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.Comment: 45 pages with 1 figure, added reference

Gauge Theories on Open Lie Algebra Non-Commutative Spaces

It is shown that non-commutative spaces, which are quotients of associative algebras by ideals generated by non-linear relations of a particular type, admit extremely simple formulae for deformed or star products. Explicit construction of these star products is carried out. Quantum gauge theories are formulated on these spaces, and the Seiberg-Witten map is worked out in detail.Comment: 11 pages, no figures, Some comments and references adde

Schwinger-Dyson operator of Yang-Mills matrix models with ghosts and derivations of the graded shuffle algebra

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 correlations. The Schwinger-Dyson operator S^i is built from the left annihilation operator, which does not satisfy the Leibnitz rule with respect to concatenation. So the loop equations are not differential equations. We show that left annihilation is a derivation of the graded shuffle product of gluon and ghost correlations. The shuffle product is the point-wise product of Wilson loops, expressed in terms of correlations. So in the limit where concatenation is approximated by shuffle products, the loop equations become differential equations. Remarkably, the Schwinger-Dyson operator as a whole is also a derivation of the graded shuffle product. This allows us to turn the loop equations into linear equations for the shuffle reciprocal, which might serve as a starting point for an approximation scheme.Comment: 13 pages, added discussion & references, title changed, minor corrections, published versio

Possible large-N fixed-points and naturalness for O(N) scalar fields

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 find a line of non-trivial UV fixed-points in the large-N limit, parameterized by a dimensionless coupling. They reduce to classical la phi^4 theory when hbar -> 0. For hbar non-zero, neither action nor measure is scale-invariant, but the effective action is. Scale invariance makes it natural to set a mass deformation to zero. The model has phases where O(N) invariance is unbroken or spontaneously broken. Masses of the lightest excitations above the unbroken vacuum are found. We derive a non-linear equation for oscillations about the broken vacuum. The interaction potential is shown to have a locality property at large-N. In 3d, our construction reduces to the line of large-N fixed-points in |phi|^6 theory.Comment: 23 page

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.Comment: 25 pages, late

Schwinger-Dyson operators as invariant vector fields on a matrix-model analogue of the group of loops

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 function on G and satisfies quadratic equations in convolution. These factorized Schwinger-Dyson or loop equations involve a collection of Schwinger-Dyson operators, which are shown to be right-invariant vector fields on G, one for each linearly independent primitive of the Hopf algebra. A large class of formal matrix models satisfying these properties are identified, including as special cases, the zero momentum limits of the Gaussian, Chern-Simons and Yang-Mills field theories. Moreover, the Schwinger-Dyson operators of the continuum Yang-Mills action are shown to be right-invariant derivations of the shuffle-deconcatenation Hopf algebra generated by sources labeled by position and polarization.Comment: 24 page

Non-anomalous `Ward' identities to supplement large-N multi-matrix loop equations for correlations

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 and action. The former are analogous to Makeenko-Migdal equations and the latter to Slavnov-Taylor identities. LE correspond to leading large-N SDE. WI correspond to 1/N2 suppressed SDE. But they become leading equations since LE for non-anomalous vector fields are vacuous. We show that symmetries at N = ∞ persist at finite N, preventing mixing with multi-trace correlations. For 1 matrix, there are no non-anomalous infinitesimal symmetries. For 2 or more matrices, measure preserving vector fields form an infinite dimensional graded Lie algebra, and non-anomalous action preserving ones a subalgebra. For Gaussian, Chern-Simons and Yang-Mills models we identify up to cubic non-anomalous vector fields, though they can be arbitrarily non-linear. WI are homogeneous linear equations. We use them with the LE to determine some correlations of these models. WI alleviate the underdeterminacy of LE. Non-anomalous symmetries give a naturalness-type explanation for why several linear combinations of correlations in these models vanish