Unitary spacetime translation operator

[omephy]

Srednicki eqn. (2.23) and (2.24) states: We can make this a little fancier by defining the unitary spacetime translation operator

$$T(a) \equiv \exp(-iP^\mu a_\mu/ \hbar)$$

Then we have
$$T(a)^{-1} \phi(x) T(a) = \phi(x-a)$$

How do we get the second equation from the first equation?

 

[The_Duck]

If you believe eq. (2.22) just above this, then you can plug $\phi(x) = e^{-iPx/\hbar}\phi(0)e^{+iPx/\hbar}$ into eq. (2.24). Then using the definition of $T(a)$ you can verify that eq. (2.24) holds.

 

[omephy]

Lovely answer.