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

[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?

[arkajad]

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.