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Consider Wick's rotation from Minkowski to Euclidean space in QFT. What is the connection between $O(4)$-invariance in Euclidean space and Lorentz invariance in Minkowski space? If we define a quantity which is $O(4)$-invariant in Euclidean space, is it guaranteed, that it will become Lorentz invariant after analytical continuation back to Minkowski space?

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Yes, you are right. In Minkowski space $(t,x,y,z)$, the space-time interval: $$ds^2=dt^2-dx^2-dy^2-dz^2$$ If we define $t=-i\tau$, then we will have $$ds^2=-d{\tau}^2-dx^2-dy^2-dz^2=-(d{\tau}^2+dx^2+dy^2+dz^2)=-ds_E^2$$ where $ds_E^2$ denotes the Euclidean interval in $4D$ Euclidean space $(x,y,z,\tau)$.

As you can see, once the transformation between $t$ and $\tau$ is fixed, we can go back and forth between the two representations which are invariant under Lorentz and Euclidean rotations respectively.

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  • $\begingroup$ Thank you for your reply. Does the same refer to all physical quantities, for instance built up in an invariant way from fields like $F^{\mu\nu}$ in QED? Does not the compactness of O(4) and noncompactness of Lorentz group influence the conclusion? $\endgroup$ – Bambinomio Jun 15 '15 at 9:14
  • $\begingroup$ you don't have to worry too much about the other quantities. Once you have done the transformation for the time, then naturally quantities with certain $t$-dependence will probably have a different $\tau$-dependence. The purpose of Wick rotation is to make the integrations converge and after that we will have to rotate back to the normal $t$, i.e. replacing the $\tau$-dependence by the corresponding $t$-dependence. There are indeed cases where this method fails. I don't think the compactness matters because the rotation is just a technique for doing functional integrations $\endgroup$ – M. Zeng Jun 15 '15 at 9:31
  • $\begingroup$ I am interested not only in performing cetrain integrals, but in the formulation of the euclidean field theory as a whole. In such theories there are certain quantities which are O(4) invariant. They are for instance connected with topological properties of fields and with their behaviour at euclidean infinity. However the topology of euclidean space and Minkowski space is entirely different. Can one be sure that these O(4)-invariants will become Lorentz-invariants after back Wick's rotation? Im am not sure, if for instance a winding number continues to have sense in Minkowski space. $\endgroup$ – Bambinomio Jun 15 '15 at 9:51
  • $\begingroup$ math.columbia.edu/~woit/wordpress/?p=160&cpage=1 and motls.blogspot.sg/2005/02/wick-rotation.html, I just found this page, the author, who used to be a SE user, had a nice discussion regarding Wick rotation in his blog. Unfortunately, some of the content is beyond my reach so far. hope this can give you some clue $\endgroup$ – M. Zeng Jun 15 '15 at 9:52
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Three of the angles from the $4$-dimensional rotations, $\mathrm{SO}(4)$, become imaginary, leading to the group called the Lorentz group. The imaginary angles correspond to a transformation called a boost, and the angles are called rapidities. The Lorentz group, at least the proper orthochronous part of it, has the property of being broken into three regions that are mutually isolated and correspond in Minkowski space to the surfaces at $\tau = \sqrt{(ct)^2 - r^2} = 1$ with $t > 0$, $\tau = 1$ with $t < 0$, and $\tau = \sqrt{-1}$. That is, those three hyper-surfaces cannot be reached by a continuous path defined by changing $3$-d rotation angles and boosts.

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  • $\begingroup$ Wait. Is the isolation of those regions a property of the group, or a property of Minkowski space-time? I think I may be wrong and that is a property of the space on which the group acts, not the group itself. $\endgroup$ – Sean E. Lake Sep 21 '16 at 11:00

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