Wick's rotation on a complex vector space

[Heidi]

I read this in the wiki article about Wick rotation:

Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the inner product, as in this case the rotation would cancel out and have no effect.

I do not see why. could you help me to understand what would cancel.
thanks

 

[vanhees71]

I have no clue what this should mean. Usually the Wick rotation is a formal analytic continuation of time from the real axis to the complex plane. This has nothing to do with the underlying Hilbert-space structure of QM or inner-product spaces.

There are two standard applications of the Wick rotation in the Q(F)T literature. One is for "vacuum QFT" and the evaluation of all kinds of Green's and proper vertex functions in perturbation theory ("Feynman diagrams"). Here the Wick rotation is usually done for the time-ordered vacuum Green's functions, which are in this case where you calculate vacuum expectation values, identical with the Feynman propagator and thus also closely related to the retarded propagator.

The other application is in many-body equilibrium QFT, where the canonical/grand-canonical statistical operator $\propto \exp(-\beta \hat{H})$ looks formally like a time-evolution operator $\propto \exp(-\mathrm{i} \hat{H} t)$ with $t \rightarrow -\mathrm{i} \tau$. Here $\tau \in (0,\beta)$ and the bosonic (fermionic) fields are subject to symmetric (antisymmetric) boundary conditions, which comes from taking the trace to evaluate equilbrium expectation values, among them the Green's function. The corresponding imaginary-time (Matsubara) propagator is then the analytic continuation of the retarded propagator.

Alternatively you can do equilibrium QFT also in terms of the Schwinger-Keldysh closed-time path formalism (extended by a vertical part of the contour). For details, see

https://itp.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

and references cited therein.

 

[George Jones]

I read this in the wiki article about Wick rotation:

Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the inner product, as in this case the rotation would cancel out and have no effect.

I do not see why. could you help me to understand what would cancel.
thanks

Here is my guess on what on the intended meaning of this passage.

A Wick rotation on a Minkowski vector space changes the Lorentz "inner product" to a Euclidean inner product.

Now, assume that $V$ is a vector space over $\mathbb{C}$ with a conventional inner product $\left<,\right>$, which is linear in one slot and conjugate-linear in the other slot. Let $v'=iv$ with $v$ in $V$. Then, $\left<v',v'\right> = (i)(-i)\left<v,v\right> = \left<v,v\right>$.

 

[Heidi]

thanks
i thought that the remark contained more than that...