Mathematics is big, people are small. So why should we learn the obscure techniques of David Hestenes's Geometric Algebra? And why should an old forklift driver type up a justification for learning the language when he could instead be more carefully watching the 1953 baseball movie The Kid from Left Field, a delightful movie for multiple reasons?

What drags an old gray haired guy out of retirement to put gnarled fingers to keyboard is the hope of making progress in the ancient dream of unifying gravity and quantum mechanics; the unified field theory.

For two theories that both work perfectly in their domain of applicability, GR and QM are amazingly different in the way they describe reality. While GR is fairly well defined, QM has several very different interpretations. But despite being different, these various interpretations of quantum mechanics do not differ in their predictions. They are all, therefore, equally unhappy in the Tolstoy interpretation.

A primary difference between the theories is that QM is a statistical theory, GR is not. The version of QM that minimizes this problem is Bohmian Mechanics, and it narrows some other differences as well.

A subtle difference between GR and QM is that in QM, velocity is hardly used at all, while in GR, and particularly in special relativity, velocity is very important. Bohmian mechanics takes the probability current of QM, and interprets it as a velocity vector field. The trajectories of the particles are then defined by the vector field.

Thus Bohmian mechanics takes the wave and particle and wraps it into a combined package. The wave defines the possible trajectories of the particle, and the paths not taken are trimmed by their not being used by the particle.

While Bohmian mechanics reduces the difference between QM and GR in terms of the probability interpretation and the wave / particle duality, there is another area where it increases the difference, at least in the way that Bohm and his coworkers described the theory. To explain how this problem arises requires that we delve into philosophical problems that physicists are no better prepared to face than engineers.

It is possible to complete an education in physics with a belief that the usual QM describes reality exactly, (i.e. the world is a Hilbert space), and it is also possible to complete a physics education with the belief that QM simply provides a method for calculating probabilities. The attraction of Bohmian mechanics is to the second sort of physicist.

Bohmian mechanics is an ontological theory, QM is not. The difference is that QM does not attempt to ascribe reality to the elements of which it is composed. Instead it is simply a method of calculating probabilities. Bohmian mechanics attempts to describe the elements of reality.

[An aside. Now the AMC channel is playing another great movie from 1953, Niagara. Also highly entertaining, it includes my buddy's big heart-throb, Marilyn Monroe. This is why I don't own a TV. Ooooh. The end of the movie reminds me of an unfortunate experience of riding in his boat on Lake Chelan. The next movie is Against All Odds].

Since Bohmian mechanics claims to be an ontological theory, the particle paths it associates with a given quantum state have to be a real part of the world. This means that they cannot depend on the frame of reference chosen to derive them. But quantum waves are non local, and consequently when one uses them to define trajectories, the trajectories are influenced by activity over space-like separations. This violates the basic philosophy of the special theory of relativity.

Outside of the foundations of physics, most branches of science have made great progress in the last few centuries by going to ontological theories. For example, biologists suppose that biology is built from chemicals, and that chemicals exist in the real world. Chemists suppose that chemicals are made up of atoms, which they also treat as real objects. None of the higher sciences confuse reality with a probability density -- but none of them have to deal with an uncertainty principle.

Physicists of the 18th century naively expected that they could build simple experiments to detect changes in the speed of light due to the earth's motion through space. I say that their ideas were naive in that they forgot (a) that the equipment they were using to measure the speed of light was built from matter held together by electromagnetic fields which would also be influenced by the earth's motion, cancelling the difference, and (b) the chemistry of their bodies depend on electromagnetic fields and if these fields depended on the motion of the earth they would not be alive.

Forced to choose between an ontological interpretation of quantum mechanics and the special theory of relativity, Bohm chose ontology. This meant that he had to assume a preferred frame of reference. Since his interpretation gives identical results as the usual QM, that frame of reference is not detectable, but according to the ontology of Bohmian mechanics, it must exist. Bohm discusses this at length in the classic Bohmian mechanics text The Undivided Universe. These efforts are in the opposite direction of the efforts on background independence.

I should admit here that I do not believe the ontology of quantum mechanics. I think that Bohmian mechanics is wrong in the way it relates wave and particle. Suppose we discuss an experiment made on January 31, 2002. In the language of relativity, we can call this an event. Bohmian mechanics associates both a wave and a particle with the event. In my view, in 2001 the event should be described by a wave, in 2003, it requires a particle. Thus QM and GR are both wrong in failing to ascribe a value to the present, or to the age of the universe. But this peculiar heresy need not concern us further here, we can take the Bohmian viewpoint.

The existence of a preferred frame of reference is not just a violation of the special theory of relativity, it is a complete violation. It cuts directly to the spirit of the theory. However, it turns out that there is a very minor change to the special theory that makes it completely compatible with a preferred reference frame. Einstein wrote his famous paper in 1905. The year before, Lorentz wrote a paper that gave, essentially, the same equations, but assumed a preferred reference frame. This was what is now known as Lorentz Ether Theory.

One can define a version of Lorentz ether theory that is fully consistent with the predictions of relativity by simply assuming all the postulates of relativity, and adding one more, that a preferred reference frame exists, but that it cannot be detected classically. This is the path that Bohm took.

In 1916, Einstein extended special relativity to GR, which is closer to the topic of this essay. If one is to assume a preferred reference frame for special relativity, one must also assume a preferred reference frame for general relativity. And even if one does not believe in Bohmian mechanics per se, one could also notice that QM has an ontological interpretation that requires a preferred reference frame, and one could then postulate that in porting QM to GR, one should end up with a theory that, in ontological terms, requires GR to have a preferred reference frame. In either case, Bohmian mechanics or just a mysterious coincidence, one has some motivation to look for a preferred reference frame in GR.

As with special relativity, adding a preferred reference frame to GR is a violation of a basic assumption that the theory is based upon, the principle of general covariance. This is more than most physicists are willing to put up with. So let us now discuss geometric algebra. Conveniently, geometric algebra was first applied to QM rather than GR, so we can kiss the beauty before the beast.

In 1928, Paul Dirac made the unpleasantly nonlinear Klein-Gordon equation linear by replacing four scalars with matrices. The four scalars chosen multiply the partial derivatives with respect to the four dimensions of spacetime. A geometric algebraic way of describing what Dirac did is to say that he promoted the partial derivatives of the Klein-Gordon equation from commuting operators to anticommuting operators. The matrices are just there for convenience, the heart and soul of the Dirac technique are the defining relations.

At this point we should distinguish some notation. It is possible to remove the partial derivatives from the Dirac operators. We call the behavior of these sorts of objects a Clifford algebra. In the standard QM curriculum, the gamma matrices are a 4x4 matrix representation of a Clifford algebra.

In splitting the anticommuting gamma matrices from the commuting partial derivatives, we did two sins, one obvious and one somewhat subtle. The obvious sin is that we wrote physics in arbitrary 4x4 matrices. This allowed us the convenience of writing quantum states as 4x1 spinors. These objects are called vectors but they are not 4-vectors in the usual relativistic sense.

The second, more subtle sin is in splitting the Clifford algebra from the associated partial derivatives. If one instead leaves these two closely associated objects together, one discovers a theory much more general and useful than Clifford algebra. It was David Hestenes who explored this subject in the 1960s and he called it the Geometric Calculus.

Complex analysis is wonderfully useful for calculus problems on two dimensions. One can convert an integral over a 2-dimensional region into an integral over its 1-dimensional boundary. The same sort of thing can be done in higher dimensions using Stoke's theorem on differential forms. And this is the subject of geometric calculus.

One can use geometric calculus while ignoring the derivative part. Then the anticommutators are still there, and are attached to the geometry. I believe that this is what Hestenes means by "geometric algebra" as opposed to geometric calculus. I admit to using "geometric algebra" to describe both systems, but I suspect that I am not alone.

The arbitrary choice of matrix reminds one of the preferred reference frame problem. Physics does not depend on the choice of matrix. Following the same sort of logic that Einstein used to eliminate the aether with special relativity, it would be better to describe QM in a language that does not involve arbitrarily chosen matrices.

Since the 4x4 matrices are a representation of a Clifford algebra, one might expect that replacing that representation with the Clifford algebra itself would eliminate the matrices. And indeed it does, but it leaves a problem in that it is not obvious what to do with the 4x1 spinors. It is this problem, the "geometrization of spinors", that David Hestenes solved in 1967, using the Geometric Calculus.

[Aside: The AMC channel has just completed Against All Odds, and now is beginning Black Widow (1987). It was running yesterday while I worked on my buddy's new website, liquacorp.com.]

As it turns out, there is no single solution to the spinor geometrization problem. A good paper illustrating the difficulties is that of Baylis. This is in contrast to the geometrization of operators, which suggests that the density operator form is the correct place to geometrize QM. And interestingly, B. J. Hiley, coauthor of Bohm on "The Undivided Universe", has recently published papers showing that density matrices can be given a Bohmian interpretation more elegantly than spinors. While this story is of great interest to this author, we need not explore it in order to continue with the description of how geometric algebra helps in unification. The only point we need to continue is that geometric calculus or algebra can be used to rewrite QM in a more elegant fashion.

In order to do calculus on a manifold one must be able to subtract vectors at different points on the manifold. The easiest way to do this is to embed the manifold in a higher dimensional flat space. In a flat space one knows how to add vectors and the problem is solved. But this is rather inelegant, especially from an ontological point of view.

The alternative is to define addition on vectors at different points by defining a connexion between the points. This adds an extra object to the geometric calculus, the connexion, and it is a rather complicated thing.

Before one adds the connexion, each point in the manifold carries its own copy of the geometric algebra. The connexion allows all these to be compared. We can pick an arbitrary point in the manifold, say P, and to subtract any two tangent vectors, we first transport them to P. This means that while we started with an infinite set of tangent spaces, the connexion reduces them all back down to that of P.

The nature of geometric algebra is that all calculations for curvature can be written within the algebra. Given a curvature compatible with Einstein's gravitational curvature equations, one might suppose that one could begin with the tangent space at P and, by taking advantage of coordinate freedom, extend that tangent space to the whole space, thereby eliminating the curvature. In 1998, A. N. Lasenby, C. J. L. Doran and S. F. Gull did just that. This defines the equations of general relativity on a flat space.

The choice of which point P to use as the common tangent space is analogous to the usual choice of preferred reference frame in special relativity. That point P has a natural flat Minkowski geometry, and the methods given in the above paper extend that geometry to the remaining points of the manifold. Since the Minkowski geometry does not have an apparent preferred reference frame, the problem of picking out the preferred reference frame remains.

The universe is a very big place and Einstein's curvature equations have not been tested over cosmological distances. But for short distances, it is quite natural to find that space is flat. In fact, the relationship of special relativity to general relativity is precisely this. By putting general relativity onto a flat space we are not rejecting Einstein's insights so much as applying them at a larger level.

The use of flat space is good news for those who would unify GR with QM from a QM point of view. The other forces are also defined on a flat space. The natural conclusion is that gravity is a force like any other. The methods used to decipher the other forces should work on gravity as well.

The simplest solution of Einstein's equations is Schwarzschild metric. A natural question to ask is what coordinates turn describe this metric in a flat space. The answer to this question turns out to be Painleve coordinates. Doran later generalized the geometric algebra to the Kerr metric.

One of the advantages of ontological over mystical explanations is that tend to be easier to understand from an intuitive level. Starting with the flat space coordinates for the Kerr metric, Andrew J. S. Hamilton and Jason P. Lisle found a model of black holes which treats them as rivers in space-time. This is an elegant paper and all should read it.

The idea of spacetime flowing like a river, to be sucked down the maw of a black hole is intuitively attractive, but it is not the opinion of this author that this is ontologically correct. For one thing, where does all that space go. More importantly, the flat space solution suggests that gravity should be treated as a flat space force just like any other force so accurately described in quantum mechanics. The river metaphor is just a metaphor, but it is instructive that it occurs on the same coordinates as are needed for geometric algebra.

Newtonian gravitation exists on a flat space as well. And Newtonian gravitation has a particularly simple equation of motion. This raises the question "what are the equations of motion for Painleve coordinates?" If we knew those equations, we would be able to compare them with the Newtonian force law and perhaps learn something about the nature of the quantum forces that lead to gravitation. The simplest coordinates for flat space are Cartesian, and Newton's gravitation rules are especially simple in these coordinates. Consequently, the coordinates chosen for flat space general relativity is frequently "Cartesian".

In general relativity, the "equations of motion" are always written with the time derivative taken with respect to proper time. To have equations of motion that we can compare to Newton's laws as an equation on flat space (and therefore look for the quantum version), we need to write the equations of motion in Cartesian form, that is, with the time derivative taken with respect to coordinate time. The bias towards proper time is so strong that if you announce that you are undertaking this effort, you will get much advice that you are foolish.

In addition to the advantages of comparing the flat GR gravity force with the flat Newtonian gravity force, equations of motion written in coordinate time have other advantages over proper time. There is no passage of time on the trajectory of a massless object so these objects have to be handled in a different manner in the usual way. A first step is to convert the Painleve coordinates into Cartesian coordinates. After that, we can generalize to the Kerr metric, and perhaps the charged ones as well. This author is fairly lazy, and not particularly interested in gravitation, so the task has been rather drawn out.