This question focuses on just free fields. The point is, studying Merzbacher's Quantum Mechanics book, in the chapter about identical particles, the author shows how Quantum Fields appear naturally when developing that theory inside Quantum Mechanics.

In that setting the symmetry constraint on the wavefunctions appear and this leads to the Fock space as the natural Hilbert space for a system of identical particles. Furthermore, the construction allows one to deal with systems with varying number of particles.

On the other hand, I've read that when restricting attention to free fields, in that context, the quantum fields act on a Fock space.

In that case, Quantum Field Theory for free fields is just Quantum Mechanics of free particles? Is that true?

So that we can, for instance, considering free electromagnetic field, say that the electromagnetic field is indeed just a system of photons like is usually said in basic modern physics courses? And the electron field is just a system of electrons?

  • $\begingroup$ No, we can't. "And even the free theory is provably unable to describe particles localized at or around points. The most obvious alternative – a “field interpretation” – has been widely advocated and has so far met with little in the way of criticism... But since wavefunctional space is unitarily equivalent to many-particle Fock space, two of the most powerful arguments against particle interpretations also undermine this form of field interpretation". See Baker $\endgroup$
    – Conifold
    Commented Nov 2, 2016 at 21:02

1 Answer 1


What we can say is that Quantum Field theory for free fields includes the relativistic formulation of free Quantum Mechanics of systems of arbitrarily many identical particles. Here, several well-known theoretical problems appear regarding relativistic localization and position operators. For low energies, if compared with the mass of the particles (this is not possible for photons), everything collapses to the more familiar description based on the non-relativistic Schroedinger equation.

In some (actually very reductive) sense a free field is a system of particles. However some classically familiar features of fields can be obtained only for bosons and for very particular states called coherent states.

"And the electron field is just a system of electrons"

Instead this is quite a dangerous statement as the Dirac field is not an observable.


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