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The bosonic path integral computes transition amplitudes. E.g. for a scalar field $\phi$, the amplitude between state $|\phi_1\rangle$ on Cauchy surface $\Sigma_1$ and $|\phi_2\rangle$ on $\Sigma_2$ is given by \begin{equation} \langle \phi_2|U_{\Sigma_1\to\Sigma_2}|\phi_1\rangle=\int_{\phi|_{\Sigma_1}=\phi_1}^{\phi|_{\Sigma_2}=\phi_2}D\phi e^{iS[\phi]}.\tag{1} \end{equation} (I'm writing $U_{\Sigma_1\to\Sigma_2}$ for the unitary evolution between the Cauchy surfaces, and $S$ for the action).

I'd like to know whether the fermionic path integral admits a similar interpretation. More precisely, if $\psi_1, \psi_2$ are Grassman-valued fields on $\Sigma_1,\Sigma_2$ (resp.), let us define: \begin{equation} Z[\psi_1,\psi_2]\equiv \int_{\psi|_{\Sigma_1}=\psi_1}^{\psi|_{\Sigma_2}=\psi_2}D\psi D\bar{\psi}e^{iS[\psi,\bar{\psi}]},\tag{2} \end{equation} where the path integral is understood to be fermionic.

What is the meaning of $Z[\psi_1,\psi_2]$? It's not clear how it can be a transition amplitude, since $\psi_1$ and $\psi_2$ don't seem to label states in the Hilbert space in any obvious way. (Compare this with the scalar case, where $\phi_1$ and $\phi_2$ label corresponding field eigenstates). But perhaps there is some nice way to associate $\psi_1$ and $\psi_2$ with states in Hilbert space, in such a way that $Z[\psi_1,\psi_2]$ gives the amplitude between the associated states.

If there is no interpretation of $Z[\psi_1,\psi_2]$ as a transition amplitude, then my question becomes: what is the reason for introducing such path integrals at all?

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    $\begingroup$ Is this from a reference? Which page? $\endgroup$
    – Qmechanic
    Dec 13, 2021 at 17:28
  • $\begingroup$ The second equation is my own definition - it is not standard notation. Perhaps I should edit the question to make this clearer? $\endgroup$ Dec 13, 2021 at 18:38
  • $\begingroup$ The fermionic integral is more or less an algebraic symbol. It cannot be compared with the real integrals in bosonic quantum field theory. Thus it does not make sense to impose the boundary conditions for the fermionic integral. $\endgroup$ Jun 6 at 12:14

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OP provides no references, so the context of OP's eq. (2) is not completely clear, but let us make the following general warning about supernumbers:

  1. An observable/measurable quantity can only consist of ordinary numbers (belonging to $\mathbb{C}$). It does not make sense to measure a soul-valued output in an actual physical experiment, cf. e.g. this related Phys.SE post.

  2. The soul-part of a supernumber [and in particular a Grassmann-odd variable like $\psi_1$ and $\psi_2$ in OP's eq. (2)] is an indeterminate/a placeholder/has no value. So e.g. (the absolute square of) OP's eq. (2) has no meaning as a (relative) probability as it stands.

  3. The only way to achieve a measurable quantity from a Grassmann-odd variable is to integrate it out, cf. e.g. this related Phys.SE post.

  4. In other words, OP's fermionic construction (2) should eventually include the Berezin integrations $\int\!\mathrm{d}\psi_1\int\!\mathrm{d}\psi_2$ in order to produce a physically measurable quantity, like an overlap or a probability.

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    $\begingroup$ What does it mean that it has no value? Of course it not a real or a complex number, it is just a different set. Still, I do not see what is wrong in saying that Psi(t) at a certain time t_0 equals a certain fixed Grassman variable. $\endgroup$ Dec 14, 2021 at 14:12
  • $\begingroup$ Supernumbers are more subtle. I updated the answer. $\endgroup$
    – Qmechanic
    Dec 15, 2021 at 14:23
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    $\begingroup$ The integral is over functions from space-time to Grassman variables, and it is a Berezin integral over such maps (a functional Berezin integral). Still, there is nothing wrong in saying that a map from spacetime to Grassman variables takes some particular (Grassman) value at some particular time. The output of such an integral will be a function(al) of the specified Grassman value. $\endgroup$ Dec 15, 2021 at 20:06
  • $\begingroup$ In fact I think (though I do not know of a reference with the details spelled out) that the wave-function(al) in a fermionic (field) theory should be writable as a function(al) of a Grassman variable, which is just a way to package several ordinary functions (taking into account of spin d.o.f.). $\endgroup$ Dec 15, 2021 at 20:12

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