arXiv subjects

Joshua Feinberg

Publications and source records attributed to Joshua Feinberg.

44 records · Page 3Linked to original sources

arXiv1996

Renormalizing Rectangles and Other Topics in Random Matrix Theory

We consider random Hermitian matrices made of complex or real $M\times N$ rectangular blocks, where the blocks are drawn from various ensembles. These matrices have $N$ pairs of opposite real nonvanishing eigenvalues, as well as $M-N$ zero eigenvalues (for $M>N$.) These zero eigenvalues are ``kinematical" in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large $N,M$ limit, in which the ``rectangularity" $r={M\over N}$ is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large $N$ renormalization techniques. In addition to the kinematical $δ$-function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if $|r-1|$ is held fixed as $N\rightarrow\infty$, the $N$ non-zero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the non-zero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. As $r\rightarrow 1$ the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal that $r\rightarrow 1$ drives a cross over to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.

cond-mat↗

arXiv1995

Self-Adjoint Wheeler-DeWitt Operators, the Problem of Time and the Wave Function of the Universe

We discuss minisuperspace aspects a non empty Robertson-Walker universe containing scalar matter field. The requirement that the Wheeler-DeWitt (WDW) operator be self adjoint is a key ingredient in constructing the physical Hilbert space and has non-trivial cosmological implications since it is related with the problem of time in quantum cosmology. Namely, if time is parametrized by matter fields we find two types of domains for the self adjoint WDW operator: a non trivial domain is comprised of zero current (Hartle-Hawking type) wave functions and is parametrized by two new parameters, whereas the domain of a self adjoint WDW operator acting on tunneling (Vilenkin type) wave functions is a {\em single} ray. On the other hand, if time is parametrized by the scale factor both types of wave functions give rise to non trivial domains for the self adjoint WDW operators, and no new parameters appear in them.

hep-th↗

arXiv1994

On Kinks and Bound States in the Gross-Neveu Model

We investigate static space dependent $\sigx=\lag\barψψ\rag$ saddle point configurations in the two dimensional Gross-Neveu model in the large N limit. We solve the saddle point condition for $\sigx$ explicitly by employing supersymmetric quantum mechanics and using simple properties of the diagonal resolvent of one dimensional Schrödinger operators rather than inverse scattering techniques. The resulting solutions in the sector of unbroken supersymmetry are the Callan-Coleman-Gross-Zee kink configurations. We thus provide a direct and clean construction of these kinks. In the sector of broken supersymmetry we derive the DHN saddle point configurations. Our method of finding such non-trivial static configurations may be applied also in other two dimensional field theories.

hep-th↗

arXiv1994

Eigenvalue Integro-Differential Equations for Orthogonal Polynomials on the Real Line

The one-dimensional harmonic oscillator wave functions are solutions to a Sturm-Liouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm-Liouville problem on the whole real line. In this paper we show how to characterize an arbitrary set of polynomials orthogonal on $(-\infty,\infty)$ in terms of a system of integro-differential equations of Hartree-Fock type. This system replaces and generalizes the linear differential equation associated with a Sturm-Liouville problem. We demonstrate our results for the special case of Hahn-Meixner polynomials.

hep-th↗

arXiv1994

Non Trivial Saddle Points and Band Structure of Bound States of the Two Dimensional O(N) Vector Model

We discuss O(N) invariant scalar field theories in 0+1 and 1+1 space-time dimensions. Combining ordinary ``Large N" saddle point techniques and simple properties of the diagonal resolvent of one dimensional Schrödinger operators we find {\it exact} non-trivial (space dependent) solutions to the saddle point equations of these models in addition to the saddle point describing the ground state of the theory. We interpret these novel saddle points as collective O(N) singlet excitations of the field theory, each embracing a host of finer quantum states arranged in O(N) multiplets, in an analogous manner to the band structure of molecular spectra. We comment on the relation of our results to the classical work of Dashen, Hasslacher and Neveu and to a previous analysis of bound states in the O(N) model by Abbott.}

hep-th↗

arXiv1993

Dissipation of the String Embedding Dimension in the Singlet Sector of a Matrix Model at Large $α^\prime$

The one dimensional Fermi gas of matrix eigenvalues of the Marinari-Parisi model at positive values of the cosmological constant is generalised.The number of matrix eigenvalues (i.e. gas particles) is varied while keeping the effective potential fixed. This model exhibits a transition from a phase whose continuum behaviour is that of $c=1$ conformal matter coupled to gravity to the well known pure gravity phase of the original model.The former phase is character- ised by an extremely large Regge slope $α^\prime$ which scales as $β^{2/5}$ causing the scaling regions of the two phases to overlap. In this way a continuous flow from one phase to the other is made poss- ible. This phase transition occurs in the singlet sector of the matrix model. The density of states and the two puncture correlator at non zero momenta are calculated on the sphere and are found to behave very differ- ently in the two phases,a fact which demonstrates the phase transition. We comment on a possible relation between this transition and large α^\prime$ semiclassical expansions in the continuum.

hep-th↗

arXiv1992

Stabilised Matrix Models for Non-Perturbative Two Dimensional Quantum Gravity

A thorough analysis of stochastically stabilised hermitian one matrix models for two dimensional quantum gravity at all its $(2,2k-1)$ multicritical points is made. It is stressed that only the zero fermion sector of the supersymmetric hamiltonian, i.e., the forward Fokker-Planck hamiltonian, is relevant for the analysis of bosonic matter coupled to two dimensional gravity. Therefore, supersymmetry breaking is not the physical mechanism that creates non perturbative effects in the case of points of even multicriticality $k$. Non perturbative effects in the string coupling constant $g_{str}$ result in a loss of any explicit relation to the KdV hierarchy equations in the latter case, while maintaining the perturbative genus expansion. As a by-product of our analysis it is explicitly proved that polynomials orthogonal relative to an arbitrary weight $\exp (-βV(x))$ along the whole real line obey an Hartree-Fock equation.

hep-th↗

arXiv1992

String Field Theory for d \leq 0 Matrix Models via Marinari-Parisi

We generalize the Marinari-Parisi definition for pure two dimensional quantum gravity $(k = 2)$ to all non unitary minimal multicritical points $(k \geq 3)$. The resulting interacting Fermi gas theory is treated in the collective field framework. Making use of the fact that the matrices evolve in Langevin time, the Jacobian from matrix coordinates to collective modes is similar to the corresponding Jacobian in $d = 1$ matrix models. The collective field theory is analyzed in the planar limit. The saddle point eigenvalue distribution is the one that defines the original multicritical point and therefore exhibits the appropriate scaling behaviour. Some comments on the nonperturbative properties of the collective field theory as well as comments on the Virasoro constraints associated with the loop equations are made at the end of this letter. There we also make some remarks on the fermionic formulation of the model and its integrability, as a nonlocal version of the non linear Schrödinger model.

hep-th↗