How does the Wilson Line change under time reversal?

[Chenkb]

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??
$-ig\int_{-\vec a_\perp}^{-\vec b_\perp} dx_\perp \cdot [-A_\perp(x_\perp)]$

 

[AndyW]

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)\