Quantum complex scalar fields and noncommutativity

TL;DR

This paper develops a new quantum framework for complex scalar fields incorporating noncommutativity as independent degrees of freedom, extending the DFR algebra and Poincaré symmetry, with explicit solutions via Green's functions.

Contribution

It introduces a novel formalism where noncommutativity parameters are operator degrees of freedom, extending existing algebraic structures and symmetries in quantum field theory.

Findings

Extended Poincaré generators explicitly constructed.

The algebra is realized through generalized Heisenberg relations.

General solutions for scalar fields obtained using Green's functions.

Abstract

In this work we analyze complex scalar fields using a new framework where the object of noncommutativity $\theta^{\mu\nu}$ represents independent degrees of freedom. In a first quantized formalism, $\theta^{\mu\nu}$ and its canonical momentum $\pi_{\mu\nu}$ are seen as operators living in some Hilbert space. This structure is compatible with the minimal canonical extension of the Doplicher-Fredenhagen-Roberts (DFR) algebra and is invariant under an extended Poincar\'e group of symmetry. In a second quantized formalism perspective, we present an explicit form for the extended Poincar\'e generators and the same algebra is generated via generalized Heisenberg relations. We also introduce a source term and construct the general solution for the complex scalar fields using the Green's function technique.