Is "quantizing" a field different from "quantizing" a particle?

Question

As I understand it, quantum mechanics for particles was developed to replace classical mechanics for particles. In essence, we realized that particle cannot be given an exact place and momentum but must be described using probabilities and more specifically wave functions.

Is quantizing a field the same idea? That is, suppose I am considering an electric field $\mathbf E(\mathbf x,t)$. Does quantizing a field mean that we no longer believe the field can have a definite value at points in spacetime? Does this mean we are looking to find a wave function analogue for fields?

Answer 1

Both quantizing a "particle" and quantizing a "field" follow the same, completely general ideas:

From the Hamiltonian point of view, we start with a classical phase space (finite-dimensional for a "particle", infinite-dimensional for a "field"), and quantization means essentially implementing the classical Poisson algebra on the phase space as the operator algebra of a Hilbert space such that the commutator of the quantum operators coincides with the classical Poisson bracket. That is, to every classical phase space function $f$ we associate an operator $\hat{f}$, and we would like that $[\hat{f},\hat{g}] = \mathrm{i}\hbar\{f,g\}$ and $p(\hat{f}) = \widehat{p(f)}$ for any polynomial $p$.¹

For the finite-dimensional case, the Stone-von Neumann theorem guarantees that the canonical Poisson bracket $\{x,p\} = 1$ leads to the unique representation of $\hat{x}$ as multiplication and $\hat{p}$ as differentiation in the position basis, but this fails for the infinite-dimensional case, which makes the canonical quantization of fields more difficult, and their analogue of wavefunctions - so-called wavefunctionals - less useful than in the particle case.

Nevertheless, quantizing particles and quantizing field is not so different - every phase space coordinate $x,p$ is promoted to a quantum operator $\hat{x},\hat{p}$, just like every field $\phi(x)$ and its canonically conjugate momentum density $\pi(x)$ are promoted to operator-valued functions $\hat{\phi}(x),\hat{\pi}(x)$ (or, more properly, operator-valued distributions) whose commutation relation is their classical Poisson bracket.

It's "just" that the infinite-dimensional phase space gives much more trouble than the finite-dimensional case, so quantizing fields is "harder".

Another way to see that quantizing fields and particles is not really different is to observe that if one thinks of the position of a particle as $x(t)$, this is a field on a 0+1 dimensional spacetime, and then usual QM becomes just the special case of a 0+1 dimensional QFT, which is best seen in the path integral approach.

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^{1In general, this is not possible due to the Groenewold-van Hove theorem. This leads to using the Moyal bracket instead of the Poisson bracket in the deforomation quantization approach.}

Answer 2

Quantization was developed to explain the behavior of wave phenomena that seemingly could only exchange energy in form of discrete quanta. This began with Planck's 1900 insight that the thermal spectrum of black bodies could only be explained if the energy spectrum was discretized. Einstein explained the photoelectric effect in terms of particles in 1905. Both effects start with electromagnetic radiation, which classically is a wave phenomenon.

Bohr added a semi-classical model of the atom in 1913 and in 1923 De Broglie came up with the concept of matter waves. In 1926 Schroedinger added his wave equation and already in 1927 people began working on the explicit quantization of classical fields, which bore fruit for the new field of high energy physics in the 1930s and 1940s.

So while modern teaching usually starts out with the discussion of a single particle wave function to teach students the general structure of quantum mechanics, one can't say that early 20th century physicists came up with it because they had a strong need to quantize single particles. The need was to quantize fields from the get-go, but that turned out to be too hard, so the program lingered around a very limited understanding of simple, non-relativistic one-particle systems for a couple decades before the generalization to relativistic multi-particle systems was finally successful.

After the 1940s the field splits. A small but non-trivial number of physicists stuck with structural questions that are most easily explored in quantum-optical experiments using light and excited atomic systems. There has been near zero progress on the physics side in that program over the past 80 years, or so. Everything that can be achieved with the phenomenology of these systems seems to pertain to the theoretically important question of what happens microscopically during the measurement process. The most important achievements there are the development of density matrix theory and the explanation of strong quantum measurements trough decoherence. This makes quantum theory an entirely self-consistent theory and it may lead to significant advanced in computing thanks to the possibility of quantum algorithms.

The far greater progress in terms of actually "new" physics has been achieved in relativistic quantum field theory, both in terms of experiments and the development of quantum field theoretical methods. Given the enormous complexity of this field it is probably still in its early stages and there are many more insights to come, especially pertaining to quantum gravity and cosmology.

Having said all of this, it is painfully obvious that I just wrote my own history of quantum mechanics down, didn't I? In hindsight this doesn't belong into the physics section... if we apply the rules correctly. Apologies to the community. I will delete this if being asked to do so or the moderators can decide this on their own and I will take no offense, whatsoever. I fully admit that we should not answer questions this way.

Answer 3

When we look for the way to construct relativistic theory, the most convenient way to introduce quantum approach is symmetry approach. In terns of this method we start from the requirement of the underlying Poincare symmetry. In terms of it, the particle is mathematically defined as irreducible representation of the Poincare group. I.e., one-particle state is a state with definite mass and square of spin (or helicity for massless case). The basis of states of noninteracting particles is represented by Fock basis, with creation-destruction operators. They are obeyed definite law of transformation under Poincare group. And when we want to construct scalars from the Fock space operators, we must to contract these operators with some coefficient functions in momentum space. Corresponding objects are called relativistic fields. Coefficient functions must satisfy definite transformation law. This fact result in a statement that we may extract from relativistic field the "parts" which define given one-particle state; we do this by imposing the restriction on relativistic field that the Casimir operators (in differential form) acting on it give the mass and the square of spin of the particle (or helicity). These restrictions are called relativistic wave equations. They may be derived from first principles. Examples are Maxwell equations, Dirac equation, Pauli-Fierz equations etc.

In a such sense "quantization of the field" (imposing relativistic wave equations) is nothing but "quantization of particle" (requirement that elementary particle is irreducible representation of the Poincare group).