✎ Wick Rotation [The Physics Travel Guide]

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====== Wick Rotation ======

<blockquote>
Field Theory and Statistical Mechanics are closely
connected. A Wick rotation t ➝ i /(kT) will take you from one
to the other.

<cite>http://jfi.uchicago.edu/~leop/TALKS/Phase%20TransitionsV2.4Dirac.pdf</cite>
</blockquote>

* [[http://jakob.physicsnotes.org/quantum_field_theory/methods/non_perturbative_qft#why_imaginary_time|Tunneling phenomena]] are described most easily by performing a Wick rotation. 
  * To classify all [[http://jakob.physicsnotes.org/the_standard_model/poincare_group#representations_of_the_lorentz_group|irreducible representations of the Lorentz group]], we must perform a Wick rotation. In this context, the Wick rotation is often called "Weyl's unitary trick".
  * Path integrals only make sense after we perform a Wick rotation:

<blockquote>
In flat space-time, the situation is well-understood: if your Hamiltonian has good positivity properties you can analytically continue to imaginary values of time, and **when you do this you end up with “Euclidean” path integrals, which actually make sense, unlike QFT path integrals expressed on Minkowski space, which don’t.** You can see the problem even in free field theory: the propagator is given by an integral that goes through two poles, so is ill-defined. The correct way to define it to get causal propagation for a theory with positive energies is to go above one pole, below the other, which is equivalent to “Wick rotating” the integration contour 90 degrees to lie on the imaginary time axis.

<cite>https://www.math.columbia.edu/~woit/wordpress/archives/000160.html</cite>
</blockquote>

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Student>

<note tip>
In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
</note>
 
<tabbox Researcher>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

--> Do we really understand Wick rotations?#

Although a Wick Rotation is a standard tool in QFT not all aspects seem to be sufficiently understood:

<blockquote>
Another peculiarity of chiral theories arises when one tries to understand how they behave under Wick rotation. Non-perturbative QFT calculations are well-defined not in Minkowski space, but in Euclidean space, with physical observables recovered by analytic continuation. But the behavior of spinors in Minkowski and Euclidean space is quite different, leading to a very confusing situation. Despite several attempts over the years to sort this out for myself, I remain confused, and can’t help suspecting that there is more to this than a purely technical problem. One natural mathematical setting for trying to think about this is the twistor formalism, where complexified, compactified Minkowski space is the Grassmanian of complex 2-planes in complex 4-space. The problem though is that thinking this way requires taking as basic variables holomorphic quantities, and how this fits into the standard QFT formalism is unclear. Perhaps the current vogue for twistor methods to study gauge-theory amplitudes will shed some light on this. On the general problem of Wick rotation, about the deepest thinking that I’ve seen has been that of Graeme Segal, who deals with the issue in the 2d context in his famous manuscript “The Definition of Conformal Field Theory”.

<cite>https://www.math.columbia.edu/~woit/wordpress/?p=2876</cite>
</blockquote>

<blockquote>
I’ve always thought this whole confusion is an important clue that there is something about the relation of QFT and geometry that we don’t understand. Things are even more confusing than just worrying about Minkowski vs. Euclidean metrics. To define spinors, we need not just a metric, but a spin connection. In Minkowski space this is a connection on a Spin(3,1)=SL(2,C) bundle, in Euclidean space on a Spin(4)=SU(2)xSU(2) bundle, and these are quite different things, with associated spinor fields with quite different properties. **So the whole “Wick Rotation” question is very confusing even in flat space-time when one is dealing with spinors.**

<cite>https://www.math.columbia.edu/~woit/wordpress/archives/000160.html</cite>
</blockquote>

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