Is the non-abelian gauge field A real or complex valued?

[popwolf]

If it's complex how could one calculate the 2 points function of A(+) and A? Thanks

 

[Javier]

Gauge fields are real. They are in the adjoint representations of the gauge groups, which are real representations. On a related note, you can show that if you have a collection of fieldstrengths that can only be expressed in the Lagrangian as complex combinations, then you cannot define vector potentials corresponding to these complex fieldstrengths. Therefore, you cannot have gauge fields in a theory if the corresponding fieldstrengths in the theory are complex.

 

[R0man]

The usual non-abelian gauge field A is typically complex. This is because the gauge group in non-abelian gauge theories, such as the Standard Model of particle physics, is a non-commutative group. This means that the generators of the gauge group do not commute with each other, and therefore the gauge field must be a complex field in order to properly describe the interactions between particles.

To calculate the 2-point function of A(+) and A, one would use the standard techniques of quantum field theory. This involves performing perturbative calculations using Feynman diagrams, where the complex gauge field is represented as a propagating particle. The 2-point function can then be obtained by summing over all possible Feynman diagrams that contribute to this quantity.

It is important to note that the complex nature of the gauge field does not affect the physical observables in the theory. These observables, such as particle masses and scattering cross-sections, are always real quantities. The complex nature of the gauge field is simply a mathematical tool used to accurately describe the underlying physics of non-abelian gauge theories.