Light

[przyk]

IMHO there's an interesting paper on this called How Long is a Photon? by Drozdov and Stahlhofen, see http://arxiv.org/abs/0803.2596. It discusses the photon as a single pulse, and the conclusion says:

"A critical review of the well known concept of a photon frequency reveals inconsistencies in the conventional picture and interpretation of the photon structure. A treatment of the photon as a single pulse of the electromagnetic field is possible as argued above without raising contradictions with basic principles of quantum mechanics and electrodynamics..."

Well don't believe everything you read. There's no inconsistency in the "conventional" description of the photon - the authors just don't seem to be familiar with it. According to them, the "conventional" description of a photon is:

[...] a particle with a certain coordinate, energy, momentum and velocity and simultaneously the spatial delocalization of order $$\lambda$$.

which simply isn't true. They don't seem to be aware that quantum mechanics has its own idea of what a particle is. All they say about it in the following sentence is:

The declaration of the ”wave-particle dualism” appointed to save the classical imagination of particle [...]

In other words, the authors seem to be under the impression that the modern view of the photon is as a "point-like particle" with "wave-like" properties attributed to it in an ad-hoc and inconsistent manner, and quantum mechanics is just some sort of explicit acceptance of this dualism. It isn't. Quantum mechanics explicitly threw out the "classical imagination of the particle" and replaced it with something new. Under certain conditions, a quantum particle can behave in a way that resembles a classical "point-like" one, and in certain respects it can exhibit what you might like to call "wave-like" behaviour, especially in interference patterns. But:

• quantum mechanics provides a single consistent description of the behaviour of a particle which reproduces these special cases.
• the full, general behaviour of a quantum particle simply doesn't have a classical analogue, so you're missing out on a lot if you obsess over the "particle-like" or "wave-like" aspects of particle behaviour.

Maybe I've missed something, but the authors just don't seem to understand the quantum mechanical concept of a particle, and set out to solve a problem that's never existed since quantum mechanics was formulated. After this false start, the rest of the article just doesn't seem to have anything to do with any real physics. It's like reading the stuff I've recently had to toil through at Physics Myths. They have an overly classical view of everything. Ironically, the authors seem to have the sort of misconception about particles you think "a lot of people" do.

 

[Farsight]

Well don't believe everything you read. There's no inconsistency in the "conventional" description of the photon...

...the authors just don't seem to understand the quantum mechanical concept of a particle, and set out to solve a problem that's never existed since quantum mechanics was formulated. After this false start, the rest of the article just doesn't seem to have anything to do with any real physics...

To make sure we're all clear on this, give us the "conventional" description of the photon, followed by the quantum mechanical concept of a particle, and finish off with a description of the photon. If that's too much to ask, just give a description of the photon. If your description merely refers to things that are in themselves not described, and is protected by "quantum cop-out", then we can discuss the issue of real physics further.

 

[przyk]

To make sure we're all clear on this, give us the "conventional" description of the photon, followed by the quantum mechanical concept of a particle, and finish off with a description of the photon. If that's too much to ask, just give a description of the photon.

Well for a brief description, then compared with the description of classical particles, there's a change in notation/terminology and a couple of twists. As far as the notation/terminology goes, we talk about a particle being in a certain "state" which is noted in Dirac ket notation. For example if a particle is located in a specific position $$\bar{x}$$, we'd say its state is $$| \bar{x} \rangle$$. If it has a momentum $$\bar{p}$$ we'd write its state as $$| \bar{p} \rangle$$.

The first "twist" in quantum mechanics is that a particle can exist in a superposition of states. You'd write this as something like:

$$
| \Psi \rangle \,=\, \int \text{d}^{3} x \, \Psi(\bar{x}) \, | \bar{x} \rangle
$$​

where $$\Psi(\bar{x})$$ is the particle's wavefunction. An important property of the wavefuction is that it is a complex quantity - it carries both amplitude and phase information.

The second "twist" is that a particle in a definite momentum state is in a superposition of position states. Specifically:

$$
| \bar{p} \rangle \,=\, \frac{1}{(2\pi \hbar)^{\frac{3}{2}}} \int \text{d}^{3} x \, e^{\frac{i}{\hbar} \bar{p} \cdot \bar{x} } \, | \bar{x} \rangle
$$​

This is just saying that a momentum state is a de Broglie wave in position space.

Finally, how quantum states evolve is governed by the Schrödinger equation (which plays the role in QM that F = ma does in classical mechanics).

As I've said I've been brief and I've restricted this to the description of single particle states (be reasonable - there's a reason entire textbooks get written on quantum theory; I don't know what you're hoping to accomplish by asking for a description of QM on an internet forum), but I've said enough to demonstrate a point: QM gives a unified description of particle states which includes all the "particle-like" and "wave-like" aspects of a particle in a single mathematical description. It also contains Heisenberg uncertainty: the momentum state above *is* a completely delocalised particle for example. QM simply doesn't provide you with any way of describing a particle with both a definite position and a definite momentum. As I said, maybe I'm missing something, but the authors of the article you dug up just seem to ignore all of this. As far as I'm concerned QM as it was formulated over 70 years ago solves all their problems.

If your description merely refers to things that are in themselves not described

Everything is ultimately explained in terms of things that are not described. For example I've told you the quantum mechanical description of a particle but I haven't really told you what a particle "is". Quantum field theory can give an answer to that (in QFT the single particle momentum states are excitation states of an underlying quantum field) - but then you've just substituted one fundamental object (the particle) for another (the quantum field). I doubt there's any way of giving a more fundamental account of what a wavefunction is. The important point is that everything that appears in the QM description of particles is relevant to making predictions about them.

then we can discuss the issue of real physics further.

Your idea of "real physics" or the physics community's idea of "real physics"? You aren't going to win any friends if you go around telling career physicists what their job "actually" is about and what they "really" should be researching.

 

[Quantum Quack]

As I've said I've been brief and I've restricted this to the description of single particle states (be reasonable - there's a reason entire textbooks get written on quantum theory; I don't know what you're hoping to accomplish by asking for a description of QM on an internet forum),

IMO this would have to be the most profound thing said so far...thanks for the inspiration....
the limitations of internet forums are both a blessing and a curse....

 

[AlphaNumeric]

No, it isn't a point-particle.

While I have no issue with the proposition that the photon isn't a point I wouldn't say your declaration is supported by evidence.

Long wave radio waves are not made up of a blizzard of point particles, and nor is visible light or any other electromagnetic radiation.

Experiments say otherwise.

Not me. Refraction depends on wavelength.

I said length, as in the photon is an object with length. I seem to remember you saying you think a radio wave is an object of length its wavelength.

No problem with that. But you can't accelerate a region of vacuum like you can accelerate a spaceship. Hence the usual description of inertial mass does not apply.

Where did I mention accelerating the vacuum? That's a meaningless concept. You accelerate yourself and you see a thermal distribution from the vacuum.

This is incorrect I'm afraid. A field is a particular disposition of spatial energy. Whilst it might be modelled in a quantized fashion via virtual particles, these particles are virtual. They aren't real particles. For example the electromagnetic field of an electron comprises energy and has a mass-equivalence. In QED it's modelled very successfully via virtual-photon exchange particles. But the only particle actually present is the electron. There are no actual photons flitting back and forth in the space between an electron and a proton. In similar vein people attempt to model gravity via gravitons, but there are no actual particles flitting back and forth in the space between two gravitating bodies.

Farsight, please don't try to pretend you know any QED. What I said was correction. The vacuum energy of a field in space-time is obtained from the ground state Hamiltonian expectation value. For instance, $$E_{0} = \frac{\langle \psi_{0} | H | \psi_{0} \rangle}{\langle \psi_{0} | \psi_{0} \rangle}$$ and for simple fields you get things like $$\int \frac{d^{3}p}{(2\pi)^{3}} \frac{\omega_{p}}{2}\; \[ a_{p},a_{p}^{\dag} \]$$

And if you think I'm wrong I just got that from Pg 19 of 'An Introduction to QFT' by Peskin and Schroder. They go on to apply such methods to QED, electroweak and QCD processes. Try reading some QED before telling people who have done QED what it involves.

You can't say what isn't actually present because you have no ability to determine the energy in short time frames precisely and thus you can't get sure how many particles are involved, thus allowing for the QFT notion of virtual particles (which don't exist in QM). And they have clear measurable effects like beta decay. A W boson flitters into existence and then decays into particles we do see. If other particles weren't created how'd a neutron turn into a proton, electron and antineutrino? Something creates them and they are observed.

 

[Acitnoids]

QQ,
If you're still having a problem with the classical representation of a photon then may I suggest this abstract demonstration:
.
Materials Needed
Sheet of paper, Scissors, Pencil and a piece of tape.
.
Steps
1. Cut a small circle out of the paper (about 4 cm which is microwave size).
2. Cut the circle in half.
3. Discard one of the two halves.
4. Tape the other half of the circle to the pencil. Make sure that the flat end is running along the length of the pencil.
5. Hold both ends of the pencil with the half circle pointing at the ceiling (This is called the peak).
6. Rotate the pencil ninety degrees so that the half circle disappears behind the pencil (I think this is called the ground but, I'm not sure).
7. Rotate the pencil another ninety degrees so that the half circle is pointing at the floor (This is called the valley).
8. Rotate the pencil another ninety degrees so that you see the half circle edge on (Once again you return to the ground).
9. Rotate the pencil another ninety degrees for a total of three hundred and sixty degrees (Once again you return to the peak).
.
The total amount of times you can do this in one second is called the frequency (In particle form this comes from the photons' spin) and this ability depends on the size of your half circle. The shorter the "wave" the higher the frequency/energy. Likewise, the longer the "wave" the lower the frequency/energy. If I'm not mistaken, even wavefunctions depend on this fact.
.
*NOTE: All experiments should be preformed with the supervision of an adult and all appropriate safety precautions should be adhered to.

 

[superluminal]

Richard Feynman, "QED". Get it. Read it. It's thin. It's good.

 

[James R]

Fraggle:

There's a problem with your absorption-re-emission model of light travelling through a transparent substance like diamond.

Transparent substances are transparent precisely because they don't absorb light. That means that photons that pass through the substance do not have energies corresponding to any permitted transitions in the atoms of the substance. Therefore, absorption does not happen.

So, an alternative explanation is needed for the delay in propagation.

Also, if there was absorption, then there would be spontaneous emission in random directions from the substance, and yet a laser beam will happily pass through diamond, say, and retain its coherence and ray-like directionality.

 

[superluminal]

Again, Feynman explains this phenomenon nicely in his little book.

 

[prometheus]

QQ,
If you're still having a problem with the classical representation of a photon then may I suggest this abstract demonstration:
.
Materials Needed
Sheet of paper, Scissors, Pencil and a piece of tape.
.
Steps
1. Cut a small circle out of the paper (about 4 cm which is microwave size).
2. Cut the circle in half.
3. Discard one of the two halves.
4. Tape the other half of the circle to the pencil. Make sure that the flat end is running along the length of the pencil.
5. Hold both ends of the pencil with the half circle pointing at the ceiling (This is called the peak).
6. Rotate the pencil ninety degrees so that the half circle disappears behind the pencil (I think this is called the ground but, I'm not sure).
7. Rotate the pencil another ninety degrees so that the half circle is pointing at the floor (This is called the valley).
8. Rotate the pencil another ninety degrees so that you see the half circle edge on (Once again you return to the ground).
9. Rotate the pencil another ninety degrees for a total of three hundred and sixty degrees (Once again you return to the peak).
.
The total amount of times you can do this in one second is called the frequency (In particle form this comes from the photons' spin) and this ability depends on the size of your half circle. The shorter the "wave" the higher the frequency/energy. Likewise, the longer the "wave" the lower the frequency/energy. If I'm not mistaken, even wavefunctions depend on this fact.
.
*NOTE: All experiments should be preformed with the supervision of an adult and all appropriate safety precautions should be adhered to.

May I suggest a similar demonstration:


Materials Needed
scissors

Steps

1. Stand with your arms loosly by your sides
2. extend your favoured arm holding the scissors in the closed position in a fist with the blades pointing upwards
3. Accelerate your fist sharply into your forehead.
Note: do not be alarmed by moderate pain and loss of consciousness.

 

[Quantum Quack]

May I suggest a similar demonstration:


Materials Needed
scissors

Steps

1. Stand with your arms loosly by your sides
2. extend your favoured arm holding the scissors in the closed position in a fist with the blades pointing upwards
3. Accelerate your fist sharply into your forehead.
Note: do not be alarmed by moderate pain and loss of consciousness.

uhm ....was that in Richard Feynmans book if so what page?

 

[Farsight]

Well for a brief description, then compared with the description of classical particles, there's a change in notation/terminology and a couple of twists. As far as the notation/terminology goes, we talk about a particle being in a certain "state" which is noted in Dirac ket notation...

Thanks for the sincere and detailed response.

Everything is ultimately explained in terms of things that are not described. For example I've told you the quantum mechanical description of a particle but I haven't really told you what a particle "is".

This is what I was driving at.

Quantum field theory can give an answer to that (in QFT the single particle momentum states are excitation states of an underlying quantum field) - but then you've just substituted one fundamental object (the particle) for another (the quantum field). I doubt there's any way of giving a more fundamental account of what a wavefunction is.

I think there is, associated with delocalised particle and a momentum state is a de Broglie wave in position space.

The important point is that everything that appears in the QM description of particles is relevant to making predictions about them.

Yes, it's important, but understanding the underlying reality is important too. That's one of the main reasons why people "do" physics, and start threads like this one.

Your idea of "real physics" or the physics community's idea of "real physics"? You aren't going to win any friends if you go around telling career physicists what their job "actually" is about and what they "really" should be researching.

With respect, I didn't. You raised the term "real physics" and dissed somebody else's research.

 

[Farsight]

While I have no issue with the proposition that the photon isn't a point I wouldn't say your declaration is supported by evidence.

You said "a photon is certainly localised into a small region of space, even to a single point". It isn't. It's delocalised. It has a wavelength. A radio-frequency photon is more delocalised than a gamma-frequency photon. My declaration is supported by evidence and experiment, stop arguing about what you seem to remember me saying.

Where did I mention accelerating the vacuum? That's a meaningless concept.

You didn't, I did in the context of vacuum energy and mass-equivalence, and I said you can't.

Farsight, please don't try to pretend you know any QED. What I said was correct...

No it wasn't. You're assigning a reality to QED virtual particles, one that misses the quite crucial point that they're virtual.

You can't say what isn't actually present because you have no ability to determine the energy in short time frames precisely and thus you can't get sure how many particles are involved...

I gave a scenario where there was an electron present. One particle, with its electromagnetic field. The latter is modelled using virtual particles, but this doesn't mean they're actually present as real photons rattling back and forth between the electron and some othe particle such as a proton that we then introduce to create a hydrogen atom.

...If other particles weren't created how'd a neutron turn into a proton, electron and antineutrino? Something creates them and they are observed.

The neutron undergoes beta decay, the proton doesn't. Virtual particles don't cause decay, and they don't create the proton electron and antineutrino. The latter are neutron decay products. It isn't stable, it breaks up.

 

[przyk]

I think there is, associated with delocalised particle and a momentum state is a de Broglie wave in position space.

Well delocalisation is part of what a wavefunction describes but there's more to it than that. As I said, the wavefunction is a complex quantity with both an amplitude and a phase (represented as a "length" and an angle in the complex plane), and only the amplitude comes into the delocalisation.

For example, the wavefunction of a particle with a definite momentum (in one dimension) is proportional to $$e^{i k x} = \cos(kx) + i \sin(kx)$$, where $$p = \hbar k$$. The "presence" of the particle in any given place is given by the norm squared of the wavefunction. This is just a constant for a de Broglie wave (the amplitude of $$e^{i k x}$$ is 1) and you're equally likely to find the particle anywhere in the universe. You don't see that a particle of well-defined momentum has any "wave-like" behaviour in this sense - it's completely contained in the phase part of the wavefunction. You'll only "see" it in position space in interference effects (eg. if you send your particle through a double slit).

With respect, I didn't. You raised the term "real physics" and dissed somebody else's research.

Well I don't particularly like saying negative things about people or their work (I wish we could all just get along) but I'm not going to state a falsely positive opinion. To me, it looks like the authors were, in 2008, suggesting that we think of the photon as just a classical wave pulse - really nothing more than a small electromagnetic wavepacket. If I haven't misinterpreted anything, then to state my opinion bluntly: that's a really naive idea. For starters, it misses the point of even the earliest concept of the photon (a smallest "unit" of electromagnetic field), which is something that's been amply confirmed in quantum optics experiment. For instance, single photons are always either transmitted or reflected through beam-splitters; they're never split in two like a classical pulse is.

A second reason, even if they can explain why a light pulse would hold together and act as an indivisible particle, is that a classical electromagnetic pulse is just that - a classical object. We have a way (via Bell's theorem) of distinguishing quantitatively between "classical" and "quantum" behaviour, and the results of Bell experiments involving photons render the idea of viewing photons as classical light pulses (or any other type of classical object) rather implausible.

Again, this is if I've understood the article correctly. I keep adding this caveat because the idea of a photon as a classical pulse is a really surprising one to see proposed by what are apparently two working physicists.

 

[AlphaNumeric]

It isn't. It's delocalised. It has a wavelength. A radio-frequency photon is more delocalised than a gamma-frequency photon. My declaration is supported by evidence and experiment, stop arguing about what you seem to remember me saying.

Evidence says otherwise. Wavelength is not the physical size of a photon, its the length of the photon path which the photon moves along over one period. Just stick your finger out infront of you and wave it up and down as your arm moves side to side, it'll map out a wave path whose wavelength is much larger than the size of your finger tip.

And evidence doesn't support you. We can make shutters which open and close in micro or nanoseconds and if you were right they'd be able to cut radio wave photons in two.

You didn't, I did in the context of vacuum energy and mass-equivalence, and I said you can't.

You haven't 'done' anything so raising the issue of accelerating the vacuum seems a bit superfluous.

No it wasn't. You're assigning a reality to QED virtual particles, one that misses the quite crucial point that they're virtual.

Farsight I suggest you get your information on high level physics from some source other than pop science books and your uninformed assumptions. I commented in my previous post how its silly for you to argue QED when you don't know any and you just demonstrated you don't know what virtual particles are (which arise in all QFTs). A 'normal' particle is known as 'on shell', which means its 4-momentum satisfies $$-m^{2} = p_{\mu}p^{\mu} = -E^{2} + \mathbf{p}\cdot\mathbf{p}$$. A 'virtual' particle doesn't. That's all there is to it, it doesn't obey the mass-energy-momentum relationship and that's why you have to integrate over momentum when doing loop level scattering processes. Well I say 'you' but I don't mean you as you've never done such things or even read about them it would seem.

I gave a scenario where there was an electron present. One particle, with its electromagnetic field. The latter is modelled using virtual particles, but this doesn't mean they're actually present as real photons rattling back and forth between the electron and some othe particle such as a proton that we then introduce to create a hydrogen atom.

You can't stipulate the number of particles in a QFT system, you can only do that in non-relativistic quantum mechanics. In QFT all physical processes involve additional particles, the virtual ones, as they are the quantum corrections. Loop diagrams arise when virtual particles are made and they represent corrections to the non-relativistic results. The determination of the g-2 value for the electron using such corrections is one of the most accurate results in all of physics. And the virtual particles don't need other particles to be around, an electron can create virtual photons and they in turn make virtual electrons or muons etc and this all contributes to what is known as the self energy of a particle. If you don't include such things you find you get the wrong predictions. And if you look closely enough you find the electron does jiggle about because you're looking closely enough to detect quantum fluctuations. These are indeed what you'd expect via the uncertainty principle as if you're looking closely you're measuring the position very precisely so your knowledge of the momentum goes up.

The neutron undergoes beta decay, the proton doesn't.

I said that.

Virtual particles don't cause decay, and they don't create the proton electron and antineutrino. The latter are neutron decay products. It isn't stable, it breaks up.

Alpha decay is caused by the $$He^{2+}$$ tunnelling out of the nucleus's potential well. Beta decay is caused by a quark having enough energy, over a tiny time scale, to turn into a W and a different quark, the W then decays into a neutrino and an electron. You are just waving your arms and saying "Oh they are decay products". Yes but what is the actual process? Standard model models of such processes have met with enormous success. I guess you're still waiting for your success..... In the mean time I suggest you actually open a book on QFT before being silly enough to argue it with people you know full well have opened a book on it. Clearly age has yet to bestow wisdom on you.

 

[Quantum Quack]

eh! why the contradiction?

in answer to the question about the size of a photon:

by JamesR:

“ You can think of it as being roughly the size of its wavelength, if you like.

vs

by Alphanumeric:
snip....Wavelength is not the physical size of a photon, its the length of the photon path which the photon moves along over one period.....

.....And evidence doesn't support you. We can make shutters which open and close in micro or nanoseconds and if you were right they'd be able to cut radio wave photons in two.

This is what I was trying to get at with my post earlier, that if a photon has length in distance it must have length in time as that distance is over time.
Alphanumeric has now confirmed that concern as being valid.

so? why the confusion?

 

[AlphaNumeric]

so? why the confusion?

Probably because you and Farsight haven't tried to learn anything relevant.

James's reply was not "It is the size of its wavelength", it was a comment about how you could think about it like that if you find it easier to grasp that way. It's not literally true, none of the quantum models of light have it as anything other than a point particle but given you, QQ, are never going to read or do any physics more advanced than tying your shoelaces if you find it convenient to think of it as a long object, so be it. Its just you shouldn't cling to that and try to apply it to things, as Farsight has tried. And failed.

If you find answers you're given to be lacking or unclear try opening a book on the subject. There's a reason every university in the world has a library, books are excellent learning aids. If all you're willing to do is be spoon fed by people don't be surprised if you fail to understand much, it is not the fault of the people who spoon fed you from time to time, its your fault.

 

[Quantum Quack]

Probably because you and Farsight haven't tried to learn anything relevant.

James's reply was not "It is the size of its wavelength", it was a comment about how you could think about it like that if you find it easier to grasp that way. It's not literally true, none of the quantum models of light have it as anything other than a point particle but given you, QQ, are never going to read or do any physics more advanced than tying your shoelaces if you find it convenient to think of it as a long object, so be it. Its just you shouldn't cling to that and try to apply it to things, as Farsight has tried. And failed.

If you find answers you're given to be lacking or unclear try opening a book on the subject. There's a reason every university in the world has a library, books are excellent learning aids. If all you're willing to do is be spoon fed by people don't be surprised if you fail to understand much, it is not the fault of the people who spoon fed you from time to time, its your fault.

well "Strike a Ferret on the kneecap"
JamesR is not to be taken literally?

Am I supposed to apologise for my confusion generated by the contradition?
especially after I was ridiculed for agree ing with you Alphanumeric.... "unfair", she cried, "just simply unfair" :bawl:
[what do I mean by temporal? ha]

As QQ pulls the scissors out of his head as suggested by prometheus
and attaches a bloody rose to them and hands them back to prometheus with a smile and a kiss to his forehead. ~ inspiration Shakespeare

 

[Farsight]

You've been spoonfed on point particles, Alphanumeric. Your textbooks cut no ice, we're writing new textbooks, and the quantum of quantum mechanics just isn't like that. It refers to E=hf where h is Planck's constant of action. Action is "kick", in the photon it's spatial momentum, and the dimensionality of action is momentum x distance. HUP applies because the photon isn't some point-particle where you have a "probability" of determining its location, but because it's an extended entity. It's delocalised. Think of it as "spacewarp", like a gravitational wave. See LIGO re length-change, though there are immersive scale change issues re measurement.

przyk: I'll get back to you properly tomorrow. Meanwhile check out Joy Christian re Bell's Theorem at:

http://arxiv.org/find/grp_physics/1/au:+christian_joy/0/1/0/all/0/1

Maybe we need a new thread on this. IMHO if one doesn't have a conceptual grasp of the photon and thence the electron, and thus a handle on the underlying reality of QED, one cannot make secure progress.

 

[przyk]

przyk: I'll get back to you properly tomorrow. Meanwhile check out Joy Christian re Bell's Theorem at:

http://arxiv.org/find/grp_physics/1/au:+christian_joy/0/1/0/all/0/1

I've had a quick look at one of his articles and I don't think his approach is going to be very helpful to you. His idea is basically to circumvent the limits of Bell's theorem by representing spin states with Clifford algebra variables (I'm not very familiar with these at all, but the essential point is that they're abstract variables with non-convential multiplication rules). To me that's a bit like saying you can violate the rule that "my bank balance squared is non-negative" by representing my bank balance with an imaginary variable. All the quantities that appear in Bell inequalities are supposed to be attributed real numbers. For instance, the fact that the variables $$A_{\bar{a}}(\lambda)$$ and $$B_{\bar{b}}(\lambda)$$ take on the values +1 and -1 is a choice: those values are attributed to each of two possible complementary experimental outcomes. You can also alternatively define the correlator more explicitly in terms of joint conditional probabilities (as is commonly done):

$$
\xi(a,\,b) \,=\, P(+,\,+ \,|\, a,\,b) \,+\, P(-,\,- \,|\, a,\,b) \,-\, P(+,\,- \,|\, a,\,b) \,-\, P(-,\,+ \,|\, a,\,b) \;,
$$​

where in a local hidden variables theory the joint probabilities are supposed to admit a factorisation of the form:

$$
P(A,\,B \,|\, a,\,b) \,=\, \int \text{d} \lambda \, \rho(\lambda) \, P(A \,|\, a;\,\lambda) \, P(B \,|\, b;\,\lambda) \;.
$$​

Bell inequalities can be proved from this factorisation, and since probabilities are by definition real numbers, Joy Christian has no opportunity to introduce Clifford algebra variables here.

Finally even if he's somehow right, he hasn't exactly disproved Bell's theorem in a way that would help you, since his counter-example really isn't the sort of thing you'd call a "classical" theory anyway. With him, instead of grappling with quantum mechanics and entanglement, you've got to grapple with what it means for spin states (and presumably other degrees of freedom including time-frequency, since we've observed entanglement there too) to be represented by Clifford algebra variables.

Basically, I really wouldn't go down that road if I were you.