# Govind Sudarshan KrishnaswamiChennai Mathematical Institute · Physics

Govind Sudarshan Krishnaswami

Doctor of Philosophy

## About

64

Publications

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Introduction

Govind Krishnaswami is a theoretical physicist working at the Chennai Mathematical Institute. Webpage: http://www.cmi.ac.in/~govind

## Publications

Publications (64)

The Rajeev–Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber k in a scalar field theory pseudodual to the 1 + 1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable. Here, we interpret the model as a novel 3D cylindrically symmetric quartic oscillator with an additional rotatio...

The Rajeev-Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber $k$ in a scalar field theory pseudodual to the 1+1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable. Here, we interpret the model as a novel 3D cylindrically symmetric quartic oscillator with an additional rotatio...

In Part I [1], we introduced the idea of a Lax pair and explained how it could be used to obtain conserved quantities for systems of particles. Here, we extend these ideas to continuum mechanical systems of fields such as the linear wave equation for vibrations of a stretched string and the Kortewegde Vries (KdV) equation for water waves. Unlike th...

Conserved quantities can help to understand and solve the equations of motion of various dynamical systems. Lax pairs are a useful tool to find conserved quantities of some dynamical systems. We give a motivated introduction to the idea of a Lax pair using examples such as the linear harmonic oscillator, Toda chain and Eulerian rigid body. A key st...

A rigid body accelerated through an inviscid, incompressible fluid appears to gain mass, which is encoded in an added mass tensor. Swimmers, air bubbles, submarines and airships are slowed down by the associated `added mass' force proportional to their acceleration, which is distinct from viscous drag and buoyancy. In particle physics, otherwise ma...

Lax pairs are a useful tool in finding conserved quantities of some dynamical systems. In this expository article, we give a motivated introduction to the idea of a Lax pair of matrices $(L,A)$, first for mechanical systems such as the linear harmonic oscillator, Toda chain, Eulerian rigid body and the Rajeev-Ranken model. This is then extended to...

In the classical three rotor problem, three equal point masses move on a circle subject to attractive cosine potentials of strength g. In the center of mass frame, energy E is the only known conserved quantity. In earlier works [Krishnaswami and Senapati, Indian Acad. Sci. Conf. Ser. 2(1), 139 (2019), and Chaos 29(12), 123121 (2019)], an order–chao...

Ideal gas dynamics can develop shock-like singularities with discontinuous density. Viscosity typically regularizes such singularities and leads to a shock structure. On the other hand, in one dimension, singularities in the Hopf equation can be non-dissipatively smoothed via Korteweg–de Vries (KdV) dispersion. In this paper, we develop a minimal c...

A rigid body accelerated through a frictionless fluid appears to gain mass. Swimmers, air bubbles, submarines and airships are slowed down by the associated 'added mass' force which is distinct from viscous drag and buoyancy. In particle physics, an otherwise massless electron, quark, W or Z boson, moving through the Higgs field acquires a mass. In...

This paper concerns the classical dynamics of three coupled rotors: equal masses moving on a circle subject to attractive cosine interparticle potentials. This system arises as the classical limit of a model of coupled Josephson junctions. In appropriate units, the non-negative energy E of the relative motion is the only free parameter. We find fam...

Ideal gas dynamics can develop shock-like singularities with discontinuous density. Viscosity typically regularizes such singularities and leads to a shock structure. On the other hand, in 1d, singularities in the Hopf equation can be non-dissipatively smoothed via KdV dispersion. Here, we develop a minimal conservative regularization of 3d ideal a...

In the classical three rotor problem, three equal point masses move on a circle subject to attractive cosine potentials of strength g. In the center of mass frame, energy E is the only known conserved quantity. In earlier work [arXiv:1811.05807], an order-chaos-order transition was discovered in this system along with a band of global chaos for 5.3...

We study the classical Rajeev-Ranken model, a Hamiltonian system with three degrees of freedom describing nonlinear continuous waves in a 1 + 1-dimensional nilpotent scalar field theory pseudodual to the SU(2) principal chiral model. While it loosely resembles the Neumann and Kirchhoff models, its equations may be viewed as the Euler equations for...

We study the classical Rajeev-Ranken model, a Hamiltonian system with three degrees of freedom describing nonlinear continuous waves in a 1+1-dimensional nilpotent scalar field theory pseudodual to the SU(2) principal chiral model. While it loosely resembles the Neumann and Kirchhoff models, its equations may be viewed as the Euler equations for a...

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the PCM is 'pseudodual' to a scalar field theory introduced by Zakharov and Mikhailov and Nappi that is strongly coupled in the ultraviolet and could serve...

The classical three-body problem arose in an attempt to understand the effect of the Sun on the Moon's Keplerian orbit around the Earth. It has attracted the attention of some of the best physicists and mathematicians and led to the discovery of chaos. We survey the three-body problem in its historical context and use it to introduce several ideas...

This paper concerns the classical dynamics of three coupled rotors: equal masses moving on a circle subject to cosine inter-particle potentials. It is a simpler variant of the gravitational three body problem. Moreover, the quantized system of n-rotors has been used to model coupled Josephson junctions. Unlike in the gravitational problem, there ar...

We study the equal-mass classical three rotor problem, a variant of the three body problem of celestial mechanics. The quantum N-rotor problem arises via a partial continuum limit of the Wick rotated XY model. In units of the coupling, the energy serves as a control parameter. We find periodic `pendulum' and `breather' orbits at all energies and ch...

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the PCM is 'pseudo-dual' to a scalar field theory introduced by Zakharov and Mikhailov and Nappi that is strongly coupled in the ultraviolet and could serv...

This paper extends our earlier approach [cf. A. Thyaharaja, Phys. Plasmas 17, 032503 (2010) and Krishnaswami et al., Phys. Plasmas 23, 022308 (2016)] to obtaining à priori bounds on enstrophy in neutral fluids and ideal magnetohydrodynamics. This results in a far-reaching local, three-dimensional, non-linear, dispersive generalization of a KdV-type...

This paper extends our earlier approach [cf. Phys. Plasmas 17, 032503 (2010), 23, 022308 (2016)] to obtaining \`a priori bounds on enstrophy in neutral fluids (R-Euler) and ideal magnetohydrodynamics (R-MHD). This results in a far-reaching local, 3-dimensional, non-linear, dispersive generalization of a KdV-type regularization to compressible dissi...

The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar three-body problem with both Newtonian and attractive inverse-square potentials. The associated JM metrics possess translation and rotation is...

The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar three-body problem with both Newtonian and inverse-square potentials. The associated JM metrics possess translation and rotation isometries in...

Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the...

Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three-dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the...

Ideal systems like MHD and Euler flow may develop singularities in vorticity (w = curl v). Viscosity and resistivity provide dissipative regularizations of the singularities. In this paper we propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the...

Ideal systems like magnetohydrodynamics (MHD) and Euler flow may develop singularities in vorticity (w=∇×v). Viscosity and resistivity provide dissipative regularizations of the singularities. In this paper, we propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV re...

Ideal Eulerian flow may develop singularities in vorticity w. Navier-Stokes
viscosity provides a dissipative regularization. We find a local, conservative
regularization - lambda^2 w times curl(w) of compressible flow and compressible
MHD: a three dimensional analogue of the KdV regularization of the one
dimensional kinematic wave equation. The reg...

Certain recent semi-classical theories of spin-half quantum plasmas are
examined with regard to their internal consistency, physical applicability and
relevance to fusion, astrophysical and condensed matter plasmas. It is shown
that the derivations and some of the results obtained in these theories are
internally inconsistent and contradict well-es...

In the Higgs mechanism, mediators of the weak force acquire masses by
interacting with the Higgs condensate, leading to a vector boson mass matrix.
On the other hand, a rigid body accelerated through an inviscid, incompressible
and irrotational fluid feels an opposing force linearly related to its
acceleration, via an added-mass tensor. We uncover...

A comment on the Letter by S. Braun, F. A. Asenjo and S. M. Mahajan, Phys.
Rev. Lett., 109, 175003 (2012). We show that recent arguments for light
amplification driven by inhomogeneous quantum spin fields in low temperature
electron plasmas in metals are invalid. In essence, a neglect of Pauli
`blocking' led the authors to over-estimate the effects...

In this Letter certain fundamental physics issues relating to recent theories
of so-called `spin quantum plasmas' are examined. It is shown that the
derivations and some of the results obtained in these theories contradict
well-established principles of quantum mechanics, especially in their treatment
of fermions and spin. The analysis presented su...

We formulate the planar "large N limit" of matrix models with a continuously infinite number of matrices directly in terms of U(N) invariant variables. Noncommutative probability theory is found to be a good language to describe this formulation. The change of variables from matrix elements to invariants induces an extra term in the Hamiltonian, wh...

We study a possibly integrable model of Abelian gauge fields on a two-dimensional surface M, with volume form μ. It has the same phase-space as ideal hydrodynamics, a coadjoint orbit of the volume-preserving diffeomorphism group of M. Gauge field Poisson brackets differ from the Heisenberg algebra, but are reminiscent of Yang–Mills theory on a null...

We show that, in 't Hooft's large N limit, matrix models can be formulated as a classical theory whose equations of motion are the factorized Schwinger–Dyson equations. We discover an action principle for this classical theory. This action contains a universal term describing the entropy of the noncommutative probability distributions. We show that...

We discuss a new non-linear PDE, u_t + (2 u_xx/u) u_x = epsilon u_xxx,
invariant under scaling of dependent variable and referred to here as SIdV. It
is one of the simplest such translation and space-time reflection-symmetric
first order advection-dispersion equations. This PDE (with dispersion
coefficient unity) was discovered in a genetic program...

We study baryons in multicolour QCD1 + 1 via Rajeev's gauge-invariant reformulation as a nonlinear classical theory of a bilocal meson field constrained to lie on a Grassmannian. It is known to reproduce 't Hooft's meson spectrum via small oscillations around the vacuum, while baryons arise as topological solitons. The lightest baryon has zero mass...

We try to use scale-invariance and the large-N limit to find a non-trivial 4d O(N) scalar field model with controlled UV behavior and naturally light scalar excitations. The principle is to fix interactions by requiring the effective action for space-time dependent background fields to be finite and scale-invariant when regulators are removed. We f...

For a class of large-N multi-matrix models, we identify a group G that plays the same role as the group of loops on space-time does for Yang-Mills theory. G is the spectrum of a commutative shuffle-deconcatenation Hopf algebra that we associate to correlations. G is the exponential of the free Lie algebra. The generating series of correlations is a...

We consider large-N multi-matrix models whose action closely mimics that of Yang-Mills theory, including gauge-fixing and ghost terms. We show that the factorized Schwinger-Dyson loop equations, expressed in terms of the generating series of gluon and ghost correlations G(xi), are quadratic equations S^i G = G xi^i G in concatenation of correlation...

We try to use scale-invariance and the 1/N expansion to construct a non-trivial 4d O(N) scalar field model with controlled UV behavior and naturally light scalar excitations. The principle is to fix interactions at each order in 1/N by requiring the effective action for arbitrary background fields to be scale-invariant. We find a line of non-trivia...

This work concerns single-trace correlations of Euclidean multi-matrix models. In the large-N limit we show that Schwinger-Dyson equations (SDE) imply loop equations (LE) and non-anomalous Ward identities (WI). LE are associated to generic infinitesimal changes of matrix variables (vector fields). WI correspond to vector fields preserving measure a...

Large-N multi-matrix loop equations are formulated as quadratic difference equations in concatenation of gluon correlations. Though non-linear, they involve highest rank correlations linearly. They are underdetermined in many cases. Additional linear equations for gluon correlations, associated to symmetries of action and measure are found. Loop eq...

We study the competing effects of gluon self-coupling and their interactions with quarks in a baryon, using the very simple setting of a hermitian 1-matrix model with action tr A^4 - log det(nu + A^2). The logarithmic term comes from integrating out N quarks. The model is a caricature of 2d QCD coupled to adjoint scalars, which are the transversely...

We study a possibly integrable model of Abelian gauge fields on a
two-dimensional surface M, with volume form μ. It has the same
phase-space as ideal hydrodynamics, a coadjoint orbit of the
volume-preserving diffeomorphism group of M. Gauge field Poisson
brackets differ from the Heisenberg algebra, but are reminiscent of
Yang-Mills theory on a null...

This thesis concerns the large-N limit, a classical limit where fluctuations in gauge-invariant variables vanish. The large dimension limit for rotation-invariant variables in atoms is given as an example of a classical limit other than hbar vanishing. Part I concerns the baryon in Rajeev's reformulation of 2d QCD in the large-N limit: a non-linear...

We draw an analogy of the spectrum of rapidly rotating Bose Einstein Condensates with that of the Integral quantum Hall effect. This perspective yields fresh insight into the nucleation of the lattice and the dissipation of normal modes. We shall also discuss the core structure and finite temperature effects.

In recent work, we have developed a variational principle for large N multi-matrix models based on the extremization of non-commutative entropy. Here, we test the simplest variational ansatz for our entropic variational principle with Monte-Carlo measurements. In particular, we study the two matrix model with action Tr[{m^2 \over 2} (A_1^2 + A_2^2)...

In this paper we study a 1+1 dimensional relativistic parton model for the structure of baryons. The quarks and anti-quarks interact through a linear potential. We obtain an analytic formula for the isospin averaged valence quark distribution in the chiral and large Nc limits. The leading and non-zero current quark mass corrections are estimated. T...

We study a previously introduced bi-local gauge invariant reformulation of two-dimensional QCD, called 2d hadron dynamics. The baryon arises as a topological soliton in hadron dynamics. We derive an interacting parton model from the soliton model, thus reconciling these two seemingly different points of view. The valence quark model is obtained as...

We interpret the action for 0+1-dimensional large N matrix models in the context of noncommutative probability theory. The actions of both 0-dimensional and 0+1-dimensional matrix models contain universal terms, free entropy and free Fisher information respectively. Their minimization properties are essential for the solution of matrix models. We a...

Let p be an odd prime and let nu-bar denote the multiplicative inverse of nu (mod p). Also let NN = {nu (mod p) : M < nu ≤ M + N } , where M ≥ 0 and N ≥ 1 are integers such that (M,M + N] ⊆ (0,p). To study the distribution of the elements of NN in intervals, we let f(m,H) be the number of n ∈ (m,m + H] with n(mod p) ∈ NN. Our main result is an esti...

We derive a variational principle for large N matrix models. The partition function and vacuum green functions are determined by the principle of minimization of a free energy. The Schwinger-Dyson equations are the conditions for the free energy to be an extremum. We obtain a parametric representation for the greens functions and ground state energ...

We report on the response of a prototype CMS hadron calorimeter module to charged particle beams of pions, muons, and electrons with momenta up to 375 GeV/c. The data were taken at the H2 and H4 beamlines at CERN in 1995 and 1996. The prototype sampling calorimeter used copper absorber plates and scintillator tiles with wavelength shifting fibers f...

We study the large Nc limit of a previously introduced reformulation of 2d QCD, HadronDynamics. This model is used for an effective description of the baryon in Deep Inelastic Scattering, when transverse momenta of partons are ignored. This allows us to determine the non-perturbative initial condition for Q2 evolution equations: xB dependence of st...

Two dimensional Quantum HadronDynamics is a bi-local gauge invariant reformulation of two dimensional QCD, introduced by one of us. The baryon arises as a topological soliton in this picture. Here we study the baryon number one sector of HadronDynamics. Starting from the soliton picture, we derive an interacting quark model, thus reconciling these...

We report on the response of a prototype CMS hadron calorimeter module to charged particle beams of pions, muons, and electrons with momenta up to 375 GeV/c. The data were taken at the H2 and H4 beamlines at CERN in 1995 and 1996. The prototype sampling calorimeter used copper absorber plates and scintillator tiles with wavelength shifting fibers f...

NuTeV is a neutrino–nucleon deep-inelastic scattering experiment at Fermilab. The detector consists of an iron-scintillator sampling calorimeter interspersed with drift chambers, followed by a muon toroidal spectrometer. We present determinations of response and resolution functions of the NuTeV calorimeter for electrons, hadrons, and muons over an...

We present a model for the structure of baryons in which the valence
partons interact through a linear potential. This model can be derived
from QCD in the approximation where transverse momenta are ignored. We
compare the valence quark structure function xF3 predicted by our model
with experimental measurements from neutrino-nucleon Deep Inelastic...

We derive the Deep Inelastic anti-quark distribution in a baryon at a low
value of Q^2 using the variational principle of Quantum HadronDynamics, an
alternative formulation of Quantum ChromoDynamics. It is determined by a
variational approach generalizing the ``valence'' quark approximation of
earlier papers. We find that the ``primordial'' anti-qu...

We use the variational principle of Quantum HadronDynamics, an alternative formulation of Quantum ChromoDynamics, to determine the wavefunction of valence quarks in a baryon at a low value of Q^2. This can be used to predict the structure function xF_3(x,Q^2) at higher values of Q^2 using the evolution equations of perturbative QCD. This prediction...

NuTeV is a neutrino-nucleon deep inelastic scattering experiment at Fermilab. The NuTeV detector is a traditional heavy target neutrino detector which consists of an iron/liquid scintillator sampling calorimeter followed by a muon spectrometer. The calorimeter response to hadrons, muons and electrons has been measured in an in situ calibration beam...

We present a model for the structure of baryons in which the valence partons interact through a linear potential. This model can be derived from QCD in the approximation where the transverse momenta are ignored. We compare the parton structure function predicted by our model with a standard global fit to Deep Inelastic Scattering data. The only par...

Data with electrons and hadrons were taken with the CCFR/NuTeV iron-scintillator sampling calorimeter. Measurements on the ratio of the response to electrons and hadrons and the non-linearity in the response to hadrons are compared to expectations from various models (Gheisha, Fluka and Gcalor).

## Projects

Project (1)