Friday, July 25, 2014

Cartan structural equations

Elie Cartan was a French mathematician who made fundamental contributions in differential geometry and group theory.

Today I want to continue the discussion regarding curvature and talk about Cartan's modern approach to differential geometry. Cartan died in 1951, three years before Yang and Mills wrote their seminal paper on gauge theory and he did not see how his ideas are used by Nature in all the four fundamental interactions: gravity, electromagnetism, weak force, and strong force.

Last time I presented the Christoffel symbols and I noted that they depend on the coordinate system used. Riemann uncovered an intrinsic geometric object called Riemann curvature tensor which expressed in terms of the Cristoffel symbols is:

\( R^{\delta}_{\alpha \beta \gamma} = \partial_\alpha \Gamma^{\delta}_{\beta \gamma} - \partial_\beta \Gamma^{\delta}_{\alpha \gamma} + \Gamma^{\delta}_{\alpha \mu} \Gamma^{\mu}_{\beta \gamma} - \Gamma^{\delta}_{\beta \mu} \Gamma^{\mu}_{\alpha \gamma} \)

Now recall the exterior product: \( a \wedge b = a\otimes b -b \otimes a\) which is skew-symmetric: \( a\wedge b = - b \wedge a\) 

\( \omega^\delta_\gamma = \Gamma^{\delta}_{\beta \gamma} dx^\beta \)

and compute \( d \omega^\delta_\gamma \) and \( \omega^\delta_\mu \wedge \omega^\mu_\gamma \) and compare the sum with the curvature equation in terms of \( \Gamma \) we get Cartan's local structural equation:

\( \Omega^\delta_\gamma = d \omega^\delta_\gamma + \omega^\delta_\mu \wedge \omega^\mu_\gamma \)

where \( \Omega^\delta_\gamma = \frac{1}{2} R^{\delta}_{\alpha \beta \gamma} d x^\alpha \wedge d x^\beta \)

In matrix form: \( A = \omega^\delta_\gamma \) is the local connection, and \( F = \Omega^\delta_\gamma \) is the local curvature.  

Cartan's local structural equation is:

\( F = d A + A \wedge A\)

which differentiated yields Bianchi's local identity:

\( d F = F \wedge A - A \wedge F \)

Now those two equations are a thing of maximal physical beauty. Sure, they look alien and vaguely interesting in this abstract mathematical form which Cartan derived, but inside them hides Maxwell's equations, Yang-Mills gauge theory, and Einstein's general relativity. In the next posts it will be our job to extract the physics from them.  

For example if \( A \) is the electromagnetic 4-potential because the electromagnetic gauge theory is abelian, \( A \wedge A = 0 \) and the two equations from above are nothing but Maxwell's equations as we will show later. Please stay tuned.


  1. Florin,

    Your posts are good, and I presume you are leading to something with these discussions on differential geometry.

    Given that you worked out theorems involving tensor products in QM I would like to draw your attention to this:

    There may be a big problem with this. However I would like to see your opinion on this prior to raising my question.

    Cheers LC

  2. Thanks Lawrence.

    Indeed ultimately I am driving to explain the Standard Model. Thanks for the link, I have read the archive paper; it will most likely appear on Nature Physics by the way is written. Looks flashy but it does not contain any deep QM insights other than superposition and its counter-intuitive consequences.

    If you follow the archive, I submitted my QM reconstruction paper late last night and if all goes well it should be announced tonight. I picked Foundation of Physics as the journal and last week was all about formatting the beast Springer style. The bulk of the work was on using BibTex for the massive reference sections.

    I am thinking to pause the Standard Model series for a while and do some publicity for the paper. The result is a really big deal.

    1. I just tried posting here and I think it went wrong. I tried recently to submit a paper to Foundations of Physics. I used the template for the Springer publications, but I just could not get the thing to work., How do you get this to work?

      Cheers LC

    2. It was a major pain. First I had to install the style file in a bizarre place so that MikTex can find it. Then it was not so bad until I had to convert the references: I used their template and copied the content section by section into it. But when I reached the references it took me a long time to make it work. I took the references out of the file and uploaded on the archive as a separate bbl file.

      You can download the source from to see how it is put together. If you like the paper, vote for it on

      I did not submit the paper yet on Foundations of Physics, I'll do it in about a month after I collect some more feedback.

  3. You wrote on the FQXI blog that you were interested in the question on what is time. I have come to a few insights on this. Quantum mechanics or really quantum gravity has no meaning to time. The Wheeler DeWitt equation illustrates this. Also in a general manifold there is no Gaussian surface where one can identify “mass-energy is in there.” Yet locally time can exist. This implies that processes such as Hawking radiation can only be unitary if time is local. Globally there is no such thing as time.

    I can write more if you are interested. I would also be interested in knowing more about how you got that *&^*$*#@ submission thing to work as FOOP. I could work it in an editor with the Springer format, but on the FOOP website everything went the way of entropy.

    Cheers LC

    1. Sorry, I forgot to say: please do write more, I am interested.

  4. PS, your paper appears to be a revision to the paper I read last September or October. The main idea appears to be much the same. If I remember properly it did appear that what you wrote was a foundation for quantum entanglements.


  5. Lawrence, are you sure was not one of my older preprints? Initially I had this but that was incomplete on two counts: how to properly deal with the hyperbolic case, and how to prove the skew symmetry of the Lie algebra product. That forced me to use more (now unnecessary) assumptions at that time.

    Time is a very tricky business. If it is emergent, why does it exist? From what principles should be derived? Don't know it yet.

    For Springer start with their sample file and make it work as is. After working all its kinks start replacing its content with your content one piece at a time and rebuild the output after every change. Then if something breaks you have the problem isolated and can try to understand why it no longer works. It took me a week to do this process.