When solving for instanton solutions in a 1+1d abelian Higgs model, it's convenient to work in Euclidean space using the substitution(adsbygoogle = window.adsbygoogle || []).push({});

[tex]x^0 \rightarrow -ix_4^E,\quad x^1 \rightarrow x_1^E[/tex]

The corresponding substitution for the covariant derivative is

[tex]D^0 \rightarrow iD_4^E,\quad D^1 \rightarrow D_1^E[/tex]

Now, many sources will write out the Euclidean action that you get from this substitution, and I am able to reproduce the gauge kinetic term and the potential term, but I'm doing something stupid with the scalar kinetic term. In real space, we have

[tex]\frac{1}{2}\left(D_{\mu}\phi\right)^*\left(D^{\mu}\phi\right)[/tex]

doing the Wick rotation,

[tex]\frac{1}{2}\left(\left(D_0\phi\right)^*\left(D_0\phi\right) - \left(D_1\phi\right)^*\left(D_1\phi\right)\right)[/tex]

[tex]=\frac{1}{2}\left(\left(-iD_4^E\phi\right)^*\left(-iD_4^E\phi\right) - \left(D_1^E\phi\right)^*\left(D_1^E\phi\right)\right)[/tex]

[tex]=\frac{1}{2}\left((i)\left(D_4^E\phi\right)^*(-i)\left(D_4^E\phi\right) - \left(D_1^E\phi\right)^*\left(D_1^E\phi\right)\right)[/tex]

[tex]=\frac{1}{2}\left(\left(D_4^E\phi\right)^*\left(D_4^E\phi\right) - \left(D_1^E\phi\right)^*\left(D_1^E\phi\right)\right)[/tex]

but in order to reproduce what I see in various sources, I should be getting

[tex]-\frac{1}{2}\left|D_{\mu}^E\phi\right|^2[/tex]

[tex]=\frac{1}{2}\left(-\left(D_4^E\phi\right)^*\left(D_4^E\phi\right) - \left(D_1^E\phi\right)^*\left(D_1^E\phi\right)\right)[/tex]

This is a very straightforward process, clearly I am missing something very obvious or doing something completely wrong, but I've stared at this for a while and I'm just not sure what it is.

Thanks for you help!

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# Trouble with Wick rotation in 1+1d abelian Higgs model

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