Krdytkyu

I was recently having a Twitter conversation with a UC Riverside Prof. John Carlos Baez about Geometric Quantizatoin, and he said (about his work) that

"Right. For example, you can get the quantum spin-1/2 particle by quantizing the classical spin-1/2 particle - something your mother probably didn't tell you about."

I then asked him to clarify, and his response was

"I was talking about a plain old nonrelativistic spin-1/2 particle, whose Hilbert space is $mathbb{C}^2$: the spin-1/2 representation of $SU(2)$."

Now, this confused me. In his second quote he seems to be talking about precisely what I understood to be a quantum spin-1/2 particle.

This led to the following question:

Question: What exactly is a classical spin-1/2 particle? And how does it differ from a quantum spin-1/2 particle?

My Guess: is that a classical spin-1/2 2-particle system (with "classical" spinors in the fundamental of SU(2) $psi, chi$) is described by the state $Psi = chi otimes psi$, where $otimes$ is just the usual direct product. Such a state in general is not anti symmetric under the exchange $psi leftrightarrow chi $, which if we imposed anti commutation relations between $chi$ and $psi$ would then become quantum spin 1/2 particles.

Counterexamples to my guess:

Example 1: When writing down the generating functional for QED theory we have that

$$Z[J] = int [dpsi][dbar{psi}]e^{iint d^4x i bar{Psi}(not{partial}-m)Psi - ie bar{Psi} gamma^mu A_mu Psi} $$

where $Psi$ are referred to as classical Dirac spinors. However, these are always defined to be Grassman valued, and so satisfy the proper anti commutation relations, which leads me to believe that my guess cannot be correct on some level (as it essentially puts all the "quantum" in the anti commuting nature of the spinors).

Example 2: My understanding is that it was a group theoretical fact that

$$ 2otimes 2 = 3 oplus 1. $$

I see know reason why this should not hold for two classical spinors (i.e. 2 of SU(2)). But then it seems that we are able to derive the addition of angular momentum (what I thought was a quantum result) from classical spinors.

Edit: As @knzhou pointed out in the comments, Baez may have been just referring to a single spin 1/2 particle. So I will also pose the question What is the difference between a spinor $psi_c$ that describes a classical spin 1/2 particle, and a spinor $psi_q$ which describes a quantum spin 1/2 particle?

Talking about Dirac spinors is a distraction; the classical Dirac field has little to do with a single classical particle, just as the classical Klein-Gordan field doesn't have much to do with a single classical spinless particle.

What is the difference between a spinor $psi_c$ that describes a classical spin 1/2 particle, and a spinor $psi_q$ which describes a quantum spin 1/2 particle?

Since Baez is talking about quantization, presumably his 'classical' and 'quantum' just refer to classical mechanics and quantum mechanics as usual. That is, a classical system is described by a configuration space manifold and a Lagrangian, or by a symplectic manifold and a Hamiltonian. For a quantum system, the state space is instead a Hilbert space and the Hamiltonian is an operator on that space.

A quantum spin $1/2$ particle has Hilbert space $mathbb{C}^2$ and lives in the fundamental representation of rotational $SU(2)$. The classical description of a spin $1/2$ particle is not nearly as well known, with some books even claiming that it's downright impossible, because spin is an 'inherently quantum' phenomenon. However, such statements are incorrect; spin is only historically associated with quantum mechanics because both were discovered around the same time. For example, this paper covers the subject primarily in the Hamiltonian formalism, while section 3.3 of Altland and Simons reaches the classical spinor by constructing a path integral for a spin $1/2$ particle and taking the classical limit.

This might just be a language issue. In the context of quantum theory, the word "classical" is used with at least three related-but-different meanings:

• First meaning: A model may be called "classical" if its observables all commute with each other and it is a good approximation to a given quantum model under a specified set of conditions. Example: "Classical electrodynamics."




• Second meaning: A model may be called "classical" if its observables all commute with each other, whether or not it is a good approximation to any useful quantum model. Examples: "Classical Yang-Mills theory," and "canonical quantization of a classical model."




• Third meaning: A field (or other dynamic variable) may be called "classical" if it is used to construct the observables in a classical model (second meaning). Example: the integration "variables" in the QED generating functional are "classical" fields.

Apparently, Baez was using the word "classical" in the second and/or third sense. For example, given the action
$$
Ssim int d^4x overlinepsi (igammapartial-egamma A)psi,
tag{1}
$$
the Euler-Lagrange equation associated with $psi$ is the Dirac equation. This is a classical model (second meaning) involving a classical field (third meaning). This sounds oxymoronic, because the Dirac equation is often treated as a Schrödinger-like equation in which $psi$ is the wavefunction, but it can also be treated as the Heisenberg equation of motion for a field operator $psi$, and this is the sense in which the model defined by (1) is "classical." I don't know if this is the specific model Baez had in mind, but I'm guessing that this example at least illustrates how he was using the language.

Example 1 in the OP shows the generating functional for QED, which has the form
$$
int [d(text{fields})] exp(iS[text{fields}] ).
$$
The action $S$ in the integrand can be regarded as the action of a classical model (second meaning) involving classical fermion fields (third meaning). The model's observables all involve products of an even number of these fields, so the observables are mutually commuting. For this to work, the fermion fields should anti-commute with each other always, not just when they are spacelike separated, just like the observables in a classical model should commute with each other always, not just when they are spacelike separated.

Example 2 in the OP illustrates yet another twist. In this case, the spinors might not be functions of space or time at all; they are not spinor fields (or dynamic variables of any kind). They're just spinors. This is sufficient for introducing things like the relationship between spinors and Clifford algebra and things like the rules for decomposing reducible representations.

By the way, when people talk about classical spinors or classical spinor fields, they might be either commuting or anti-commuting. These are not equivalent, but the word "classical" is used in both cases. This is one of those details that needs to be checked from the context whenever reading about "classical spinors," such as when reading about things like identities involving products of multiple spinor fields.