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The super-Poisson bracket follows from a super-version of the Dirac-Bergmann or the Faddeev-Jackiw procedure. Diligent care must be taken to achieve consistent sign conventions when dealing with Grassmann-odd variables, see e.g. my Phys.SE answer here. The singular Legendre transformation for fermions is also discussed in my Phys.SE answer here. In OP's ...


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In classical field theory, the ground state is also called minimizer (of the energy functional); and just to prove its existence is already a quite difficult task, from a mathematical standpoint (as you can imagine, much more difficult is to write eventually its explicit form). Often you can only have minimizing sequences, i.e. sequences of classical states ...



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