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

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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

µ

is defined by the four vector ε

µ

satisfying ε

µ

k

µ

= 0. In the Coulomb

gauge we instead use the transverse polarization ε

⊥µ

= (0, ~ ε

⊥

). Show that the conditions ε

⊥0

= 0 and ε

⊥i

k

i

= 0 are

not Lorentz invariant (namely, the Lorentz transform of ε

⊥µ

has in general ε

0⊥0

6= 0 and ε

0⊥i

k

0i

6= 0), but it is still

possible to find a vector ˜ ε

⊥µ

= ε

0⊥µ

+ αk

0µ

, i.e. equal to ε

0⊥µ

up to a longitudinal component, which is satisfying the

two conditions.

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