QFT

We show that the calculation of Berezin integrals over anticommuting variables can be reduced to the evaluation of expectations of functionals of Poisson processes via an appropriate Feynman Kac formula. In this way the tools of ordinary analysis can be applied to Berezin integrals and, as an example, we prove a simple upper bound. Possible applications of our results are briefly mentioned.

We present a simple derivation of a Feynman-Kac type formula to study fermionic systems. In this approach the real time or the imaginary time dynamics is expressed in terms of the evolution of a collection of Poisson processes. This formula leads to a family of algorithms parametrized by the values of the jump rates of the Poisson processes. From these an optimal algorithm can be chosen which coincides with the Green Function Monte Carlo method in the limit when the latter becomes exact.

A new method for approximating a quantum-mechanical partition function by an effective classical partition function is proposed. The associated effective classical potential is found by using a variational procedure and a Gaussian Ansatz to estimate the statistical weight of paths starting and ending at the same point in the path-integral representation of the partition function. This method can be generalized to a variational-convergent calculation of path-integral. This systematic technique involves neither perturbation nor stochastic processes. Numerical results are presented and discussed for quartic, sextic, mixed quartic-sextic, and double-well anharmonic oscillators.

In models of (non-relativistic and pseudo-relativistic)electrons interacting with static nuclei and with the (ultraviolet-cutoff)quantized radiation field,the existence of asymptotic electromagnetic fields is established.Our results yield some mathe- matically rigorous understanding of Rayleigh scattering and of the phenomenon of relaxation of isolated atoms to their ground states.Our proofs are based on prop- agation estimates for electrons inspired by similar estimates known from N -body scattering theory.

In models of (non-relativistic and pseudo-relativistic)electrons interacting with static nuclei and with the (ultraviolet-cutoff)quantized radiation field,the existence of asymptotic electromagnetic fields is established.Our results yield some mathe- matically rigorous understanding of Rayleigh scattering and of the phenomenon of relaxation of isolated atoms to their ground states.Our proofs are based on prop- agation estimates for electrons inspired by similar estimates known from N -body scattering theory.

One of the major developments of twentieth-century physics has been the gradual recognition that a common feature of the known fundamental interactions is their gauge structure. In this article the authors review the early history of gauge theory, from Einstein's theory of gravitation to the appearance of non-Abelian gauge theories in the fifties. The authors also review the early history of dimensional reduction, which played an important role in the development of gauge theory. A description is given of how, in recent times, the ideas of gauge theory and dimensional reduction have emerged naturally in the context of string theory and noncommutative geometry.

The effective action for the current and density is shown to satisfy an evolution equation, constructed by the analogy of the functional renormalization group, which describes the dependence of the one-particle irreducible vertex functions on the strength of the disorder and the Coulomb interaction. No small parameter is assumed in deriving the evolution equation. The case of the non-interacting electron gas is discussed in more details. The effective action is constructed in the initial condition for weakly fluctuating impurity field in the framework of the gradient expansion. It is conjectured that non-linearities should drive the transport coefficients to zero at finite strength of the disorder in a manner analogous to a formal diffusion process and the quartic order terms of the gradient expansion would be the key to the localized phase.

The functional renormalization group method is used to take into account the vacuum polarization around localized bound states generated by external potential. The application to Atomic Physics leads to improved Hartree-Fock and Kohn-Sham equations in a systematic manner within the framework of the Density Functional Theory. Another application to Condensed Matter Physics consists of an algorithm to compute quenched averages with or without Coulomb interaction in a non-perturbative manner.

The introduction of spinor and other massive fields by quantizing particles (corpuscles) is conceptually misleading. Only spatial fields must be postulated to form the fundamental objects to be quantized (that is, to define a formal basis for all quantum states), while apparent particles are a mere consequence of decoherence. This conclusion is also supported by the nature of gauge fields.