Spin vs Dirac Equation

Gold Member
I believe I understand the mathematical derivation of the Dirac equation. I understand how the four 4X4 matrices, and their relation to the 2X2 Pauli Matrices, arise from that derivation. I understand that the 3 spin observables for Fermions are ALSO represented by the 3 Pauli Matrices.

But, I do not understand why the 3 Pauli Matrices that arise from the Dirac equation derivation MUST be physical spin. How do we know that it is not just a mathematical coincidence that the 3 Pauli Matrices embedded in the Dirac equation also happen to be the 3 spin observables/operators?

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kith
But, I do not understand why the 3 Pauli Matrices that arise from the Dirac equation derivation MUST be physical spin. How do we know that it is not just a mathematical coincidence that the 3 Pauli Matrices embedded in the Dirac equation also happen to be the 3 spin observables/operators?
How do you know that the 3 spin operators are "physical spin"?

Gold Member
How do you know that the 3 spin operators are "physical spin"?
In non-relativistic QM, the spin property (for Fermions) is represented by a 2-dimensional Hilbert Space. So the spin states can be represented by 2-dim column vectors. Based on experimental results, the 3 Pauli Matrices work for measuring the spin in the 3 spatial dimensions.

Yes, those same 3 Pauli Matrices appear in the derivation of the Dirac equation. But I do not see the following PHYSICAL connection: Dirac Equation --> Spin. The Dirac equation was created to be the relativistic counterpart of the Schrodinger equation. That equation relates to energy, linear motion and linear momentum. Why should its relativistic counterpart (the Dirac equation) lead to intrinsic ANGULAR momentum?

WannabeNewton
Why should its relativistic counterpart (the Dirac equation) lead to intrinsic ANGULAR momentum?
Relativistic theories are constrained through Lorentz covariance. The object (Dirac spinor) entering the Dirac equation must then transform under a certain representation of the Lorentz group. Thus the intrinsic angular momentum, in the form of the gamma matrices, enters the equation through the Lorentz generators which include boosts and rotations.

See: http://web.mit.edu/8.323/sp12/Lecture notes and slides/SP2012/8-flat.pdf

Gold Member
Relativistic theories are constrained through Lorentz covariance. The object (Dirac spinor) entering the Dirac equation must then transform under a certain representation of the Lorentz group. Thus the intrinsic angular momentum, in the form of the gamma matrices, enters the equation through the Lorentz generators which include boosts and rotations.

See: http://web.mit.edu/8.323/sp12/Lecture notes and slides/SP2012/8-flat.pdf
Are you saying that the Dirac equation is Lorentz covariant ONLY if we assume that the particle has an intrinsic angular momentum?

stevendaryl
Staff Emeritus
I believe I understand the mathematical derivation of the Dirac equation. I understand how the four 4X4 matrices, and their relation to the 2X2 Pauli Matrices, arise from that derivation. I understand that the 3 spin observables for Fermions are ALSO represented by the 3 Pauli Matrices.

But, I do not understand why the 3 Pauli Matrices that arise from the Dirac equation derivation MUST be physical spin. How do we know that it is not just a mathematical coincidence that the 3 Pauli Matrices embedded in the Dirac equation also happen to be the 3 spin observables/operators?

Well, you can prove that the combination $\vec{J} = \vec{L} + \vec{S}$ is conserved, while $\vec{L}$ by itself is not.

This is just an amateur answer, but perhaps you may find some help with my "terrenal" description!

First, the spin manifests to us with the electromagnetic interaction, so, we should demonstrate that S is the spin if, when we join the fermion field (operator, lagrangian, whatever) with the electromagnetic field (operator...) we can reproduce electromagnetic experiments.

I think that the answer is that when you add U1 gauge invariance to the lagrangian associated to fermions (which is derived so as to have a maximum when the field satisfies the dirac equation), this leads naturally to a new lagrangian (with an operator A that couples (is multiplied in the lagrangian) to the fermion operator) which reproduces all the electromagnetic phenomena.

The Dirac Matrices has eingenstates (associated with some eigenvalues -lets called them "s"-) that, when you make it evolve with the lagrangian mentioned, it reproduces the features seen when a particle (system in general) is prepaired in order to have spin "s".

Im sure that I may not be using the appropiate words but I think that you may get the big picture. Please, if anybody finds any mistake let us know!

WannabeNewton
Are you saying that the Dirac equation is Lorentz covariant ONLY if we assume that the particle has an intrinsic angular momentum?
No that's not the correct train of thought. Nowhere do we assume that particles have intrinsic angular momenta. This fact falls out of the representation theory of the Lorentz group. This is not unique to relativistic QM or QFT. A similar treatment applies in non-relativistic QM through the representation theory of the Galilei group.

Did you see the derivation of the Dirac equation in the slides I linked? I don't think it can get clearer than what is in those slides. If you want a more detailed treatment then read chapter 2 of Ryder's QFT text.

Gold Member
No that's not the correct train of thought. Nowhere do we assume that particles have intrinsic angular momenta. This fact falls out of the representation theory of the Lorentz group. This is not unique to relativistic QM or QFT. A similar treatment applies in non-relativistic QM through the representation theory of the Galilei group.

Did you see the derivation of the Dirac equation in the slides I linked? I don't think it can get clearer than what is in those slides. If you want a more detailed treatment then read chapter 2 of Ryder's QFT text.
I will check out those slides. I also have Ryder's book. Thanks.

kith
In non-relativistic QM, the spin property (for Fermions) is represented by a 2-dimensional Hilbert Space. So the spin states can be represented by 2-dim column vectors. Based on experimental results, the 3 Pauli Matrices work for measuring the spin in the 3 spatial dimensions.
Not only for measuring but for general spin dynamics. When Pauli introduced his matrices in the 1927 paper, he also introduced the Pauli equation which explains all non-relativistic spin phenomena of a particle with spin-1/2. So looking at it from this angle, spin is the same in both domains because the Dirac equation reduces to the Pauli equation in the non-relativistic limit.

king vitamin
Gold Member
While the slides linked by WannabeNewton are nice, they look like they take as their starting point having already derived the irreducible representations of the Lorentz group (or at least the Weyl/Majorana representations), as well as some knowledge of simpler quantum field theories. I'm afraid the OP might be using the "one-particle" approximation which Dirac was originally using (probably unbeknownst to himself) which is still sometimes taught at the end of a graduate quantum 2 course, where the Dirac equation is satisfied by some one-particle wavefunction.

At this level, it might be easiest to simply take a non-relativistic approximation and see that it gives you the Schrödinger-Pauli equation, which the OP apparently accepts as a description of non-relativistic spin. Try looking at the Wiki section on deriving Pauli theory from the Dirac equation. It's actually a bit messy, you need to square the Hamiltonian, square-root the resulting diagonal matrix, and do an expansion in pc/mc^2, but you immediately get the main results. I highly recommend doing this on your own. You obtain the non-relativistic electron coupled to an electromagnetic field with no electric dipole moment, and a fixed magnetic dipole moment of exactly twice the Bohr magneton.

EDIT: kith beat me to it

Gold Member
Thank you all for your valuable comments. I will be exploring this subject further. My education is in Math, not Physics. Most of my physics knowledge has come from watching the Leonard Susskind Stanford U web courses. His course on QFT consists of 10, 2-hour lectures, so obviously it is just an overview of a vary extensive, deep subject.