Confusion about gamma matrices in Euclidean spacetime

Question

I have encountered a number of sources with differing definitions of the transition from Minkowski spacetime to Euclidean spacetime.

I'd like some clarification as to how to go from Minkowski to Euclidean spacetime; in particular, how the Dirac gamma matrices change (and how one knows this).

Here's what I've found so far:

Source 1: Fujikawa: Path integrals and Quantum anomalies, p50:

(Minkowski metric $(+,-,-,-)$). Write the time component of coordinates as

\begin{equation} x^0 := -ix^4. \end{equation}

The gamma matrix is defined as

\begin{equation} \gamma^0 := -i\gamma^4. \end{equation}

The gamma matrices are then anti-hermitian $\left(\gamma^{\mu}\right)^{\dagger}=-\gamma^{\mu}$.

The Euclidean metric is then $(-,-,-,-)$.

Source 2: arXiv https://arxiv.org/abs/0912.2560

(Minkowski metric $(+,-,-,-)$). In going to the Euclidean metric $\eta^{\mu \nu}=\delta_{\mu \nu}$, one takes

\begin{equation} \partial_{0}^{M} \rightarrow i \partial_{0}^{E}, \quad \partial_{i}^{M} \rightarrow \partial_{i}^{E} \end{equation}

and defines

\begin{align} \gamma_{M}^{0}=\gamma_{E}^{0}, \quad \gamma_{M}^{i}=i \gamma_{E}^{i} \end{align}

such that

\begin{align} \left(\gamma_{E}^{\mu}\right)^{\dagger}=\gamma_{E}^{\mu}. \end{align}

So the Dirac gamma matrices in Euclidean spacetime are hermitian according to this source.

Source 3: The answer in SE post Hermitian properties of Dirac operator

I quote from the accepted answer: "[in the Euclidean theory]... $\bar{\gamma}^{\mu}=\gamma^{\mu}$, not $\gamma^{\mu \dagger}=\gamma^{\mu}$, which is false."

So this source says that $\bar \gamma^\mu:= (\gamma^\mu)^\dagger \gamma^0 = \gamma^\mu$ in the Euclidean theory (if my definition of overbar is consistent with that used in the answer).

A comment on Fujiwaka: I was under the impression that the Euclidean metric (in 4D) is uniquely given by the signature $(+,+,+,+)$, viz. http://mathworld.wolfram.com/EuclideanMetric.html.

Again, all these sources are in contradiction in some way or another, from my perspective. Therefore, I'd like to know how these seemingly contradictory definitions arise.

Answer 1

Since no one else tried to given an answer I will give it a try. I will manly follow the conventions and reasoning put forward in C.Wetterich, 2011, Spinors in euclidean field theory, complex structures and discrete symmetries which I think is one of the better references when it comes to this topic.

In the following we will consider an arbitrary number of dimensions $d$ with signature $s$ meaning we consider a diagonal metric $g^{mn}$ with $d-s$ eigenvalues $+1$ and $s$ eigenvalues $-1$:

• Case $s=0$: Euclidean space with signature $(+,\ldots,+)$,
• Case $s=1$: Minkowski space with signature $(-+,\ldots,+)$ -- "mostly plus", "East Coast" or "Pauli" convention,
• Case $s=d-1$: Minkowski space with signature $(+-,\ldots,-)$ -- "mostly minus", "West Coast" or "Bjorken and Drell" convention.

For conversion tables, a historical review and comments on the differences between the two different conventions for Minkowski space I can recommend App. E of Burgess, C., & Moore, G. (2006). The Standard Model: A Primer.

We use ${2^{\lfloor d/2\rfloor}}$ dimensional Dirac spinors described by two associated elements $\psi$ and $\bar\psi$ of a Grasmann algebra which transform under infinitesimal $\mathrm{SO}(s,d-s)$ transformations of the generalized Lorenz group as $$ \begin{align} \delta\psi^{{\alpha}} &= -\tfrac{1}{2}\epsilon_{mn}(\Sigma^{mn})^{{\alpha}}{}_{{\beta}} \psi^{{\beta}} \, ,\\ \delta\bar\psi_{{\alpha}} &= \tfrac{1}{2}\epsilon_{mn}\bar\psi_{{\beta}}(\Sigma^{mn})^{{\beta}}{}_{{\alpha}} \, , \end{align} $$ with ${ \epsilon_{mn}=-\epsilon_{nm}=\epsilon_{mn}^*}$, the spinor indices ${\alpha}$ and ${\beta}$ and where we suppressed the corresponding transformations of the generalized Lorenz group of coordinates or momenta in our notation. The generators $$ \Sigma^{mn} = \frac{1}{4}[\gamma^m,\gamma^n] $$ of the $\mathrm{SO}(s,d-s)$ group can be constructed using the elements $\gamma^\mu$ of the matrix representation of the Clifford algebra $$ \begin{align} \{\gamma^m,\gamma^n\}=2 g^{mn},\tag{1} \end{align} $$ with hermitian $(\gamma^m)^\dagger=\gamma^m$ for $g^{mm}=1$ and ant-hermitian $(\gamma^m)^\dagger=-\gamma^m$ for $g^{mm}=-1$. This signature specific hermiticity is related to the signature of the respective metric component and ultimately required this way to guaranty a hermitian Dirac Hamiltonian. So far we have introduced Gamma matrices for a metric with arbitrary signature.

With a set of Gamma matrices fulfilling Eq. (1) for an associated metric $g^{mn}$ with euclidean signature ($s=0$) the action of a free massless Dirac spinor is given by $$ S_E=\mathrm{i}\int\mathrm{d}^d x e\bar\psi \gamma^m e_m{}^\mu \partial_\mu \psi, $$ with the vielbein $e_\mu{}^m=\delta_\mu{}^m$, inverse vielbein $e_m{}^\mu$ defined by $e_m{}^\mu e_\mu{}^n=\delta_m{}^n$ and $e=\det(e_\mu{}^m)$. One can use the vielbein as a free variable to change to a different metric/signature. This approach based on a general change of a vielbein is more flexible than the usual notion of analytic continuation of a time component. We define $$ \begin{align} \gamma^\mu&= \gamma^m e_m{}^\mu \tag{2}\\[.75em] \Rightarrow \{\gamma^\mu,\gamma^\nu\}&= e_m{}^\mu e_n{}^\nu \delta^{\mu\nu}=g^{\mu\nu}\tag{3} \end{align} $$

Using $ e_m{}^\mu e_n{}^\nu \delta^{\mu\nu}=g^{\mu\nu}$ one can specify a set of $\{ e_m{}^\mu \}$ to realize a metric with arbitrary signature. To convert from euclidean ($s=0$) to mostly plus ($s=1$) Minkowski signature we can use $$ e_{m}{}^0=-\mathrm{i}\,\delta_m{}^0, \quad e_{m}{}^k=\delta_{m}^k \quad\text{and}\quad e=\mathrm{i},\tag{4.1} $$ which gives for the Gamma matrices $$ \gamma_{M(s=1)}^0=-\mathrm{i}\,\gamma_E, \quad \text{and}\quad \gamma_{M(s=1)}^k=\gamma_E^k.\tag{4.2} $$

Conversely to convert from euclidean ($s=0$) to mostly minus ($s=d-1$) Minkowski signature we can use $$ e_{m}{}^0=\delta_m{}^0, \quad e_{m}{}^k=-\mathrm{i}\,\delta_{m}^k \quad\text{and}\quad e=\mathrm{i}^{d-1},\tag{5.1} $$ which gives for the Gamma matrices $$ \gamma_{M(s=d-1)}^0=\gamma_E, \quad \text{and}\quad \gamma_{M(s=d-1)}^k=-\mathrm{i}\,\gamma_E^k.\tag{5.2} $$

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With this discussion in mind a few comments regarding the sources 1 to 3 from the question:

1. Using the mostly minus Minkowski metric they continue to a $s=d$ "Euclidiean" space with signature $(-,\ldots,-)$ which strikes me as very odd but apart from maybe some global signs it should be usable. The Hermiticity is inline with the notion put forward in this answer.
2. This convention can be realized with an alternative rotation $$e_{m}{}^0=\delta_m{}^0, \quad e_{m}{}^k=+\mathrm{i}\,\delta_{m}^k \quad\text{and}\quad e=(-\mathrm{i})^{d-1}$$ leading to $$\gamma_{M(s=d-1)}^0=\gamma_E, \quad \text{and}\quad \gamma_{M(s=d-1)}^k=+\mathrm{i}\,\gamma_E^k$$. Note however the difference in sign of $e$ for even $d$ leading to a different prefactor for the Minkowski action.
3. I am not sure but I think this statement refers to Minkowski signature not euclidean signature.

In summary once a metric is chosen the Clifford algebra (1) and hermicity constraints from the Hamiltonian fix the relevant properties of the gamma matrices. Analytical continuation or more general transformations to different metrics can be performed using a vielbein formalism which in principle makes it very clear how to translate the gamma matrices and also where to put which prefactors ($e$). For more details see C.Wetterich, 2011, Spinors in euclidean field theory, complex structures and discrete symmetries.