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.]

-drl

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

http://arxiv.org/abs/hep-th/0005039
http://arxiv.org/abs/hep-th/0106273

-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)

-drl

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:

Relativity in binary systems as root of quantum mechanics and space-time
(hep-th/0408116)

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.

-drl

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?