I Interpreting ##A^{\mu}(x)|0\rangle## and ##\psi (x) |0\rangle##

Ryder Rude
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I can understand how ##\phi (x)|0\rangle## represents the wavefunction of a single boson localised near ##x##.I don't understand how the same logic appies to ##A^{\mu}(x)|0\rangle## and ##\psi |0\rangle##. Both of these operators return a four component wavefunction when operated on the vaccuum, because of the vector/spinor indices in the expansion of these operators. However, the wavefunction of a photon or an electron is described by a one-component wavefunction as I show below:

A wavefunction of a single electron is written as ##\sum_s \int C_s(p) |p, s\rangle dp##. The ##s## label represents the two spin states. A wavefunction of a single photon is written as ##\sum_r \int C_r(p) |p,r\rangle dp##, ##r## labels polarisations.

The wavefunction returned by ##\psi (x)|0\rangle## is of the form ##\sum_{\alpha} \sum_{s} \int C_{s,\alpha} (p) |p,s\rangle dp##. This has an extra index ##\alpha## which runs from 0 to 3.

##C## is just a complex number in all of the above
 
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You're right, the state ##\psi(x)|0\rangle## represents a specific superposition of localized fermion states which is exactly why the LSZ theorem for fermions includes factors of ##\overline{u}_{s}(k)## and ##\overline{v}_{s}(k)## to project out your desired states. It's probably better to think of objects like ##\langle 0|\overline{\psi}(x)\psi(y)| 0 \rangle## as the correlation function of the field rather than some transition function between one-particle states.
 
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Ryder Rude said:
However, the wavefunction of a photon or an electron is described by a one-component wavefunction
No it isn't. You should distinguish the state in the Hilbert space from the wave function. The former is a vector so is a single object, the latter is one of components of the vector. Even for zero spin, the vector in the Hilbert space ##|\psi\rangle## has infinitely many components ##\psi_x=\psi(x)=\langle x|\psi\rangle##.
 
HomogenousCow said:
You're right, the state ##\psi(x)|0\rangle## represents a specific superposition of localized fermion states which is exactly why the LSZ theorem for fermions includes factors of ##\overline{u}_{s}(k)## and ##\overline{v}_{s}(k)## to project out your desired states. It's probably better to think of objects like ##\langle 0|\overline{\psi}(x)\psi(y)| 0 \rangle## as the correlation function of the field rather than some transition function between one-particle states.

So I should think of ##\psi (x) |0\rangle## as just an absract object with spinor indices rather than a particle at a position ##x##, right? And the same logic applies to ##A^{\mu} (x)|0\rangle##?

Also, can you please explain your interpretation of this as a correlation function? Why is it called a correlation function? It's measuring correlation between what?

Should I think of ##\langle 0| \phi(y) \phi(x) |0\rangle## also as a correlation function, rather than an inner product between localised particle wavefunctions?
 
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Demystifier said:
No it isn't. You should distinguish the state in the Hilbert space from the wave function. The former is a vector so is a single object, the latter is one of components of the vector. Even for zero spin, the vector in the Hilbert space ##|\psi\rangle## has infinitely many components ##\psi_x=\psi(x)=\langle x|\psi\rangle##.

Okay. But how should I interpret ##\psi (x)|0\rangle## and ##A^{\mu} (x) |0\rangle##? Both of these objects have have extra index ##\mu## or ##\alpha##, which really shouldn't be there in the state-vector describing a single photon or an electron/positron.
 
Ryder Rude said:
Okay. But how should I interpret ##\psi (x)|0\rangle## and ##A^{\mu} (x) |0\rangle##? Both of these objects have have extra index ##\mu## or ##\alpha##, which really shouldn't be there in the state-vector describing a single photon or an electron/positron.
Strictly speaking, in a definition of state you should have both an integration over ##x## and a sum over spin indices. Something like
$$|\psi\rangle=\int d^3x\, c_{\mu}({\bf x}) A^{\mu}({\bf x}) |0\rangle$$
But you can still get a dependence on ##{\bf x}## and ##{\mu}## if ##c_{\mu}({\bf x})## is a Krorencker ##\delta## in the discrete label and Dirac ##\delta## in the continuous one.
 
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