# Chern Simons Spin density as hidden variable.

1. Dec 9, 2007

### Hans de Vries

The electromagnetic Chern Simons spin density

is only know since the nineteen seventies from advanced Quantum Field Theory
on the chiral anomalies. It is the correct form of the electromagnetic spin-
density of the vacuum. It can be expressed as a 4-vector as follows:

$$\mbox{Chern Simons Spin Density:}\qquad \vec{\cal S}\ =\ \textsf{D} \times \vec{A}\ \ +\ \ \textsf{H}\ \Phi\ ,\qquad {\cal S}^o \ =\ \frac{1}{c}(\textsf{D}\cdot\vec{A})\ \quad$$

Which are just the familiar electromagnetic potentials and fields.
It's still virtually unknown to the wider audience, hidden as it is in the
more advanced QFT texts in a less accessible form. Especially interesting
are the electromagnetic spin density fields of the electron and the photon.

For instance:

Linear polarized photons, originating from spin 1 transitions, don't carry net
spin, however, they still contain the information of the original spin sign within
the canceling, non zero, components of the EM spin density from electric and
magnetic vacuum polarization. Linear photons therefor come in two types
which might be physically distinguishable in entanglement experiments.

$$\begin{array}{|l|c|c|c|c|} \hline &&&& \\ \mbox{polarization} &\ \ \mbox{orbit spin}\ \ &\ \mbox{EM spin}\ \vec{S}\ &\ \textsf{ D} \times \vec{A}\ & \ \ \ \textsf{ H}\ \Phi\ \ \\ &&&& \\ \hline &&&& \\ \mbox{Linear} & +\hbar & 0 & +\hbar & -\hbar \\ \mbox{Linear} & -\hbar & 0 & -\hbar & +\hbar \\ \mbox{Circular} & +\hbar & +\hbar & +\hbar & \ \ 0 \\ \mbox{Circular} & -\hbar & -\hbar & -\hbar & \ \ 0 \\ &&&& \\ \hline \end{array}$$

The current experimental status suggests that we have to either, give up
locality and/or reality, or show that Malus law can be violated in polarizing
beam splitters. The two types of linear polarized photons might open the
door to the latter possibility.

The derivations (which I had to do myself since somehow one can't find
these anywhere) and many details can be found in my paper, here:

The electro magnetic Chern Simons spin density

Regards, Hans

Last edited: Dec 9, 2007
2. Dec 10, 2007

### strangerep

In your paper, you say that
I could not find that exact wording in M&W. At the start of their section 10.6 they talk
about angular momentum of the (classical) electromagnetic field in vacuum. And later,
around eqn(10.6-10) they talk about the term $\epsilon_0 E\times A$ as the intrinsic
(spin) angular momentum of the EM field. But I couldn't find where they talk about
angular momentum of the (quantum) vacuum.

Could you please give me a more precise reference?

TIA.

3. Dec 11, 2007

### Phred101.2

Isn't this model being used in superconducting nanodevice research by somebody?

4. Dec 11, 2007

### Hans de Vries

Hi, strangerep

In 10.6.3 they discuss the (integrated) term Js in terms of annihilation and
creation operators. Js is defined in 10.6.12

Mandel & Wolf start in section 10.6.2 with the (orbital) angular momentum
density and at some stage $\epsilon_0(E\times A)$, which is a spin-angular momentum
density, occurs in the 3rd and last term.

At that stage however the second term "hides" a term $-\epsilon_0 (E\times A)$ with
opposite sign. This second term is subsequently ignored as a surface term
after integration. Thus, $\epsilon_0(E\times A)$ is not part of the orbital angular momentum
density but it contributes to the total integrated angular momentum if the
surface term can be ignored.

So it's a little bit tricky but the end conclusion about $\epsilon_0(E\times A)$ is right.
It's however not the full contravariant 4-vector which is the Chern Simons
current. For circular polarized photons viewed form the source's restframe
it's OK since $\frac{1}{\mu_o}B\Phi$ vanishes in this case.

Regards, Hans

5. Dec 11, 2007

### Phred101.2

Chern–Simons theory and BCS superconductivity

Manuel Asorey, Fernando Falceto and Germán Sierra
Departamento de Física Teórica, Univ. Zaragoza, Spain
Instituto de Matemáticas y Física Fundamental, CSIC, Spain
Received 30 October 2001; accepted 30 November 2001. Available online 11 December 2001.

Abstract

We study the relationship between the holomorphic unitary connection of Chern–Simons theory with temporal Wilson lines and the Richardson's exact solution of the reduced BCS Hamiltonian. We derive the integrals of motion of the BCS model, their eigenvalues and eigenvectors as a limiting case of the Chern–Simons theory.

--sciencedirect.com

"They're" able to build nanodevices that manipulate a BCS, I can recall reading somewhere...
It's just amazing what you can do with google:

The Tomonaga-Luttinger Model and the Chern-Simons Theory for the Edges of Multi-layer Fractional Quantum Hall Systems
Categories: physics.mes-hall Mesoscopic Systems and Quantum Hall Effect

Abstract: Wen's chiral Tomonaga-Luttinger model for the edge of an m-layer quantum Hall system of total filling factor nu=m/(pm +- 1) with even p, is derived as a random-phase approximation of the Chern-Simons theory for these states. The theory allows for a description of edges both in and out of equilibrium, including their collective excitation spectrum and the tunneling exponent into the edge. While the tunneling exponent is insensitive to the details of a nu=m/(pm + 1) edge, it tends to decrease when a nu=m/(pm - 1) edge is taken out of equilibrium. The applicability of the theory to fractional quantum Hall states in a single layer is discussed.

Version 1: Sun, 30 Jul 2006 09:50:04 GMT
http://front.math.ucdavis.edu/0607.0796

Last edited: Dec 11, 2007
6. Dec 11, 2007

### strangerep

I still don't understand...

Do you mean M&W's eqn(10.6-8)? The 2nd term therein is

$$- \epsilon_0 \oint \underline{E} [(\underline{r} - \underline{r_0})\times \underline{A}] \bullet d\underline{s}$$

I don't see how a "$-\epsilon_0(E\times A)$" is "hidden" in that term.

I'm not sure you understood my original problem. I was objecting to the
word "of" in your "angular momentum ... of the vacuum". (I have no problem with the
usual notion of $J_s$ as an intrinsic spin contribution to the total angular
momentum of a non-vacuum EM field. That's standard fare.)

Also note that $<0|J_s|0> \,= 0$, where $J_s$ is given by M&W's eqn(10.6.16), i.e.,

$$\underline{J}_s = i \sum_k \hbar \kappa (a^\dagger_{k2}a_{k1} - a^\dagger_{k1}a_{k2})$$

7. Dec 12, 2007

### Hans de Vries

Ah, OK I agree with you here. I guess the word vacuum slipped in via the frequent
use of the terms "vacuum polarization" and "vacuum fluctuations" So I used the
word vacuum as in "in absence of matter", the Chern Simons current is the spin
density of the electromagnetic field in absence of matter. If there is matter then
it is "co-conserved" according to the chiral analomy.

It's in the term below via the chain rule and $\nabla_i(r_m-r_{0m})=\delta_{im}$

$$-\varepsilon_0\epsilon_{lmn}\nabla_i(r_m-r_{0m})E_iA_n$$

Regards, Hans

8. Dec 12, 2007

### RandallB

“Chern Simons current is the spin density of the electromagnetic field in absence of matter”
Oddly this does make clearer your view of “Chern Simons Spin Density” or CSSD as I’ll call it here. I’m interested in your objective or opinion in how you define this Chern Simons approach.

As I understand what you’re saying the CSSD should obey conservations laws “if there is matter”. I’ll extend that just a bit by saying CSSD should maintain conservation if it affects and reacts to matter even if there is no matter in CSSD.

Do you consider the goal of this CSSD approach to satisfy the local & realistic expectations of Einstein even if CSSD is at odds with typical Local Realists?
That is; CSSD is it LR or Non-Local in your opinion?

LR meaning CSSD holds out the hope of defining a version of Local Realism that (although not deterministic) would be determinate enough in the local treatment of variables to deny the completeness of Non-Local Theories like QM & BM etc.

9. Dec 12, 2007

### Hans de Vries

The Chern Simons current, the electromagnetic spin density, comes from
Quantum Field Theory and is therefor local. It mixes with the spin density
(the axial current) of the Dirac electron as well as with a mass related term.

These three terms are conserved locally and globally together. Historically
it is a descendant of the discovery of the chiral anomaly by Adler, Jackiw and
Bell, (The John Bell from the EPR experiments) This was around 1969. This
discovery followed from theoretical work on the electromagnetic decay of
the pion and had in fact nothing to do with John Bell's inequalities.

There's for instance this recollection from Bell's partner Jackiw presented
at Bell's memorial in 2000 Vienna here:
http://arxiv.org/PS_cache/hep-th/pdf/0011/0011274v1.pdf

Regards, Hans

Last edited: Dec 12, 2007
10. Dec 12, 2007

### RandallB

Thanks Hans
Great reference it helped put in perspective their (Bell, Jackiw & Adler) views, such as “The symmetry breaking in question is a quantum phenomenon that violates the correspondence principle; it arises from the necessary infinities of quantum field theory.”

From the “necessary infinities of quantum field theory” I interpret that Bell would say QFT fails the “Classically Realistic” part of Einstein Local which requires both Local and Realistic.
Thus, CSSD or Chern Simons current - electromagnetic spin density, as a part of QFT; should not be thought of as an attempt to reach a “Local Realist” vision of reality.
But a “non-local” one that may only need the “unrealistic” part of “non-local”.
Therefore not in disagreement with the “non-local” conclusions of EPR-Bell.

Last edited: Dec 13, 2007
11. Dec 13, 2007

### Hans de Vries

Randall, You puzzle me with what you are saying here and how you come to all
these conclusions....

Renormalization of infinities has nothing to do with locality or "realism"

The Chern Simons current propagation strictly respects locality but it was never
part of any EPR discussion as far as I know nor was it conceived as an attempt
to prove anything in that area.

What I did in (my paper) was to calculate and visualize the spin density fields
for electrons and photon of arbitrary polarization and show that there are
two distinctly different types of linear polarized photons which might open the
door to a systematic violation of Malus law necessary to explain the correlations
with a hidden parameter model.

Calculation and Visualization of the EM Spin Density fields of electrons and photons

Regards, Hans

Last edited: Dec 13, 2007
12. Dec 13, 2007

### RandallB

Since you didn’t directly answer my question I was trying to state what you seemed to imply in your answer.
Einstein always claimed Bohr’s QM was incomplete and when you use "hidden variable and EPR correlations” in your title I was trying to understand what you intended by it. Are you implying:
1) QFT with CSSD may be more complete that Bohr’s OQM could be.
And if so
2) do YOU consider such a “systematic violation of Malus law necessary to explain the correlations with a hidden parameter model” if successful as “Einstein local and realistic” or no.

That is all I was asking you make clear.

RB

13. Dec 13, 2007

### Hans de Vries

OK, sorry for not being clear.

Yes, It looks inevitable with all the experimental correlation results,
The models where the hidden parameter is an input for Malus law can't
explain the correlations seen.

Thus if you want to uphold locality then you need to find physics
which causes systematic violations of Malus law in polarizing beam
splitters.

For me this was for the reason to investigate in the Chern Simons spin
density of photons. First to show that this current indeed qualifies as
the electromagnetic spin density and then to show that there are
differences in linear polarized photons which might cause systematic
violations of Malus law.

Regards, Hans

14. Dec 14, 2007

### RandallB

Sorry, maybe we have a language or syntax problem in communicating;
I’m still not able to tell if you are saying “Yes or NO”.

In one line I see you say :
YES Chern Simons spin density (CSSD) as a identified physics of systematic violation of Malus law “looks inevitable” to be a successful ‘Einstein local and realistic’ solution for reality; based “all the experimental correlation results”. [If this is what you meant to say certainly on point 1 CSSD as a complete local solution would be more complete than orthodox QM by Bohr.]

However, you complete the line by saying:
“The models where the hidden parameter is an input for Malus law can't explain the correlations seen.”

Which I translate as NO
with CSSD being one of those models that “can't explain the correlations seen” Therefore CSSD cannot yet be viewed as a complete ‘Einstein local and realistic’ solution for reality.

So I still cannot tell if:
Y or N
1) Do you consider QFT (CSSD) capable of being more complete than ‘orthodox QM’.
And
Y or N
2) If the intent of your paper is to point towards a solution that MIGHT be so much better than ‘orthodox QM’ that with more work it could describe a hidden variable acceptable as a complete ‘Einstein local and realistic’ solution.

Just looking to clearly understand your opinion or intent, I do not expect you to prove the opinion here and now.

15. Dec 15, 2007

### Hans de Vries

Hi, Randall

The intent is to find an explanation of the correlations while upholding locality.

It is clear from the experimental data that such an explanation must explain
systematic violations of Malus law one way or the other. That is, Malus law
should be valid as an average over spins states, but it should be different
if the input photons are of only one of the two spin types given in the table
in post #1

The two types should be physically different, now, since the Chern Simons
current is co-conserved with the axial current of the Dirac electron $J_A$ one
could expect that it influences the so called vector current $J_V$ of the Dirac
electron as well, $J_V$ is the charge/current density of the electron.

A further requirement seems to be that the two spin components from ExA
and BV are propagated differently in two-axial birefringent dielectrics such
as the ones used in polarizing beam splitters.

Regards, Hans

16. Dec 15, 2007

### monish

I'm going to make a suggestion here that I hope someone can make sense of. I have always wondered what makes a two-photon emission different from a single-photon emission. When people talk about the EPR experiment, they sometimes invoke the decay of positronium as the source of entangled photons. I have trouble with this because I don't know how to visualize a detailed mechanism for the creation of light in this situation.

But isn't there a simpler case that gives us two entangled photons? I'm thinking of the 2s-1s decay of the hydrogen atom. In this case I can ALMOST visualize a mechanism for the generation of light. The reason I say "almost" is that there's a problem with this being one of the so-called "classically forbidden" transitions. The "allowed" transition (like the
2p-1s transition) is really quite easy to picture semi-classically, and the emission of light in this case is really no more mysterious than the transmission of a radio wave from a small dipole antenna.

My suggestion is to focus on the 2s-1s transition as a possible source of these new kinds of emissions, because you have a case where it might be simple enough to follow the time-evolution of the system in detail and actually understand what's going on. The interesting possibility would be if you could show that the light emitted from this system was really different from classical e-m radiation.

17. Dec 17, 2007

### RandallB

You could eliminate the issue of interpreting how light emanates from atoms and use your complete photon description on the PDC creation of two photons from one photon. Although you don’t have the problems of dealing with atoms producing photons like 2s-1s decay, you still need an entirely complete classical description of a photon. Orthodox QM using a point particle and HUP description is as complete as it gets according to Bohr. Your effort would require a much more complete classical description than OQM claims is possible.

To do what you want requires you define a more complete description an individual photon than OQM allows. That is you need to displace and replace OQM first and that is not simple at all.

18. Dec 17, 2007

### monish

Originally Posted by monish
I'm going to make a suggestion here .... a simpler case that gives us two entangled photons .... 2s-1s decay of the hydrogen ..

... really quite easy to picture semi-classically, really no more mysterious than the transmission of a radio wave ....

... it might be simple enough to follow the time-evolution of the system in detail and actually understand what's going on. The interesting possibility would be if you could show that the light emitted from this system was really different from classical e-m radiation.

You've selectively edited my quote to say something different from what I originally said. I didn't say the 2s-1s decay was easy to picture classically, I said it was forbidden classically. It's the 2p-1s transition that looks like an ordinary (classical) dipole antenna.

19. Dec 17, 2007

### RandallB

But that still does not answer my question on what do you mean by “upholding locality” with QFT.

Especially when you use the term “Hidden Variable”.
Other non-local theories like BM and MWI sometimes claim a version of local. But not the same kind of “local” required to build a “Local & Realistic” HV as defined by Einstein needed to solve something like EPR-Bell. Thus in that context they are still “Non-Local” and no more complete than OQM. See thread: Is BM “Bohmian Local” actually Local

I and most recognize QFT as a non-local interpretation basically equivalent to OQM. Meaning the “local fields” as defined within the QFT interpretation is not the same as LOCAL required to build a classical HV as needed to satisfy the Einstein argument against the completeness of OQM.

So again is it your intent in describing CSSD as potentially a HV in QFT that the version of local used in the QFT interpretation, could be upgraded to the point where it might become a more complete description than OQM can be?
And thus show Bohr wrong about OQM being complete.
Essentially, this would mean a classically understandable photon description, more complete than OQM, something close to what monish is looking to use.

I’m just trying to find out if your use of the term HV is intended to define the possibility of a more complete solution in QFT, than OQM claims is possible.

For myself I view the spacelike separations of different parts of a single ‘local’ field defined by QFT marks it as a non-local theory in the Einstein “Local & Realistic” definition of “local” as it applied to things like EPR-Bell.
I’m willing to consider how CSSD might change that view of QFT.
Which is why I ask if your CSSD is intended to achieve that kind of changed view by QFT as superior to OQM view.

20. Dec 17, 2007

### Hans de Vries

The point is that the interference term in a superposition of the 1s and 2s state
does not produce a rotating charge density. You need a superposition of two
states with a different quantum number m for that.

Regards, Hans.