Causal Poisson bracket via deformation quantization

TL;DR

This paper develops a causal Poisson bracket for classical field theories using deformation quantization, connecting it to the Moyal product, and demonstrates its application through examples including scalar fields, strings, and nonlinear particles.

Contribution

It introduces a causal Poisson structure via deformation quantization, relating it to the Moyal product and extending classical brackets with causal features.

Findings

Derived a causal star-product related to the Moyal product.

Obtained causal generalizations of creation/annihilation relations.

Constructed a causal Virasoro algebra for the bosonic string.

Abstract

Starting with the well-defined product of quantum fields at two spacetime points, we explore an associated Poisson structure for classical field theories within the deformation quantization formalism. We realize that the induced star-product is naturally related to the standard Moyal product through an appropriate causal Green's functions connecting points in the space of classical solutions to the equations of motion. Our results resemble the Peierls-DeWitt bracket analyzed in the multisymplectic context. Once our star-product is defined we are able to apply the Wigner-Weyl map in order to introduce a generalized version of Wick's theorem. Finally, we include some examples to explicitly test our method: the real scalar field, the bosonic string and a physically motivated nonlinear particle model. For the field theoretic models we have encountered causal generalizations of the creation/annihilation relations, and also a causal generalization of the Virasoro algebra for the bosonic string. For the nonlinear particle case, we use the approximate solution in terms of the Green's function in order to construct a well-behaved causal bracket.