Global unitary fixing and matrix-valued correlations in matrix models

We consider the partition function for a matrix model with a global unitary invariant energy function. We show that the averages over the partition function of global unitary invariant trace polynomials of the matrix variables are the same when calculated with any choice of a global unitary fixing, while averages of such polynomials without a trace define matrix-valued correlation functions, that depend on the choice of unitary fixing. The unitary fixing is formulated within the standard Faddeev-Popov framework, in which the squared Vandermonde determinant emerges as a factor of the complete Faddeev-Popov determinant. We give the ghost representation for the FP determinant, and the corresponding BRST invariance of the unitary-fixed partition function. The formalism is relevant for deriving Ward identities obeyed by matrix-valued correlation functions.Comment: Tex, 22 page

Hamiltonian Map to Conformal Modification of Spacetime Metric: Kaluza-Klein and TeVeS

It has been shown that the orbits of motion for a wide class of nonrelativistic Hamiltonian systems can be described as geodesic flows on a manifold and an associated dual. This method can be applied to a four dimensional manifold of orbits in spacetime associated with a relativistic system. We show that a relativistic Hamiltonian which generates Einstein geodesics, with the addition of a world scalar field, can be put into correspondence with another Hamiltonian with conformally modified metric. Such a construction could account for part of the requirements of Bekenstein for achieving the MOND theory of Milgrom in the post-Newtonian limit. The constraints on the MOND theory imposed by the galactic rotation curves, through this correspondence, would then imply constraints on the structure of the world scalar field. We then use the fact that a Hamiltonian with vector gauge fields results, through such a conformal map, in a Kaluza-Klein type theory, and indicate how the TeVeS structure of Bekenstein and Sanders can be put into this framework. We exhibit a class of infinitesimal gauge transformations on the gauge fields ${\cal U}_\mu$ which preserve the Bekenstein-Sanders condition ${\cal U}_\mu {\cal U}^\mu = -1$. The underlying quantum structure giving rise to these gauge fields is a Hilbert bundle, and the gauge transformations induce a non-commutative behavior to the fields, i.e., they become of Yang-Mills type. Working in the infinitesimal gauge neighborhood of the initial Abelian theory, we show that in the Abelian limit the Yang-Mills field equations provide nonlinear terms which may avoid the caustic singularity found by Contaldi, et al.Comment: Plain TeX, 8 pages. Proceedings of Conference of International Association for Relativistic Dynamics, Thessaloniki, Greece, June 2008. Revision includes discussion of field norm preserving gauge on Hilbert bundle and nonlinear contributions to field equations in Abelian limi