# How to understand the time reversal symmetry of position operator?

How to understand the fact that position operator is symmetric under time reversal? I can visualize the momentum and magnetic field being odd under time reversal. Got the same doubt for Electric field as well.

You can easily find out which quantities are even and which are odd upon time-reversal by considering a video showing physical processes and then comparing it to the time-reversed video showing the same processes backwards.

See also "T-symmetry - Effect of time reversal on some variable of classical physics" for lists of even and odd physical quantities.

### Even quantities

Obviously the spatial position $$\vec{x}$$ doesn't change sign in the time-reversed video. (A body being on the right side in the original video is still on the right side in the time-reversed video, not on the left side.)

Also the acceleration $$\vec{a}$$ doesn't change sign. (In the original video a ball thrown up and then falling down is accelerated downwards. Viewed in the reversed video, the ball is again accelerated downwards, not upwards.)

From Newton's $$\vec{F}=m\vec{a}$$ you can conclude, force $$\vec{F}$$ also doesn't change sign.

And then from the definition of electric field ($$\vec{F}=q\vec{E}$$), you can further conclude, the electric field $$\vec{E}$$ doesn't change sign.

### Odd quantities

Obviously in the reversed video time $$t$$ and velocity $$\vec{v}$$ have the opposite sign as compared to the original video.

The magnetic field $$\vec{B}$$ is defined by the Lorentz force ($$\vec{F}=q\vec{v}\times\vec{B}$$). Because force $$\vec{F}$$ is even and velocity $$\vec{v}$$ is odd, you can conclude the magnetic field $$\vec{B}$$ must be odd.

• Great help. In the part where you talk about acceleration, If we shot a video of the ball falling down only, then the reversed video would be showing the ball go up. This looks like acceleration changed sign for me. Sorry, I am very confused about these things. Thanks anyway. Sep 27 at 9:40
• @NIKHILJOSEPHJOY When you have a ball falling down, its velocity increases, hence accelerated downwards. When you view this in time-reversed video, you have a ball thrown up, with decreasing velocity, hence accelerated downwards again. So acceleration does not change sign. Sep 27 at 10:57

Position is ondependent of time. The magnetic field is even under time reversal, the electric field odd. I wouldn't know how to visualize this.