Hidden role of antiunitary operators in Fierz transformation

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.

6 pages; minor typos corrected, close to final version; to appear in Phys. Rev. D

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• Spontaneous breaking of the $\text{SO}(2N)$ symmetry in the Gross-Neveu model
• Triplet-Superconductivity in Triple-Band Crossings
• p-wave superconductivity and the axi-planar phase of triple-point fermions
• Superconductivity in Luttinger semimetals near the SU(4) limit