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I have difficulties with the conceptualization of the fundamental nature of fermions in the path integral approach (for me this approach seems more intuitive than canonical quantization). I will try to present my train of thought sequentially. In path integral approach:

  1. We write an action for field theory, postulating certain symmetries. The requirement of space-time symmetry leads to the fact that our fields can only be representations of the Lorentz group. That is, from a geometric point of view, fields can only be scalars, spinors, vectors, etc.

  2. If we are interested in classical field theory - we minimize the action, obtaining classical equations of motion that describe the classical evolution of fields.

  3. If we are interested in quantum field theory - we integrate the exponent of the action over all field histories, obtaining the amplitude of the quantum transition between the initial and final field states.

In this case, scalar and vector fields (which are complex- or real-valued) are integrated as is. However, spinor fields are integrated not as complex-valued (as spinor representations of SL(2,C) should be), but as Grassmann-valued.

The usual argument (eg in E.Zee, "QFT in a Nutshell") is that commuting (complex-valued) spinor fields lead to negative-definite energy and, as a consequence, there is no stable vacuum in field theory. Therefore, spinor fields must be anticommuting (Grassmann-valued). Over time, the system must transition to a vacuum state, because it corresponds to the minimum free energy at zero temperature (maximum entropy). If the energy is negative-definite, then the system will fall down infinitely.

This is essentially the spin-statistics theorem. However, this argument seems to be taken from a completely different area of ​​theory - statistical physics. Where did the talk about temperature, equilibrium, maximum entropy come from, when we initially talked about the representations of the Lorentz group?

  • Is there a more direct explanation (that doesn't involve a whole other branch of theory) why spinor representations should be Grassmann-valued rather than complex-valued?
  • Is this explanation motivated mathematically rather than physically? For example, maybe Grassmannian-valuedness comes from special geometric properties of spinor representations?
  • Or maybe there really is a very deep mathematical connection between the idea of ​​entropy and the representations of the Lorentz group?
  • Or is there no connection at all and any Universes are realized (including violation of the spin-statistics theorem)? And then the anthropic principle comes into play.
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  • $\begingroup$ Have you considered a more orthodox treatment, from one of the many textbooks, or consulted the standard go-to book on the topic? $\endgroup$
    – naturallyInconsistent
    Commented Sep 24 at 15:37
  • $\begingroup$ Related: Why are anticommutators needed in quantization of Dirac fields? , In QFT, why do fermions have to anticommute in order to insure causality? and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Sep 24 at 15:37
  • $\begingroup$ Quantum mechanics is inherently (in the standard Dirac-von Neumann axiomatization) a statistical theory, so that no wonder the Pauli-Lüders theorem (in the standard Whiteman axiomatization) follows. $\endgroup$
    – DanielC
    Commented Sep 24 at 15:38
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    $\begingroup$ Can you give a specific page reference for this argument in Zee? $\endgroup$
    – alanf
    Commented Sep 24 at 21:42
  • $\begingroup$ In Zee I meant the result (13) in section II.5 and subsequent conclusions In Peskin and Shredder (in canonical formalism) analogous result is (3.90) and subsequent conclusions. $\endgroup$
    – ValkerN
    Commented Sep 28 at 15:55

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