Mathematics is big, people are small. So why should we
learn the obscure techniques of
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
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
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
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
[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
Lake Chelan. The next movie is Against
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
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
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
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
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
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,
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
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
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
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.