Can the mass term be responsible for creation and destruction of particles?

Question

In an interacting quantum field theory, for example, QED, the Dirac mass $m\bar{\psi}\psi$ is a piece of the free Dirac Lagrangian. On the other hand, the interaction term $j^\mu A_\mu=e\bar{\psi}\gamma^\mu\psi A_\mu$ is supposed to be solely responsible for creating or destroying a particle or an antiparticle.

However, the answer here by Lubos Motl states

"...the Dirac mass term destroys a particle and creates a new one, or destroys/creates a particle-antiparticle pair, or destroys an anti-particle and creates a new one."

In free theory, the particle number must be conserved. So my question is whether the interpretation of Lubos is valid.

Answer 1

You would be right that a mass term cannot create or destroy particles in free theories in the flat spacetime. The Hilbert space is a Fock space, a natural basis is composed of states that simply combine some particles with given momenta and these momenta are even individually conserved, and so is the number of particles.

But more generally, the particle number $N$ is only conserved if it commutes with the Hamiltonian $$[N,H]=NH - HN = 0$$ This condition isn't equivalent to the theory's being "free". Free field theory is usually defined as a theory with a quadratic or bilinear Hamiltonian. But if $H$ is bilinear in $a,a^\dagger$, it doesn't mean that it conserves $N$ because it may also include terms $aa$ and their Hermitian conjugates of the form $a^\dagger a^\dagger$.

When it does, the Hamiltonian may change the number of particles by $\pm 2$. An example is a Hamiltonian transformed to an accelerating coordinate system, like in the Unruh effect. In that case, the original Hamiltonian in the flat space which only had $a^\dagger a$ terms gets modified so that $aa$ and their conjugates are also present. In this coordinate system, one sees "particle production" and "particle destruction". Also, the ground state (vacuum) in the original flat space is reinterpreted as a state with excitations (not a well-defined number, not an eigenstate of $N$) or vice versa.

The particle production obtained from an accelerating reference frame is known as the Unruh effect, and it may be considered a limiting and simplified description of what happens near the black hole event horizon in the Hawking radiation.

Cubic terms like $A\bar\psi\psi$ are the terms that we normally associate with the production of particle pairs – from a virtual photon, in this case. But what is needed for the particle production is

1. the presence of the bilinear expression of the form $aa$

2. non-conservation of the particle number $N$

The non-conservation of the particle number $N$ may be guaranteed by inserting another quantum field, like the operator for the electromagnetic potential. But it may also be guaranteed by replacing $A_\mu$ in your cubic term by a classical, space-time-dependent field $A_\mu$, or any other spacetime-dependent field (including a scalar or tensor/gravitational one). In all these cases, the pair production/destruction of pairs becomes possible.

The only point I am making is that one doesn't really need to describe $A_\mu$ as a full-blown quantum field, a classical limit for this $A_\mu$ is sufficient, and it doesn't have to be a spin-one field – any integer-spin field or their product is enough.