Creation/Ann operators acting on <x|p>

[LostConjugate]

What does it mean for a creation or annihilation operator to act on the state <x|p>. For example:

$$a_p e^{ip \cdot x}$$

 

[haael]

<x|p>

This is not a state. It's just a number. In the second formula you have an operator multiplied by a number.

 

[LostConjugate]

This is not a state. It's just a number. In the second formula you have an operator multiplied by a number.

Ok, will try to give it in context.

What does it mean to write a free field scalar as a linear sum of creation and annihilation operators like this?

$$<br /> \int \frac{d^3 p}{{(2\pi)}^3}\frac{1}{\sqrt{2\omega_p}} [ a_p e^{ip \cdot x} + a_p^\dagger e^{-ip \cdot x} ]$$

What is the creation / annihilation operator role in this, it ends up acting on the amplitude...

$$a = \sqrt{\frac{\omega}{2}}q + \frac{i}{\sqrt{2\omega}}p$$

 

[haael]

What does it mean to write a free field scalar as a linear sum of creation and annihilation operators like this?

This is a functional transform, much just like the Fourier transform. You have a bunch of creation/annihilation operators that depend on a parameter p. You transform them into another operator set that depends on x.

Annihilation operators here do not "act" on anything. Rather, they are a subject of transformation. The result of the transformation is another operator that may act on something.