# How do a and a^\dagger transform under time reversal transformation?

• wdlang
In summary, the operator a remains unchanged under a time reversal transformation, while the operator a^\dagger changes sign and becomes -a^\dagger. The commutation relation between a and a^\dagger remains unchanged. The time reversal transformation is unitary and changes the sign of all physical observables.
wdlang
it is know that x goes to x while p goes to -p under time reversal transform

so a=x+ip whill transform to a=x-i(-p)=x+ip=a?

so a^\dagger to a^\dagger?

Indeed. You have

x = (a^dagger + a)/2
p= i(a^dagger - a)/2

which transform to x and -p if a and a^dagger are fixed by time inversion.

## 1. How does the operator a change under a time reversal transformation?

The operator a remains unchanged under a time reversal transformation.

## 2. What happens to the operator a^\dagger under a time reversal transformation?

The operator a^\dagger changes sign under a time reversal transformation, becoming -a^\dagger.

## 3. How does the commutation relation between a and a^\dagger change under a time reversal transformation?

The commutation relation between a and a^\dagger remains unchanged under a time reversal transformation.

## 4. Is the time reversal transformation unitary?

Yes, the time reversal transformation is unitary, meaning that it preserves inner products and norms.

## 5. What is the effect of a time reversal transformation on physical observables?

A time reversal transformation changes the sign of all physical observables, indicating a reversal in the direction of time.

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