# Can field interact with itself?

I was recently reading the Nobel lecture by Feynman, and there he talks about this idea of fields interacting with itself. Now according to classical theory interaction with itself imply r=0 and since force is 1/r^2, then F= infinity.

Are there any ways we can understand this type of interaction? Is this idea completely discarded by physicists in all possible theory ever existed?

• – Avantgarde Nov 10 '18 at 15:12
• "Is this idea completely discarded by physicists in all possible theory ever existed?" Quite the contrary. Self interaction is essential to field theory. – my2cts Nov 10 '18 at 15:57

Yes. Here are six examples, four in quantum field theory and two in General Relativity.

In the Standard Model, the Higgs particle, the gluons, and the W and Z bosons all have self-interactions. They can be “renormalized” to handle the infinities. Self-interactions are an intrinsic feature of nob-abelian gauge theories.

In General Relativity, a point charge at rest outside a black hole experiences a finite electrostatic self-force away from the hole. And General Relativity is nonlinear so gravity by itself is a self-interacting theory.

All particles have self interaction. A particle cannot have a field without a self interaction term in its Lagrangian.

• In quantum field theory, a quadratic kinetic term is not considered a self-interaction. There are fields, such as the electromagnetic field, which have a quadratic kinetic term but whose interactions are only with other fields (i.e., those corresponding to charged particles). Similarly, quarks do not interact directly with quarks, electrons do not interact directly with electrons, etc. – G. Smith Nov 10 '18 at 23:32
• @G. Smith I should clarify. By selfinteraction I mean the interaction of a particle with its own e. g. electromagnetic field. – my2cts Nov 11 '18 at 0:02
• @my2cts The term "self-interaction" means that you'd have a term in the Lagrangian that is of order cubic or higher in the same field, for example for gauge bosons in non-Abelian gauge theory, where you have cubic and quartic (self) couplings of the gauge boson. Also, if in your last comment 'particle' means electron and 'electromagnetic field' means the Maxwell field, then the interaction is described by the primitive QED vertex term $e \bar{\psi} \gamma^\mu \psi A_\mu$ in the Lagrangian. This is not a self-interaction term; it tells you how the photon couples to fermions. – Avantgarde Nov 11 '18 at 5:16
• @Avantgarde This term also implies that an electron interacts with itself via its own field, which is self-interaction. – my2cts Nov 11 '18 at 10:31
• @my2cts The terminology as it is used today does not call the electron-photon-electron interaction term as the "self-interaction" of the electron field. – Avantgarde Nov 12 '18 at 2:33