## “On Space and Time,” a book review – 1

While tidying up the ‘math center’ (a set of shelves in the dining room full of book, supplies and mathematical manipulatives) I came across a book I had apparently checked out of the library some time ago but have forgotten about.  The title is “On Space and Time” and its a collections of essays put together by the editor Shahn Majid.

I recognized two names, Connes and Penrose but decided to start at the beginning with Andrew Taylor’s essay which does a nice job of laying out current cosmological theories and evidence.  It was serendipitous that I was reading this when BICEP2 announced their results!

I had never heard of Shahn Majid (which means nothing of course) but am quite enamored with him.  His contribution to the book is an essay called ‘Quantum Spacetime and Physical Reality.’  It’s clear from the outset that he’s interested in a noncommutative approach to spacetime.  I can’t say that I know what this is yet, even months after developed an academic crush on Alain Connes who also is interested in fundamental physics via his noncommutative geometry.

I’m an unfortunate student pulled to the center of three areas of academics.  I find mathematics, philosophy and physics all very interesting, in particular where they overlap.  I love the rigor of pure mathematics (I’m a happy Platonist), I love the way physics weaves the ideas and language of mathematics into the siren song of the elegant universe we simpletons inhabit, and I love the deep questions philosophers of science and mathematics pose (the sorts of questions that cause normal sane folk to screw up their faces in disgust).

Majid seems to also be floating in that nexus of physics, mathematics and philosophy, though obviously more productively than I am.  I’d like to take this post and talk about his contribution to the book and why it excites me so.  I confess not having any deep understanding of his plan or vision, it’s written for a educated but nontechnical audience and I’m trying to focus on investigating previous ideas without becoming sniped by his many claims and assertions.

Majid has been playing with various ideas and models about quantum gravity and apparently has had some success in 3d spacetime.  His current goals are to determine what such a theory ought to look like, what attributes must it have?  As many people attempt to work on bits and pieces of the puzzle, what will the “final” theory look like and how will it reduce to those many pieces on which researches are finding success?  Majid covers up the category theory until the very end but he’s certainly looking forwards at a category theoretic physics, much as Baez is also.   In what follows I’ll put a bunch of quotes up and some commentary.  It may not appeal to you as it does me, folks have different aesthetics.

“Along the way, the essay provides at least some insight into such theoretical questions as ‘why are things quantized’, ‘why is there gravity’ and ‘why is there a cosmological constant’. “

Majid includes an interested plot of size vs. mass-energy.  On it he uses Compton’s equation $\lambda = h/mc$ and the Schwarzschild radius $r=GM/c^2$ and indicates regions of the plot that are forbidden by quantum theory and areas that are forbidden by general relativity.  There are two lines that intersect at Planck scale (Planck mass $2.18\times 10^{-5}g$ and length $1.62\times 10^{-33}m$).  He has this to say,

“…approaching which we would need a theory of quantum gravity to understand.  What is happening here is that as you try to probe smaller and smaller length scales using more and more massive quantum particles…you eventually need particles so big  that they form black holes… and mass up the very geometry you are trying to measure.  What this means is that distances less than $10^{-33}$cm are intrinsically unknowable.”

Majid spends some time commenting on the assumption of using a continuum model for space and time.  It’s a holdover, he claims, from previous physics which was set on scales based on our experiences.  Extrapolating to the very large and very small  with this notion of the continuum he takes issue with.  It is also here that he introduces the notion of duality, positing that if there is a minimum length scale, say Planck length, then there ought to be a minimum momentum scale which he produces a la de Broglie, $p_min \approx 2\times 10^{-55}$ g cm/s.  This becomes a recurring theme which I don’t think we can expect to understand or follow given the level of writing.  Roughly it seems as though the length scale is related to quantum phenomena while the momentum scale is related to gravitational phenomena a la stress energy tensor.  Majid suspects that the eventual theory of quantum gravity will have a split personality of sorts.  Depending on the length scale it will manifest the necessary attributes to explain observations but will smoothly transition between those scales.

A process I’d like to know more about is the process of taking a classical phase space and turning it into whatever it becomes quantum mechanically, an operator algebra?  Majid talks of q-deformations of groups that, in a limit, return to the original group.  I’ve read stuff like this also in Atiyah’s article on TQFTs.  I’m in the dark how its done but would love to know.  The way Majid ties this into quantum gravity is to suggest that the noncommutativity of spacetime is scale dependent.  It could be something like $q-1\approx 2\pi (l_{planck}/r)$ to first order.  For scales close to Planck length this noncommutativity would be obvious while for classical scales it would essentially vanish.

Proposition: in the context of quantum phase spaces which happen to be quantum groups, there is possible a dual interpretation in which the roles of observables and states are reversed. The existence of this possibility is a compatibility constraint between quantum theory and gravity in so far as expressed in the curvature of phase space.  The roles of position and momentum in the dual model are also reversed.

Yeah, so what does that mean?  I don’t understand the notion of duality he’s talking about.  If you read the essay and do understand it I’d love to hear your thoughts.  It sounds really interesting to be able, in essence or structure, to have one object display traits commonly found in quantum phenomena or gravitational phenomena.  Mind you, I’m not sure what those traits are in this context and I’ve become even less sure as I read this essay.

Majid is also a champion of pure mathematics,

“But Nature does not necessarily use the maths already in maths books, hence theoretical physicists should be prepared to explore the tableau of all of pure mathematics and pick out objectively what is needed, not merely the bits of structures they stumble upon by accident or fashion (which is what tends to happen).  This requires a very different mind set from the usual one in physics.  It is true that theoretical physicists can eventually take on new mathematical structures, for example quantum groups. But even as this new machinery becomes ‘absorbed’ the mindset in theoretical physics is to seek to apply it and not to understand the conceptual issues and freedoms.  By contrast a pure mathematician is sensitive to deeper structural issues and to the subtle interplay between definition and fact.”

Brazen no?  The final idea I want to chat about and quote concerns the end of the essay which comes a bit closer to unveiling some of the specifics of his research, though not too much.  It’s clear he’s looking for structure via category theory but what’s also clear is that he’s acting almost as a mathematical archaeologist sifting through various categories for evidence of physics.   I do not know what a self-dual category is, but apparently this what Majid is after and is related to the earlier bits about duality and representation theoretic models.  He has an interesting graphic which organizes various categories.  Crossing the image horizontally is a line of self dual categories.  Below it are categories that deal, in some way, with gravity and above the horizontal, categories dealing with quantum mechanics.  Just try reading this next bit without feeling heavy.

“Next, above the axis moving to Heyting algebras and beyond takes us into intuitionistic logic and ultimately into an axiomatic framework for quantum field theory.  A Heyting algebra describes logics in which one drops the ‘law of the excluded middle’.  This generalization is also the essential feature of the logical structure of quantum mechanics because in quantum theory a physical observable does not have to be either this or that value such as true or false, it can be in a mixed quantum state like Schrodinger’s cat.  Less familiar but dual to this is the notion of co-Heyting algebra and co-intuitionistic logic in which one drops the axiom that the intersection of a proposition and its negation is always false.  It has been argued by F. W. Lawvere and his school that this intersection $d(A)=A\cap \bar{A}$ is like the ‘boundary’ of the proposition, and, hence, that these co-Heyting algebras are the ‘birth’ of geometry… Thus we see that $d(A\cap B)=(A\cap dB)\cup (dA\cap B)$

This is awesome.  I’m interested in other logics.  It’s unsettling that mathematical truths are dependent on the logical system we’re in and that there are a couple different systems out there which very clever people have thought about.  In intuitionist logic, for example, it can be shown that every function from the reals to the reals is continuous. Every one.  That’s a very different result than what you’d find in classical analysis.  Which is the right logic?  I don’t think science necessarily settles the question.  Mathematical theorems have a scope that dwarfs reality.  They’re written and proven to resist any manner of monstrous counterexample that could crawl out of the dark corners of the Platonic realm.  Counterexamples that don’t show up in particle colliders or x-ray telescopes.  So while the implications of standard mathematical truths yields tools that physicists use to describe experiments in a consistent and predictive manner, it doesn’t imply that the mathematical truth is ‘true’ only that the use in science hasn’t strained the domain of the theorem.  I apologize for the awkwardness of the writing here.  These thoughts are hard to express and could simply be wrong, but they’re worrisome at the moment.

The above passage by Majid demonstrates the utility of two fairly odd algebras.  Algebras without the excluded middle, a tautology in classic logic and the co-algebra where “A and not-A” is no longer automatically a contradiction!  The idea that these two alternative logics could be categories for the birth of quantum and gravitational physics is profound and it’s hard to remain objective.  I would like it to be the case!

I’ll end with one more, which wraps up the above,

“What comes as the next even more general self-dual category of objects?  Undoubtedly something, but note that the last one considered is already speaking about categories of categories.  One cannot get too much more general that this without running out of mathematics itself.  In that sense physics is getting so abstract, our understanding of it so general, that we are nearing the ‘end of physics’ as we currently know it.  This explains from our point of view why quantum gravity already forces us to the edge of metaphysics and forces us to face the really ‘big’ questions.”

Mind you, Majid isn’t claiming that science stops, he notes that there’s always more to do.  From here he launches into his own bit of philosophy he calls relative realism which I don’t care to try and summarize.  All in all the article is fairly exciting, really brings some new ideas to the table and is a good read before taking a nice long walk.