Abstract:
The flow of couplings under anisotropic scaling of momenta is computed in $\phi^3$ theory in 6 dimensions. It is shown that the coupling decreases as momenta of two of the particles become large, keeping the third momentum fixed, but at a slower rate than the decrease of the coupling if all three momenta become large simultaneously. This effect serves as a simple test of effective theories of high energy scattering, since such theories should reproduce these deviations from the usual logarithmic scale dependence.

Abstract:
Curci and Ferrari found a unique BRS-invariant action for non-Abelian gauge theories which includes a mass term for the gauge bosons. I analyze this action. While the BRS operator is not nilpotent, the Zinn-Justin equation generalizes in a simple way so that the renormalization of the theory is consistent with the infrared regularization provided by the mass---infrared singularities and ultraviolet infinities are therefore clearly separated. Relations between renormalization constants are derived in dimensional regularization with minimal subtraction. Additional new symmetries allow a simple characterization of physical operators. A new formula is given for the gauge parameter dependence of physical operators.

Abstract:
The nontrivial transformation of the phase space path integral measure under certain discretized analogues of canonical transformations is computed. This Jacobian is used to derive a quantum analogue of the Hamilton-Jacobi equation for the generating function of a canonical transformation that maps any quantum system to a system with a vanishing Hamiltonian. A formal perturbative solution of the quantum Hamilton-Jacobi equation is given.

Abstract:
Halpern and Huang recently showed that there are relevant directions in the space of interactions at the Gaussian fixed point. I show that their result can be derived from Polchinski's form of the Wilson renormalization group. The derivation shows that the existence of these directions is independent of the cutoff function used.

Abstract:
The free energy is shown to decrease along Wilson renormalization group trajectories, in a dimension-independent fashion, for $d>2.$ The argument assumes the monotonicity of the cutoff function, and positivity of a spectral representation of the two point function. The argument is valid to all orders in perturbation theory.

Abstract:
It is shown that the world-line can be eliminated in the matrix quantum mechanics conjectured by Banks, Fischler, Shenker and Susskind to describe the light-cone physics of M theory. The resulting matrix model has a form that suggests origins in the reduction to a point of a Yang-Mills theory. The reduction of the Nishino-Sezgin 10+2 dimensional supersymmetric Yang-Mills theory to a point gives a matrix model with the appropriate features: Lorentz invariance in 9+1 dimensions, supersymmetry, and the correct number of physical degrees of freedom.

Abstract:
Crossing symmetry appears in Dbrane-anti-Dbrane dynamics in the form of an analytic continuation from U(N) for N brane amplitudes to U(N-p,p) for the interactions of N-p branes with p anti-branes. I consider the consequences for supersymmetry and brane-anti-brane forces.

Abstract:
The flow of the action induced by changing $N$ is computed in large $N$ matrix models. It is shown that the change in the action is non-analytic. This non-analyticity appears at the origin of the space of matrices if the action is even.

Abstract:
The exact free energy of SU($N$) Chern-Simons theory at level $k$ is expanded in powers of $(N+k)^{-2}.$ This expansion keeps rank-level duality manifest, and simplifies as $k$ becomes large, keeping $N$ fixed (or vice versa)---this is the weak-coupling (strong-coupling) limit. With the standard normalization, the free energy on the three-sphere in this limit is shown to be the generating function of the Euler characteristics of the moduli spaces of surfaces of genus $g,$ providing a string interpretation for the perturbative expansion. A similar expansion is found for the three-torus, with differences that shed light on contributions from different spacetime topologies in string theory.