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The Lagrangian of fermions is first order both in space-derivatives and time-derivatives. In the variation of the action one usually fixes both the initial point and end point. I have the following questions:

  1. How does the variational principle for fermions formally work so that it's mathematically correct. I do not want to know how to derive the Euler-Lagrange equation from the Dirac Lagrangian. I know how to do that. I don't know how one even gets these equations since a solution to the Euler-Lagrange equations will not generally connect these chosen points.

  2. If I have an initial and end point that are not connected by an on-shell path (one that solves the Dirac equation), then how do I calculate the path that minimizes the action?

Qmechanic
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WIMP
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2 Answers2

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We integrate over virtual paths in the fermionic path integral. Without integrating over fermionic field configurations, the stationary and virtual paths (in the fermionic action principle) are rendered ill-defined for various reasons:

  1. The fields are Grasmann-odd indeterminates, cf. e.g. this Phys.SE post.

  2. The set of boundary conditions (BCs) $$\psi(t_i\!=\!0)~=~0, \qquad \psi^{\dagger}(t_i\!=\!0)~=~0,\qquad\psi(t_f\!=\!0)~=~0, \qquad \psi^{\dagger}(t_f\!=\!0)~=~0, $$ are quantum mechanically incompatible$^1$ with CARs. [Classically (meaning when $\hbar\to 0$), the first two initial BCs are equivalent. Similarly, the last two final BCs are classically equivalent.]

  3. There is a mismatch$^1$ between the number of BCs and the first order nature of the Dirac equation, meaning that the stationary path is overdetermined.

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$^1$ These issues also appear in the Hamiltonian bosonic coherent state path integral, cf. e.g. this Phys.SE post.

Qmechanic
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The way of doing path-integration in fermions is ussing the grassman numbers. A good link for you will be http://www.int.washington.edu/users/dbkaplan/571_14/Fermion_Path_Integration.pdf This was what you were asking for? I hope you like it.

The path doesn't really matter. If you are still interested in it, I think that you can calculate it ussing the propagator and putting the restriction of final and initial conditions that you want on the first-quantized wavefunction.

Iliado Odiseo
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