Trace-class property without the auxiliary spinor trace

Prove or disprove that the first- and second-order resolvent-expansion terms of the finite-temperature Pauli–Villars-regularised Dirac free energy are trace-class in the presence of a nonzero electrostatic potential without first applying the \(\mathbb{C}^4\)-trace.

Background

In proving well-posedness, the paper applies the finite-dimensional C4\mathbb{C}^4-trace to eliminate non-trace-class contributions arising from a generic electrostatic potential. When the electrostatic potential vanishes, the problematic terms vanish and the auxiliary trace is unnecessary.

For a nonzero electrostatic potential, the author states that the trace-class property of the first- and second-order terms cannot be proved without taking the C4\mathbb{C}^4-trace. This leaves unresolved whether the terms themselves are trace-class before that operation.

References

In particular, we are unable to prove that the $n=1$ and $n=2$ terms are trace-class in the presence of an electrostatic potential without first taking the $\C4$--trace; however, if $V\equiv0$, it is easy to see that the problematic terms vanish (see Equations~(3.28)~and~(3.32)), so that taking the $\C4$--trace is no longer necessary to show that they belong to $S{1}$.

— Recent Advances on a Model for the Electromagnetic Dirac Field at Thermal Equilibrium  (2609.11283 - Morellini, 10 Sep 2026) in Section 4.1, Proof of Theorem~\ref{stat:definition-FPV}