 Solution

ID: past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2016/2/c/a/solution

ε = ℏ ω

. Their total four-momentum is

P = \frac{ε}{c} (2, 1 + cos θ, sin θ, 0) .

Its squared invariant is

P^{2} = 2 ε^{2} (1 - cos θ) / c^{2}

. Therefore the system's rest energy is

M c^{2} = E_{tot}^{2} - c^{2} ∣ p_{tot} ∣^{2} = 2ℏ ω sin \frac{θ}{2} .

Two individually massless particles can thus have positive total invariant mass. Back-to-back photons give rest energy

2ℏ ω

, whereas exactly parallel photons have zero invariant mass and no massive-system rest frame.

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