# Quantum(izing) fields & what else?

1. Dec 30, 2015

### jlcd

After the success of QM. They want to extend it to fields like electric field, electromagnetic field.. hence the initial attempt at quantum field theory (QED).. next they think particles like electrons can be treated the same.. hence the so called second quantization. Besides fields.. what else are there that must be quantized (or quantumized)? They tried to quantize geometry but can't seem to be successful. What is the criteria before things can be quantized.. like why do they not quantize economics or others? Why only fields? Nothing else? Is reality this simple?

2. Dec 31, 2015

### Staff: Mentor

3. Dec 31, 2015

### jlcd

If physics history didn't go to the path of second quantization (where electrons already quantum become quanta of electron field). Is there other calculative way to derive at QCD, QED of electron and photon.. let's say physics just stop after quantizing the pure electromagnetic field?

I know second quantization is encouraged because of de Broglie success with seeing particle and wave as different manifestation of the same thing. Hence they got encourage and thought electron particle and electron field is manifestation of the same thing. But can you say that by treating QM of electron relativistically alone without the idea of second quantization.. you end up with the same equations?

4. Dec 31, 2015

### vanhees71

First of all the name "second quantization" is a misnomer. It's a historical name. There is only one quantum theory as a general concept. Then there is the non-relativistic approximation, which is from a heuristical point of view simpler than the full relativistic theory. The reason is that in the non-relativistic approximation many systems can be described as one with a fixed number of particles, e.g., atomic and low-energy nuclear physics, interacting via action-at-a-distance potentials. In this case you can work with a wave function $\psi(t,\vec{x}_1,\ldots,\vec{x}_N)$, where $N$ is the fixed number of particles, and the dynamics is described by the Schrödinger equation. If some or all of the particles are indistinguishable you have to symmetrize or antisymmetrize the wave function with respect to interchange of the arguments, depending on whether the particles are bosons or fermions, respectively. The observables are represented by self-adjoint operators, which are determined from the symmetry group of non-relativistic space-time. For spinless particles everything can be derived from the Heisenberg algebra of position and momentum, and heuristically you can find this algebra of operators, which in this case are all functions of the operators for position and momentum, by assuming that the Poisson brackets of the Hamilton formalism of classical mechanics translate into commutators of the operators (with an appropriate imaginary factor). This is what's historically called "first quantization" or "canonical quantization". To understand the appearance of spin in quantum theory from first principles you have to dig deeper into the representation theory of the Galilei group. For a good treatment of this, see

L. Ballentine, Quantum Mechanics - A modern development

In the relativistic case these concepts do not work anymore, because relativistic interactions of particles by definition means that the collision energies become large compared to the masses of the involved particles. The relativistic symmetry group of space-time, the Poincare group, is different from the Galilei group in the sense that you cannot define relativistic equations of motion for the wave function (analogous to the Schrödinger equation for the non-relativistic case) which leads to a positive definite conserved charge, which in the non-relativistic case is $|\psi|^2$ and can be used to interpret the wave function as probability amplitude. The physical reason for that "failure" (in fact, it's a "feature" :-)) is that there are no conserved particle numbers in the relativistic realm. It's always possible that in a reaction new particles are produced and/or particles get destroyed. The only constraints are the fundamental conservation laws, which are also governed by symmetries. Space-time symmetry (Poincare symmetry) leads to the conservation of energy, momentum, and angular momentum, and intrinsic symmetries (mostly local gauge symmetries) lead to conservation laws of various charges (like the electric charge, baryon and lepton number in the strong interactions, baryon minus lepton number in the weak interactions). Due to this possibility of production and annihilation of particles in collisions at relativistic energies, the "1st quantization" doesn't work anymore and you have to work with a formalism that allows for these particle-number changing processes. The formalism for this was very early after the discovery of modern quantum theory developed by Dirac in dealing with the "creation" of photons in atomic transitions. It turned out that the most convenient way to deal with such processes is quantum field theory. Heuristically you can take some relativistic field theory, formulate it in the Hamilton least-action formalism and treat the fields as infinitely many dynamical degrees of freedom and quantize canonically (modulo trouble with gauge invariance in the case of the electromagnetic field). That's why QFT was called "second quantization", because you can do this for non-relativistic particles too by "quantizing" the Schrödinger wave-function. Since the Schrödinger wave-function was seen as the "first quantization", they called the QFT formalism of non-relativistic many-body theory "second quantization". For the relativistic case, it is pretty complicated not to formulate it as QFT, and that's why it's a misnomer to call it "2nd quantization" in this case, because there is not really another formalism. In the very beginning Quantum Electrodynamics was formulated by Dirac in a pretty complicated way as the socalled "hole theory", which is not very consistent in itself but it's equivalent to modern QED, which is formulated as a relativistic quantum field theory. In the most simple version you take electrons and positrons (described by a quantized Dirac field) and photons (described by a massless vector field) and apply (modulo complications because of gauge invariance) and apply the rules of "canonical quantization". You can use perturbation theory in terms of Feynman diagrams to deal with the evaluation of S-matrix elements. Later the Standard Model of elementary particle physics was discovered, which includes not only the electromagnetic but also the strong and weak interactions (with electromagnetism and weak interaction combined in a kind of "unification" in Quantum Flavor Dyamics (also known as the Glashow-Salam-Weinberg model)).