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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)]##

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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