# Local Fields without Restrictions on the Spectrum of 4-Momentum Operator and Relativistic Lindblad Equation

## TL;DR

This paper develops a framework for Lorentz invariant local scalar fields with unrestricted 4-momentum spectra, including CPT-violating cases, and extends the Lindblad equation for such fields to describe quantum evolution.

## Contribution

It introduces a generalized Kallen-Lehmann representation and constructs local fields that violate CPT invariance, expanding the theoretical understanding of quantum fields and their dynamics.

## Key findings

- Generalized propagator representation for unrestricted spectra
- Construction of CPT-violating local fields that annihilate the vacuum
- A relativistic Lindblad equation applicable to these fields

## Abstract

Quantum theory of Lorentz invariant local scalar fields without restrictions on 4-momentum spectrum is considered. The mass spectrum may be both discrete and continues and the square of mass as well as the energy may be positive or negative. Such fields can exist as part of a hidden matter in the Universe if they interact with ordinary fields very weakly. Generalization of Kallen-Lehmann representation for propagators of these fields is found. The considered generalized fields may violate CPT- invariance. Restrictions on mass-spectrum of CPT-violating fields are found. Local fields that annihilate vacuum state and violate CPT- invariance are constructed in this scope. Correct local relativistic generalization of Lindblad equation for density matrix is written for such fields. This generalization is particulary needed to describe the evolution of quantum system and measurement process in a unique way. Difficulties arising when the field annihilating the vacuum interacts with ordinary fields are discussed.