# Mass gap for photons

I am puzzled by the answers to the question:

What is a mass gap?

There, Ron Maimon's answer gives a clear-cut definition, which I suppose applies to any quantum field theory with Hamiltonian $H$, that the theory has a mass gap if there is a positive constant $A$ such that $$\langle \psi| H |\psi \rangle\geq \langle 0 |H | 0 \rangle +A$$ for all nonzero (normalized) $\psi$.

But then, Arnold Neumaier says

QED has no mass gap, as observable photons are massless states.

I would quite appreciate a brief explanation of this statement. The definition is concerned with the minimum possible energy for non-zero states. So I don't see why the photons having zero mass would imply the absence of a mass gap.

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to : "for all nonzero (normalized) ψ" : must be "for all normalized ψ that are orthogonal to the ground state" – jjcale May 18 '13 at 18:17

The point is that if you have particles of zero mass, there are states with an arbitrary positive mass. The reason is that n-particle states made up of particles with momentum $p_1,...,p_n$ the total momentum is $p=p_1+...+p_n$, which is a state of positive mass $m=\sqrt{p^2}$. If all of the $p_k$ come from a photon, it is a simple mathematical exercise to see that $m$ can take any positive value. Thus the mass spectrum has no gap.