Stochastic Quantization of Massive Fermions

Abstract

We consider a general solution of the Langevin equation describing massive fermions to an appropriate boundary problem. Assuming existence of such solution we show that its correlators coincide with the Schwinger functions of corresponding Euclidean Quantum Field Theory.

Appendices

Appendix A: U _r and $\mathsf {U}_{\mathsf {l}}^{T}$

Since in a general situation the potential V includes all required counterterms which are fine-tuned to cancel singularities we still use here the same notation as if we have the whole action L. Furthermore all derivatives below are left derivatives.

$$\begin{array}{@{}rcl@{}} \mathsf{U}_{\mathsf{r}}&=&\left( \begin{array}{lll}-\frac{\delta^{2} L}{\delta \bar{\psi}_{i} \delta \psi_{j}}&-\frac{\delta^{2} L}{\delta \bar{\psi}_{i} \delta \bar{\psi}_{j}}&\frac{\delta^{2} L}{\delta \bar{\psi}_{i} \delta \phi_{j}}\\ \frac{\delta^{2} L}{\delta \psi_{i} \delta \psi_{j}}&\frac{\delta^{2} L}{\delta \psi_{i} \delta \bar{\psi}_{j}}&-\frac{\delta^{2} L}{\delta \psi_{i} \delta \phi_{j}}\\ -\frac{\delta^{2} L}{\delta \phi_{i} \delta \psi_{j}}&-\frac{\delta^{2} L}{\delta \phi_{i} \delta \bar{\psi}_{j}}&\frac{\delta^{2} L}{\delta \phi_{i} \delta \phi_{j}} \end{array}\right)=\left( \begin{array}{lll} u_{11}& u_{12}& u_{13} \\ u_{21}&u_{22}&u_{23}\\u_{31}&u_{32}&u_{33} \end{array}\right) \end{array} $$

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$$\begin{array}{@{}rcl@{}} \mathsf{U}_{\mathsf{l}}^{T}&=&\left( \begin{array}{lll}\frac{\delta^{2} L}{\delta \psi_{i} \delta \bar{\psi}_{j}}&-\frac{\delta^{2} L}{\delta \psi_{i} \delta \psi_{j}}&\frac{\delta^{2} L}{\delta \psi_{i} \delta \phi_{j}}\\ \frac{\delta^{2} L}{\delta \bar{\psi}_{i} \delta \bar{\psi}_{j}}&-\frac{\delta^{2} L}{\delta \bar{\psi}_{i} \delta \psi_{j}}&\frac{\delta^{2} L}{\delta \bar{\psi}_{i} \delta \phi_{j}}\\ \frac{\delta^{2} L}{\delta \phi_{i} \delta \bar{\psi}_{j}}&-\frac{\delta^{2} L}{\delta \phi_{i} \delta \psi_{j}}&\frac{\delta^{2} L}{\delta \phi_{i} \delta \phi_{j}} \end{array}\right)=\left( \begin{array}{lll} u_{22}& -u_{21}& -u^{T}_{31} \\ -u_{12}&u_{11}&-u^{T}_{32}\\u^{T}_{13}&u^{T}_{23}&u_{33} \end{array}\right) \end{array} $$

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The equation $Q^{T} \mathsf {U}_{\mathsf {l}}^{T}=\mathsf {U}_{\mathsf {r}} Q^{T}$ is equivalent to $u_{31}=u^{T}_{23}$ and $u_{13}=-u^{T}_{32}$.

Appendix B: Symmetries of the action

We define as usual the time and parity reversal operators

$$\begin{array}{@{}rcl@{}} \mathcal{T}:\left( \begin{array}{ll} x_{1} , x_{2} \end{array}\right) \mapsto \left( \begin{array}{ll} -x_{1} , x_{2} \end{array}\right), \quad \mathcal{P}:\left( \begin{array}{ll} x_{1} , x_{2} \end{array}\right) \mapsto \left( \begin{array}{ll} x_{1} , - x_{2} \end{array}\right), \end{array} $$

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and corresponding transformations for the spinor ψ

$$\begin{array}{@{}rcl@{}} P: \psi(x) \mapsto \gamma_{1} \psi(\mathcal{P} x), \quad T: \psi(x) \mapsto \gamma_{3} \psi(\mathcal{T} x), \quad C: \psi(x) \mapsto \gamma_{1} \bar{\psi}(x), \end{array} $$

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where T is anti-linear, i.e. TαψT^− 1 = α^∗TψT^− 1. The action L, see (5), is invariant under P. Invariance under CT can be obtained if one simultaneous makes inversion of the sign of m

$$\begin{array}{@{}rcl@{}} \psi(x) \mapsto \gamma_{2} \bar{\psi}(\mathcal{T}x), \quad \bar{\psi}(x)\mapsto \psi(\mathcal{T}x) \gamma_{2},\quad m\mapsto -m. \end{array} $$

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