Peskin Page 284

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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 [itex]\phi_S(x_1)|\phi_1\rangle=\phi_1(x_1)|\phi_1\rangle[/itex]. I tried hard to prove this relation but still can't get it right. Could anyone give me some hints? Thanks.
 
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vanhees71
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It's a bit strangely formulated. The generalized kets [itex]|\varphi \rangle[/itex] are defined as generalized eigenstates of the field operators (here in the Schrödinger picture of time evolution), i.e.,
[tex]\hat{\phi}_S(\vec{x}_1) |\varphi \rangle = \varphi(x) |\varphi \rangle.[/tex]
Note that [itex]\hat{\phi}_S[/itex] is a field operator in the Schrödinger picture while [itex]\varphi[/itex] is a (complex or real-valued, depending on whether you describe charged or strictly neutral scalar bosons) c-number field.
 
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Bill_K
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Note that [tex]\hat{\phi}_S[/itex] is a field operator in the Schrödinger picture while [itex]\varphi[/itex] is a (complex or real-valued, depending on whether you describe charged or strictly neutral scalar bosons) c-number field.
This is difficult to read. Please format your LaTeX correctly, or use UTF. :wink:
 
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vanhees71
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This is difficult to read. Please format your LaTeX correctly, or use UTF. :wink:
Done (butt no UTF, which is hard to read either ;-)).
 
  • #5
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It's a bit strangely formulated. The generalized kets [itex]|\varphi \rangle[/itex] are defined as generalized eigenstates of the field operators (here in the Schrödinger picture of time evolution), i.e.,
[tex]\hat{\phi}_S(\vec{x}_1) |\varphi \rangle = \varphi(x) |\varphi \rangle.[/tex]
Note that [itex]\hat{\phi}_S[/itex] is a field operator in the Schrödinger picture while [itex]\varphi[/itex] 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.
 
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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 [itex]|\phi_1\rangle[/itex]. We're trying to pick out the state in the Hilbert space that, when you hit it with the field operator at any position [itex]x[/itex], will give you an eigenvalue equal to the c-number [itex]\phi(x)[/itex]. 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.
 
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