What is the simplest, yet not simplistic, way to explain what is a "quantum field" and how does it interact with matter? originally appeared on Quora, the place to gain and share knowledge, empowering people to learn from others and better understand the world. You can follow Quora on Twitter, Facebook, and Google Plus.
Quantum fields are the quantum theoretical generalizations of classical fields. The two archetypal classical fields are Maxwell’s electromagnetic field and Einstein’s metric field of gravitation. One way to think about the process of quantization is that we first reformulate the (still classical) field equations in terms of mathematical operators replacing some numerical quantities (this part is pure algebra/calculus, no new physics is introduced yet); but then, we “solve” the resulting operator-valued equations, including solutions that do not appear in the classical theory, and make the assertion (validated by observation) that these new, “nonsensical” (in an intuitive, not in a mathematical sense) solutions accurately describe Nature, including all the observed quantum behavior that contradict the classical theory.
There are several rationales for using a quantum field theory. First, it is a natural generalization of classical field theories, which are our most successful (non-quantum) theories of Nature. Second, a quantum field theory can account for the (observed, well-studied) creation and annihilation of particles, processes that do not exist in quantum mechanics. Third, quantum field theory is inherently relativistic, and “magically” (not really, just elegant math) resolves issues of causality that plague even relativistic quantum particle theories.
But no, quantum fields do not interact with matter. Quantum fields are matter. In a quantum field theory, what we perceive as particles are excitations of the quantum field itself.
The simplest “practical” quantum field theory is quantum electromagnetism. In it, two fields exist: the electromagnetic field and the “electron field”. These two fields continuously interact with each other, energy and momentum are transferred, and excitations are created or destroyed. So for instance, what we picture intuitively as an electron absorbing a photon is, in quantum electrodynamics, a specific interaction between the electromagnetic field and the electron field, in which the electromagnetic field loses one excitation quantum, and the electron field gains its energy, momentum and angular momentum.
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