Sum of two 3-momenta that are equal and opposite is zero. Can the sum of two nonzero 4-momenta be zero?
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he sum of two 3-momenta p⃗ 1p→1 and p⃗ 2p→2 can be zero if their magnitudes are equal and they are directed opposite to each other.
What about the sum of two 4-momenta pμ1p1μ and pμ2p2μ?
Let's assume that the sum can be zero which implies that their individual components must add up to zero. In particular, pμ1+pμ2=0p1μ+p2μ=0 implies for the zeroth components that E1+E2=0E1+E2=0. Can this condition ever be satisfied other than the trivial case when E1=E2=0E1=E2=0? Is this possible for a particle-antiparticle pair because antiparticles have negative energy.
What about the sum of two 4-momenta pμ1p1μ and pμ2p2μ?
Let's assume that the sum can be zero which implies that their individual components must add up to zero. In particular, pμ1+pμ2=0p1μ+p2μ=0 implies for the zeroth components that E1+E2=0E1+E2=0. Can this condition ever be satisfied other than the trivial case when E1=E2=0E1=E2=0? Is this possible for a particle-antiparticle pair because antiparticles have negative energy.
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