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Consider classical fermionic field. We have it's action, equations of motion and so we can get it's solutions, right? For example, we can consider gravitational solutions with fermions (in particular, in cosmology - see something like https://inspirehep.net/literature/919514) and it seems to be ok, isn't it? I don't see any difference between bosonic and fermionic fields in this case.

Nonetheless, many people say that classical fermionic fields don't exist because of pauli-exclusion-principle. Fermions can't be in the same state so they can't construct global field. Speaking about gravity, people always consider supergravity backgrounds without fermions. But I don't understand them:

  1. Why, speaking about classical fermionic solutions, argument from quantum theory plays a role? I would be happy if someone could explain that effect in detail.
  2. Is there any classical fundamental difference between fermionic and bosonic solutions if we just consider their actions?
  3. Isn't our ordinary matter made up of atoms in fact classical fermionic fields?
  4. To sum it all up, what classical fermionic field configurations are impossible and what are possible?
Qmechanic
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1 Answers1

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Note that the soul-part of a supernumber [and in particular a Grassmann-odd variable] is an indeterminate/a placeholder/has no value. This is fine as long as we make manipulations within the framework of supernumber mathematics. The problem arises when we try to extract physical observables. It does not make sense to measure a soul-valued output in an actual physical experiment.

See also e.g. this Phys.SE post and links therein.

Qmechanic
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