Unbroken Symmetry of Fermion Grading

We prove unbroken symmetry of grading transformations multiplying all fermion field operators by −1. We give a criterion of spontaneously symmetry breaking for general quasi-local systems with any statistics called macroscopic spontaneously symmetry breaking, or for short MSSB. It is formulated based on our thermodynamical viewpoint that each pair of distinct phases (appeared in spontaneous symmetry breaking) should be disjoint not only for the total system but also for every outside system of a local region specified by the given local structure. The first theorem shows the absence of MSSB for the above fermion grading transformations in an absolutely model independent way which encompasses both lattice and continuous, also fermion, fermion-boson and fermion-spin systems. The second theorem asserts that the fermion grading symmetry is perfectly unbroken for non-zero temperature equilibrium states of the lattice systems, i.e. they vanish all odd elements without any technical assumption on the dynamics or the potentials. As an immediate consequence of the second theorem, we show that our local thermodynamical stability condition (which for a given local region takes the subsystem on its complement region as its outside system) is equivalent to the KMS condition.