How does the Wilson Line change under time reversal?

In summary: P} \exp\{ig\int_{-\vec b_\perp}^{-\vec a_\perp} dx_\perp \cdot A_\perp(-x_\perp)\}##In summary, under time reversal, the four vector changes as ##x^\mu \to \tilde x^\mu = x_\mu##, and ##i \to -i##. The gluon field changes as ##A^\mu(x) \to A_\mu(-x)##. The Wilson line also changes accordingly, with the spatial limits and gluon field being flipped in sign.
  • #1
Chenkb
41
1
Textbooks tell us that a four vector under time reversal changes as ##x^\mu \to \tilde x^\mu = x_\mu##, and ##i \to -i##. The gluon field changes as ##A^\mu(x) \to A_\mu(-\tilde x)##.
My question is how does the following integral (the wilson line in the perpendicular direction) change unter time reversal?
##\mathcal{P} \exp\{ig\int_{\vec a_\perp}^{\vec b_\perp} dx_\perp \cdot A_\perp(x_\perp)\}##
Can it be the following way??
##-ig\int_{-\vec a_\perp}^{-\vec b_\perp} dx_\perp \cdot [-A_\perp(x_\perp)]##
 
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  • #2


I would like to clarify and expand on the concepts mentioned in the forum post.

Firstly, it is important to note that time reversal is a mathematical operation that changes the direction of time in a physical system. It is represented by the symbol ##\mathcal{T}## and is defined as follows:

##\mathcal{T} \psi(t) = \psi(-t)##

where ##\psi(t)## is a wavefunction or field at time ##t##.

In the context of special relativity, we can define a four vector ##x^\mu = (ct, x, y, z)##, where ##c## is the speed of light and ##x, y, z## are the spatial coordinates. Under time reversal, this four vector changes to ##\tilde x^\mu = (ct, -x, -y, -z)##. This means that the time component remains the same, but the spatial components are reversed.

Similarly, for a complex number ##z = x + iy##, under time reversal, it changes to ##\tilde z = x - iy##. This is where the statement ##i \to -i## comes from in the forum post.

Now, let's look at the gluon field ##A^\mu(x)##, which is a four vector field in space-time. Under time reversal, it changes to ##\tilde A^\mu(\tilde x) = A_\mu(-x)##. This means that the field at the reversed spatial coordinates is the same, but with the sign of the spatial components flipped.

Coming to the integral in question, the Wilson line is a gauge invariant quantity in quantum chromodynamics (QCD) and is defined as follows:

##\mathcal{P} \exp\{ig\int_{\vec a_\perp}^{\vec b_\perp} dx_\perp \cdot A_\perp(x_\perp)\}##

where ##g## is the coupling constant and ##A_\perp(x_\perp)## is the gluon field in the perpendicular direction.

Under time reversal, the Wilson line changes as follows:

##\mathcal{T} \mathcal{P} \exp\{ig\int_{\vec a_\perp}^{\vec b_\perp} dx_\perp \cdot A_\perp(x_\perp)\
 

1. How does the Wilson Line change under time reversal?

Under time reversal, the Wilson Line changes by reversing the direction of all particles and antiparticles involved in the process. This means that all momenta and spins of the particles are also reversed.

2. Does the Wilson Line have a specific symmetry under time reversal?

No, the Wilson Line does not have a specific symmetry under time reversal. This is because the Wilson Line is defined as the trace of a path-ordered exponential of a gauge field, which is not invariant under time reversal.

3. How does the change in the Wilson Line under time reversal affect physical observables?

The change in the Wilson Line under time reversal does not affect physical observables, as the time reversal transformation also affects the other operators involved in the calculation of physical observables, canceling out the effect of the Wilson Line.

4. Is the Wilson Line conserved under time reversal?

No, the Wilson Line is not conserved under time reversal. This is because time reversal reverses the direction of the path integration in the definition of the Wilson Line, resulting in a change in the overall value of the Wilson Line.

5. Can the Wilson Line be used to detect violations of time reversal symmetry?

Yes, the Wilson Line can be used to detect violations of time reversal symmetry in certain cases. For example, if the Wilson Line is not invariant under time reversal, it may indicate a violation of time reversal symmetry in the underlying theory. However, this is not always the case and further analysis is needed to confirm any violation.

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