Why Is a Given State an Eigenstate of Field Operators in Quantum Field Theory?

[chientewu]

Hi,

I am studying Peskin's An Introduction To Quantum Field Theory. On the beginning of page 284, the authors say We can turn the field $\phi_S(x_1)|\phi_1\rangle=\phi_1(x_1)|\phi_1\rangle$. I tried hard to prove this relation but still can't get it right. Could anyone give me some hints? Thanks.

 

[vanhees71]

It's a bit strangely formulated. The generalized kets $|\varphi \rangle$ are defined as generalized eigenstates of the field operators (here in the Schrödinger picture of time evolution), i.e.,
$$\hat{\phi}_S(\vec{x}_1) |\varphi \rangle = \varphi(x) |\varphi \rangle.$$
Note that $\hat{\phi}_S$ is a field operator in the Schrödinger picture while $\varphi$ is a (complex or real-valued, depending on whether you describe charged or strictly neutral scalar bosons) c-number field.

 

[Bill_K]

Note that [tex]\hat{\phi}_S[/itex] is a field operator in the Schrödinger picture while $\varphi$ is a (complex or real-valued, depending on whether you describe charged or strictly neutral scalar bosons) c-number field.[/tex]

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[vanhees71]

This is difficult to read. Please format your LaTeX correctly, or use UTF.

Done (butt no UTF, which is hard to read either ;-)).

 

[chientewu]

It's a bit strangely formulated. The generalized kets $|\varphi \rangle$ are defined as generalized eigenstates of the field operators (here in the Schrödinger picture of time evolution), i.e.,
$$\hat{\phi}_S(\vec{x}_1) |\varphi \rangle = \varphi(x) |\varphi \rangle.$$
Note that $\hat{\phi}_S$ is a field operator in the Schrödinger picture while $\varphi$ is a (complex or real-valued, depending on whether you describe charged or strictly neutral scalar bosons) c-number field.

Thanks! That makes sense but I still don't understand why this given state is an eigenstate of field operators with eigenvalue being the field amplitudes at some specific position.

 

[Chopin]

Thanks! That makes sense but I still don't understand why this given state is an eigenstate of field operators with eigenvalue being the field amplitudes at some specific position.

Because that's how we're defining the state $|\phi_1\rangle$. We're trying to pick out the state in the Hilbert space that, when you hit it with the field operator at any position $x$, will give you an eigenvalue equal to the c-number $\phi(x)$. It's a way of going from states in a Hilbert space to simple c-number functions, so that you can perform the functional integral over them.