# Discovery of Maxwell field's spin

1. Oct 31, 2014

### jjustinn

After coming across an article by Schwinger where he tries to deduce the spin of a neutron, I started to wonder when the same thing was done for the photon...
Specifically, I've been trying to locate "the" article where the spin of the electromagnetic field was theoretically determined to be 1 (though I'd also settle for "an" article). I know that photons were known to obey Bose-Einstein statistics even before those statistics had a name, but as I understand it, it wasn't until Pauli's 1940 paper that the connection between spin/statistics was established, and even then that would only establish that it had an integral value.

I say "theoretically" because presumably it could have been determined experimentally in early experiments (ala Compton) from momentum conservation, but that's not very interesting; we could have known the same for electrons, but we'd still cite Dirac as the one who established it theoretically; so I'm looking for X where Electron Spin: Dirac :: Photon Spin:X.

The trouble is most articles now treat the photon spin of 1 as a self-evident fact based upon the vector-nature if the wave equation, or via the irreducible representation theory of the Lorentz group...but that doesn't seem like it would have been that obvious in the 20's/30's (though if it was, I'd love to see the papers).

2. Oct 31, 2014

### samalkhaiat

Electrodynamics and very basic QM are all what you need to calculate the spin of the photon. Ok, I am going to make you derive it yourself. The energy density (Hamiltonian density) and the “spin” angular momentum density of an EM field are given by
$$\mathcal{ E } = F^{ 0 \mu } \partial^{ 0 } A_{ \mu } - \frac{ 1 }{ 4 } F^{ 2 } ,$$
$$\mathcal{ S }_{ i j 0 } = F_{ i 0 } A_{ j } - F_{ j 0 } A_{ i } .$$
Your aim is to calculate the “spin” vector density
$$S_{ k } = \frac{ 1 }{ 2 } \epsilon_{ k i j } \mathcal{ S }_{ i j 0 } .$$
For that use the following (circularly polarized) plane wave (moving in the z-direction)
$$A^{ \mu } = ( 0 , a \cos \omega ( z - t ) , \pm a \sin \omega ( z - t ) , 0 ) .$$
After you substitute this plane wave in the above expressions, you find
$$\mathcal{ E } = a^{ 2 } \omega^{ 2 } , \ \ \ \ S_{ z } = \pm a^{ 2 } \omega .$$
Now, think of the EM wave as a beam of particles (photons) each carrying $\omega \hbar$ amount of energy. So, if there are $n$ particles per unit volume then $\mathcal{ E } = n \omega \hbar$. Therefore
$$S_{ z } = \pm n \hbar .$$
So, left circularly polarized photon moving in the z-direction carries spin angular momentum $+ \hbar \hat{ z }$ and each right polarized photon has Helicity $- \hbar \hat{ z }$
As you see, It is an undergraduate material, yet not many graduate students know how to do it!

3. Oct 31, 2014

### jjustinn

Hey samalkhaiat,

That's probably the most concise derivation I've seen (and definitely undergrad level, which is also a big plus), so thanks for that...but, I'm actually looking for the historical record -- how it was originally derived without the benefit of a century of quantum mechanics. Now, that derivation does actually look like it *could* have been written with just the tools available in the late '20s...but I'm trying to track down an actual contemporaneous article.

4. Oct 31, 2014

### samalkhaiat

I don’t know the history. However, in 1865 Maxwell knew about the equations
$$\mathcal{ E } = a^{ 2 } \omega^{ 2 } , \ \ \ \ S_{ z } = \pm a^{ 2 } \omega ,$$
We also know that the Planck-Einstein relation $\mathcal{ E } = n \omega \hbar$ was known to physicists in 1901-1904. So, one can guess that $S_{ z } = \pm n \hbar$ was also known in the beginning of the 20-th century. I Know that experimental proof of the photon spin was made in 1931.

5. Nov 1, 2014

### dlgoff

Richard Feynman touches on this history in his lecture,

.

6. Nov 1, 2014

### vanhees71

Well, the above derivation is somewhat misleading! In relativistic quantum theory it does not make much sense to distinguish between orbital angular momentum and spin as it is in non-relativistic quantum mechanics.

The photon is described in the Standard Model as a massless spin-1 field. A careful analysis of the unitary ray representations of the Poincare group shows that such a field is necessarily an Abelian gauge field, if you assume that there are no continuous intrinsic "spinlike degrees of freedom" for the simple reason that no such thing is observed. Since the field has integer spin, it's a boson in order to have a local QFT with stable ground state (Pauli-Lüders spin-statistics theorem).

Further, since the photon is massless there are only 2 helicity states not 3 spin-$z$ eigenstates as for a massive vector boson. The expression in #2 is sometimes referred to as spin part of the angular-momentum density, but that's misleading, because the expression is obviously gauge dependent and thus cannot be a physical observable.

The correct way to define angular momentum for the electromagnetic field is via Noether's theorem with the additional demand that the energy-momentum tensor of the theory should be a symmetric 2nd-rank tensor and gauge independent. This leads to
$$T_{\mu \nu}={F_{\mu}}^{\rho} F_{\rho \nu}+\frac{1}{4} F_{\rho\sigma} F^{\rho \sigma} g_{\mu \nu}.$$
The angular-momentum tensor then is
$$J_{\mu \alpha \beta}=x^{\alpha} T^{\beta \mu}-x^{\beta} T^{\alpha \mu}.$$
The momentum-helicity eigenmodes of the electromagnetic quantum field are given by the left- and right-circular polarized plane waves, and you can calculate for an appropriate wave packet that it caries helicity $h=\vec{p} \cdot \vec{J}/|\vec{p}|=\pm \hbar$.

7. Nov 1, 2014

### dextercioby

Well, I think it was Proca who computed first the spin angular momentum of the em. field. But I'll dig to find out for sure. EDIT: See the comments in post#9 below.

Last edited: Nov 1, 2014
8. Nov 1, 2014

### jjustinn

@dlgoff, I watched the first half hour of so; I'd forgotten how much of a badass Feynman was...but I didn't catch the part in question.

@vanhees71, I know that was just intended to correvt the record, but it does nicely highlight some of the difficulties that early theorists would have had to deal with (no spin-statistics theorem to tell them that boson -> integer spin, no gauge theory, etc), though the derivation of the helicity (projection of the classical angular momentum tensor onto the direction of the momentum), though I suppose technically not "spin", should have been accessible even to Einstein in 1905 (though I guess not in the operator language which seems to be needed)...

@dextercioby, that's the first lead I've had, so thanks for that; it looks like he published frequently in Comptes rendus de l'Académie des sciences, whose archive is online at gallica.bnf.fr (which would be awesome if I could read French).

But, it looks like Proca started publishing in 1931, and didn't really pick up until 1934 -- I suppose it's possible that there was no theoretical proof of photon spin until the '30s, but it seems like there should have been something sooner, if only a non-relativistic/gauge-dependent formulation ala samalkhaiat.

9. Nov 1, 2014

### dextercioby

A covariant formulation of the classical em field was available by 1915, when Hilbert wrote this famous article on the foundations of GR. Noether's theorem was a creation of 1918. The first quatization of the e-m field was done by Dirac in 1927*. But the first relativistic computation of the em field angular momentum was due to Belinfante and Rosenfeld**, according to this source: http://www.hep.princeton.edu/~mcdonald/examples/spin.pdf, page 2.
I don't have access to these 3 quoted papers, so I can't tell if they also <quantized> the spin part of the angular momentum, or kept the treatment purely classical.

* A quote from Jauch & Roehrlich (2nd Ed.) is:
The quantum theory of the radiation field was developed by P. A. M. Dirac,
Proc. Roy. Soc. 112, 661 (1926); 114, 243, 710 (1927). P. Jordan and W. Pauli,
Z. Physik 47, 151 (1928). W. Heisenberg and W. Pauli, Z. Physik 56, 1 (1929);
59, 169 (1930).

The split of total angular momentum into spin and orbital, as calculated by Belinfante and Rosenfeld is known to be problematic. See the comment also from Jauch and Rohrlich:

2-8 The spin of the photon. The spin of the photon is usually assumed
to be 1. This statement is in need of elaboration in view of the fact,
pointed out in connection with Eq. (2-55), that the separation of the total
angular-momentum operator into spin and orbital parts is not gauge in-
variant. Since only gauge-invariant quantities are observable, the concept
of the photon spin, as a physical quantity, is obscure.

**
See below in post# 11

Last edited: Nov 1, 2014
10. Nov 1, 2014

### aleazk

11. Nov 1, 2014

### dextercioby

This continues post#9:

Following the note in the appendix II of W. Heitler's QED book, one finds a note about the spin angular momentum of light in Darwin's PRSL article, PRSL Vol. 136, p. 36, 1932, section 3.

The first relativistic (in terms of space-time tensors in flat spacetime) computation of the energy-momentum 4-tensor of the free em field is known to be due to Minkowski 1908, Gött. Nachrichten, according to Whittaker, A history of theories of ether and electricity, page 66 of Vol.2

So the comment above in post# 9 about Belinfante and Rosenfeld leads to a weird conclusion: the covariant form of the em field's angular momentum tensor was computed only some 30 years later than the energy momentum one, even though covariant electromagnetism was pretty much ordinary subject all this time.

12. Nov 1, 2014

### jjustinn

The prize goes to dextercioby! I'd downloaded that Darwin paper just this week (If all journals were as open as the Royal Society, the world would be a more wonderful place), but hadn't gotten to it yet (since it was one of a few hundred), but that's definitely a derivation...but, there's a twist: he notes that his derivation agrees with Dirac's textbook, which gives values of +/-h, though he does sound as if he believes his derivation (if not the result) is novel. So it goes back to at least 1932 (the papers MacDonald cited were 1940)...

Also, re: those Rosenfeld papers, there's an an English translation of [4] here: http://neo-classical-physics.info/u...rosenfeld_-_on_the_energy-momentum_tensor.pdf

@aleazk, I'm monolingual as well, but I'm definitely bookmarking that retro.seals.ch site.

As an aside, that KT Macdonald site that dextercioby linked has a ton of historical papers (though only the experimental physics section doesn't require a password; but even that section has more than I could reasonably get through in a lifetime).

Last edited: Nov 1, 2014
13. Nov 2, 2014

### samalkhaiat

Are the classical and quantum results of that derivation wrong? Which result is not gauge-invariant, $\mathcal{E} = \omega S_{ z }$ or $S_{ z } = \pm \hbar$?
It is not easy, but it does make sense. See attached PDF and references therein.
See the attached PDF. Results from COMPASS, STAR, PHENIX and HERMES show that what we actually measure coincides with the gauge non-invariant photon/gluon spin evaluated in a particular gauge. This is exactly what I did in post #2.
On general grounds, one can show that the Poincare’ generators CANNOT be gauge invariant! Does this make the use of the Poincare generators “misleading” and “does not make much sense”? It is not necessary to insist on gauge-invariant operators as long as you can show that the MATRIX ELEMENTS of those operators between arbitrary physical states are gauge-invariant.
O yes. That is what I used: The CANONICAL Noether current.
There is no fundamental principle which requires that "additional demand". The Belinfante-Rosenfield tensors may be gauge-invariant, but they also FAIL to generate the correct Poincare algebra and that is very bad. Even in general relativity, the symmetry of $T_{ \mu \nu }$ DOES NOT follow from strict physical requirements, rather it is a convenient assumption. For example, in Einstein-Cartan theory $T_{ \mu \nu }$ has an antisymmetric part that is coupled to the torsion.
Forget about "the em quantum field" and “an appropriate wave packet” I want you to use this (gauge-invariant, modified Belinfante-Rosenfield) angular momentum tensor to calculate the angular momentum carried by the circularly polarized plane wave of post#2. The canonical “spin” part of the Noether current in post#2 led to gauge invariant, universally true expression and that was $\langle \mathcal{ E } \rangle / \langle J \rangle = \omega$: that is the frequency of the wave is determined by the ratio of energy to angular momentum. Can the Belinfante-Rosenfield tensor reproduce that ratio?

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14. Nov 2, 2014

### dextercioby

Leader's article is known to me. Unfortnately it propagates the idea from his book and I quote: <
In non-relativistic Quantum Mechanics, the spin of a particle is introduced as an additional, independent degree
of freedom.
> This is utterly wrong. It should have read <
In non-relativistic Quantum Mechanics, the spin of a particle is introduced as an automatic degree
of freedom, just like in Dirac's theory of 1928
>.

15. Nov 2, 2014

### samalkhaiat

I am not going to say who is "utterly wrong" about this UTTERLY irrelevant issue. The article is a review of the so-called "Spin Controversy" and examins all possible solutions which have been offered by different people in the last 40 years.

16. Nov 2, 2014

### dextercioby

A review paper of 90 pages may very well be flawless for 99,9% of its content, but propagating a false idea even after 50 years after Levy Leblond's work still counts. And about its relevance, this is a purely subjective matter. You can consider it irrelevant, but I think it otherwise.

Our PF member @Demystifier wrote an article http://arxiv.org/pdf/quant-ph/0609163.pdf on myths in quantum physics to which I make (maybe I've already done this in the past, at least with respect to the 2nd point below) 2 objections:

1. It's missing the myth <Spin is a consequence of the successful marriage of special relativity with the quantum mechanics of Schrödinger done by Dirac in 1928, while in non-specially relativistic quantum mechanics it's a(n) (convenient) assumption*, not a natural consequence of the formalism>.

2. It invokes Pauli's mathematically flawed argument to show that putting T as an operator on an equal footing with H is wrong, since it implies the lack of lower bond on H's spectrum.

*typically presented in textbooks as the <spin hypothesis>, something that Pauli put in by hand in 1927 to account for the experiments showing the influence of a magnetic field upon a beam of electrons.

Last edited: Nov 2, 2014
17. Nov 6, 2014

### vanhees71

Right now I'm at a workshop in Catania and can't read the above cited review in detail. I'll do so at home next week.

What I can say already now is the following: Concerning the representation theory of the Poincare group there is no problem with the canonical or gauge-invariant symmetric energy-momentum tensor since there only the total energy and momentum and angular momentum and center-momentum cordinates enter as the generators of the group operations, and these are gauge invariant and the same when integrating either the canonical or Belinfante energy-momentum tensor.

Further of course also for classical waves you have to use wave packets of finite energy and momentum to evaluate the these quantities and the angular momentum. It turns of course out that for a wave packet that is sufficiently close to a plane wave the circularly polarized states are those of helicity $\pm 1$ and that thus the relation you mentioned is correct. This is all well-known classical field theory and tranlates with a grain of salt to QED.