# Exploring Analogue of Spin Angular Momentum for Classical Dirac Fields

• jostpuur
In summary, the conversation discusses the concept of spin in classical and quantum theories. It is mentioned that while the classical theory can incorporate spin as an intrinsic property, it cannot explain it or the quantization of it. The difference between classical and quantum theories is also highlighted, with the use of complex numbers and spinors in quantum mechanics. It is noted that spinors can be understood as the polarization of the wave function and the spin operator as rotations of that polarization. Additionally, it is mentioned that the spin of a particle can be introduced to classical field theory using a second rank anti-symmetric tensor or a four vector. The conversation ends with a discussion about the representation of spin in classical and quantum theories.

#### jostpuur

I've had some trouble with the argument "spin angular mometum has no classical analogue", that everyone seems to be repeating. I used Noether's theorem to calculate angular momentum of a classical Dirac's field, and found that it has internal angular mometum density, that cannot be written as a cross produc of position and momentum. To be more presice, if I rotate Dirac's field around an angle $$(\theta_1,\theta_2,\theta_3)=\theta(n_1,n_2,n_3)$$, I calculated $$d\psi/d\theta$$ to be (in Weyl representation)

$$n_k\Big((\boldsymbol{x}\times\nabla)^k + \frac{i}{2}\left[\begin{array}{cc}\sigma^k & 0 \\ 0 & \sigma^k \\ \end{array}\right]\Big)\psi$$

The first term is kind of transformation that also scalar fields have, and the second term comes from property of Dirac's field. Ignoring the first term, I then solved a vector

$$\frac{\hbar c}{2}\psi^\dagger \left[\begin{array}{cc}\boldsymbol{\sigma} & 0 \\ 0 & \boldsymbol{\sigma} \\ \end{array}\right]\psi$$

to be a density of a conserved quantity. Isn't it justified to call this the internal angular mometum of classical Dirac's field? I also noted, that I can get spin operators by substituing $$\psi$$ operators into this expression, so it seems very much that this internal angular mometum is quite analogous to the spin of electrons.

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Yes, that's the internal angular momentum of the Dirac field. The key observation here is that the Dirac field has no physical signification at classical level, and only becomes significant if quantized. After quantization, this angular momentum contribution becomes the spin of the Dirac field (massive spin 1/2 fermion)

After this I would guess, that any classical field that has nontrivial rotation transformation, has internal angular momentum? So what about the electromagnetic field? I did the same calculation to it, and found k-component of the internal angular momentum to be $$F^0{}_i (R^k)^{ij}A_j$$, or $$\boldsymbol{A}\times\boldsymbol{E}$$. (Here $$(R^1,R^2,R^3)$$ is a vector that has 3x3-matrixes, that generate so(3), as components.) I've never heard of internal angular momentum of classical electromagnetic field, but is this it? It would make sense. And isn't this then analogous to the spin of photons?

Yes, it's true, it's the classical analogue of the spin of the photon. This time the em field is significant classically and what you have computed is the internal angular momentum for the electromagnetic field.

Spin is an intrinsic property of CLASSICAL FIELD theories. However, spin does not have a classical PARTICLE analogue. Spin is an intrinsically quantum property only if classical FIELDS are reinterpreted as quantum wave functions of PARTICLES.

A related questions is: What is more fundamental, particles or fields?
For a discussion with o poll see:

Jostpuur, forgive my ignorance, but am I right in asuming that the A in your expression in post #3 is the vector potential ?

Yes, you are. It can't be something else.

Thanks, Dexter. I've never seen that expression . I'm going to have a play with it.

The problem with the spin classically is there in not reason to exist. You can incorporate the spin in the classical theory, classical field theory is one of the theories that incorporate the spin.. But the theory can't explain the spin or the quantization of it. In a classical theory the spin is just an postulate, is an unexplained intrinsic property of matter that follows rules that are not explained by the theory. The problem of trying to explain the spin like the internal angular momentum is that, there is no reason for that internal angular momentum to behave like the the spin behaves. If you do a model of the spin like a classical internal angular momentum of the particle, that mean that like you can generate the spin, also you can stop the spin of the particle. Then the spin classically is just an intrinsic property and can't be explained using any mechanism. That's the reason that people said that the spin have no analogue classically, because can't be explained or understood classically, not because is impossible to integrate it to a classical description.

Let me explain myself, there's no reason from the classical physics postulate that a particle have a intrinsic spin. But in an quantum theory there is not reason why the particle should not have an intrinsic spin. That's the difference ibetween the theories. That mean that any theory that follows the classical postulate and deal with an intrinsic spin is not a classical theory, if not is a semi classical theory.

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Isn't numerical representation an issue also in a difference between classical and quantum models?

Classically, even though any particle is allowed to spin (and produce a magnetic field if it contains charge) it is sufficient to use real numbers to represent position and momentum.

But QM requires complex numbers to quantify position and momentum. And that requires spinor or quaterion mathematics to handle transformations in more than 2 dimensions.

Yes in QM the wave function and the operators can be complex and only the expectation values should be real. In QM spinors can be understood like the polarization of the wave function and the spin operator like the rotations of that polarization.

We can deal with spinors in the classical description because is possible to make an isomorphism between Lorentz transform and the complex 2X2 unimodular group, and we can define the spinors like the two dimensional vectors that transform under the representation of the Lorentz transform in this space. The spinors are complex in general in this representation. Is possible to develop Classical Field Theory in this space (Electrodynamics and Classical Theory of Fields and Particles Barut). Because of the pseudo metric is not that strange to see complex numbers in Classical Field Theory. Also is the expectation value of the spin what matters in a classical description and that value is real.

Is easily to introduce the spin of a particle to the classical field theory. You can define a second rank anti-symmetric tensor which components can be relate to the spin. Also you can define a four vector which spatial components in the rest frame are equal to the spin and the four vector is orthogonal to the four velocity. From this follows the BMT equation, for example. Also using the Thomas precession frequency is also possible to describe the spin and external field interactions in an arbitrary inertia reference frame.

The problem is not to introduce the spin in the Classical theory, the problem is that we need to introduce it. The spin is not a dynamical variable, it doesn't depend of q and p, then the spin have no classical analogue because can't be defined naturally from the theory. Most of the operators in QM can be introduced from the Classical theory to the Quantum theory by inspection, but the spin is an exception. This concept is purely QM and should be introduce to a Classical theory from a Quantum Theory.

Spin is a Casimir invariant of the Poincare group, namely the angular momentum in the rest frame of the particle. This does not preclude that spin is a purely classical quantity. E.g. the usual angular momentum L of a composite particle like a hydrogen atom or a neutron is a really a spin according to this definition.

The Galilei group has three Casimirs, one of them which you could identify as some sort of "classical spin".

Yes, there has also been some work by Levy Leblond showing that the same method which Dirac used to linearize the Klein Gordon equation, when applied to the Schroedinger equation, leads to the Pauli equation and a gyromagnetic ratio of 2. So spin is also not bound to relativity.

well I didn't know that you can calculate the gyromagnetic ratio of an electron without a quantum-relativity theory. That's cool.

Also I suck in Lie algebra, but in the classical theory I believe the generators of the Lorentz group can be related to a second rank anti-symmetric tensor which can be show that correspond to the angular momentum (homogeneous Lorentz transform). Also you can relate three of them with the boost and three of the with the spatial rotations. Then I don't know why the spin should be part of this algebra. All this generators can be expressed like function of q and p, where p is the four momentum or the translation operator and x is the four position vector.

In QM the spin is an generator of rotation of the spinors and the total angular momentum is the real generator of rotations. Then the spin are part of the lie algebra because of the polarization of the wave function.

In classical mechanics this polarization doesn't exist, then there is not need for the spin like a generator.

Also I'm not familiar with the concept of Casimir, I'm an ignorant in many of this concepts, I'm just in my first year of graduate studies. But for what I found the Casimirs are the member of the algebra that commute with all the generators. In that case is possible to see a connection between the spin an one of Casimirs invariants. I resist the idea that the spin can be identify classically with one of the Casimirs Invariants, because should exist another operators that commute with all the members of the algebra, for example in the SO(3) I understand that the L^2 is a casimir, which is in agreement with the classical theory.

At the moment all the classical models that I Know of the spin, follows directly from quantum mechanics and not from the classical theory. For me classical field theory is an incomplete theory, the spin can be added to the theory for the porpoise of completeness. Then the theory can deal with particles with spin from an classical point of view, but the spin is an alien of the theory. For me is the same that the Gibbs Factor in statical mechanics, which make non-sense in a classical theory because identical particles can be distinguish, but have the roots in the Quantum theory where identical particles are indistinguishable.

Here's the link to the still very interesting article by Levy Leblond:

Journal Title - Communications in Mathematical Physics
Article Title - Nonrelativistic particles and wave equations
Volume - Volume 6
Issue - 4
First Page - 286
Last Page - 311
Issue Cover Date - 1967-12-01

Author - Jean-Marc Lévy-Leblond
DOI - 10.1007/BF01646020

Thanks for the article of Levy Leblond.

The problem is that to define that pseudo-vector in a classical theory and relate it to the spin we need first to introduce an anti-symmetric rank two tensor which components are related to the spin. They assume that in the case of a free particle the components of this new tensor in the rest frame are equally to zero except for three components that are equally to the spin of the particle (well six components, but like is antisymmetric reduce to three components). After that you can define the pseudo-vector, which should be orthogonal to the four momentum because the "spin tensor" is antisymmetric. After that the introduction of the covariant Hamiltonian is easy and then is possible to find the equations of motion.

In most of the books, the argument of this "spin tensor" follows from the dipole tensor and that we can relate the magnetic moment to the spin of the particle. Also assume that what should be conserved for a free particle is no the angular momentum, but the total angular momentum which is composed by the angular momentum tensor and the spin tensor. In other words the introduction of the pseudo-vector needs the assumption that the spin exist and that the total angular momentum is the sum of the angular momentum and the spin tensor. Then this description is not natural for the theory, is just a way to make assure the connection between the classical description and the quantum description holds.

## 1. What is the analogue of spin angular momentum for classical Dirac fields?

The analogue of spin angular momentum for classical Dirac fields is known as the "Dirac orbital angular momentum". It is a conserved quantity that describes the rotation of classical Dirac fields around a central point.

## 2. How is the analogue of spin angular momentum for classical Dirac fields different from spin angular momentum in quantum mechanics?

In quantum mechanics, spin angular momentum is an intrinsic property of particles and is quantized into discrete values. However, in classical Dirac fields, the analogue of spin angular momentum is a continuous variable and is not limited to specific values.

## 3. What is the physical significance of the analogue of spin angular momentum for classical Dirac fields?

The analogue of spin angular momentum for classical Dirac fields plays an important role in describing the rotation and dynamics of these fields in a classical framework. It also has implications in fields such as electromagnetism and fluid mechanics.

## 4. How is the analogue of spin angular momentum for classical Dirac fields calculated?

The analogue of spin angular momentum for classical Dirac fields is calculated using the classical fields themselves, as well as their derivatives. This is in contrast to spin angular momentum in quantum mechanics, which is calculated using quantum operators.

## 5. What are the applications of the analogue of spin angular momentum for classical Dirac fields?

The analogue of spin angular momentum for classical Dirac fields has applications in various fields, such as optics, acoustics, and fluid dynamics. It is also used in the study of topological insulators and other condensed matter systems.