Hestenes' Geometric Algebra. What good is it?

[CarlB]

Why don't we discuss the Geometric algebra and how it differs from other Clifford algebras?

For introduction, here's Hestenes' home page on Geometric calculus:

http://modelingnts.la.asu.edu/

This is an easy reading introduction:

(1) GA seamlessly integrates the properties of vectors and complex numbers to enable a completely coordinate-free treatment of 2D physics.

(2) GA articulates seamlessly with standard vector algebra to enable easy contact with standard literature and mathematical methods.

(3) GA Reduces \grad, div, curl and all that" to a single vector derivative that, among other things, combines the standard set of four Maxwell equations into a single equation and provides new methods to solve it.

(4) The GA formulation of spinors facilitates the treatment of rotations and rotational dynamics in both classical and quantum mechanics without coordinates or matrices.

(5) GA provides fresh insights into the geometric structure of quantum mechanics with implications for its physical interpretation. All of this generalizes smoothly to a completely coordinate-free language for spacetime physics and general relativity to be introduced in subsequent papers.
http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf

Carl

 

[Hurkyl]

It's not really a new thing, is it? I got the impression that his thesis is that it's a good thing to teach early in one's education, and that it's a generally good idea to phrase things in this language.

 

[CarlB]

From what I understand, the basics for GA dates to the mid 19th century, more or less. But Hestenes' articles on calculus and the like are from around the 1980s. He's a professor emeritus at Arizona State University in Tempe (outside Phoenix). I met him last summer when he kindly found time to talk to me at his office for most of an hour.

The reason I love the stuff is that it is very intuitive and easy to use.
There's a part of Hestenes' program that I disagree with and that's his attempt to avoid coordinates. Where possible, anything I write is knee deep in rectangular coordinates getting a workout. For example, if I want to prove something I will typically assume a convenient coordinate system and then show that the result is independent of the choice of coordinates. Hestenes will tend to give the proof without ever getting his hands dirty but I find it less convincing and more confusing.

One thing you quickly discover with the GA is that it has a lot of degrees of freedom. I generally work with one hidden dimension and a complexified GA, so there are 64 real degrees of freedom. Calculations can get wearing, so I wrote a Java applet calculator that I use all the time. The applet is here (along with a crystal drawing applet I wrote):
http://www.brannenworks.com

The calculator works by executing commands you type in. I'm all the time wanting new features so I just add them in and then delete them when I run out of command letters. So the source code is messy and the program is not bug free. But with that warning, I think that the following link (look for the file GAixyzst.jpx.zip ) works for anyone:
http://groups.yahoo.com/group/euclideanrelativity/files/

If someone wants to learn the GA, I think that they should come up with ideas for interesting things to do with it, and then spend a few weeks with a calculator trying to make it work.

Not knowing the "spectral decomposition theorem", I spent about a week looking for idempotents that had a real part of 1/3. This was in an attempt to get SU(3)'s quark irreps to fit into the GA naturally, but with idempotents. I eventually ended up writing code that found idempotents by using stuff similar to Newton's approximation but no matter what I tried, all the idempotents ended up with real parts of the form n/8 where n was a small integer.

By the way, Jose Almeida fit SU(3)xSU(2)xSU(1) into the complexified GA in this paper:
http://arxiv.org/abs/math.GM/0307165
Carl

 

[Haelfix]

Its hard for me to completely judge this body of work, as its really a different language, but there are a number of things that bother me about it.

He's complexified Maxwells equations, leading ot one equation, with diff forms you get 2. Ok so its not hard to see you get 1 equation out.

However therein lies a problem. Complexification is both topologically restricting and depends on the choice of complex structure. His formalism therefore, in general is not unique, and restrictive again to a subclass of physical theories.

Of course in that subclass he will have the full power of complex analysis, more or less built in, so its not surprising it could lead to quicker computation.

 

[CarlB]

However therein lies a problem. Complexification is both topologically restricting and depends on the choice of complex structure. His formalism therefore, in general is not unique, and restrictive again to a subclass of physical theories.

This is a very astute observation.

One of the things I like about the Geometric Algebra paradigm is the rejection of complex numbers per se. Where Hestenes uses "i" it is standing for an element of the geometric algebra that, when squared, gives -1.

My own interest in physics is in elementary particles and fields and from that point of view, E and B fields are just bulk properties of matter that are better described by QED. So while I agree that Hestenes' complexification of E&M is hokey, it doesn't bother me.

What does bother me is his use of an "i" in his conversion of the Dirac / Pauli / Schroedinger equations to GA form. I think that when he did this he erred. There are several other versions of QM on GA out in the literature by various authors, but I disagree with them too.

I'm writing up a version of QFT that uses Hestenes' GA as the underlying algebra for the field theory. The major difference between my theory and the rest is that I am basing it on the density operator (density matrix) formalism instead of the spinor formalism.

If you read Hestenes' several versions of QM on the GA, it will be obvious that the unphysical "i" that he uses, which basically shows up as a gauge freedom to choose the bivector that is interpreted as "i", disappears when you move from spinor form to density operator form. Here's an example of the problem from a Hestenes paper. I think this is the original 1981 paper that started all the QM on GA (small) industry. Look at eqn (5.7) on page 13. Ah what the heck, I'll type it in:

<<
The symbol

[tex]i = \gamma_1 \gamma_2 = \sigma_1 \sigma_2 = i\sigma_3 \;\;\;\;\;\;[/tex] (5.7)

emphasizes that this bivector plays the role of the imaginary [tex]i'[/tex] that appears explicitly in the matrix form (5.4) of the Dirac equation. To interpret the theory, it is crucial to note that the bivector [tex]i[/tex] has a definite geometrical interpretation while [tex]i'[/tex] does not.
http://modelingnts.la.asu.edu/pdf/Geom_Dirac.pdf

In the above, [tex]i'[/tex] is the usual imaginary constant in the Dirac equation while [tex]i = \sigma_1\sigma_2\sigma_3[/tex].

So what is happening in Hestenes' work is that he has to use [tex]i\sigma_3[/tex] and that is a gauge choice. It corresponds to the choice, in a 2-element Pauli type spinor, of which direction the two complex degrees of freedom are oriented with respect to. Typically, spinors are defined around the z direction hence the [tex]\sigma_3[/tex] in Hestenes' equation. This gives spinors where (1,0) and (0,1) transposed, are the eigenvectors for spin measured in the z direction (each eigenvector multiplied by anbitrary phase of course).

Now consider what happens to this when you pass to the density matrix formalism. When you "square" your spinors to turn them into density operators the arbitrary phase disappears (globally but not locally!). In the GA, the result is that the gauge choice [tex]\sigma_3[/tex] becomes squared to give [tex](\sigma_3)^2 =1[/tex]. In other words, the arbitary global gauge freedom melts away and Hestenes' formalism loses the arbitrary choice of imaginary constant. Does this happen as well in E&M? Being an elementary particles and fields hobbiest only, I don't know and care little. I never took undergraduate E&M and I think my intuition there is lacking. Perhaps a reader with a traditional physics education can comment.

So I agree with your misgivings. I felt them when I saw Hestenes' work as well. But I liked the idea so I figured out how to fix it. I'll put up a paper on the subject in a few days. It will be interesting to see if Arxiv accepts it or if I have to publish it under their math section or even self publish. I was very fortunate (or perhaps subtly pulled the right string) to get invited to the Particles and Nuclei International Conference in Santa Fe Oct 23-28 and will present the paper at their poster session.

Carl

 

[laserblue]

I know that Dr. Pertti Lounesto (http://users.tkk.fi/~ppuska/mirror/Lounesto/) has expressed some disagreement with Dr. Hestenes regarding Geometric Algebra and Geometric Calculus, but one can't walk three roads at the same time and Dr. Hestenes has done a good job of designing a mathematical language that can be used from fermis to light years. J.W. Gibbs tried to do something similar by cutting scalars off quaternions and advocating only vectors for use in physics. Clifford algebras are a part of quantum mechanics but they need to be a part of a unified mathematical language for all of physics. As Dr. Hestenes points out in the papers listed below, there is a lot of redundancy in mathematical physics which needs to be handled. In designing a better language, one has to take decisions and make tradeoffs. Is there some feature of other clifford algebras you would like to see in a better mathematical language for physics?
When I was young I read “Gravity” by Misner, Thorne and Wheeler (1979) and it used a good half dozen formalisms (vectors, tensors, differential forms, spinors etc.) and in the physics journals there sometimes seems to be one for each researcher (Van der Waerden symbols, tetrads etc.). Over the years I've seen Maxwell's equations expressed in terms of scalars, vectors, tensors, differential forms, quaternions, spinors, twistors and Geometric Algebra/Calculus.
Physicists have sometimes invented their own math and interpretations of the math and sometimes empirically spliced two different formalisms together such as with the use of the Pauli matrices in Quantum Mechanics. The Geometric Algebra/Calculus of Dr. Hestene was designed as a unified mathematical language and allows one a different viewpoint on the physical interpretation of the math. Sometimes seemingly different equations describe similar things as John Shive points out in "Similarities in Physics" or as Schrodinger showed regarding Heisenberg's Matrix Mechanics and Schrodinger's Wave Mechanics. Geometric Algebra allows one to see some similarities more easily and get General Relativity and Quantum Mechanics speaking the same mathematical language.
One of the problems in quantum mechanics involves complex amplitudes that has made some suggest a need for a complex probability calculus but the problem is really in interpreting what the appearance of the complex numbers mean. For example, several physicists have been intrigued by the idea of the Schrodinger equation as a diffusion equation with complex number elements. What do Hamilton-Jacobi theory and Hamilton's optico-mechanical analogy look like in Geometric Calculus? What is the source of the infinities in QED that needs renormalization procedures to avoid them? Physicists can get the right answers to 11 decimal places but they need to be more explicit in step two where they just replace infinite values with known values. Sure it works, but why?

Oersted Medal Lecture 2002: Reforming the Mathematical Langauge of Physics
by David Hestenes
(http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf )

“My purpose is to lay bare some serious misconceptions that complicate quantum
mechanics and obscure its relation to classical mechanics. The most basic of
these misconceptions is that the Pauli matrices are intrinsically related to spin.
On the contrary, I claim that their physical significance is derived solely from
their correspondence with orthogonal directions in space. The representation of
σi by 2×2 matrices is irrelevant to physics. That being so, it should be possible
to eliminate matrices altogether and make the geometric structure of quantum
mechanics explicit through direct formulation in terms of GA. How to do that
is explained below. For the moment, we note the potential for this change in
perspective to bring classical mechanics and quantum mechanics closer together.”

Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics
by David Hestenes
In the American Journal of Physics, Vol. 39/9, 1013-1027, September 1971.

"Though the geometric algebra discussed here is isomorphic to the so-called
"Pauli (matrix) algebra," the interpretations of the two systems differ considerably,
and the practical consequences of this difference are not trivial. Thus, questions of
the representation of Pauli matrices and of transformations among representations
never arise in geometric algebra, because they are irrelevant. Matrix algebra was
invented to describe linear transformations. So it should not be surprising to find
that matrices have irrelevant features when they are used to represent objects of
a different nature. From the geometric viewpoint of geometric algebra, matrices
are seen to arise in the theory of linear geometric functions; that is, geometrics
are taken to be more fundamental than matrices, rather than the other around.
Simplifications which result from this reversal of viewpoint are manifest in text and
references of this paper."

MATHEMATICAL VIRUSES*
by David Hestenes
http://modelingnts.la.asu.edu/pdf/MathViruses.pdf

* In: A.Micali et al., Clifford Algebras and their Applications in Mathematical Physics,
3-16. 1992 Kluwer Academic Publishers.

CLIFFORD ALGEBRA AND THE INTERPRETATION
OF QUANTUM MECHANICS
by David Hestenes
http://modelingnts.la.asu.edu/pdf/caiqm.pdf

In: J.S.R. Chisholm/A.K. Commons (Eds.), Clifford Algebras and their Applications in
Mathematical Physics. Reidel, Dordrecht/Boston (1986), 321-346.

A UNIFIED LANGUAGE FOR MATHEMATICS AND PHYSICS
by DAVID HESTENES
http://modelingnts.la.asu.edu/html/GeoCalc.html

In: J.S.R. Chisholm/A.K. Commons (Eds.), Clifford Algebras and their Applications in
Mathematical Physics. Reidel, Dordrecht/Boston (1986), 1-23.


THE VECTOR CALCULUS GAP:
Mathematics does not = Physics
by Tevian Dray and Corinne A. Manogue
(24 September 1998)
(http://www.math.oregonstate.edu/bridge/papers/calculus.pdf)

BRIDGING THE VECTOR (CALCULUS) GAP
TEVIAN DRAY and CORINNE A. MANOGUE
(http://www.physics.orst.edu/bridge/papers/pathways.pdf)

Bridging the Vector Calculus Gap Workshop
(http://www.math.oregonstate.edu/bridge/)

 

[Surprised]

I would have thought this question (What good is it?) was moot.

Geometric Algebra together with its associated calculus is successful and widely used in many areas including Physics.

 

[LukeD]

Woah, what a timely Zombie Thread. I started reading about GA about a week ago.
As far as I can tell (which isn't too far as I haven't been reading about it for too long), GA is exactly the same as Clifford Algebra. The only difference is where Hestenes puts his emphasis.
Personally, I really like it because it provides a clear geometric picture for any expression in a real Clifford Algebra.

I suppose I disagree with Hestenes' banishment of the Complex numbers because they can be very useful. However, as far as I can tell, no one has provided a very clear geometric picture of the complex Clifford Algebras. I believe that such a thing would make the Geometric Algebra all the more powerful (no longer being limited to real Clifford Algebras)

So I'm wondering, does anyone have a geometric picture for a Complex Clifford Algebra? I was trying to construct one by introducing axial vectors and a preferred orientation. I'm thinking though that this might be exactly the same as just defining i = unit volume element.

 

[LukeD]

Hestenes' view seems to be that whenever the imaginary unit i is seen in an equation, we can instead see it as an element of a real Clifford algebra. I suppose that this answers my call for a geometric view of complex Clifford algebras. However, it looks to me (and I've just begun studying Clifford algebras, so I may be very wrong), that this isn't a unique process. It seems that we can get extra structure in this new real Clifford algebra that was not present in the original complex Clifford algebra. Am I justified in suspecting this? Is there some algebraic construction that can get rid of the extra structure?

 

[Surprised]

Hi Luke

If you are interested in a complexified geometric algebra, have a look here:

http://www.garretstar.com/gsobczyk.pdf (Unitary Geometric Algebra).

Although algebraically a Clifford algebra, the GA view is essentially that you may always seek a physical interpretation of any imaginary unit that appears.

In fact, complex numbers have not been banished so much as replaced by the realization that there are multiple algebraic entities that square to -1 (indeed +1 as well).

Anything you can do with complex numbers and quaternions can be handled in the real algebra; if needed translation between the usual forms and GA is simple.

After a while, you will see that the "extra structure" causes no difficulty as you can easily split into even/odd components.

Once you get into the swing of marrying up your (classical) geometric instincts with the the algebraic formulation things become a lot clearer.

 

[hansen97]

I'm writing up a version of QFT that uses Hestenes' GA as the underlying algebra for the field theory. The major difference between my theory and the rest is that I am basing it on the density operator (density matrix) formalism instead of the spinor formalism...

So I agree with your misgivings. I felt them when I saw Hestenes' work as well. But I liked the idea so I figured out how to fix it. I'll put up a paper on the subject in a few days. It will be interesting to see if Arxiv accepts it or if I have to publish it under their math section or even self publish. I was very fortunate (or perhaps subtly pulled the right string) to get invited to the Particles and Nuclei International Conference in Santa Fe Oct 23-28 and will present the paper at their poster session.

Carl

Have you finished you formulation of QFT in terms of GA? If so, I would very much like to see it.

On a different note, I don't understand the confusion as to the uniqueness of Hestenes' GA formulation. Of course it's not unique. If it were we would just be discovering QM and other such things GA can be used to describe (the different descriptions of the same things are called isomorphisms). Having a formulation that is background, or coordinate, independent allows for the theory to be generalized and much easier to handle (i.e. showing that your formulation is invariant to rotations, etc.). This is one problem physics is running into with gravity-QM unification, the one is background independent whereas the other is not.

One thing I've found useful when trying to understand what is stated above is a Jewish proverb "'For example...' is not proof."