### Light Nostalgia

Louise Riofrio continues with excellent cosmology posts, and now Carl Brannen also weighs in on the subject. I was wondering what originally got me very interested in the subject of a varying $c$, and I decided it probably happened around 1995, when I spent a few months studying the early physics papers on quantum group fiber bundles.

I seem to recall that these papers were not particularly mathematically sophisticated, but one element stood out: whereas a classical principle bundle looks the same at every point, the deformation parameter in a quantum bundle may easily vary from point to point. Even in those days, people thought a lot about relating deformation parameters to $\hbar$. This was all just a mathematical curiosity, until it became clear that some tough (and extremely interesting) algebraic geometry, and other mathematics, lay at the bottom of it. (Of course, all roads led to category theory in the end).

Algebraic geometers love spaces with extra structure which varies from point to point. They talk about spectra (usually of rings) and we need not be afraid of these gadgets because they are naturally specified by a functor from a suitable category of algebras into a category of spaces. And it turns out that this functor is best understood from the point of view of a special topos, because the weird topologies that algebraic geometers like to use are neatly encoded by axioms of Grothendieck. (In fact, this is where the idea of a topos comes from in the first place).

At the time, I believe it was Zamolodchikov who advised me to ditch lattice gauge theory (which I was supposed to be doing) for something more interesting. In the end, I did give up the lattice gauge theory, but I can't say it was because I listened to anybody's advice. (And as it turns out, lattice gauge theory has actually done rather well over the last decade).

I seem to recall that these papers were not particularly mathematically sophisticated, but one element stood out: whereas a classical principle bundle looks the same at every point, the deformation parameter in a quantum bundle may easily vary from point to point. Even in those days, people thought a lot about relating deformation parameters to $\hbar$. This was all just a mathematical curiosity, until it became clear that some tough (and extremely interesting) algebraic geometry, and other mathematics, lay at the bottom of it. (Of course, all roads led to category theory in the end).

Algebraic geometers love spaces with extra structure which varies from point to point. They talk about spectra (usually of rings) and we need not be afraid of these gadgets because they are naturally specified by a functor from a suitable category of algebras into a category of spaces. And it turns out that this functor is best understood from the point of view of a special topos, because the weird topologies that algebraic geometers like to use are neatly encoded by axioms of Grothendieck. (In fact, this is where the idea of a topos comes from in the first place).

At the time, I believe it was Zamolodchikov who advised me to ditch lattice gauge theory (which I was supposed to be doing) for something more interesting. In the end, I did give up the lattice gauge theory, but I can't say it was because I listened to anybody's advice. (And as it turns out, lattice gauge theory has actually done rather well over the last decade).

## 12 Comments:

As always, your support is appreciated. As someone commented on the blog, suspecting that c is changing shows that great minds do think alike.

And thank you for your support also, Louise. You are very kind.

By its very nature c=1 and cannot change. It represents the scale of space vs. time in a locally pseudo-Euclidean metric space. Technically the distance of two points has the form

D = k log R

where R is a cross-ratio of 4 points formed by joining the points by a line and then letting that line cut the fundamental quadric in 2 more points. The constant k is arbitrary, but fixed. For Euclidean geometry is it imaginary. For Minkowski geometry it is real. There is nothing complicated about this. VSL is a canard which arises from a misunderstanding of the role of C in relativity. It is not so much an actual speed (despite having the dimensions of velocity) as it is an arbitrary scale factor for measuring space vs. time, that is, for converting seconds to meters.

-drl

DRL, please do not confuse popular VSL theories with Louise's cosmology, in which you may well think of c as a scale factor.

There can be no variation of c, just as there can be no variation of i. The latter is trivially obvious on some level, but really is not - the ultimate origin of i in geometry comes from the invariant characterization of Euclidean geometry as a projective-affine geometry with the fundamental quadric

x1^2 + .. + xN^2 = 0

Similarly for c. It's not that it's just a scale factor - it's simply a consequence of the choice of units. I did not mean scale factor in a conformal sense. If you wish to interpret conformal geometry as VSL, then this is a matter of interpretation, but again, one is forcing the invariant characterization of the geometry into the wrong box.

A similar issue arises with other misinterpretations of SR (e.g. pole in barn, Lorentz contraction, moving mirrors etc.) The VSL interpretation is simply wrong, and for simple, irrefutable reasons.

-drl

Of course it's a question of interpretation. If you are willing to criticize a theory without putting any effort into understanding it, this discussion is over.

My point is that this is not a matter of interpretation - it's a matter of what this constant means in the context of geometry. The "speed of light" is not a speed as such, but a constant characterizing the geometry. That is has the dimensions of a velocity is incidental to its actual role in characterizing the geometry. This is not an arcane or trivial point. It goes to the very heart of the meaning of pseudo-Euclidean geometry and its physical implications.

-drl

DRL, why do you think I am mentioning algebraic geometry here? Your point is the whole point of this post. And c changes.

This is why the modern world is so interminably frustrating.

What I am talking about was known, with perfect understanding, to Felix Klein, at the turn of the next-to-last century. However, no one knows this work - it is ignorance of history that is ruining our world, in all its details. There is no point in talking, because people have no context in which to listen. I could explain, but there is no desire to learn.

-drl

Speak for yourself.

Your assumption of my (and others') ignorance is the hallmark of somebody who is not willing to listen. More has happened in Geometry since the days of Klein.

Once a problem is solved there is no need to invent more complex explanations. The wonders of algebraic geometry and principal bundles are completely irrelevant to this problem. That is my point. The reason physics has gotten itself in the shape it's now in, is that people have forgotten how to actually solve problems. They are supplied with an enormous arsenal of powerful techniques which are wielded with no skill, and so they create a wasteland of logic chopping and endless discussion of nearly trivial points that were settled decades ago. Anyone with any sense of history would be dismayed. I certainly am.

-drl

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