Wick rotation and imaginary number

In summary, Wick rotation can be used to turn any real variable into an imaginary one, not necessarily time, in order to make an integral converge. It is a change of variables that is often used in quantum field theory to connect it with statistical mechanics. It does not require a flat space-time, but can also be applied in general relativity. There are various resources available for understanding and using Wick rotation, such as the book "QFT in a Nutshell" by Anthony Zee and Wikipedia's article on the topic.
  • #1
touqra
287
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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.
 
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  • #2
touqra said:
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
 
  • #3
can you give the name of a "self-respecting" intro QFT book please? :redface:
 
  • #4
MadMax said:
can you give the name of a "self-respecting" intro QFT book please? :redface:

"QFT in a Nutshell" by Anthony Zee

marlon
 
  • #5
touqra said:
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 [tex]\tau = it[/tex] has, but that might be trying to give physical meaning to too many things when you're just wanting to crunch some numbers.
touqra said:
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. :)
 
  • #7
AlphaNumeric said:
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 [tex]\tau = it[/tex] 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?
 
  • #8
MeJennifer said:
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
 
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  • #9
marlon said:
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
 
  • #10

Related to Wick rotation and imaginary number

What is wick rotation and how does it relate to imaginary numbers?

Wick rotation is a mathematical technique used in theoretical physics to convert calculations involving real numbers into calculations involving imaginary numbers. It involves rotating the coordinates of a problem from real time to imaginary time. This is useful because calculations involving imaginary numbers can be easier to solve and can also connect to other areas of mathematics.

Why are imaginary numbers used in wick rotation?

Imaginary numbers are used in wick rotation because they allow for complex numbers to be used in calculations. This is important because many physical problems involve oscillations or periodic behavior, which can be described using complex numbers. Additionally, using imaginary numbers allows for the use of other mathematical tools and techniques that can simplify calculations.

What are the applications of wick rotation and imaginary numbers in science?

Wick rotation and imaginary numbers have various applications in science, particularly in theoretical physics. They are commonly used in quantum field theory, where they simplify calculations and make it easier to study certain phenomena. They are also used in statistical mechanics, string theory, and other areas of physics.

Can wick rotation and imaginary numbers be visualized?

While imaginary numbers cannot be visualized in the same way that real numbers can, wick rotation can be visualized as a rotation in the complex plane. This can be helpful in understanding the concept and how it relates to real and imaginary numbers.

Is wick rotation and imaginary numbers a widely accepted concept in the scientific community?

Yes, wick rotation and imaginary numbers are widely accepted concepts in the scientific community. They have been used for decades in various areas of physics and have been proven to be effective in simplifying calculations and providing insights into complex problems.

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