arXiv subjects

S. G. Rajeev

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

At least 55 records · Page 3Linked to original sources

arXiv2002

Collective potential for large N hamiltonian matrix models and free Fisher information

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. Non-commutative 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,which is crucual in determining the ground state. We find that this collective potential has a natural meaning in terms of non-commutative probability theory:it is the `free Fisher information' discovered by Voiculescu. This formulation allows us to find a variational principle for the classical theory described by such large N limits. We then use the variational principle to study models more complex than the one describing the quantum mechanics of a single hermitian matrix (i.e., go beyond the so called D=1 barrier). We carry out approximate variational calculations for a few models and find excellent agreement with known results where such comparisons are possible. We also discover a lower bound for the ground state by using the non-commutative analogue of the Cramer-Rao inequality.

hep-th↗

arXiv2002

A Model of Two Dimensional Turbulence Using Random Matrix Theory

We derive a formula for the entropy of two dimensional incompressible inviscid flow, by determining the volume of the space of vorticity distributions with fixed values for the moments Q_k= \int_w(x)^k d^2 x. This space is approximated by a sequence of spaces of finite volume, by using a regularization of the system that is geometrically natural and connected with the theory of random matrices. In taking the limit we get a simple formula for the entropy of a vortex field. We predict vorticity distributions of maximum entropy with given mean vorticity and enstrophy; also we predict the cylindrically symmetric vortex field with maximum entropy. This could be an approximate description of a hurricane.

physics.flu-dyn↗

arXiv2002

A Cohomological Interpretation of the Migdal-Makeenko Equations

The equations of motion of quantum Yang - Mills theory (in the planar `large N' limit), when formulated in Loop-space are shown to have an anomalous term, which makes them analogous to the equations of motion of WZW models. The anomaly is the Jacobian of the change of variables from the usual ones i.e. the connection one form $A $, to the holonomy $U$. An infinite dimensional Lie algebra related to this change of variables (the Lie algebra of loop substitutions) is developed, and the anomaly is interpreted as an element of the first cohomology of this Lie algebra. The Migdal-Makeenko equations are shown to be the condition for the invariance of the Yang-Mills generating functional $Z$ under the action of the generators of this Lie algebra. Connections of this formalism to the collective field approach of Jevicki and Sakita are also discussed.

hep-th↗

arXiv2001

Entropy of Operator-valued Random Variables: A Variational Principle for Large N Matrix Models

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 non-commutative probability distributions. We show that this entropy is a nontrivial 1-cocycle of the non-commutative analogue of the diffeomorphism group and derive an explicit formula for it. The action principle allows us to solve matrix models using novel variational approximation methods; in the simple cases where comparisons with other methods are possible, we get reasonable agreement.

hep-th↗

arXiv2000

Superconductivity in a Dilute Two Dimensional Electron System

We develop a finite temperature mean field theory in the path integral picture for an extremely dilute system of interacting Fermions in a plane. In the limit of short ranged interactions, the system is shown to undergo a phase transition to a superconducting regime. Unlike the well known BCS transition for metals, this phase transition is found to be very sensitive to the two dimensional nature of the problem. A quantitative estimate of this sensitivity is carried out by repeating the analysis at non-zero thickness. The validity of the mean field results are also proved using renormalization group techniques.

cond-mat.str-el↗

arXiv1999

Valence Parton Distribution Functions from Quantum ChromoDynamics

We show that two dimensional QCD can, to a good approximation, describe the hadronic structure functions measured in Deep Inelastic Scattering. We transform this theory into a new form, Quantum HadronDynamics (QHD), whose semi-classical approximation is closer to nature. The Baryon is then a topological soliton, and its structure function can be predicted by a variational principle. This prediction can be tested by comparison with measurements of neutrino scattering cross-sections.

hep-ph↗

arXiv1999

The x-dependence of Parton Distributions Compared with Neutrino Data

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 is compared to the measurements of neutrino scattering cross-section by the CDHS and CCFR experiments. The agreement is quite good, confirming the validity of QHD as a way of studying hadronic structure.

hep-ph↗

arXiv1999

The Anti-quark Distribution Function of the Baryon

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-quarks carry less than a percent of the baryon momentum. In the limit of chiral symmetry and N_c to infinity, we show that the anti-quark content of the proton vanishes at low Q^2

hep-ph↗

arXiv1999

Large-N Yang-Mills Theory as Classical Mechanics

To formulate two-dimensional Yang-Mills theory with adjoint matter fields in the large-N limit as classical mechanics, we derive a Poisson algebra for the color-invariant observables involving adjoint matter fields. We showed rigorously in J. Math. Phys. 40, 1870 (1999) that different quantum orderings of the observables produce essentially the same Poisson algebra. Here we explain, in a less precise but more pedagogical manner, the crucial topological graphical observations underlying the formal proof.

hep-th↗

arXiv1999

Symmetry Algebras of Large-N Matrix Models for Open Strings

We have discovered that the gauge invariant observables of matrix models invariant under U($N$) form a Lie algebra, in the planar large-N limit. These models include Quantum Chromodynamics and the M(atrix)-Theory of strings. We study here the gauge invariant states corresponding to open strings (`mesons'). We find that the algebra is an extension of a remarkable new Lie algebra ${\cal V}_Λ$ by a product of more well-known algebras such as $gl_{\infty}$ and the Cuntz algebra. ${\cal V}_Λ$ appears to be a generalization of the Lie algebra of vector fields on the circle to non-commutative geometry. We also use a representation of our Lie algebra to establish an isomorphism between certain matrix models (those that preserve `gluon number') and open quantum spin chains. Using known results on quantum spin chains, we are able to identify some exactly solvable matrix models. Finally, the Hamiltonian of a dimensionally reduced QCD model is expressed explicitly as an element of our Lie algebra.

hep-th↗

arXiv1999

A Lie Algebra for Closed Strings, Spin Chains and Gauge Theories

We consider quantum dynamical systems whose degrees of freedom are described by $N \times N$ matrices, in the planar limit $N \to \infty$. Examples are gauge theoires and the M(atrix)-theory of strings. States invariant under U(N) are `closed strings', modelled by traces of products of matrices. We have discovered that the U(N)-invariant opertors acting on both open and closed string states form a remarkable new Lie algebra which we will call the heterix algebra. (The simplest special case, with one degree of freedom, is an extension of the Virasoro algebra by the infinite-dimensional general linear algebra.) Furthermore, these operators acting on closed string states only form a quotient algebra of the heterix algebra. We will call this quotient algebra the cyclix algebra. We express the Hamiltonian of some gauge field theories (like those with adjoint matter fields and dimensionally reduced pure QCD models) as elements of this Lie algebra. Finally, we apply this cyclix algebra to establish an isomorphism between certain planar matrix models and quantum spin chain systems. Thus we obtain some matrix models solvable in the planar limit; e.g., matrix models associated with the Ising model, the XYZ model, models satisfying the Dolan-Grady condition and the chiral Potts model. Thus our cyclix Lie algebra described the dynamical symmetries of quantum spin chain systems, large-N gauge field theories, and the M(atrix)-theory of strings.

hep-th↗

arXiv1999

A Review of Symmetry Algebras of Quantum Matrix Models in the Large-N Limit

This is a review article in which we will introduce, in a unifying fashion and with more intermediate steps in some difficult calculations, two infinite-dimensional Lie algebras of quantum matrix models, one for the open string sector and one for the closed string sector. Physical observables of quantum matrix models in the large-N limit can be expressed as elements of these Lie algebras. We will see that both algebras arise as quotient algebras of a larger Lie algebra. We will also discuss some properties of these Lie algebras not published elsewhere yet, and briefly review their relationship with well-known algebras like the Cuntz algebra, the Witt algebra and the Virasoro algebra. We will also review how Yang--Mills theory, various low energy effective models of string theory, quantum gravity, string-bit models, and quantum spin chain models can be formulated as quantum matrix models. Studying these algebras thus help us understand the common symmetry of these physical systems.

hep-th↗

arXiv1999

A Condensation of Interacting Bosons in Two Dimensional Space

We develop a theory of non-relativistic bosons in two spatial dimensions with a weak short range attractive interaction. In the limit as the range of the interaction becomes small, there is an ultra-violet divergence in the problem. We devise a scheme to remove this divergence and produce a completely finite formulation of the theory. This involves reformulating the dynamics in terms of a new operator whose eigenvalues give the {\it logarithm} of the energy levels. Then, a mean field theory is developed which allows us to describe the limit of a large number of bosons. The ground state is a new kind of condensate (soliton) of bosons that breaks translation invariance spontaneously. The ground state energy is negative and its magnitude grows {\it exponentially} with the number of particles, rather than like a power law as for conventional many body systems.

hep-th↗

arXiv1999

Derivation of Hadron Structure Functions from Quantum ChromoDynamics

We solve a long-standing problem in particle physics: that of deriving the Deep Inelastic structure functions of the proton from the fundamental theory of strong interactions, Quantum ChromoDynamics (QCD). In the Bjorken limit, the momenta of the constituents of the proton (the partons) can be assumed to be in a two-dimensional plane in Minkowski space: a dimensional reduction of QCD to two space-time dimensions. Two dimensional QCD is then shown to be equivalent for all energies and values of number of colors \m{N} to a new theory of hadrons, Quantum HadronDynamics (QHD). The phase space of QHD is the Grassmannian (set of subspaces) of the complex Hilbert space L^2(R). The natural symplectic form along with a hamiltonian define a classical dynamical system, which is equivalent to the large N limit of QCD. 't Hooft's planar limit is the linear approximation to our theory: we recover his integral equation for the meson spectrum but also all the interactions of the mesons. The Grassmannian is a union of connected components labelled by an integer (the renormalized dimension of the subspace) which has the physical meaning of baryon number. The proton is the topological soliton: the minimum of the energy in the sector with baryon number one gives the structure functions of the proton. We solve the resulting integral equations numerically; the agreement with experimental data is quite good for values of the Bjorken variable x>0.2.

hep-th↗

arXiv1999

Bound States in Models of Asymtotic Freedom

We describe a new formalism which expresses asymtotically free thories in a manifestly finite way, after renormalization and dimensional transmutation. The time evolution is NOT differentiable in these systems, so the hamiltonian does not exist. Instead, there is a new operator, the `Principal Operator', (which is roughly speaking the logarithm of the hamiltonian) which is finite (cut-off independent). We construct the Principal operator in several examples, including the Many body Problem of bosons in two dimensions with a short range attractive interaction. This allows us to estimate the ground state energy of a two-dimensional Bose condensate (with an attractive interaction). The ground state energy depends exponentially on the number of particles.

hep-th↗

arXiv1998

Poisson Brackets of Normal-Ordered Wilson Loops

We formulate Yang-Mills theory in terms of the large-N limit, viewed as a classical limit, of gauge-invariant dynamical variables, which are closely related to Wilson loops, via deformation quantization. We obtain a Poisson algebra of these dynamical variables corresponding to normal-ordered quantum (at a finite value of $\hbar$) operators. Comparing with a Poisson algebra one of us introduced in the past for Weyl-ordered quantum operators, we find, using ideas closly related to topological graph theory, that these two Poisson algebras are, roughly speaking, the same. More precisely speaking, there exists an invertible Poisson morphism between them.

hep-th↗

arXiv1998

A Model of Interacting Partons for Hadronic Structure Functions

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 parameter we can adjust is the fraction of the baryon momentum carried by the valence partons. Our prediction agrees well with data except for small values of the Bjorken scaling variable.

hep-ph↗

arXiv1998

Integrability of Supersymmetric Quantum Matrix Models in the Large-N Limit

Many physical systems like supersymmetric Yang-Mills theories are formulated as quantum matrix models. We discuss how to apply the Beth ansatz to exactly solve some supersymmetric quantum matrix models in the large-N limit. Toy models are constructed out of the one-dimensional Hubbard and t-J models as illustrations.

hep-th↗