- #1
kof9595995
- 679
- 2
This question is going to be a bit vague and might lead to nowhere, but still I'll take the risk and try to ask it here.
I know in general how to quantize a field, and from the quantized field one gets the quantized Hamitonian thus the time-evolution operator. However, I wonder what're the precise relations between the classical and quantum time evolution. I can think of one, i.e.
[tex]U(t)\langle\alpha|\hat{\psi}(x)|\alpha\rangle= \langle\alpha|e^{i\hat{H}t}\hat{\psi}(x)e^{-i\hat{H}t}|\alpha\rangle[/tex]
Where [itex]U(t)[/itex] is the time evolution on the classical field, [itex]\hat{\psi}(x)[/itex] is the field operator, and [itex]|\alpha\rangle[/itex] is some quantum state.
However if one looks into more details, there is some thing odd(I think) happening: Take a free field for example, the eigenvectors of quantum time-evolution are Fock states, but the field expectation value of Fock states is 0, which is not an eigenvector of the classical time-evolution, since the eigenvetors of classical evolution are plane-waves. So what is happening here mathematically?
Also I'd like to know more about the relations classical and quantum time-evolution, any reference is also appreciated.
I know in general how to quantize a field, and from the quantized field one gets the quantized Hamitonian thus the time-evolution operator. However, I wonder what're the precise relations between the classical and quantum time evolution. I can think of one, i.e.
[tex]U(t)\langle\alpha|\hat{\psi}(x)|\alpha\rangle= \langle\alpha|e^{i\hat{H}t}\hat{\psi}(x)e^{-i\hat{H}t}|\alpha\rangle[/tex]
Where [itex]U(t)[/itex] is the time evolution on the classical field, [itex]\hat{\psi}(x)[/itex] is the field operator, and [itex]|\alpha\rangle[/itex] is some quantum state.
However if one looks into more details, there is some thing odd(I think) happening: Take a free field for example, the eigenvectors of quantum time-evolution are Fock states, but the field expectation value of Fock states is 0, which is not an eigenvector of the classical time-evolution, since the eigenvetors of classical evolution are plane-waves. So what is happening here mathematically?
Also I'd like to know more about the relations classical and quantum time-evolution, any reference is also appreciated.