How do different fields interact with each other?

Recently I've seen a few talks and lectures about Quantum Field Theory. They explained what a "particle" means in a field, and that a large enough excitement in a certain field can excite another field (effectively creating a new particle), but they didn't mention how that interaction happens.

Searching around I've heard that the interaction is being transmitted by "virtual particles", but won't these particles require their own field (because a particle is just an specific ripple in a field)? And if those virtual particles do have their own field, then how does that field interact with the 2 other fields that it's mediating?

Thank you!

2 Answers

Using the description of field theory, and it's true both in classical field theory and quantum field theory, fields interact because of specific terms in the Lagrangian or the Hamiltonian that depend on several fields. Or, equivalently, they interact because terms depending on one field appear in the equations of motion for another field.

For example, the Dirac field interacts with the electromagnetic field due to the triple product, cubic term $$e A_\mu \bar\Psi \gamma^\mu \Psi$$ in the Lagrangian or in the Hamiltonian. This has the effect of adding the Dirac current $\bar\Psi \gamma^\mu\Psi$ on the right hand side of the Maxwell's equation for the electromagnetic field, $\partial_\nu F^{\mu\nu}=\dots$. Similarly, the Dirac equation for the Dirac field gets modified because the partial derivatives get "decorated" by the gauge potential $A_\mu$ to become covariant derivatives.

Consequently, simple wave (or similar simple) solutions for the non-interacting fields are no longer solutions if these interactions are present or enabled. The nontrivial values of one field imply a different evolution of another field, and so on. For example, an electromagnetic wave gets redirected when it hits a charged particle (or a packet of a charged field) etc.

If one insists on field theory, there is no "deeper" explanation why the terms are present in the Hamiltonian, Lagrangian, or equations of motion – those pretty much define the theory at the deepest level. In string theory or another (hypothetical) underlying theory, the terms in the Lagrangian may be derived from a deeper principle.

Virtual particles or fields enable interactions between other fields – e.g. the virtual photons (virtual quanta of the electromagnetic field) cause the electrostatic repulsion between two Dirac or other charged fields. But that's just a special type of "composite" interaction. At the microscopic level, the elementary interaction is the interaction between the electromagnetic and charged fields, and this interaction is "direct" and doesn't depend on any additional virtual particles.

To illustrate the concept of interactions, consider a free Dirac fermion, governed by the Lagrangian,

$$\mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi$$

as well as a $U(1)$ gauge field $A_\mu$ governed by the standard Maxwell Lagrangian,

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where $F_{\mu\nu}$ is the field strength tensor. We can couple these fields to each other, in a renormalizable interaction term, which will allow the particles which these fields give rise to, to interact:

$$\mathcal{L}_{\mathrm{int}}=e\bar{\psi}\gamma^\mu A_\mu \psi$$

The term gives rise to a Feynman diagram vertex of a positron ($\bar{\psi}$), electron ($\psi$) and photon $(\gamma^\mu A_ \mu)$.

The electromagnetic force is 'mediated' by virtual photons, as one can construct by combining two interaction vertices, namely:

For the internal photon line $\gamma$, we include a factor of the photon propagator, and an integration over momenta (depending on how we formulate your Feynman rules). The photon may be off-shell; that signifies simply that it does not need to obey the relativistic dispersion relation,

$$E^2=p^2+m^2$$

in units $c=\hbar=1$. We do not need to add a new field to describe the virtual photon.