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• Show that the Bose symmetry of the photon imposes a constraint on the wave function.


Academic year: 2022

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Quantum Field Theory

Set 17

Exercise 1: Multiple photons states

Consider the wave function for a state consisting of two photons:

• Show that the Bose symmetry of the photon imposes a constraint on the wave function.

• Find the condition on the wave function in order to have a physical state.

• Find the condition on the wave function in order to have a state with positive norm.

Exercise 2: Transformation properties of transverse photons

The polarization of a photon of momentum k


is defined by the four vector ε


satisfying ε




= 0. In the Coulomb

gauge we instead use the transverse polarization ε


= (0, ~ ε

). Show that the conditions ε


= 0 and ε




= 0 are

not Lorentz invariant (namely, the Lorentz transform of ε


has in general ε


6= 0 and ε




6= 0), but it is still

possible to find a vector ˜ ε


= ε


+ αk


, i.e. equal to ε


up to a longitudinal component, which is satisfying the

two conditions.


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