Wick rotation and imaginary number

[touqra]

Can you use Wick rotation to turn any real variable to an imaginary one, not necessary time, such that your integration converges, and then, return back to the real? I'm not really sure how to use Wick rotation.

 

[marlon]

Can you use Wick rotation to turn any real variable to an imaginary one, not necessary time, such that your integration converges, and then, return back to the real? I'm not really sure how to use Wick rotation.

You are not sure to "use" a Wick rotation ? Well, what is the problem ? In any self respecting intro QFT study book, you will find a nice illustration of the Wick rotation and how/why it is used. Knowing that, will also answer your first question.

marlon

 

[MadMax]

can you give the name of a "self-respecting" intro QFT book please?

 

[marlon]

can you give the name of a "self-respecting" intro QFT book please?

"QFT in a Nutshell" by Anthony Zee

marlon

 

[AlphaNumeric]

Can you use Wick rotation to turn any real variable to an imaginary one, not necessary time, such that your integration converges, and then, return back to the real? I'm not really sure how to use Wick rotation.

If memory serves, it crops up in relativity now and again since it's essentially a way of 'Euclideanising' a metric. In some general relativity cases, it's not the time coordinate which is time-like so you'd perform a Wick rotation on the radial coordinate perhaps.

It's nothing more than a change of variables to allow you to compute the integral. Some people have reservations about it because they question what physical meaning $$\tau = it$$ has, but that might be trying to give physical meaning to too many things when you're just wanting to crunch some numbers.

can you give the name of a "self-respecting" intro QFT book please?

"An Introduction to Quantum Field Theory" - Peskin & Schroeder gets my vote. It's the beginners QFT bible in plenty of UK unis. :)

 

[humanino]

My two cents :

For a quick reference, Wikipedia's article on Wick's rotation links to this nice introduction

 

[MeJennifer]

If memory serves, it crops up in relativity now and again since it's essentially a way of 'Euclideanising' a metric. In some general relativity cases, it's not the time coordinate which is time-like so you'd perform a Wick rotation on the radial coordinate perhaps.

It's nothing more than a change of variables to allow you to compute the integral. Some people have reservations about it because they question what physical meaning $$\tau = it$$ has, but that might be trying to give physical meaning to too many things when you're just wanting to crunch some numbers.

It seems to me that such a rotation changes the results when space-time is no longer flat. Am I perhaps mistaken in that?

 

[marlon]

It seems to me that such a rotation changes the results when space-time is no longer flat. Am I perhaps mistaken in that?

Err, a non flat space time is not an ingredient of QFT, which is the formalism where this Wick rotation is very often used and which is indeed the context withint which the OP was asking the question.

The main reason why this rotation is used in QFT is that it connects quantum (field) theory to statistical mechanics. So the equations from both formalisms are linked to each other and one of the two formalisms can be used to describe a fenomenon in the other.

marlon

 

[humanino]

Err, a non flat space time is not an ingredient of QFT, which is the formalism where this Wick rotation is very often used and which is indeed the context withint which the OP was asking the question.

I agree that "Wick rotation" refers usually to QFT in flat spacetime. There are however interesting studies in GR using the "Wick rotation". See for instance : From Euclidean to Lorentzian General Relativity: The Real Way

 

[humanino]

a non flat space time is not an ingredient of QFT

I may be off topic, but... I remember that you were fond of LQG at some point. As I was browsing google, I found this interesting article : Constructing Hamiltonian quantum theories from path integrals in a diffeomorphism invariant context in which (among other things) the Lorentzian metric is recovered in a diffeomorphism invariant manner.