arXiv subjects

S. G. Rajeev

Publications and source records attributed to S. G. Rajeev.

At least 19 recordsLinked to original sources

arXiv2024

Gauge Theories With Infinite Multiplets of Fermions

We study the coupling constant renormalization of gauge theories with an infinite multiplet of fermions, using the zeta function method to make sense of the infinite sums over fermions. If the gauge group K is the maximal compact subgroup of a simple non-compact group G, such infinite multiplets can arise naturally, as reductions of discrete series unitary representations of G. The example K=U(1) and G=SU(1,1) will be studied in detail. Surprisingly, there are abelian gauge theories which are asymptotically free; and others that are UV finite.

hep-th↗

arXiv2022

The Mass Hyperboloid as a Poisson-Lie Group

The light cone formalism of a massive scalar field has been shown by Dirac to have many advantages. But it is not manifestly Lorentz invariant. We will show that this is a feature not a bug: Lorentz invariance is indeed a symmetry, but in a different sense defined by Drinfel'd. The key idea is that the mass shell (mass hyperboloid) is a Poisson-Lie group: there is a non-abelian group multiplication and non-zero Poisson brackets between components of four-momentum. Rotations form the dual group of the hyperboloid in the sense of Drinfel'd. Infinitesimal Lorentz transformations form a Lie bi-algebra.

hep-th↗

arXiv2021

Lie Bi-Algebras on the Non-Commutative Torus

Infinitesimal symmetries of a classical mechanical system are usually described by a Lie algebra acting on the phase space, preserving the Poisson brackets. We propose that a quantum analogue is the action of a Lie bi-algebra on the associative $*$-algebra of observables. The latter can be thought of as functions on some underlying non-commutative manifold. We illustrate this for the non-commutative torus $\mathbb{T}^2_θ$. The canonical trace defines a Manin triple from which a Lie bi-algebra can be constructed. In the special case of rational $θ=\frac{M}{N}$ this Lie bi-algebra is $\underline{GL}(N)=\underline{U}(N)\oplus \underline{B}(N)$, corresponding to unitary and upper triangular matrices. The Lie bi-algebra has a remnant in the classical limit $N\to\infty$: the elements of $\underline{U}(N)$ tend to real functions while $\underline{B}(N)$ tends to a space of complex analytic functions.

math-ph↗

arXiv2020

An Infinitesimal Quantum Group Underlies Classical Fluid Mechanics

Arnold showed that the Euler equations of an ideal fluid describe geodesics in the Lie algebra of incompressible vector fields. We will show that helicity induces a splitting of the Lie algebra into two isotropic subspaces, forming a Manin triple. Viewed another way, this shows that there is an infinitesimal quantum group (a.k.a. Lie bi-algebra) underlying classical fluid mechanics.

math-ph↗

arXiv2017

Curvature in Hamiltonian Mechanics And The Einstein-Maxwell-Dilaton Action

Riemannian geometry is a particular case of Hamiltonian mechanics: the orbits of the hamiltonian $H=\frac{1}{2}g^{ij}p_{i}p_{j}$ are the geodesics. Given a symplectic manifold (Γ,ω), a hamiltonian $H:Γ\to\mathbb{R}$ and a Lagrangian sub-manifold $M\subsetΓ$ we find a generalization of the notion of curvature. The particular case $H=\frac{1}{2}g^{ij}\left[p_{i}-A_{i}\right]\left[p_{j}-A_{j}\right]+ϕ$ of a particle moving in a gravitational, electromagnetic and scalar fields is studied in more detail. The integral of the generalized Ricci tensor w.r.t. the Boltzmann weight reduces to the action principle $\int\left[R+\frac{1}{4}F_{ik}F_{jl}g^{kl}g^{ij}-g^{ij}\partial_{i}ϕ\partial_{j}ϕ\right]e^{-ϕ}\sqrt{g}d^{n}q$ for the scalar, vector and tensor fields.

math-ph↗

arXiv2016

Highly nonlinear wave solutions in a dual to the chiral model

We consider a two-dimensional scalar field theory with a nilpotent current algebra, which is dual to the Principal Chiral Model. The quantum theory is renormalizable and not asymptotically free: the theory is strongly coupled at short distances (encountering a Landau pole). We suggest it can serve as a toy model for $λϕ^{4}$ theory in four dimensions, just as the principal chiral model is a useful toy model for Yang-Mills theory. We find some classical wave solutions that survive the strong coupling limit and quantize them by the collective variable method. They describe excitations with an unusual dispersion relation $ω\propto|k|^{\frac{2}{3}}$ . Perhaps they are the "preons" at strong coupling, whose bound states form massless particles over long distances.

hep-th↗

arXiv2015

Continuity of Scalar Fields With Logarithmic Correlations

We apply select ideas from the modern theory of stochastic processes in order to study the continuity/roughness of scalar quantum fields. A scalar field with logarithmic correlations (such as a massless field in 1+1 spacetime dimensions) has the mildest of singularities, making it a logical starting point. Instead of the usual inner product of the field with a smooth function, we introduce a moving average on an interval which allows us to obtain explicit results and has a simple physical interpretation. Using the mathematical work of Dudley, we prove that the averaged random process is in fact continuous, and give a precise modulus of continuity bounding the short-distance variation.

math-ph↗

arXiv2013

A Hierarchical Finite Element Method for Quantum Field Theory

We study a model of scalar quantum field theory in which space-time is a discrete set of points obtained by repeatedly subdividing a triangle into three triangles at the centroid. By integrating out the field variable at the centroid we get a renormalized action on the original triangle. The exact renormalization map between the angles of the triangles is obtained as well. A fixed point of this map happens to be the cotangent formula of Finite Element Method which approximates the Laplacian in two dimensions.

math-ph↗

arXiv2012

A Non-Riemannian Metric on Space-Time Emergent From Scalar Quantum Field Theory

We show that the two-point function σ(x,x')=\sqrt{<[ϕ(x)-ϕ(x')]^{2}>} of a scalar quantum field theory is a metric (i.e., a symmetric positive function satisfying the triangle inequality) on space-time (with imaginary time). It is very different from the Euclidean metric |x-x'| at large distances, yet agrees with it at short distances. For example, space-time has finite diameter which is not universal. The Lipschitz equivalence class of the metric is independent of the cutoff. σ(x,x') is not the length of the geodesic in any Riemannian metric. Nevertheless, it is possible to embed space-time in a higher dimensional space so that σ(x,x') is the length of the geodesic in the ambient space. σ(x,x') should be useful in constructing the continuum limit of quantum field theory with fundamental scalar particles.

hep-th↗

arXiv2011

Scalar Quantum Field Theory on Fractals

We construct a family of measures for random fields based on the iterated subdivision of simple geometric shapes (triangles, squares, tetrahedrons) into a finite number of similar shapes. The intent is to construct continuum limits of scale invariant scalar field theories, by imitating Wiener's construction of the measure on the space of functions of one variable. These are Gaussian measures, except for one example of a non-Gaussian fixed point for the Ising model on a fractal. In the continuum limits what we construct have correlation functions that vary as a power of distance. In most cases this is a positive power (as for the Wiener measure) but we also find a few examples with negative exponent. In all cases the exponent is an irrational number, which depends on the particular subdivision scheme used. This suggests that the continuum limits corresponds to quantum field theories (random fields) on spaces of fractional dimension.

hep-th↗

arXiv2010

A Dispersion Relation for the Density of States With Application to the Casimir Effect

The trace of a function of a Schrodinger operator minus the same for the Laplacian can be expressed in terms of the determinant of its scattering matrix. The naive formula for this determinant is divergent. Using a dispersion relation, we find another expression for it which is convergent, but needs one piece of information beyond the scattering matrix. Except for this `anomaly', we can express the Casimir energy of a compact body in terms of its optical scattering matrix, without assuming any rotational symmetry for its shape.

quant-ph↗

arXiv2010

On The Relativistic Classical Motion of a Radiating Spinning Particle in a Magnetic Field

We propose classical equations of motion for a charged particle with magnetic moment, taking radiation reaction into account. This generalizes the Landau-Lifshitz equations for the spinless case. In the special case of spin-polarized motion in a constant magnetic field (synchrotron motion) we verify that the particle does lose energy. Previous proposals did not predict dissipation of energy and also suffered from runaway solutions analogous to those of the Lorentz-Dirac equations of motion.

hep-th↗

arXiv2010

Nonconcentration of energy for a semilinear Skyrme model

We continue our investigation of a model introduced by Adkins and Nappi, in which omega mesons stabilize chiral solitons. The aim of this article is to show that the energy associated to equivariant solutions does not concentrate.

math.AP↗

arXiv2010

A continuity argument for a semilinear Skyrme model

We investigate a semilinear modification for the wave map problem proposed by Adkins and Nappi, and prove that in the equivariant case the solution remain continuous at the first possible singularity.

math.AP↗

arXiv2010

The Geometry of Non-Ideal Fluids

Arnold showed that the Euler equations of an ideal fluid describe geodesics on the Lie algebra of incompressible vector fields. We generalize this to fluids with dissipation and Gaussian random forcing. The dynamics is determined by the structure constants of a Lie algebra, along with inner products defining kinetic energy, Ohmic dissipation and the covariance of the forces. This allows us to construct tractable toy models for fluid mechanics with a finite number of degrees of freedom. We solve one of them to show how symmetries can be broken spontaneously.In another direction, we derive a deterministic equation that describes the most likely path connecting two points in the phase space of a randomly forced system: this is a WKB approximation to the Fokker-Plank-Kramer equation, analogous to the instantons of quantum theory. Applied to hydrodynamics, we derive a PDE system for Navier-Stokes instantons.

math-ph↗

arXiv2009

A Lie-Algebraic Approach To the Kondo Problem

The Kondo problem is studied using the unitary Lie algebra of spin-singlet fermion bilinears. In the limit when the number of values of the spin $N$ goes to infinity the theory approaches a classical limit, which still requires a renormalization. We determine the ground state of this renormalized theory. Then we construct a quantum theory around this classical limit, which amounts to recovering the case of finite $N$.

math-ph↗

arXiv2009

Topological Dark Matter in the Little Higgs Models

We show that certain little Higgs models with symmetry breaking SU(N) -> SO(N) for N >= 4 admit topologically stable solitons that may contribute to cosmological dark matter. We have constructed a spherically symmetric soliton and estimated its mass in the case of SU(5) -> SO(5). Its lower bound is found to be around 10.3 TeV. Whether this particle is a fermion or a boson depends on the value of an integer-valued parameter of the underlying theory, analogous to the number of colors of QCD. In either case, the particle is neutral. If it is a fermion, it is a Majorana particle, which could take part in a seesaw mechanism for neutrino masses.

hep-ph↗

arXiv2009

The Maxwell-Boltzmann Distribution is not the Equilibrium on a Hyperboloid

We give a geometric formulation of the Fokker-Planck-Kramer equations for a particle moving on a Lie algebra under the influence of a dissipative and a random force. Special cases of interest are fluid mechanics, the Stochastic Loewner Equation and the rigid body. We find that the Boltzmann distribution, although a static solution, is not normalizable when the algebra is not unimodular. This is because the invariant measure of integration in momentum space is not the standard one. We solve the special case of the upper half-plane (hyperboloid) explicitly: there is another equilibrium solution to the Fokker-Planck equation, which is integrable. It breaks rotation invariance; moreover, the most likely value for velocity is not zero.

math-ph↗