How to Perform Wick Rotation in the Lagrangian of a Gauge Theory (like QCD)?

Question

I'm studying Lattice QCD and got stuck in understanding the process of going from a Minkowski space-time to an Euclidean space-time. My procedure is the following:

I considered the Wick rotation in quantum mechanics $x_0 \to -i x_4$. From this, I thought it would be reasonable to assume that for the potential vector, the Wick rotation would be $A_0 \to -i A_4$, since $A_\mu$ is a four-vector like $x_\mu$. This implies $F_{0 i}F^{0 i} \to -F_{4 i}F_{4 i}$ and assuming a metric $g^{\mu \nu} = \;\mbox{diag}(1,-1,-1,-1)$, this results in $F_{\mu \nu}F^{\mu \nu} \to -F_{\mu \nu}F_{\mu \nu}$. Now, considering that $d^4x = dt\, d^3x \to -i d\tau\, d^3x$ the action should transform as

\begin{equation} i S = -\frac{i}{2}\int d^4x \;\mbox{Tr}(F_{\mu \nu}F^{\mu \nu}) \to \frac{1}{2}\int d^4x \;\mbox{Tr}(F_{\mu \nu}F_{\mu \nu}) = S_E\,, \end{equation} where $S_E$ is the Euclidean action, which is a positive number. So, $iS \to S_E$ instead of the expected $iS \to -S_E$. I am obviously doing something wrong. I suspect it could be in the transformation of $d^4x$, but I cannot see why it would be wrong. One thing that I noticed is that if I use the metric $g^{\mu \nu} = \;\mbox{diag}(-1,1,1,1)$, then I get the proper signal. But this is changing the metric in the middle of the calculation, which would be wrong without compensating with an appropriate minus signal and then the issue would persist.

I have issues with the Fermionic sector as well. I considered $\partial_0 \to -i\partial_4$ following the transformation of $x_0$. Also, I saw in the books (Gattringer, Rothe) that it was needed that $\gamma^0 \to \gamma_4$ and $\gamma^i \to i \gamma_i$ so the definition for the $\gamma$ matrices could change from $\{\gamma^\mu,\gamma^\nu\} = 2 g^{\mu \nu} \to \{\gamma_\mu, \gamma_\nu\} = 2 \delta_{\mu \nu}$. It seens reasonable. The problem is that the transformation in the action becomes

\begin{equation} iS = i\int d^4x \; \bar{\psi}(i\gamma^\mu \partial_\mu + g_0 \gamma^\mu A_\mu - m)\psi \to \int d^4x \;\bar{\psi}(\gamma_\mu \partial_\mu - i g_0 \gamma_\mu A_\mu - m)\,, \end{equation}

which is not the Euclidean action. I tried using $A_0 \to i A_4$ in the hope I could have made some mistake in the logic above, but with no luck. So what is the prescription to perform the Wick rotation? How to figure out which transformations I should perform in a wick rotation?

Answer 1

I) Bosonic part: When we Wick-rotate, it is more natural to use signature convention $(-,+,+,+) $ for the Minkowski (M) metric $g^M_{\mu\nu}$, and $(+,+,+,+)$ for the Euclidean (E) metric $g^E_{\mu\nu}$, cf. e.g. my Phys.SE answer here. We will use Greek indices $\mu,\nu=0,1,2,3$, to denote curved spacetime indices; Roman indices $a,b=0,1,2,3$, for flat spacetime indices; and Roman indices $j,k=1,2,3$, for spatial indices. We shall not relabel $x^0=x^4$ to avoid messing with the orientation.

Standard conventions for the Wick rotation are [1]

$$\begin{align} -S_E~=~&iS_M \quad\text{action}, \cr t_E~=~&it_M \quad\text{time}, \cr -{\cal L}_E~=~&{\cal L}_M \quad\text{Lagrangian density}, \cr d^4x_E~=~&id^4x_M \quad\text{spacetime $4$-form}, \cr -\mathbb{L}_E~=~&i\mathbb{L}_M \quad\text{Lagrangian $4$-form}, \cr V_E^0~=~&iV_M^0 \quad\text{zero-comp. of contravariant vector}, \cr V^M_0~=~&iV^E_0 \quad\text{zero-comp. of covariant vector}, \end{align}\tag{1}$$

and a straightforward generalization to tensor fields $T^{\mu_1\ldots\mu_r}{}_{\nu_1\ldots\nu_s}$. (NB: The indices of the Levi-Civita symbol are not Wick-rotated, as the values of the symbol only consists of $\pm 1$ and $0$.) Both curved and flat indices are Wick-rotated. For this reason the determinant of the vielbein $$e~=~\det(e^a_{\mu})\quad\text{and}\quad\sqrt{|g|}~=~\sqrt{\det(g_{\mu\nu})\det(\eta^{ab})}\tag{2}$$ are invariant under Wick-rotation.

Example: Topological/theta/Chern-Simons term. Consider a Lagrangian 4-form term of the form of a $4$-form $$\begin{align} i\mathbb{L}_E~\stackrel{(1)}{=}~&\mathbb{L}_M~=~\Omega_M~=~d^4x_M ~\Omega^M_{0123}\cr ~\stackrel{(1)}{=}~&d^4x_E ~\Omega^E_{0123}~=~\Omega_E. \end{align}\tag{3} $$ The corresponding Lagrangian density term reads $$\begin{align} -{\cal L}_E~\stackrel{(1)}{=}~& {\cal L}_M~=~\frac{e}{4!}\varepsilon^{\mu_0\mu_1\mu_2\mu_3} \Omega^M_{\mu_0\mu_1\mu_2\mu_3}~=~ \Omega^M_{0123}\cr ~\stackrel{(1)}{=}~&i\Omega^E_{0123}~=~\frac{ie}{4!}\varepsilon^{\mu_0\mu_1\mu_2\mu_3} \Omega^E_{\mu_0\mu_1\mu_2\mu_3} .\end{align}\tag{4} $$

Example: Kinetic $F\wedge \star F$ term.

$${\cal L}_E~\stackrel{(1)}{=}~-{\cal L}_M~=~\frac{e}{4}F^M_{\mu\nu}F_M^{\mu\nu}~\stackrel{(1)}{=}~\frac{e}{4}F^E_{\mu\nu}F_E^{\mu\nu}.\tag{5}$$

Example: QED in flat space. Let us here only consider QED (abelian gauge theory), and leave it to the reader to generalize to QCD (non-abelian gauge theory). The zero-component of the gauge variables (with indices down) is a co-vector/one-form and should transform like a time derivative

$$ \frac{\partial}{\partial t_M}~\stackrel{(1)}{=}~i \frac{\partial}{\partial t_E}\tag{6}$$

under Wick rotation. This implies

$$ -A^0_M~=~A^M_0~\stackrel{(1)}{=}~iA^E_0~=~iA^0_E, \qquad F^M_{0j}~\stackrel{(1)}{=}~iF^E_{0j},\tag{7}$$

Therefore the Maxwell Lagrangian density transforms as

$$ {\cal L}_M~=~-\frac{1}{4}F^M_{\mu\nu}F_M^{\mu\nu}~=~\frac{1}{2}F^M_{0j}F^M_{0j}-\frac{1}{4}F_{jk}F_{jk}, \tag{8}$$

$$ \begin{align} {\cal L}_M~=~&{\cal T}_M-{\cal V},\cr {\cal T}_M~=~&\frac{1}{2}F^M_{0j}F^M_{0j}, \cr {\cal V}~=~&\frac{1}{4}F_{jk}F_{jk};\end{align}\tag{9}$$

and

$$ {\cal L}_E~=~\frac{1}{4}F^E_{\mu\nu}F_E^{\mu\nu}~=~\frac{1}{2}F^E_{0j}F^E_{0j}+\frac{1}{4}F_{jk}F_{jk},\tag{10}$$

$$ \begin{align} {\cal L}_E~=~&{\cal T}_E+{\cal V},\cr {\cal T}_E~=~&\frac{1}{2}F^E_{0j}F^E_{0j}, \cr {\cal V}~=~&\frac{1}{4}F_{jk}F_{jk}.\end{align}\tag{11}$$

In particular, an Euclidean Lagrangian density ${\cal L}_E$ looks like a standard Lagrangian density (i.e. kinetic term minus potential term), with an apparent potential equal to minus ${\cal V}$.

II) Fermionic part: Wick rotation of spinor fields is a well-known non-trivial problem, cf. e.g. Refs. 2-4.

References:

1. W. Siegel, Fields; p. 329.

2. P. van Nieuwenhuizen and A. Waldron, A continuous Wick rotation for spinor fields and supersymmetry in Euclidean space, arXiv:hep-th/9611043.

3. A.J. Mountain, Wick rotation and supersymmetry, 1999.

4. A. Bilal & S. Metzger, arXiv:hep-th/0307152.