TITLE: If an electron is an excitation of the electron field, what causes the excitation to be stable?
QUESTION [19 upvotes]: I won't pretend I understand even the basics of QFT, but from what I've heard about electrons, there are really two main ways of thinking about them. Quantum Mechanics describes an electron by a wave function who's squared magnitude gives the probability of finding the electron in a certain position or with a certain momentum. QFT, from what I understand, describes the electron as an excitation of the electron field. Both of these models describe the electron as some excitation of a mathematical field permeating space and time. My question is this:
If an electron really is described in this way (either using a wave function, or a field), what properties of the field, or of space-time itself, make it so the electron is stable (i.e. the excitation does not spontaneously decay, stop existing, or change) what causes the electron to continuously remain the same from one moment to the next)?

REPLY [0 votes]: There are really two main ways of thinking about [electrons]. Quantum
  Mechanics describes an electron by a wave function who's squared
  magnitude gives the probability of finding the electron in a certain
  position or with a certain momentum. QFT ... describes the electron as
  an excitation of the electron field. Both of these models describe the
  electron as some excitation of a mathematical field permeating space
  and time.

This understanding is generally correct, but the last line, “Both of these models ... permeating space and time”, is not a proper statement. In single-particle non-relativistic quantum mechanincs, or “first quantization”, electrons are eigen-states of a Hamiltonian, not a spatio-temporal field (in its proper sense). The quantum fields appear, in the precise sense, in many-body quantum mechanics or “second quantization”.

If an electron really is described in this way (either using a wave
  function, or a field), what properties of the field, or of space-time
  itself, make it so the electron is stable (i.e. the excitation does
  not spontaneously decay, stop existing, or change) what causes the
  electron to continuously remain the same from one moment to the next)?

The stability of the single-electron state is only due to the absence of interactions. In the non-interacting case, the electronic state (more precisely, the quantum state corresponding to a single electron) is an eigen-state of the Hamiltonian and therefore, remains stable. This is comparable to question on the stability of electronic orbitals in an atom < Why don't electrons crash into the nuclei they "orbit"? >; they do not loose energy or fall into the nucleus just because the corresponding states (or “orbitals”) are eigen-states of the atomic Hamiltonian. Being an eigen-state implies that the state will not be changed in the course of the evolution of the system.
Introduction of an interaction between electrons (like the Coulomb interaction) or between the electrons and other particles (like photons or phonons) would lead to an instability of such an electronic state and therefore, decay processes.
{"set_name": "stack_exchange", "score": 19, "question_id": 217589}