stevendaryl said:
Apparently, there are two different routes to get to quantum field theory from single-particle quantum mechanics: (I'm going to use nonrelativistic quantum mechanics for this discussion. I think the same issues apply in relativistic quantum mechanics.)
Route 1: Many-particle quantum mechanics
Start with single-particle QM, with the Schrodinger equation: - \frac{1}{2m} \nabla^2 \psi = i \frac{\partial}{\partial t} \psi
- Now, extend it to many (initially, noninteracting) particles: \Psi(\vec{r_1}, \vec{r_2}, ..., \vec{r_n}, t)
- Introduce creation and annihilation operators to get you from (a properly symmetrized) n-particle state to an n+1-particle state, and vice-verse.
Route 2: Second quantization
Once again, start with the single-particle wave function.
- Instead of viewing \psi as a wave function, you view it as a classical field.
- Describe that field using a Lagrangian density \mathcal{L} = i \psi^* \dot{psi} - \frac{1}{2m}|\nabla \psi|^2
- Derive the canonical momentum using \pi = \dfrac{\partial}{\partial \dot{\psi}}
- Impose the commutation rule: [\pi(\vec{r}), \psi(\vec{r'})] = -i \delta^3(\vec{r'} - \vec{r})
Conceptually, these routes are very different. The first one is just many-particle quantum mechanics re-expressed in terms of creation and annihilation operators. The second is field theory in which the field is quantized. Is it just a coincidence that the result is the same, or is there some deeper reason?
Maybe it's inevitable because there is only one QFT possible for free nonrelativistic fields satisfying Galilean symmetry. But it still seems strange, because the first route doesn't start with a general principle about commutation relations; those rules are just consequences of the way creation and annihilation operators work on Fock space, together with the symmetry/anti-symmetry rules for identical particles. It doesn't presuppose any general commutation rule relating a field to its canonical momentum.
First we have to get a bit more precise between your two routes to non-relativistic QFT (NRQFT). Route 1 is extending many-body quantum theory, expressed in terms of wave mechanics, i.e., the position representation for a single particle to the many-body case by introducing wave functions for a fixed number of particles. In position representation you have wave functions with ##N## position arguments for an ##N##-particle system.
Another important ingredient is the notion of indistinguishability of of particles of the same type, i.e., particles which have the same intrinsic quantum numbers (which usually are mass and electric charge in atomic physics) cannot be individually followed in their dynamics, which implies that it should not be possible to distinguish indidual particles. This leads, after some careful analysis, to the conclusion that in 3 spatial dimensions there can only be bosons or fermions, i.e., the wave functions must be symmetric or antisymmetric under exchange of any two spatial arguments of indistinguishable particles in the system.
Route 2 is not different at all. All you do is to represent the states a bit differently. You introduce another Hilbert space, a socalled Fock space, where the particle number is not fixed a priori, and then the natural basis are occupation-number states with respect to a given single-particle basis. Thus this approach is a somewhat more general setup, because you can describe processes, where in principle interactions can change the particle number, i.e., the creation and destruction of particles. In many cases of NR QT the interactions are, however such that the particle number is conserved, and thus given by the initial condition of the dynamics. Then everything can be described in a subspace of the Fock space with a fixed total number of particles, and in this case the occupation-number representation and the position representation for a fixed number of particles are precisely the same theory.
However, NRQFT can be very convenient when describing many-body systems in terms of socalled quasiparticles, which are not conserved. This happens all the time in condensed-matter physics and is a concept that goes back to Landau when working out the theory of liquid helium (in his case first He4, i.e., bosons and superfluidity). There you find that the system can be described most elegantly not in terms of the real particles (electrons and nuclei) but in terms of the relevant collective excitations above the ground state of the many-body system. In thermal equilibrium these quantum collective excitations behave like an ideal gas dilute gas of quasiparticles, which can be destroyed and created in interactions. You end up with Feynman rules as in relativistic QFT. Only the meaning of these symbols are different.
The final question is not specific to the one or the other route (often misleadingly called "first and second quantization") to many-body NRQT. This is applied group theory, and in my opinion, the very basis of NRQT to begin with. Usually one doesn't do this in QM1, because it you need a quite long time to introduce the necessary mathematical concepts, namely the ray representations of groups in Hilbert space, and for NRQT it's even a bit more cumbersome than in relativistic QT, because the Galileo group has a somewhat different structure than the Poincare group. You can derive this in a very similar way as for the Poincare group, and it was done in both cases by Wigner. The conclusion is that in quantum theory the classical Galileo group is substituted by a quantum version, where in addition to the 10 generators of the original group (temporal and spatial translations (4 generators for energy and momentum), rotations (3 operators for angular momentum), and Galilei boosts (3 generators)) the mass occurs as an additional central charge of the group. Instead of the classical rotation group you have to use the covering group SU(2) to get the complete realizations. An elementary non-relativistic particle is then defined such that its quantum mechanics deals with a Hilbert space where a specific irreducible representation of this quantum-Galileo group is realized by the corresponding operator algebra. These representations are classified by the Casimir operators of the quantum Galileo group, i.e., mass and spin of the particle as in the relativistic case. The only difference is that for the Galileo group the mass is a central charge, while for the Poincare group the mass squared is a Casimir operator of the Poincare group, i.e., ##p_{\mu} p^{\mu}##. Thus in non-relativistic QT you have a mass-superselection rule, which you do not have in the relativistic case. Also in NRQT particles with mass 0 do not lead to useful quantum dynamics.
Whether you describe the physics of a non-relativistic many-body system with fixed total particle number (implying that this total particle number is conserved and the corresponding operator thus commutes with the Hamiltonian) in the first or second quantization formalism doesn't matter, you always deal with the same representations of the quantum Galileo group.
For a good introduction to the group-theoretical aspects, see the textbook by Ballentine, Quantum Mechanics.