Spin in the path integral : anti - commuting versus commuting variables

Path Integrals with Generalized Grassmann Variables

We construct path integral representations for the evolution operator of q-oscillators with root of unity values of q-parameter using BargmannFock representations with commuting and non-commuting variables, the differential calculi being q-deformed in both cases. For q = −1 we obtain a new form of Grassmann-like path integral. on leave of absence from Nuclear Physics Institute, Moscow State Uni...

Integrability of quantum chains: theory and applications to the spin-1/2 XXZ chain

In this contribution we review the theory of integrability of quantum systems in one spatial dimension. We introduce the basic concepts such as the Yang-Baxter equation, commuting currents, and the algebraic Bethe ansatz. Quite extensively we present the treatment of integrable quantum systems at finite temperature on the basis of a lattice path integral formulation and a suitable transfer matr...

The QCD Abacus: A Cellular Automata Formulation for Continuous Gauge Symmetries

This talk will explain a new way to formulate statistical (or quantum eld) theories entirely in terms discrete quantum spins. Remarkably even theories with continuous symmetries such as 3-d rotations can be exactly represented in such a discrete (or binary) "computational" framework. A new application of this idea to Quantum Chromodynamics (QCD), the fundamental gauge theory for nuclear forces,...

Probability of Mutually Commuting N-Tuples in Some Classes of Compact Groups

Evaluating Grassmann Integrals

I discuss a simple numerical algorithm for the direct evaluation of multiple Grassmann integrals. The approach is exact, suffers no Fermion sign problems, and allows arbitrarily complicated interactions. Memory requirements grow exponentially with the interaction range and the transverse size of the system. Low dimensional systems of order a thousand Grassmann variables can be evaluated on a wo...