Euclidean signature and compact gauge group

In summary, the conversation discusses the need to continue the gauge field in addition to the analytic continuation to Euclidean space in order to keep the gauge group compact. This is important for maintaining a unitary and finite dimensional representation of the gauge transformation. Suggestions for further reading are mentioned, and it is noted that the gauge group remains unchanged as a compact Lie group in the transition from Minkowski space to Euclidean space.
  • #1
Einj
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Hello everyone,
I have been reading around that when performing the analytic continuation to Euclidean space ([itex]t\to-i\tau[/itex]) one also has to continue the gauge field ([itex]A_t\to iA_4[/itex]) in order to keep the gauge group compact.
I already knew that the gauge field had to be continued as well but I didn't know anything about keeping the gauge group compact. Can someone explain it to me?

Thanks!
 
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  • #2
I believe it has to do with keeping the representation of the gauge transformation unitary and finite dimensional.
 
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  • #3
Do you have any idea on how to show it or any source I could look at? Thanks for you reply!
 
  • #4
I would guess any decent grad text on field theory might cover this. I don't know of one myself. I recall reading something on the complexification in Ryder's book "Quantum Field Theory" but I don't recall him speaking of justification. I don't recall Kaku addressing it directly in his book but I haven't peeked in his text in a while and didn't read it extensively when I last did. Maybe someone else has a suggestion?
 
  • #5
The point is that you entirely go from Minkowski space with a fundamental form of signature (1,3) (or (3,1) if you come from the east coast ;-)) to Euclidean space, i.e., the proper orthochronous Lorentz group is substituted by O(4). So all four-vectors become Euclidean vectors. The gauge group stays as it is, i.e., a compact Lie group.
 
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  • #6
Oh I see! Thanks a lot
 

1. What is Euclidean signature in physics?

Euclidean signature refers to a specific mathematical convention used in physics, particularly in the study of spacetime. It involves changing the sign of the time component in the metric tensor from negative to positive, resulting in a positive-definite metric. This is done to simplify certain calculations and make them more manageable.

2. How does Euclidean signature differ from Minkowski signature?

Euclidean signature differs from Minkowski signature in terms of the sign convention used for the time component in the metric tensor. While Euclidean signature changes the sign from negative to positive, Minkowski signature keeps the sign negative. This leads to different mathematical properties and implications in the study of spacetime.

3. What is a compact gauge group in physics?

A compact gauge group is a type of symmetry group used in theoretical physics, particularly in the study of gauge theories. It is a mathematical group that describes the symmetry of a physical system, such as a particle or field, under certain transformations. Compact gauge groups have a finite number of dimensions and are closed under multiplication, making them useful for describing the laws of nature.

4. How are compact gauge groups related to the standard model of particle physics?

The standard model of particle physics uses compact gauge groups to describe the fundamental interactions between particles. Specifically, it uses the compact gauge groups SU(3), SU(2), and U(1) to describe the strong nuclear force, weak nuclear force, and electromagnetic force, respectively. These groups allow for a unified description of the fundamental forces and particles in the universe.

5. What are some applications of compact gauge groups in physics?

Compact gauge groups have numerous applications in physics, including the study of quantum field theory, particle physics, and cosmology. They are also used in the development of theories beyond the standard model, such as grand unified theories and string theory. Additionally, compact gauge groups have practical applications in technology, such as in the development of quantum computers and other quantum technologies.

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