# Clifford Modules

John Baez had a weblog long before the term was even invented, and for many years now has been consistently putting out interesting current material about math and physics under the title This Week’s Finds in Mathematical Physics. The latest edition has a beautiful explanation of the structure of modules of the Clifford algebra.

Traditionally one thinks about geometry in n-dimensions in terms of n-dimensional vectors and tensors built by taking tensor products of vectors. These are all representations of the general linear group GL(n), or if one has a metric, the othogonal group SO(n) of transformations that preserve the metric. However, it turns out that there are representations more fundamental than vectors, the spinor representations. These require a metric for their definition, and are projective representations of SO(n), or true representations of the double-cover Spin(n). When one tries to construct spinors, one quickly runs into a fundamental algebraic structure associated with a real n-dimensional vector space: the Clifford algebra C(n). Spinors occur as “modules” of the Clifford algebra, i.e. vector spaces that the Clifford algebra acts on. The structure of these possible Clifford modules is rather intricate, with a certain eight-fold periodicity. Baez gives a beautiful explanation of part of this story.

Physicists generally complexify everything in sight (i.e. assume all numbers are complex), which makes things much simpler. Then the story is periodic with period 2 instead of 8, and Clifford algebras are just one or two copies of a complex matrix algebra of k by k matrices, where k is some power of 2. Clifford modules (including the spinors) in this case are just complex vector spaces of dimension k, and tensors built out of these. One good place to read about all this, together with its relation to the index theorem, is in the book “Spin Geometry” by Lawson and Michelson, but there are by now lots of others.

If one believes in a deep relation between physics and geometry, these Clifford modules should somehow come into play in the structure of the most fundamental physical theories. To some extent this is already in evidence in the way spinors and the Dirac operator occur in the standard model. There are also tantalizing relations between the idea of supersymmetry and the Clifford algebra story. Many, many people have been motivated by this kind of idea over the years to try and use Clifford algebras to come up with a fundamental particle theory, one that would explain the structure of the standard model. While some of these attempts have very interesting features, none of them yet seems to me to have gotten to the heart of the matter and used this kind of geometry to give a really convincing explanation of how it is related to the standard model. Some crucial idea still seems to be missing.

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### 34 Responses to Clifford Modules

1. Chris W. says:

New preprint on an application of supersymmetry in a condensed matter model:

Supersymmetric Model of Spin-1/2 Fermions on a Chain

Introduction (excerpt):

For many condensed matter systems, the key to understanding the physical properties lies in the analysis of a quantum many body problem with strong correlations. For the analysis of such systems, approaches that go beyond the standard perturbative techniques are always needed. It has recently been proposed [1, 2] that, for a special class of lattice models for correlated fermions, supersymmetry can provide a tool for non-perturbative analysis. In these models, questions about the existence and degeneracies of strongly correlated ground states at zero energy are easily answered with the help of supersymmetry and elementary combinatorics. Explicit properties of these same ground states are being studied with techniques that are, in various ways, associated with supersymmetry [1, 3, 4].

Conclusion (excerpt):

We have introduced a model of interacting spin-1/2 fermions on a chain with a manifest SU(2) extended N = 4 supersymmetry. Our representation of N = 4 supersymmetry is highly non-linear, as it is entirely built from degrees of freedom that are fermionic. We have looked for a supersymmetric model where SU(2) spin symmetry is faithfully represented, and this has led us to a somewhat unusual restricted Hilbert space, with anti-ferromagnetic correlations built in from the start. The algebraic structure we have uncovered is very rich, but we are lacking a systematic mathematical framework. Such a framework will be most valuable, as it will allow us to further work out our present model and to decide on possibilities for alternative realizations of N = 4 supersymmetry.

[For some background, see (eg) hep-th/0210161.]

2. D R Lunsford says:

CW, see also the “Space Time Code” series from the 60s and 70s, which I seem to recall he thought of as a relative failure, but good practice. Your exposition reminded me strongly of that.

-drl

3. D R Lunsford says:

Yes of course, I thought that was well-known. Finkelstein’s primary object is the chronon thought of as a type of simplex, and the edges are conceived as making up a Clifford algebra.

http://www.physics.gatech.edu/people/faculty/dfinkelstein.html#research

-drl

4. Chris W. says:

DRL,

Yes, but not for a long time. I do recall that his work attracted some significant attention as far back as the 1960s. I should revisit it; I remember enough about it to see what you’re getting at.

Come to think of it, I did visit his home page a few months ago. BTW, I just went to the Quantum Relativity Group’s page at Georgia Tech, and found this:

Clifford algebra as quantum language
(hep-th/0009086)

5. D R Lunsford says:

Chris W – have you looked at Finkelstein’s work? (Starting probably with “Quantum Relativity”.)

-drl

6. Chris W. says:

Introduction

In a comment on his weblog posting “Clifford Modules” (3/11/2005), Peter Woit said the following (in a reply to Tony Smith):

…… But I still think there is a crucial idea missing here and wonder if you really strongly disagree with this. To be a little more specific, what seems to me to be missing is some deeper link between the geometry and the dynamics. In the path integral formalism, why are we integrating over a space of connections the exponential of the norm squared of the curvature of the connection? Why the determinant of the Dirac operator? I tend to think we need some insight into these mysteries in addition to more kinematical ideas about Clifford algebras.

In a brief email correspondence I had with Ray Streater 9 months ago he made the following remark in reference to Irving Segal:

I notice that Baez was a student of Irving Segal, for whom I was research assistant in 1965, and from whom I learnt a lot of things. But my sympathies are with Chandrasekhar whose face fell when Segal introduced space-time as a poset: we want physics.

From Streater’s brief memoir of the encounter (linked above):

Atiyah then invited Chandrasekhar, as the century’s most eminent astrophysicist, to open the questions. Chandra complained that gravity was nowhere mentioned, and that there were no dynamical laws in the theory. [emphasis added]

In the following I will introduce a simple (and fairly familiar) notion of dynamics that can be connected immediately with geometry in a way that may be initially surprising, but is quite natural. Furthermore, this notion will prove to be unexpectedly fruitful, in a way that is also initially surprising, but is again natural, indeed, almost obvious after some contemplation. More specifically, I will be drawing tight linkages between elementary geometrical (and topological) notions, a primitive notion of local gauge freedom which bears a provocative resemblance to supersymmetry, and causal sets, which are, of course, posets (partially ordered sets) as mentioned by Streater.

A Boolean Network as an Abstract Simplicial Complex

Consider a simple collection of binary elements or Boolean variables. Initially we assume no structure on this set other than the assumption that each of its members possess a binary state, which may change. Of course the collection (or ensemble) has a collective state, which may also change.

Assume that the changes of (and within) the collection can be recorded. We can accumulate a history:

S _ S _ S _ S _ S _ S _ S _ S _ S _ S _ S _ S
S _ S S S _ S _ S S S _ _ _ S _ S _ S _ S _ _
S _ S _ S _ S S S _ S _ S _ S _ _ _ S _ S _ S
_ _ S _ S _ S _ S _ S _ S _ S _ S S S _ S _ _
S _ S _ S S S _ S _ S S S _ S _ S S S _ S _ S
S _ S _ S _ S _ S _ S _ S _ S _ S _ S _ S _ S

Bear in mind that in representing the set this way we have assigned an implicit labeling of its members, which may be considered an integer {1, …, N} assigned from left to right. The changes in state are tracked with respect to this labeling, and we are forced—for the moment—to assume that the labeling can be meaningfully carried forward in “time”, as the state of the collection changes.

We may now ask ourselves, what are the dynamics? We can record a history, but can we predict successive states? Can we describe stable correlations among members of the collection?

We must guess at the dynamics, and then test our guesses. The simplest thing we can do that seems likely to yield non-trivial results is to associate transition functions of two variables (two input states) with each element in our collection. These functions must of course be Boolean, and the number of possibilities are very limited. In fact, there are 16 functionally complete options from which to choose, all interrelated by a familiar web of dualities:

NAND(x, y) = ~NOR(~x, ~y)
NAND(~x, y) = ~NOR(x, ~y)
NAND(x, ~y) = ~NOR(~x, y)
NAND(~x, ~y) = ~NOR(x, y)

NOR(x, y) = ~NAND(~x, ~y)
NOR(~x, y) = ~NAND(x, ~y)
NOR(x, ~y) = ~NAND(~x, y)
NOR(~x, ~y) = ~NAND(x, y)

Here “~x” or NOT(x) can of course be taken as a shorthand for NAND(x, x) = NOR(x,x).

We specify a complete transition on our set of N binary variables by selecting functions from this set of 16, assigning them to ordered triples ({c; a, b}) taken from the set of N variables; one variable receives the “output”—the “next” state—and the other two provide the inputs. The elements of the triple are not necessarily distinct.

Let us focus attention on the inputs of the transition functions. In the “typical” case they associate each member of our set of Boolean variables with another member. This makes clear that in this context the variables may be thought of as vertices, and the transition functions as edges or links. Collectively they define a simplicial complex, which is carried forward in “time” as the set of variables evolves according to the transition rules.

We have arrived at the roots—at least in part—of geometry.

A Boolean Network as a Gauge Field

There is another way to view the foregoing. Let us reconsider the brief history of our small sample collection of variables:

S _ S _ S _ S _ S _ S _ S _ S _ S _ S _ S _ S
S _ S S S _ S _ S S S _ _ _ S _ S _ S _ S _ _
S _ S _ S _ S S S _ S _ S _ S _ _ _ S _ S _ S
_ _ S _ S _ S _ S _ S _ S _ S _ S S S _ S _ _
S _ S _ S S S _ S _ S S S _ S _ S S S _ S _ S
S _ S _ S _ S _ S _ S _ S _ S _ S _ S _ S _ S

It was mentioned before that the representation of the collection as a one-dimensional array implied nothing more that a simple labeling by a set of integers. It may now be noted that the representation employs a labeling of the states of the variables which is equally arbitrary. We can globally invert the assignment of our state labels (S, _) without making any essential change to the information content of our history:

_ S _ S _ S _ S _ S _ S _ S _ S _ S _ S _ S _
_ S _ _ _ S _ S _ _ _ S S S _ S _ S _ S _ S S
_ S _ S _ S _ _ _ S _ S _ S _ S S S _ S _ S _
S S _ S _ S _ S _ S _ S _ S _ S _ _ _ S _ S S
_ S _ S _ _ _ S _ S _ _ _ S _ S _ _ _ S _ S _
_ S _ S _ S _ S _ S _ S _ S _ S _ S _ S _ S _

It is crucial that the state relabeling be global. We have no sensible way to propagate a selective relabeling of some variables throughout the history without corrupting our knowledge of the correlated changes in the collection, because we have no notion of dynamics, other than the raw fact of change reflected in our data.

Everything changes when we introduce dynamics, that is, our set of transition functions. An arbitrary local state relabeling can now be absorbed into the transition rules using the dualities that interrelate them. In this austere context we have hit upon two central ideas of gauge symmetry. (1) The variables’ states are like phases; we can detect changes and variations across the collection, but we can’t assign an invariant meaning to the states themselves. This dictates a global symmetry. (2) The introduction of dynamics allows us to turn this into a local symmetry.

A Boolean Network as a…Supersymmetric Gauge Field?

As we think about this more deeply we start to see some subtleties. There is another relabeling freedom in the collection of variables—the labeling of the variables themselves. We can permute the labeling, and as long as we do this consistently throughout our recorded history—prior to any assumption of dynamics—we change nothing essential in the record. On the other hand, if we have segments of a history that precede and follow relabelings of variables we must remember the permutations that were performed in order to compare the segments.

Now, let us consider again what happens when we perform a local relabeling of states and absorb the relabeling into the set of transition functions. The functions fall into at most 16 distinct classes or types; of course we may not have used all the types in equal proportions. The absorption of the state relabeling will transform the type of the affected functions, and the distribution of transition functions assigned to the variables among the possible classes will change, in general. Of course, in a very large Boolean network of this type with suitable statistical properties, the distribution may not change much at all. Indeed, we may merely witness what appears to be merely a shuffling—a permutation—of the positions of the affected instances of the 16 classes with respect to the labeling of the variables. It is quite conceivable that this shuffling could be undone by following the state relabeling with a relabeling of the variables, which implies a relabeling of the transition functions assigned to them.

It is clear that understanding the structure of these transformations is becoming rather delicate, and the structure—the geometry—of the simplicial complex is deeply involved. However, we can at least offer the observation that the internal gauge transformation—the state relabeling—appears to be intertwined with the “external” or positional transformation, ie, the selective relabeling of the Boolean variables. Indeed, they seem to be in some sense inseparable. This strikes me as deeply reminiscent of supersymmetry.

A Boolean Network as a Generator of a Causal Set

What precisely is being preserved in the gauge transformations we have been discussing? I hope it quite clear by now that the relabeling transformations preserve the identity of events or state transitions in our collection, and the causal connections among those events. In other words, the collection’s history has an invariant temporal “skeleton” which fits the description of a causal set. For those who are unfamiliar with causal set theory the significance of this can be better appreciated after reading a review such as this September 2003 article by Rafael Sorkin:

Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School)

Also, in January, a highly speculative but extremely interesting and detailed preprint by Seth Lloyd appeared that sets forth an attempt to ground general relativity, and by implication, quantum gravity, in quantum computation. The idea, roughly speaking, is to generate the causal and metric structure of spacetime out of the dynamics of an ensemble of qubits. Lloyd acknowledges that his approach, as an attack on the problem of quantum gravity, is closest in spirit to causal set theory among the various competing alternatives.

What of Spinors and Clifford Algebras?

I will cut this short and simply assert that the relevance and role of spinors is near at hand. The reasoning behind this is still sketchy (like most of the above) but is deeply inspired by the following two articles, to be read in the given order:

Spin foams, causal links and geometry-induced interactions
(hep-th/0403137)

Pregeometry and Entropy

In closing, I should acknowledge that the potential role of Jaynes’ Maximum Entropy principle has played a central role in my thinking about these questions for over 15 years. I recently attempted to sum this up in an email to Lee Smolin (8/5/2004) which can be taken as something of a manifesto:

Dear Dr. Smolin,

See http://www.arxiv.org/abs/nlin.AO/0408007. I thought this paper would be of particular interest to you, in light of your recent preprint (hep-th/0407213) and your past collaboration with Stuart Kauffman and long-standing interest in the structures of discrete networks and their relevance to spin networks and quantum gravity.

This new paper is to my mind relevant to the effort to put putative explanations of “anthropic” features of the universe on a sound scientific basis. The general point of view I have in mind is this:

One would like to in some sense show that:

• the fundamental theme of entropy maximization under constraints underlies the laws of physics and the structure of spacetime, and ultimately, cosmological structure and evolution,
• at the same time underlies the evolution of biological systems and, ultimately, sentient life,
• and binds the two realms together in a way that genuinely explains the familiar observations that have previously motivated the more problematic formulations of the anthropic principle.

The key element is this: Rather than suppose that there is a kind of super-universal selection of law-like alternatives (including values of fundamental constants), imagine that a sort of optimization through maximally unbiased selection constitutes, in essence, the foundation of physical law. Of course one might reasonably ask if this is so different than more conventional formulations. That is, fundamental symmetries provide the constraints, and some sort of selection sorts out the alternatives allowed by these constraints. The crucial question becomes, does this selection happen in time, along the lines suggested by biological evolution, or does it effectively happen outside of time, via a kind of timeless optimization principle? The latter is suggested by the intimate relationship of variational action principles and symmetries of an action functional. Can such extremization under variation be understood or recast as a entropy maximization under constraints?

Given the profound connection of MaxEnt principles to statistical mechanics and thermodynamics, and the equally profound “thermodynamic” character of general relativity, as witnessed for example in Ted Jacobson’s gr-qc/9504004, one wonders if this might not be the preferred avenue to explore.

However, in his article Jacobson expresses a traditional point of view. If a putatively fundamental physical law has the character of a thermodynamic equation of state, then it can’t be that fundamental. There must be a complicated microstructure (and micro-dynamics) underlying it whose governing law is obscured by statistical regularities. I am suggesting, in essence, that the statistical regularities are most of the story. The underlying microstructure is primordial, not derived (ie, not a “solution”), and possesses some primitive combinatorial symmetries and nothing more. Everything else comes out of a kind of statistical optimization (entropy maximization?) of the microstructure subject to its symmetries.

I would suggest furthermore that the symmetries follow largely from background independence of the microstructure. Because its possible configurations cannot be distinguished by differences in their “pinning” to a background, they must fall into equivalence classes — probably very large equivalence classes. This is fertile ground on which to build a potent kind of statistical mechanics.

What remains, then, is to say something more definite about the nature of the microstructure.

7. D R Lunsford says:

Everything points to y5 as the missing link.

-drl

8. Anonymous says:

I don’t get it. Why is a theory bad if it does not explain the gauge coupling strengths, the Yukawa couplings and the exact matter content? In principle, the standard model is enough to determine the unitary evolution. Why must we explain the parameters of the standard model? Why can’t they just be what they are?

9. Alejandro says:

A serious puzzle when trying to relate Clifford algebras to the real world is the existence of three generations of particles, all having equal charges. I mean, you can suspect that the fact of having fermions around could have a geometrical motivation, and you can even be happy about having four fermions which you could perhaps use to build a well oriented slice of space time or whatever geometrical construct. But 12 formions? What in the hell does Nature expects we should do with them?

10. Doug says:

Peter wrote:

If one believes in a deep relation between physics and geometry, these Clifford modules should somehow come into play in the structure of the most fundamental physical theories. To some extent this is already in evidence in the way spinors and the Dirac operator occur in the standard model. There are also tantalizing relations between the idea of supersymmetry and the Clifford algebra story. Many, many people have been motivated by this kind of idea over the years to try and use Clifford algebras to come up with a fundamental particle theory, one that would explain the structure of the standard model. While some of these attempts have very interesting features, none of them yet seems to me to have gotten to the heart of the matter and used this kind of geometry to give a really convincing explanation of how it is related to the standard model. Some crucial idea still seems to be missing.

Peter, everyone: With all due respect, trying to use the Clifford algebras to “come up with a fundamental particle theory” that “would explain the structure of the standard model,” actually could be the reason that “some crucial idea still seems to be missing.” However, please understand: in making this statement it’s not my intention to attack the greatest intellectual achievement of the 20th Century by suggesting this.

Nevertheless, I believe that it’s important to recognize that string theory exists as a direct result of trying to avoid the difficulties of a “fundamental particle theory,” and maybe we need to consider that the need to resort to the concept of fields was motivated in the same way: the idea that the existence of a fundamental particle, or set of fundamental particles, can explain nature, seems to be misguided.

However, replacing the point particle with the concept of a field, and the idea of force with the concept of interaction, does not change the basic assumption underlying a theory of fundamental particles, anymore than replacing it with the concept of a vibrating string does, even though we can point to the “spectacular” success of QED/QCD, and string theorists can point to the “astounding” beauty of their M theory. As we all know, the most important aspects of any theory for investigators is its failures, not its successes.

The most obvious failure of the standard model is its lack of an explanation of mass and the associated interaction (force) of gravity, yet we steadfastly fail to see this as a failure of a “fundamental particle theory.” Frankly, though, what this seems to be shouting at us is that we really need to look for an alternative to a fundamental particle theory. If there is indeed “a deep relation between physics and geometry,” and “these Clifford modules [actually do] come into play in the structure of the most fundamental physical theories,” shouldn’t we stop trying to force that structure into a particle theory and look for an alternative?

11. Doug says:

Quantoken wrote:

“What is time” is a question so profound that it is beyond mathematics to try to find an answer. The concept of time and time arrow inheritantly associates with the concept of causality. If there is no causal relationship between things, then time does not exist. But is the causal relationship just our perceptions of the world, or is it part of the reality?

It doesn’t have to be that hard, though. Time has the same effect on motion that space does, and space, while it’s also a problematic concept in some ways, it’s not as mysterious. However, it’s the concept of space and time in the definition of motion that is important.

Clifford algebra recognizes this in its noncommutativity: a^b = -b^a, a directed area. Clockwise rotation is different than counter-clockwise rotation. It doesn’t mean that time can run backwards, but it does mean that there is an inverse, a mirror image to all directions, and if it exists in all directions then it follows that it exists in the relationship of space and time, or motion, as well. In fact, we can see it: A decrease in time has the same effect on motion as an increase in space. They are reciprocally related. That’s the important thing to understand.

This reciprocity in turn implies a deeper symmetry: whereas time continuously progresses, so should space; and whereas space has three dimensions, so should time. Now that we can actually observe the progression of space in the receding galaxies, we are even more justified in assuming three dimensions of time to complete the symmetry, which, of course, means three independent dimensions of motion should exist as well.

In Cl3, we see a similar symmetry: 1 scalar, 3 vectors | 3 bivectors, 1 pseudoscalar. What does this mean? I’ll offer one, simple, but profound idea: The scalar magnitude of motion can easily progress continuously outward from the origin of vectors (outward translational paths diverge, think of expanding volume), but, on the other hand, it cannot continuously progress inward translationally in the same manner lest, passing zero, it again becomes outward motion (inward translational motion converges, think of shrinking volume). Therefore, for continuous inward scalar motion to exist, it must exist as a rotation, not a translation.

To find that these two fundamental modes of scalar motion are represented perfectly in Clifford algebra, is sobering.

12. Tony Smith says:

Peter, I am somewhat concerned that this message may be drifting off-topic with respect to this thread subject of Clifford algebras, so it is OK with me if you delete it, but I am initially posting it mostly to respond to a couple of comments by others.

Quantoken, you say, about my approach: “… That is not how a theory gets accepted as science. … acceptance of a science is really acceptance by the general public …”.
My objective is NOT for my model to be “… accepted … by the general public …”. It is for my model to be available in the archived records of physics so that anyone who is interested in it can study it, criticize it, and perhaps improve it or make use of it or some part of it.
You also say: “… if at the end of day, only ten people out of the whole population of this planet manage to figure out what your theory is all about, and agree with you. So you have acceptance of 10 people. Does that make your theory an accepted scientific theory? No. …”.
Actually, I would be happy if 10 people figured it out and accepted it. As I said above, my objective is NOT to make my model THE accepted theory.
In fact, I think that the current abysmal situation with superstring theory shows that it is BAD for ANY single model (mine included) to become so “accepted” that alternatives cannot be made available in the archived records of physics for evaluation by anyone with interest. The “archive” part is important, because sometimes it is decades before some useful stuff is appreciated as being useful.

Steve M, you say that you “… think your work is more suited to something like Journal of Mathematical Physics, Communications in Mathematical Physics or Advances in Theo. Math Phys. J. Math. Phys. … Phys. Rev. D is rigid and pretty conservative …”.
My comments here may have been misleading to you in that I have emphasized mathematical structures. However,
I consider the phenomenological part of my model ( not mentioned by me hereinbefore because of less connection with the topic of Clifford algebras ) to be equally important. An example is the relevance of my model to the interpretation of Fermilab’s T-quark event data, and the idea ( based on a paper by Froggatt at http://xxx.lanl.gov/abs/hep-ph/0307138 ) that the data show not one single peak for the T-quark at around 170 GeV, but also two other peaks (one higher, one lower) that can be reasonably interpreted by seeing the Fermilab T-quark data as showing three peaks coming from a T-quark – Higgs – Vacuum system. Such material seems to me to be off-topic for the mostly purely mathematical journals that you recommend. It is my opinion that the most appropriate journal ( in my home country of the USA ) that covers both the math/theory and phenomenology aspects of my model is Phys. Rev.D, which is why I chose it as the journal for my prize offer.

Tony Smith http://www.valdostamuseum.org/hamsmith/

13. Anonymous says:

“In the path integral formalism, why are we integrating over a space of connections the exponential of the norm squared of the curvature of the connection? Why the determinant of the Dirac operator?”

Renormalizability and universality classes?

14. Quantoken says:

Tony Smith said:

“Unless someone is already familiar with ALL of those things, they CANNOT understand my model without spending a lot of time and effort on learning background material that may not be of interest to them for any other reason.”

Tony, I am troubled by that and your $100,000 reward. That is not how a theory gets accepted as science. Put it this way, if at the end of day, only ten people out of the whole population of this planet manage to figure out what your theory is all about, and agree with you. So you have acceptance of 10 people. Does that make your theory an accepted scientific theory? No. You could argue that your theory is correct, it is only because 99.9999% people are unwilling to spend the time to try to understand your theory. But that is useless. Let’s look at another example. Super string theory. Seemingly this is an “accepted” theory at current time, despite of the fact that it’s unable to make any predictions. Super string theory is taught on almost every colleague campuses and many students accept it because their teachers seem to accept it. As for the general public, they are unable to understand the math involved, but they too have learned a few basic ideas like elementary particles are made of little ringy strings, etc. And they have watched scifi movies about wormholes and cosmic strings and such, and think those stuff are neat and cool. So they too, accept super string theory as a legitimate scientific theory. So, at currently time, super string theory is a theory accepted by the general public and the general scientific community, and opponents are only minority, due to the propagation. That’s a sad fact. But it could change, eventually people will say: This thing can’t make any predictions and seem to be useless, why should we accept it and continue to support it? By the time, there may still be a couple thousands “experts” in this field that continue to believe in the stuff, including all of the well known names like Witten, Lubos, Vafa etc. But they will be a minority groups and marginize and be reject as a colt group, once the GENERAL PUBLIC, I repeat, the GENERAL PUBLIC, not the elite peoples in ivory towers, become uninterested in the ideas. You can argue: “But I am the expert in the field” but it really doesn’t help you if your whole field is rejected by the general “none-experts”. So acceptance of a science is really acceptance by the general public, not just acceptance by a elite group. Another example is General Relativity. There is a saying that initially there were only 12 people in the world who can understand GR. If it stay that way, GR will never become an accepted scientific theory. The truth is even most people are unable to understand the tensor mathematics. They are perfectly capable of underdstanding, and accepting the equivalence principle, so they have no problem accepting the GR as a correct theory derived from equivalence principle. So my point is, whatever theory you have, to have any hope of being accepted. You must be able to describe the basic idea or basic principle in very simple language, to the general public. So even the general public can accept the basic principles of your theory. Then those interested can further study the detailed mathematics. Science is not an enterprise in the ivory tower. Science is an enterprise in which not only researchers exchange ideas among themselves. But they also need to actively promote their ideas to the widest audiences in the general public. That’s why it’s so important for dissident scientists like Peter et all, to turn towards the general public audience, and explain to them why there is a problem and why string theory does not work. The big problem is different opinions are SUPPRESSED in the public medias, so the public never realize that the main stream scientific theories have problems in them! Quantoken 15. Anonymous says: It seems to me that the idea that beautiful mathematics will lead to new physical principles is as likely to succeed as string theory. I think it is the other way round, mostly, physical insight bends beautiful mathematics into its service. However, there seems to be no other way to proceed, so I can only hope that someone is successful. 16. Steve M says: Hi Tony,$100,000 eh?:). I am a (theoretical) physicist and have also worked as a freelance editor/copyeditor in the past for some pretty technical stuff in physics and biology. If I could make a few points. First I think your work is more suited to something like Journal of Mathematical Physics, Communications in Mathematical Physics or Advances in Theo. Math Phys. J. Math. Phys. is particularly open to new mathematical approaches and interpretations to physical problems and for development of mathematical ideas relevant to the formulation of physical theories. They publish some really mathematically dense and technical stuff. Second, I think you should be flexible about the title. As a journal Phys. Rev. D is rigid and pretty conservative. They reject a lot of technically sound papers (probably the most interesting ones).

Generally, it is very difficult to get a “new paradigm” published and accepted by the community (dreadful word I know:) especially as a single author, and much easier to publish something interesting or minor within an ongoing and accepted research direction. At any rate, a totally professional Latex presentation is always essential nowadays. Two or three readable shorter papers is also better than a huge submission, which is offputting to reviewers. A thorough review of the existing literature relevant to the problems you are attempting to solve is always essential, explaining what you think are the shortcomings of the standard approaches and why your approach might make progress where others have failed. It is also better to be have a modest but interesting presentation style/title rather than making huge claims. Making massive claims in titles and abstracts and having a pompous and self-important style of presentation (I am not saying this applies to you)usually results in a crackpot label and instant rejection, even if the paper has a lot of substance and very good ideas. Anyway, these are just some thoughts based on my experiences both editing papers and publishing my own.
regards

17. Tony Smith says:

Hi Garrett. You say that I should use LaTex in a self-contained introductory paper. Actually, back in the days before I was blacklisted, I did put up such a paper on what is now known as arXiv. It is at http://xxx.lanl.gov/abs/hep-ph/9501252 It does not incorporate some changes and corrections that I have made in the last 10 years, but the basic ideas are similar. It is over 100 pages long, and the part about the Mayer-Trautman-Kobayashi-Nomizu material does not appear until about page 60 or so.
Even though it was extensive and in LaTex, nobody paid any (contructive) attention to it, so I really don’t think that any lack of the LaTex look is why people don’t understand it. I think that the real difficulty is that you cannot formulate the model without using such things as:
1 – Clifford algebras;
2 – Quaternions and Octonions;
3 – the Mayer-Trautman-Kobayashi-Nomizu material;
4 – theory of bounded complex domains and their Shilov boundaries, as described by L. K. Hua in his book Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains by Hua (Am. Math. Soc., 1979;
5 – generalization of MacDowell-Mansouri mechanism for gravity;
6 – I. E. Segal’s conformal gravity material; and
7 – how to generalize the Hyperfinite II1 von Neumann factor to the case of real Clifford algebras from the case of complex Clifford algebras on which the usual Hyperfinite II1 von Neumann factor is based.
8 – For the lattice version of my model, familiarity with generalization of the Feynman Checkerboard is needed ( see my paper at CERN-CDS-EXT-2004-030 ).

Unless someone is already familiar with ALL of those things, they CANNOT understand my model without spending a lot of time and effort on learning background material that may not be of interest to them for any other reason.

Since I am a lone individual with no institution that could provide grants or jobs related to studying my work, it is understandable that nobody would spend such time and effort.

It is with that in mind that I have put up a \$100,000 prize for the first person to meet its conditions, which include writing up my model, getting it posted on archives in arXiv, and getting it published in Phys. Rev. D. A statement of the prize details is at http://www.valdostamuseum.org/hamsmith/VoDouPhysicsPrizeV.html
Maybe nobody will try to do the work for the prize, but on the other hand maybe somebody will, and at least the first person who succeeds will get some money for the time and effort expended.

Tony Smith http://www.valdostamuseum.org/hamsmith/

18. Quantoken says:

Doug:

“What is time” is a question so profound that it is beyond mathematics to try to find an answer. The concept of time and time arrow inheritantly associates with the concept of causality. If there is no causal relationship between things, then time does not exist. But is the causal relationship just our perceptions of the world, or is it part of the reality?

Mathematics, on another hand, is totally incapable to understand what is causality. Because causality relationships do NOT exist in any branch of mathematics. In math you can say 2+3=5, or you can say 5-3=2. But among 2,3,5, who is the cause, and who is the result caused by the cause? The answer is none, “2+3=5” is NOT the reason why “5-3=2”, nor is it a result of “5-3=2”. They are equivalent to each other but does not depend on each other as cause and result, so there is no causal relationship.

All of our difficulties in understand spacetime is because math does NOT have a causality relationship, but causal relationship must be introduced to understand time properly.

Quantoken

19. Garrett Lisi says:

Hi Peter,
First off, thanks for keeping such a cool journal — and for being a voice of reason speaking out on the current physics emperor’s state of undress. Your posts and the poignant reactions to them have been very amusing to read. You’ve also now hit upon a subject close to my heart — it always bothered me that spinor fields seemed to be “cooked up” rather than really derived from geometry, so I got drawn into looking around for a better way. I’ve also been inspired by the attempts to derive the structure of the standard model using Clifford algebra — so far my favorite has been by Greg Trayling:

http://arxiv.org/abs/hep-th/9912231

The only cool thing I’ve found myself is a way of deriving spinors as BRST ghosts associated with the Clifford adjoint invariance of a frame in GR:

http://arxiv.org/abs/gr-qc/0212041

That’s the closest I’ve been able to get to a true geometric derivation of what spinor fields are. Using that and Kaluza-Klein to connect with Trayling’s work has been giving me fits because it’s so close but not quite there. The universe doesn’t just laugh at string theorists.

I see that Tony Smith is hanging around your journal as well (Hi Tony). I have looked around at his stuff, and found it to be very interesting and potentially good but I only wish he would spend the painful extra time necessary on exposition for people without his same eclectic background. If he put together an introductory paper to his stuff, maybe even using LaTeX… he’d probably get a much better reception, and I for one would have a better chance at really getting some of what he’s saying. He especially needs to keep in mind people are lazy and need straightforward presentations of background material all in one place, rather than just references.

Anyway, good to finally come out from lurking and post something here.

Best,
Garrett

20. Doug says:

If fundamental physics deals with the matters of space and time, and fundamental mathematics deals with the matters of geometry, then the connection between the two should not be surprising, because ultimately geometry is the study of space. When advances in mathematics enabled men to add time concepts to these geometrical aspects, mathematics became indispensable to the investigators of physics.

However, it’s the fundamentals of space and time that are the core of physics, not the mathematical expressions of space-like and time-like constructs, or various unions of these. The fact that spinors are a mathematical construct hearkens back to others, more primitive, but similar, such as Pythagoras’ theorem. Clearly, Pythagoras’ theorem tells us something about space and spinors tell us something about time, but whatever it is can be misleading if we begin to think of these properties as real. Space can have no meaning without time because you can’t draw a line without the time to do it. Time can have no meaning without space, because, no matter how much time you have, you can’t rotate it.

All that this suggests is that “the deep relationship between Clifford algebras and spinors, geometry, index theory and physics,” exists because of the deep relationship between space and time. That relationship is reciprocal and symmetrical in the equation of motion, just like the 8 hour “Clifford clock,” is reciprocal and symmetrical. What Lubos apparently doesn’t understand is that the difference between the elementary and sophisticated is not profundity, but elaboration. The profound secrets we seek to uncover are hidden in the simple, but correct relationship of space and time. Sophisticated elaboration of an incorrect understanding of this elemental relationship only serves to further obscure the error.

21. Tony Smith says:

Peter, you say that you “… wonder if you [Tony] really strongly disagree [with the thought that] … there is a crucial idea missing here … what seems to me [Peter] to be missing is some deeper link between the geometry and the dynamics …”.

I can see how what I think is an intuitively natural link between the geometry and the dynamics might appear to others as an ad hoc construction.

Roughly (again ignoring a lot of technicalities) the way that I link geometry and dynamics is:

1 – describe dynamics by a Lagrangian;

2 – define the Lagrangian in terms of Clifford algebra structure:

a – the 8-dim spacetime over which integration takes place is represented by the 8-dim vectors

b – the gauge term in the Lagrangian is based on the 28-dim bivectors

c – the Dirac term is based on the 8-dim fermions and antifermions.

3 – See what happens to the Lagrangian when the freezing/choice of a quaternionic subspace of the initial 8-dim spacetime changes it to a Lagrangian over a 4-dim spacetime. This is highly nontrivial but it does take you from a very simple-looking Lagrangian with 8-dim spacetime to a much more complicated one with 4-dim spacetime, but it turns out that the complicated 4-dim Lagrangian is actually quite realistic.
This is done by using some geometric techniques done about 20 years ago by Meinhard Mayer http://www.ps.uci.edu/physics/mayer.html who worked with A. Trautman. They used some key ideas from Kobayashi and Nomizu’s book Foundations of Differential Geometry, vol. 1 (John Wiley 1963), particularly Proposition 11.4 of chapter II, and their work in some detail can be found in Hadronic Journal 4 (1981) 108-152, and also articles in New Developments in Mathematical Physics, 20th Universitatswochen fur Kernphysik in Schladming in February 1981 (ed. by Mitter and Pittner), Springer-Verlag 1981, which articles are:
A Brief Introduction to the Geometry of Gauge Fields (written with Trautman);
The Geometry of Symmetry Breaking in Gauge Theories; and
Geometric Aspects of Quantized Gauge Theories.

It may be from some points of view regrettable that my link between geometric structures and Lagrangian dynamics is heavily dependent on really understanding that material, but that is the way it is.
If the Mayer-Troutman-Kobayashi-Nomizu material seems so natural to you that it is (to use a phrase that is sort of a joke among mathematicians) intuitively obvious, then my link might seem natural.
Otherwise, my link would probably be thought of as a missing link.

The details are in my papers and web site, but I am not the world’s best expositor (I wish I has 1/10 the talent of John Baez in that regard).

Anyhow, long story short, if anyone does not clearly understand the Mayer-Troutman-Kobayashi-Nomizu material then they will definitely not see my model as a natural construction.

On the other hand, when (years ago) I began to work through the 8-to-4 dimensional reduction part of my model using the Mayer-Troutman-Kobayashi-Nomizu material and to see how after grinding out the results that they actually look realistic, it made me think that there must be something fundamentally right/useful about the work.

I could go into more material related to dynamics of Lagrangians and path integral quantization, but this comment is already long and I thought that the most basic point was how the parts of the Clifford algebra give a realistic 4-dim Lagrangian, which is fundamentally an application of the Mayer-Troutman-Kobayashi-Nomizu material.

Tony Smith http://www.valdostamuseum.org/hamsmith/

22. quantoken says:

Peter said:

“But it definitely is one of my deepest beliefs that the most fundamental structures in mathematics are intimately connected with the most fundamental structures in physics. There seems to me to be a lot of evidence for this.”

I can almost totally agree with you, except for one small reservation (which is actually not small). That is, what is considered “most fundamental structures in mathematics” is really “in the eyes of the beholder”. i.e., it’s totally opinious unless that judgement (what is more fundamental) is guided by empirical physics experimental evidences.

Mathematics is much bigger than physics. There are infinitely many possible self-consistent theories in mathematics. It’s hard to make a judgement what is fundamental and what is not. And not all mathematics can have something to do with physics. Number theories, or the study of prime numbers or such, seems totally unrelated to physics. At least all I know, any prime numbers bigger than trivial ones like 2,3,5, never occurs in any physics formulas. The string theory, for it’s “11” dimentions, may break this rule. But then maybe not if ultimately it’s shown that string theory has nothing to do with physics:-)

Tom Smith, what you claim seems to be interesting if it is true:

“In short, you get all the particles of the Standard Model and none of the unobserved wino, gluino, squark, zino, etc particles of naive 1-1 supersymmetry, and you get reasonable values for all the otherwise ad hoc parameters of the Standard Model. ”

But I glanzed at your web in the past and could not associate its content, which is hard to read and understand, with your claim. Maybe I will try to spend a little bit time figuring out exactly what your ideas are.

Quantoken

23. Peter says:

Hi Tony,
Sure, you’re one of the people I had in mind when I wrote the posting. I’ve always been intrigued by the kind of thing you’re doing (and think it’s absurd that the arxiv happily accepts absurd string theory related papers but not yours). But I still think there is a crucial idea missing here and wonder if you really strongly disagree with this. To be a little more specific, what seems to me to be missing is some deeper link between the geometry and the dynamics. In the path integral formalism, why are we integrating over a space of connections the exponential of the norm squared of the curvature of the connection? Why the determinant of the Dirac operator? I tend to think we need some insight into these mysteries in addition to more kinematical ideas about Clifford algebras.

24. Tony Smith says:

Peter, you say:
“… none of … [the] attempts … to try and use Clifford algebras to come up with a fundamental particle theory … yet seems to me [Peter] … to give a really convincing explanation of how it is related to the standard model. …”.

I cannot disagree with that statement because “seems to me [Peter]” is a statement of subjective opinion and you [Peter] are certainly entitled to your opinion.

“… Which “ideas” exactly do you think would “explain the structure of the Standard Model”? …”.

About the only strictly relevant thing that I can say is that I am one of the “… Many, many people [who] have been motivated by this kind of idea …”, and that my efforts use structures related to Cl(8) which, as John Baez points out in his week 211 (among other writings) is, due to real 8-periodicity, a natural building block of any arbitrarily large real Clifford algebra. Roughly (ignoring many technical details) my physical interpretation is:

The 8-dim vector part of Cl(8) represents an 8-dim spacetime;

The 28-dim bivector part of Cl(8) represents 28 gauge bosons;

The 8-dim +half-spinor part of Cl(8) represents 8 first-generation fermion particles (e-; r,g,b up quarks; r,g,b down quarks; nu_e);

The 8-dim -half-spinor part of Cl(8) represents 8 first-generation fermion anti-particles (e+; r,g,b up anti-quarks; r,g,b down anti-quarks; anti-nu_e).

If you break the 8-dim spacetime into a 4-dim physical spacetime plus a 4-dim internal symmetry space,
by freezing out at low (current experimental energy levels) a particular quaternionic subspace, then:

The 28 gauge bosons split into 16 for U(2,2) which contains the conformal group SU(2,2) = Spin(2,4) which along the lines of work of I. E. Segal and MacDowell and Mansouri give you an Einstein-Hilbert lagrangian etc plus 12 for the SU(3)xSU(2)xU(1) Standard Model; and the fermion particles and antiparticles get a 3-generation structure.

If you look at the geometric structures in a way motivated by (but not identical to) the work of Armand Wyler, then you can unambigously calculate particle mass and force strength ratios, getting results that are at tree level quite realistic if you consider the quark masses to be constituent quark masses.

In short, you get all the particles of the Standard Model and none of the unobserved wino, gluino, squark, zino, etc particles of naive 1-1 supersymmetry, and you get reasonable values for all the otherwise ad hoc parameters of the Standard Model.

It is OK with me if you [Peter], Lubos, and anyone else find my structure unconvincing, because that is just an expression of your personal opinions and tastes, but
it would be interesting to imagine what the PR blitz would be if a prominent string theorist were to come up with similar results.

Details of my work exist, but they are too long for this message. However, they can be found on my web site and on a few papers such as CERN EXT-2003-087 which I was able to post before CERN terminated the EXT series (in October 2004, possibly at the behest of arXiv, which has blacklisted some people including me and may have been unhappy that CERN EXT provided a way for some of us blacklisted people to post our work where it might be preserved for posterity (note that web sites, ISPs, etc come and go, and CERN’s web site is likely to last much longer than any web site I might have)).

Tony Smith http://www.valdostamuseum.org/hamsmith/

25. Peter says:

Hi z,

I’m a bit leery of the term “beauty” in this context, what with “all in the eye of the beholder” and everything. Somewhere on the internet there’s an argument between me and Lubos about the beauty of string theory.

But it definitely is one of my deepest beliefs that the most fundamental structures in mathematics are intimately connected with the most fundamental structures in physics. There seems to me to be a lot of evidence for this. In my posting I was explaining a bit about how fundamental Clifford algebras and spinors are in geometry (and I didn’t even get into their importance in K-theory). One of the other most fundamental ideas in modern geometry is that of a connection and its curvature. The fact that all these structures show up in the standard model seems to me not a coincidence.

But neither mathematics nor physics is a finished subject. As another commenter mentioned, it’s still a matter of debate among mathematicians whether we really understand the right way to think about the geometry of spin. We know spinors are fundamental because they are the representations we can build all others out of. But there are several different ways of constructing spinors, some that don’t use Clifford algebras at all (see e.g. Graeme Segal’s Borel-Weil sort of construction as a space of holomorphic sections). Clifford algebras are part of the story, but maybe not the most fundamental one. I don’t think we know yet. Similarly, in physics we know that spinor fields, the Dirac equation, connections and the Yang-Mills functional are fundamental parts of the story, because we have strong experimental evidence for this. But there quite possibly is some more fundamental way of looking at these things, one which would give us the right idea about how to get beyond the standard model. I’m guessing that such a new idea exists, but is missing now, and that whatever it is, it will be related to some new perspective on the geometry of spin.

There’s more detailed speculation of this kind in my paper of “Quantum Field Theory and Representation Theory”. Maybe I’m wrong, and in any case, it’s certainly historically been true that theorists have needed help from experiments to figure out what the right fundamental mathematical structures are. But in these times of no help from experiment, I don’t think we have much choice but to try and see if mathematical insights can help us see which way to go with the physical theory.

26. quantoken says:

Z:

It’s a profound philosphical question that has been debated for thousands of years. That is whether fundamental physics structures, like the standard model, can be derived from fundamental mathematics structures, like say if there can be a mathematics based theory naturally leads to the particular value of all 17 free parameters of the standard model, or you can not?

Every one can have an opinion on that. But I think it’s far from being settled. So it is totally legitimate for Peter to ask that profound question as being insightful, and it is childish for Lubos to question Peter’s intelligence in raising the legitimate question.

Personally, I do believe that the standard model as we know it, must be reducible in some way. Not all 17 free parameters can be foundamental. There must be a connection between them and there must be a mathematical model describing that connection, and therefore it would allow us to reduce the standard model into something simpler with less number of free parameters.

But I do not believe physics can be reduced to pure mathematics. When the ultimate theory of everything is discovered, you would have reduced all physics parameters to just a few, or maybe just one, the famous alpha. But you can not reduce it further to zero free parameter. The very last un-explained physics constant, be it alpha, would have absolute no mathematical explanation whatsoever why it is the value it is. It is just happenedness, just the way it is in this universe, with no more question allowed to be asked why or how.

Quantoken

27. Anonymous says:

Peter said:

“Many, many people have been motivated by this kind of idea over the years to try and use Clifford algebras to come up with a fundamental particle theory, one that would explain the structure of the standard model.”

I can sympatize with these attempts–wouldn’t it be neat if some simple/nice algebraic structure were behind our “fundamental” theories? However, this kind of an approach tends to remind me that of trying to use platonic solids in planetary astoronomy. That, too, would have been neat if it worked.

I find Peter’s last sentence

“Some crucial idea still seems to be missing.”

rather strange. There seems to be a presumption here, that this idea of explaining the structure of the standard model by using Clifford algebras in some cool way is, in fact, correct.

There is no strong evidence for this. Beauty of a mathematical structure does not guarantee its relation to a field of physics of one’s choice.

I have to say I am rather amused by seeing this kind of a statement about such an issue, made by Peter.

z

28. quantoken says:

E:

Indeed this is an open question far from being settled. And peter is rightful in pointing that Lubos had said nothing intelligent but just personal attacks. But forgive him for feeling depressed a bit that people are unable to post comments on his weblog this past few days.

And why do you think Lubos is a “prominent” figure, just because he made himself famous by being active on the internet? He is just a junior assistant professor. In my judgements, Lubos is a smart guy and he may have learned a bit of math tools useful for string theory research but he is absolutely lacking in basic physics instinctions that I think any one with some good basic training in general physics should have.

One good example is he could not think of a possible method of measuring solar radiation intensity above the atmosphere, without a tool or mean to actually go above the atmosphere to measure it.

Quantoken

29. E says:

Dear Lubos,

You may be misunderstanding something… or perhaps you are simply part of a muscular new wave within the academy.

Peter is simply aligning himself on one side of a decades-old debate between some of the most distinguished living mathematicians. On one side are mathematicians, like Singer, who believe that after the rediscovery of the Dirac operator, spinors have been nearly perfectly understood. On the other side are mathematicians, like Sullivan, who have amassed evidence that we are merely nibbling around the edges of a giant poorly understood structure.

I suppose what suprises me is that someone as prominent as yourself could be simply unaware of the debate. Perhaps physicists have recently resolved this issues amongst themselves. Mathematicians are probably nowhere close to this point. Look at how long it took both groups to replace self-dual equations with SW. Think of how mysterious the spinor condition was in the role of the rigidity of the signature operator on loop space.

I myself am not sure which group is correct, but the idea of personally deriding people like Lawson, Sullivan, Woit, Bourguignon, Taubes, Hitchin, who have spoken publically about this belief in the incompleteness of spinor theory is quite bold (given the above).

Of course, I have heard Is Singer explain his convictions on this point which I suppose vaguely echo your beliefs. But even given his role in the history of the Dirac operator, he was always humbly aware that he might be wrong. Perhaps your impatience indicates that you will soon delight us with profound contributions in this area that will settle this debate conclusively. Should that happen, we will all be glad for the light shed on this fascinating topic.

I will read your blog with interest and wish you every success.

30. Peter says:

If anyone has anything intelligent to say about the topic of my posting, please do so, but I’ve already had to delete comments from people who think an intelligent response to anything I write is to personally attack me. If string theorists feel my comments about string theory are attacks on them and feel the need to personally attack me when I write about string theory, that’s fine. But I’m not going to put up with juvenile attacks based on my explanations of what a Clifford module is.

31. Lubos Motl says:

I just can’t resist. Your last paragraph, Peter, is such an incredible stupidity that I must emphasize it a bit more clearly.

Spinors and Clifford algebras are of course closely related and this relation is well understood. By “understood”, I don’t mean that everyone understands everything about it. I mean that mathematicians and physicists have understood it for decades (especially the physical consequences of it), in most aspects long time before the Standard Model was constructed. Spinors and their properties are just an elementary mathematical piece of a Standard Model, one of many technicalities. Spinors in 3 or 3+1 dimensions were understood a few months after quantum mechanics was first proposed.

Which “ideas” exactly do you think would “explain the structure of the Standard Model”? Spinor is just a spinor. It’s an almost complete triviality that the physicists today must understand as undergrads. Your “ideals” about physics are equivalent to those of a high school student who learns about spinors earlier than when she’s ready to comprehend them fully.

Of course that there are “tantalizing relations between supersymmetry and Clifford algebra” – it’s because the generators of supersymmetry transform as spinors. It’s not really tantalizing; it’s elementary.

You’re making an invisible mysterious elephant out of an ordinary gray rat.

32. Lubos Motl says:

Hi Peter,

is this article directed to the laymen, or the physicists? In the latter case, I suppose that it is a part of your plan to convince all physicists to study the Dirac operator. ðŸ˜‰

The things you write is elementary math from the early 20th century.

Best
Lubos, 2005

33. Jody says:

Hi, Peter,

There is a very nice discussion by Atiyah of the deep relationship between Clifford algebras and spinors, geometry, index theory and physics in the short article he wrote “The Dirac equation and geometry” for the book:

Paul Dirac: The Man and His Work
Edited by Peter Goddard
Cambridge 1998