Euclideanizing Action: Refs & Fermionic/Spin-1 YM Actions

[TriTertButoxy]

Is there a good reference on the procedure of Euclideanizing the action? In particular, giving a detailed account of Wick rotating in this context. I can't seem to figure out if they are supposed to give precisely the same answer (even numerically), but just that one converges better, or if they are two different quantities (just one related to the other).

Also, is there a standard way of Euclideanizing the fermionic action and the spin-1 Yang Mills action? There seem to be subtleties which seems very poorly motivated. For example, multiplying $\sigma^0$ and $A^0$ by the imaginary unit, $i$.

 

[RedX]

I don't understand Euclideanizing the action either, so I'm bumping this!

Does it even make sense to integrate over imaginary time? How would you go about doing this? - finding the anti-derivative of the integral, and plugging in imaginary +- infinity via the fundamental theorem of calculus?

Sometimes Euclideanizing the action is called a Wick rotation, which is confusing because there is this other Wick rotation used in momentum space which is definitely sound.

As for multiplying $$A^0$$ by the imaginary unit 'i', that would kind of make sense, since $$A^0$$ transforms like time. However, shouldn't it be '-i', and not '+i', since the relation is $$\tau=it$$ where $$\tau$$ is the Euclidean time.

 

[Entropia]

Thank you for your question. Euclideanizing the action is a common procedure in theoretical physics, particularly in the study of quantum field theories. It involves analytically continuing the time coordinate from the Minkowski space to Euclidean space, which allows for easier calculation and interpretation of certain physical quantities.

To answer your first question, there are several good references that discuss the procedure of Euclideanizing the action and Wick rotating in this context. Some examples include "Quantum Field Theory" by Lewis H. Ryder, "Quantum Field Theory and the Standard Model" by Matthew D. Schwartz, and "Quantum Field Theory in a Nutshell" by A. Zee. These texts provide detailed explanations of the process and its implications for different physical systems.

Regarding your second question, there is no one standard way of Euclideanizing the fermionic action and the spin-1 Yang-Mills action. The procedure may vary depending on the specific theory and the goals of the calculation. However, there are general guidelines that can be followed. For example, in the case of the fermionic action, the fermion fields can be Wick rotated along with the time coordinate, while the gauge fields may require a different procedure. These subtleties are necessary to ensure that the resulting Euclideanized theory is still physically meaningful and consistent.

As for the motivation behind multiplying \sigma^0 and A^0 by the imaginary unit, this is a mathematical convention that allows for the correct interpretation of these fields in Euclidean space. In Minkowski space, these fields have a time-like component, but in Euclidean space, they are purely spatial. Multiplying by i ensures that the correct physical interpretation is maintained.

In summary, Euclideanizing the action can be a complex process, and there may not be one standard way of doing it. However, with a good understanding of the underlying principles and some mathematical conventions, it can be a powerful tool for studying quantum field theories. I hope this helps clarify some of your questions.