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In Vafa and Witten's paper Parity Conservation in Quantum Chromodynamics, in order to show that QCD does not spontaneously break parity, they argue that any parity-odd operator $X$ must pick up a factor of $i$ under Wick rotation. Specifically, they claim

To be Lorentz invariant, $X$ is constructed from the gauge field $A^a_\mu$, the metric tensor $g_{\mu\nu}$, and the antisymmetric tensor $\epsilon_{\mu\nu\alpha\beta}$. Although $A^a_\mu$ and $g_{\mu\nu}$ remain real in Euclidean space, $\epsilon_{\mu\nu\alpha\beta}$ picks up a factor of $i$ under Wick rotation.

I can't quite follow the argument that the Levi-Civita tensor should pick up a factor of $i$ under Wick rotation. I guess it has something to do with the relation to the volume form, but I haven't been able to come up with a rigorous derivation. Moreover, most explanations of Wick rotations I've seen involve the time component of vectors picking up a factor of $i$ under Wick rotation - so how is it that $A_\mu^a$ remains real in Euclidian space?

Qmechanic
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Henry Shackleton
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1 Answers1

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  1. The Levi-Civita (LC) symbol $\varepsilon^{\mu_0\mu_1\mu_2\mu_3}$ itself is not Wick-rotated, as the values of the symbol only consist of $\pm 1$ and $0$, cf. e.g. my Phys.SE answer here. But it still has an indirect effect on the Wick rotation.

  2. A differential form $\omega$ is form-invariant under Wick rotation since it contains the same number of covariant and contravariant indices.

  3. One geometric way to track the Wick rotation is to use the fact that the Hodge star operator Wick rotates $$ \star_E~=~\color{red}{i}\star_M . \tag{1}$$ which is necessary in order to fulfill the involution property $$\star\star\omega~=~(-1)^{|\omega|(4-|\omega|)}\color{red}{{\rm sgn}(g)}\omega, \tag{2}$$ where $|\omega|$ denotes the rank of the differential form $\omega$.

  4. So if a Minkowskian Lagrangian 4-form term contains $\#(\star)$ Hodge stars $$\color{red}{i}\mathbb{L}_E~=~\mathbb{L}_M~=~\Omega_M~=~\color{red}{(-i)}^{\#(\star)}\Omega_E,\tag{3}$$ then the corresponding Euclidean Lagrangian 4-form term is$^1$ $$\mathbb{L}_E~=~\color{red}{(-i)}^{\#(\star)-1}\Omega_E.\tag{4}$$

  5. In particular, the Lagrangian density Wick rotates as $$\begin{align} {\cal L}_M~=~& \Omega^M_{0123}\cr ~\Downarrow~&\cr {\cal L}_E~=~& \color{red}{(-i)}^{\#(\star)-1}\Omega^E_{0123} ~=~\color{red}{(-i)}^{\#(\varepsilon)}\Omega^E_{0123}, \end{align}\tag{5} $$ which explains the statement about the LC symbol in Ref. 1.

  6. Concerning the gauge field $A_{\mu}$: The 0-component does Wick-rotate $$A^M_0=\color{green}{i}A^E_0;\tag{6}$$ however the gauge field $A_{\mu}$ is taken to be real at both ends of the Wick rotation. This is achieved via appropriate analytic continuation.

References:

  1. C. Vafa & E. Witten, Parity Conservation in QCD, Phys. Rev. Lett. 53 (1984) 535.

  2. D. Tong, Gauge theory lecture notes, Subsection 5.6.3 Thm 1 p. 289.

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