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

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

The effect of weathering regime on uranium decay series and osmium in two soil profiles

Two soil profiles from the United States with radically different emplacement and climatic histories were analyzed for U, Th and members of the 238U decay series (234U, 230Th, 226Ra, 210Pb), 137Cs and osmium isotopes. The arid New Mexico profile is developed on an approximately 250,000 years old colluvium while the temperate New Hampshire profile is formed on till after the last glaciation at about 10,000 years ago. Both the profiles show significant 234U/238U, 230Th/234U and 226Ra/230Th disequilibria, however, in the New Hampshire profile, the disequilibria are far more pronounced in mid-depths (20-50 cm). High Os concentration with highly radiogenic 187Os/188Os is another characteristic of the mid-depths of the New Hampshire profile. This layer, particularly at about 30-40 cm depth has the characteristics of a soil developed on black shale, as evidenced from both the high U and Os concentrations and the large excess of 230Th over 238U. This profile clearly shows that the regolith on which the contemporary soil is developing was not homogeneous. The presence of measurable excess 226Ra activity over 230Th activity in both profiles suggests the need for a source of 226Ra external to the regolith in both cases. Atmospheric deposition of 226Ra is a possible source for this 226Ra excess and brings to light the important role of atmospheric deposiion of nuclides and their transport in the soil profile in pedogenic processes. It also shows that regolith developed by glacial processes need not be homogeneous, thereby confounding the understanding of vertically modified soil profiles