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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

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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