violating normal fluids in two dimensions to first order in the gradient expansion, finding results that differ from those previously reported in the literature. As for relativistic field theories, the equality-type constraints imposed by the existence of an entropy current appear to be in one-to-one correspondence with those arising from the existence of a hydrostatic partition function. Along the way, we obtain a number of useful results about non-relativistic hydrodynamics, including a manifestly boost-invariant presentation thereof, simplified Ward identities, the systematics of redefinitions of the fluid variables, and the positivity of entropy production.

Aging logarithmic Galilean field theories

We analytically compute correlation and response functions of scalar operators for the systems with Galilean and corresponding aging symmetries for general spatial dimensions d and dynamical exponent z, along with their logarithmic and logarithmic squared extensions, using the gauge/gravity duality. These non-conformal extensions of the aging geometry are marked by two dimensionful parameters, eigenvalue M of an internal coordinate and aging parameter α.We further perform systematic investigations on two-time response functions for general d and z, and identify the growth exponent as a function of the scaling dimensions Δ of the dual field theory operators and aging parameter α in our theory. The initial growth exponent is only controlled by Δ, while its late time behavior by α as well as Δ. These behaviors are separated by a time scale order of the waiting time. We attempt to make contact our results with some field theoretical growth models, such as Kim–Kosterlitz model at higher number of spatial dimensions d.

Galilean conformal algebra in two dimensions and cosmological topologically massive gravity

We consider a realization of the Galilean conformal algebra (GCA) in two-dimensional space–time on the AdS boundary of a particular three-dimensional gravity theory, the so-called cosmological topologically massive gravity (CTMG), which includes the gravitational Chern–Simons term and the negative cosmological constant. The infinite-dimensional GCA in two dimensions is obtained from the Virasoro algebra for the relativistic CFT by taking a scaling limit t → t , x → ϵ x with ϵ → 0 . The parent relativistic CFT should have left and right central charges of order O ( 1 / ϵ ) but opposite in sign in the limit ϵ → 0 . On the other hand, by Brown–Henneaux's analysis the Virasoro algebra is realized on the boundary of AdS 3, but the left and right central charges are asymmetric only by the factor of the gravitational Chern–Simons coupling 1 / μ . If μ behaves as of order O ( ϵ ) under the corresponding limit, we have the GCA with non-trivial centers on AdS boundary of the bulk CTMG. Then we present a new entropy formula for the Galilean field theory from the bulk black hole entropy, which is a non-relativistic counterpart of the Cardy formula. It is also discussed whether it can be reproduced by the microstate counting.

Lorentz-like formulation of Galilean field theories

We review recent articles in which we have constructed nonrelativistic Lagrangian field models by enforcing Galilean covariance on a (4,1) Minkowski manifold and by reducing onto the (3,1) Newtonian space-time. This formalism provides an elegant construction of nonrelativistic Lagrangians, for example, those that describe the electric and magnetic limits of Galilean electromagnetism. We discuss the quantization of scalar and Fermi fields. Scattering amplitudes and cross sections are computed for the Coulomb interaction, as well as electron–electron and electron–positron scattering.PACS Nos.: 03.70.+k, 03.65.Nk, 11.10.Kk, 11.10.Cp

Axiomatic Foundations of Galilean Quantum Field Theories

A realistic axiomatic formulation of Galilean Quantum Field Theories is presented, from which the most important theorems of the theory can be deduced. In comparison with others formulations, the formal aspect has been improved by the use of certain mathematical theories, such as group theory and the theory of rigged Hilbert spaces. Our approach regards the fields as real things with symmetry properties. The general structure is analyzed and contrasted with relativistic theories.

Galilean field theories and nonlocal S-matrix symmetries

Any operator that commutes with the S matrix and is additive, i.e., transforms an asymptotic incoming n-particle state as a sum of its constituent one-particle states, is called a symmetry of the S matrix. The structure of local S-matrix symmetries in Galilean field theories is known. In this paper, S-matrix symmetries that are nonlocal, i.e., may transform asymptotic fields into nonlocal operators, are investigated under the assumption that there is a finite number of such symmetries in the theory.

On the particle level of Galilean Quantum field theories

The procedure devised in a previous paper to obtain the exponential decay law for an unstable non-relativistic isolated particle within a Galilean Quantum Field Theory is applied to describe the transition from the particle to its decay products, as well as the transitions occurring in scattering processes.

The embedding of unstable non-relativistic particles into Galilean quantum field theories

Some unpleasant features of the usual treatment of irreversible processes in quantum mechanics are discussed. It is shown how the description of a non-relativistic unstable particle can be cleanly embedded into a reversible Galilean quantum field theory. It is proven that in the case of stable particles the embedding procedure gives the same values for internal energy which are obtained by the usual procedure. Finally the technique is applied to the ‘Galilee’ model.