There are several things that can still be confusing about quantum field theory (QFT) even if one already knows quantum mechanics.  For example, how does QFT relate to the single-particle Schrödinger equation? For one, a single particle wave function \phi(\bold{x}) is already a continuous field, but quantum field theory refers to more than single-particle quantum mechanics. Moreover, we know that two particles are described in quantum mechanics as a six-dimensional wavefunction, yet quantum field theory discusses quantum fields that permeate all of space, in only three dimensions.  The purpose of this post is to briefly summarize the steps from single-body to many-body quantum mechanics, which clarifies these questions.  We follow the canonical quantization method.  The intended audience is someone familiar with quantum field theory calculations, but wanting a reminder of its meaning.

Our starting point is a familiarity with quantum mechanics.  One knows from quantum mechanics that a N-particle wavefunction is 3N-dimensional.  For instance, for N=3, the wavefunction is written in position basis as \phi(\bold{x_1},\bold{x_2},\bold{x_3}) where \bold{x_1} is the 3-dimensional position of particle 1.  Recall that, for two identical particles, only the phase of a wavefunction changes if two of the particles are permuted, and in fact we can accept as experimental fact that the wavefunction is either completely symmetric (bosons) or anti-symmetric (fermions) under permutation. Note that the wavefunction notation above makes reference to which particle is which, and so for identical particles there is extra, non-physical information in the above notation.  In fact, the symmetry means that it has 1/N! fewer distinct points in its domain than expected (i.e. than R^{3N}).  This suggests that there is actually a more elegant way to describe the state.  The more efficient way is in the basis of occupation number, which does not distinguish which particle is occupying each 3D-location, only how much each position is occupied.  This better basis for identical particles is called Fock space.  The Fock space vector \Psi(\bold{x}) means that there is \Psi(\bold{x}) amplitude for particles being at 3D position \bold{x}.  Rather than position basis, these vectors are often written in wavevector basis, but for conceptualizing them at this point, we can remain in position space.

Now recall the mathematical usefulness of the creation and annihilation operators of the 1D simple harmonic oscillator in quantum mechanics, a^\dagger_i and a_i. These operators create or annihilate a particle in the state i.  That is, a^\dagger_2  \lvert 0 0 0 ... \rangle= \lvert 0 1 0 ...\rangle adds one particle to the location \bold{x}=\bold{x}_2.  We can define the so-called field operators that create a particle in eigenstate \lvert \bold{x} \rangle as \Psi(\bold{x}) = \sum \phi^*_i(\bold{x}) a_i.

The Heisenberg representation of quantum mechanics moves the time-dependence of the wavefunction to the operators. Using that here we finally have arrived at quantum field theory. The fields described by quantum field theory are these fields of operators, \Psi(\bold{x}). They’re defined on 3-dimensional space, since they operate on M-dimensional Fock-space where M is the number of orthogonal multi-particle states that span the available space. These operator fields are free to fluctuate in magnitude. The operators operate on the current state of the fields. The background state is time-independent, and so all dynamics is in the field operators. (While the operators are the interesting, evolving quantities, they must ultimately be applied to the state to refer to the state of the system.) Commutation relations enforce the underlying quantization of the single-particle wavefunctions.

So this is the object that is studied by quantum field theory. The next question is, what are the dynamics of these fields?

In classical field theories, often the field equations of motion are not derived from microscopic equations, but instead they come from reasoning about symmetries and conserved quantities. For example, the Navier-Stokes equations describing fluid flow are often derived by postulating a continuous field and conserving mass, momentum, and energy, independent of any atomic picture.  However, there is also a more complicated, microscopic approach to deriving Navier-Stokes that starts from Boltzmann’s equation for particle motions and an assumed scattering kernel. The situation is the same in QFT. The equation of motion of the field operators can be derived (amazingly) from the single-particle Schrödinger equation, as long as there are no interactions between particles. In more complex cases, the approach is often to use the symmetries of the problem to determine the properties of the equation of motion. These procedures lead to the Klein-Gordon and Dirac equations for relativistic bosons and fermions.

This post comes out of discussions from the book Advanced Quantum Mechanics by Schwabl, Quantum Field Theory by Schwatrz, Wikipedia, and discussion with Zlatko T and Matt W.