# Hidden role of antiunitary operators in Fierz transformation

## TL;DR

This paper reveals how antiunitary operators influence the classification of interaction terms in field theories, simplifying the Fierz transformation process and impacting many-body physics models.

## Contribution

It introduces a theorem linking antiunitary symmetry properties to the count of independent contact interaction terms, bypassing traditional Fierz matrix calculations.

## Key findings

- The number of independent quartic terms depends on the symmetry under antiunitary operators.
- The theorem applies to various physical systems with antiunitary symmetries.
- Examples demonstrate relevance to current many-body physics research.

## Abstract

We show that whenever the symmetry group of a field theory commutes with one or more antiunitary operators $T$, which do not have to but may represent the reversal of physical time, the number of linearly independent contact two-body (quartic) terms is determined by the number of tensors that are even, or by the number of tensors that are odd, under such $T$. The choice depends on the sign of $T^2$ and on the statistics of the fields. The theorem enables one to circumvent the usual computation of the Fierz matrix in determining the independent interaction terms. Some physical examples of current interest in many-body physics are discussed.