What are quantum fields?
Relativistic QFT is a fine theory. If you hold your nose and follow textbook recipes without asking too many questions, you can finally calculate a lot of stuff (scattering cross-sections, Lamb shifts, etc.) in a remarkable agreement with experiment.
However, on this blog we are going to ask questions. Why not? For example:
- Q: What are quantum fields?
- A: They are operator functions on the Minkowski space-time.
- Q: What is their physical meaning?
- A: They are obtained by second quantization of wave functions, at least in the non-interacting theory.
- Q: Wave functions have probabilistic interpretation, and in a relativistic theory the probabilities must be invariant with respect to Poincare transformations. Is it true?
- A: Yes.
- Q: Then particle wave functions must transform by a unitary representation of the Poincare group. Right?
- A: Right.
- Q: So, if quantum fields (even non-interacting) are somehow related to particle wave functions, they also must transform unitarily. Right?
- A: That sounds plausible.
- Q: However, this is not true. Take, for example, Dirac field for electrons. Lorentz transformations of this field are represented by 4x4 matrices. However, it is well-known that there are no finite-dimensional unitary representations of the non-compact Lorentz group. So, Lorentz transformations of the Dirac field are non-unitary.
- A: So what?
- Q: This means that there is no direct connection between the electron's quantum field and electron's wave function?
- A: I am not sure.
- Q: In the quantum field ψ(x,t) the argument x is physical position and t is time. Isn't it?
- A: Yes.
- Q: Is it possible to think (as we do in quantum mechanics) that x as an eigenvalue of the position operator?
- A: I guess so.
- Q: Then we should accept the existence of the time operator as well, which is impossible .
- A: Why did you make this conclusion?
- Q: Because the Lorentz transformation of the field ψ(x,t) involves intermixing of the arguments x and t, which is only possible if x and t are eigenvalues of operators X and T that do not commute with the boost generator.
- A: I guess you may be right.
- Q: So, there is no connection between quantum field and quantum-mechanical wavefunctions. And quantum fields are just abstract operator functions on an abstract 4-dimensional space whose coordinates are not necessarily related to experimentally observed positions and times.
- A: Why do you call Minkowski space-time "abstract"?
- Q: We will talk about Minkowski space-time later. However, arguments x and t of quantum fields are just dummy integration variables without any particular physical meaning.
- A: Why it is so?
- Q: Because, as we discussed here, the only physical thing predicted by relativistic QFT is S-matrix. And in the expression for the S-matrix quantum fields enter integrated over x and t. All dependence on x and t is lost.
P.S. Everything said above refers only to relativistic quantum fields used in theories like QED. They should be separated from non-relativistic quantum fields used, for example, in condensed matter physics. The quantum field of phonons is a well-defined operator which is related to displacements of real atoms from lattice positions.