International Journal of Modern Physics A

This paper begins investigation of the concept of a “generalized τ function,” defined as a generating function of all the matrix elements of a group element g∈G in a given highest weight representation of a universal enveloping algebra . In the generic situation, the time variables correspond to the elements of maximal nilpotent subalgebras rather than Cartanian elements. Moreover, in the case of quantum groups such τ “functions” are not c numbers but take their values in noncommutative algebras (of functions on the quantum group G). Despite all these differences from the particular case of conventional τ functions of integrable (KP and Toda lattice) hierarchies [which arise when G is a Kac-Moody (one-loop) algebra of level k=1], these generic τ functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The most important applications of the formalism should be to k>1 Kac-Moody and multiloop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (zero-loop) algebra SL(2) and its quantum counterpart SL_q(2), as well as for the system of fundamental representations of SL(n).

I investigate the Kazakov-Migdal (KM) model — the Hermitian gauge-invariant matrix model on a D-dimensional lattice. I use an exact large N solution of the KM model with a logarithmic potential to examine its critical behavior. I find critical lines associated with γ_str=−1/2 and γ_str=0 as well as a tricritical point associated with a phase transition of the Kosterlitz-Thouless type. The continuum theories are constructed expanding around the critical points. The one associated with γ_str=0 coincides with a d=1 string while a phase transition of the Kosterlitz-Thouless type separates it from that with γ_str=−1/2, which is indistinguishable from pure 2D gravity for local observables but has a continuum limit for correlators of extended Wilson loops at large distances due to a singular behavior of the Itzykson-Zuber correlator of the gauge fields. I re-examine the KM model with an arbitrary potential in the large D limit and show that it reduces at large N to a one-matrix model whose potential is determined self-consistently. A relation with discretized random surfaces is established via the gauged Potts model, which is equivalent to the KM model at large N providing the coordination numbers coincide.

We present a general calculation of the corrections to the Casimir force between plates with small deviations from plane parallel geometry of the arbitrary type. These corrections are considered up to the fourth order with respect to the relative amplitudes of the deviations. The developed formalism is applied to the cases of nonparallel plates and of large scale as well as short scale deformations of the surfaces of different types. A number of particular examples are explicitly calculated. The method used is the previously developed method of additive summation of the Casimir force potentials with a subsequent renormalization. The results obtained by this method are tested by comparison with the results from the Green function method. A special investigation is performed into the role of mixed terms in the expansion with respect to the deviation amplitudes resulting from different plates. It is shown that such contributions are present already in the first nonvanishing correction to the Casimir force, so that this correction is nonadditive.

In a search for pairs of quantum systems linked by dynamical symmetries, we give a systematic analysis of novel extensions of standard one-dimensional supersymmetric quantum mechanics. The most general supercharges involving higher order derivatives are introduced, leading to an algebra which incorporates a higher order polynomial of the Hamiltonian. We investigate the condition for irreducibility of such a higher order generator to a product of standard first derivative Darboux transformations. As a new example of application of this approach we study the quantum-mechanical radial problem including the scattering amplitudes. We also investigate the links between this higher derivative SUSY and a q-deformed supersymmetric quantum mechanics and introduce the notion of self-similarity in momentum space. An explicit model for the scattering amplitude is constructed in terms of a hypergeometric function which corresponds to a reflectionless potential with infinitely many bound states.

It is argued that renormalization group flow can be interpreted as a Hamiltonian vector flow on a phase space which consists of the couplings of the theory and their conjugate “momenta,” which are the vacuum expectation values of the corresponding composite operators. The Hamiltonian is linear in the conjugate variables and can be identified with the vacuum expectation value of the trace of the energy-momentum operator. For theories with massive couplings the identity operator plays a central role and its associated coupling gives rise to a potential in the flow equations. The evolution of any quantity, such as N-point Green functions, under renormalization group flow can be obtained from its Poisson bracket with the Hamiltonian. Ward identities can be represented as constants of the motion which act as symmetry generators on the phase space via the Poisson bracket structure.