# Path integral for particle with spin and Dirac propagator

Building on the answer from @ACuriousMind I want to point out that the procedure being followed is the Faddeev-Popov gauge fixing.

The symmetry transformations can be solved to put $$e(t) = T$$, a constant and $$\chi(t) = 0$$ (on the loop) or $$\chi(t) = \theta$$, constant (on the open line). $$T$$ and $$\theta$$ represent physically distinct configurations of the worldline fields (that is, distinguish configurations not related by gauge transformations) and are known as modular parameters.

The integral, $$\int \mathscr{D}e(t) \mathscr{D}\chi(t)$$ is divergent because it vastly overcounts independent / distinct configurations related by the reparameterisation and SUSY symmetries; hence you divide out by the volume of these symmetries (also formally infinite). The Fadeev-Popov procedure deals with this by gauge fixing, where the integral becomes $$\int \mathscr{D}e(t) \mathscr{D}\chi(t) \longrightarrow \int \mathscr{D}f \int \mathscr{D}g \int dT \int d\theta \, \mu(T, \theta)$$ where $$\int \mathscr{D}f = \textrm{vol}(D)$$, $$\int \mathscr{D} g = \textrm{vol}(S)$$ give the volumes of the diffeomorphism and SUSY transformation groups that cancel the volumes on the denominator. Here, and it something that @ACuriousMind left out, the measure $$\mu(T, \theta)$$ is the measure on the moduli, and comes from the Faddeev-Popov determinant factor arising in the gauge fixing.

In this case, the Faddeev-Popov determinant for gauge fixing $$e(t) = T$$ is equal to $$1$$ on the open line and $$1 / T$$ on the loop. For fixing $$\chi(t) = \theta$$ on the line we get a determinant factor of $$1$$ and for $$\chi(t) = 0$$ on the loop we get the same factor. In other words we have $$\int \mathscr{D}e(t) \mathscr{D}\chi(t) \,\Omega[e(t), \chi(t)] \longrightarrow \textrm{vol}(D) \textrm{vol}(S) \times \begin{cases} \int dT \int d\theta \, \Omega[T, \theta] & \textrm{Line}\\ \int \frac{dT}{T}\, \Omega[T, 0] & \textrm{Loop} \end{cases}$$ for any functional $$\Omega$$ of these fields.

Good references include Appendices B and C of https://arxiv.org/abs/1410.3288 or section 1.5.1, 1.5.2 of https://arxiv.org/abs/1512.08694 or the notes at http://www-th.bo.infn.it/people/bastianelli/2-ch6-FT2-2018.pdf (second 2.1).