April 2, 1999

This Week's Finds in Mathematical Physics (Week 132)

John Baez

Today I want to talk about n-categories and quantum gravity again. For starters let me quote from a paper of mine about this stuff:

1) John Baez, Higher-dimensional algebra and Planck-scale physics, to appear in Physics Meets Philosophy at the Planck Scale, eds. Craig Callender and Nick Huggett, Cambridge U. Press. Preprint available as gr-qc/9902017.

By the way, this book should be pretty fun to read - it'll contain papers by both philosophers and physicists, including a bunch who have already graced the pages of This Week's Finds, like Barbour, Isham, Rovelli, Unruh, and Witten. I'll say more about it when it comes out.

Okay, here are some snippets from this paper. It starts out talking about the meaning of the Planck length, why it may be important in quantum gravity, and what a theory of quantum gravity should be like:

Two constants appear throughout general relativity: the speed of light c and Newton's gravitational constant G. This should be no surprise, since Einstein created general relativity to reconcile the success of Newton's theory of gravity, based on instantaneous action at a distance, with his new theory of special relativity, in which no influence travels faster than light. The constant c also appears in quantum field theory, but paired with a different partner: Planck's constant ħ. The reason is that quantum field theory takes into account special relativity and quantum theory, in which ħ sets the scale at which the uncertainty principle becomes important.

It is reasonable to suspect that any theory reconciling general relativity and quantum theory will involve all three constants c, G, and ħ. Planck noted that apart from numerical factors there is a unique way to use these constants to define units of length, time, and mass. For example, we can define the unit of length now called the `Planck length' as follows:

L = sqrt(ħ G /c^3)

This is extremely small: about 1.6 x 10^{-35} meters. Physicists have long suspected that quantum gravity will become important for understanding physics at about this scale. The reason is very simple: any calculation that predicts a length using only the constants c, G and ħ must give the Planck length, possibly multiplied by an unimportant numerical factor like 2π.

For example, quantum field theory says that associated to any mass m there is a length called its Compton wavelength, L_C, such that determining the position of a particle of mass m to within one Compton wavelength requires enough energy to create another particle of that mass. Particle creation is a quintessentially quantum-field-theoretic phenomenon. Thus we may say that the Compton wavelength sets the distance scale at which quantum field theory becomes crucial for understanding the behavior of a particle of a given mass. On the other hand, general relativity says that associated to any mass m there is a length called the Schwarzschild radius, L_S, such that compressing an object of mass m to a size smaller than this results in the formation of a black hole. The Schwarzschild radius is roughly the distance scale at which general relativity becomes crucial for understanding the behavior of an object of a given mass. Now, ignoring some numerical factors, we have

L_C = ħ/mc


L_S = Gm/c^2.

These two lengths become equal when m is the Planck mass. And when this happens, they both equal the Planck length!

At least naively, we thus expect that both general relativity and quantum field theory would be needed to understand the behavior of an object whose mass is about the Planck mass and whose radius is about the Planck length. This not only explains some of the importance of the Planck scale, but also some of the difficulties in obtaining experimental evidence about physics at this scale. Most of our information about general relativity comes from observing heavy objects like planets and stars, for which L_S >> L_C. Most of our information about quantum field theory comes from observing light objects like electrons and protons, for which L_C >> L_S. The Planck mass is intermediate between these: about the mass of a largish cell. But the Planck length is about 10^{-20} times the radius of a proton! To study a situation where both general relativity and quantum field theory are important, we could try to compress a cell to a size 10^{-20} times that of a proton. We know no reason why this is impossible in principle. But we have no idea how to actually accomplish such a feat.

There are some well-known loopholes in the above argument. The `unimportant numerical factor' I mentioned above might actually be very large, or very small. A theory of quantum gravity might make testable predictions of dimensionless quantities like the ratio of the muon and electron masses. For that matter, a theory of quantum gravity might involve physical constants other than c, G, and ħ. The latter two alternatives are especially plausible if we study quantum gravity as part of a larger theory describing other forces and particles. However, even though we cannot prove that the Planck length is significant for quantum gravity, I think we can glean some wisdom from pondering the constants c,G, and ħ - and more importantly, the physical insights that lead us to regard these constants as important.

What is the importance of the constant c? In special relativity, what matters is the appearance of this constant in the Minkowski metric

ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2

which defines the geometry of spacetime, and in particular the lightcone through each point. Stepping back from the specific formalism here, we can see several ideas at work. First, space and time form a unified whole which can be thought of geometrically. Second, the quantities whose values we seek to predict are localized. That is, we can measure them in small regions of spacetime (sometimes idealized as points). Physicists call such quantities `local degrees of freedom'. And third, to predict the value of a quantity that can be measured in some region R, we only need to use values of quantities measured in regions that stand in a certain geometrical relation to R. This relation is called the `causal structure' of spacetime. For example, in a relativistic field theory, to predict the value of the fields in some region R, it suffices to use their values in any other region that intersects every timelike path passing through R. The common way of summarizing this idea is to say that nothing travels faster than light. I prefer to say that a good theory of physics should have *local degrees of freedom propagating causally*.

In Newtonian gravity, G is simply the strength of the gravitational field. It takes on a deeper significance in general relativity, where the gravitational field is described in terms of the curvature of the spacetime metric. Unlike in special relativity, where the Minkowski metric is a `background structure' given a priori, in general relativity the metric is treated as a field which not only affects, but also is affected by, the other fields present. In other words, the geometry of spacetime becomes a local degree of freedom of the theory. Quantitatively, the interaction of the metric and other fields is described by Einstein's equation

G_{ab} = 8 π G T_{ab}

where the Einstein tensor G_{ab} depends on the curvature of the metric, while the stress-energy tensor T_{ab} describes the flow of energy and momentum due to all the other fields. The role of the constant G is thus simply to quantify how much the geometry of spacetime is affected by other fields. Over the years, people have realized that the great lesson of general relativity is that a good theory of physics should contain no geometrical structures that affect local degrees of freedom while remaining unaffected by them. Instead, all geometrical structures - and in particular the causal structure - should themselves be local degrees of freedom. For short, one says that the theory should be background-free.

The struggle to free ourselves from background structures began long before Einstein developed general relativity, and is still not complete. The conflict between Ptolemaic and Copernican cosmologies, the dispute between Newton and Leibniz concerning absolute and relative motion, and the modern arguments concerning the `problem of time' in quantum gravity - all are but chapters in the story of this struggle. I do not have room to sketch this story here, nor even to make more precise the all-important notion of `geometrical structure'. I can only point the reader towards the literature, starting perhaps with the books by Barbour and Earman, various papers by Rovelli, and the many references therein.

Finally, what of ħ? In quantum theory, this appears most prominently in the commutation relation between the momentum p and position q of a particle:

pq - qp = -i ħ,

together with similar commutation relations involving other pairs of measurable quantities. Because our ability to measure two quantities simultaneously with complete precision is limited by their failure to commute, ħ quantifies our inability to simultaneously know everything one might choose to know about the world. But there is far more to quantum theory than the uncertainty principle. In practice, ħ comes along with the whole formalism of complex Hilbert spaces and linear operators.

There is a widespread sense that the principles behind quantum theory are poorly understood compared to those of general relativity. This has led to many discussions about interpretational issues. However, I do not think that quantum theory will lose its mystery through such discussions. I believe the real challenge is to better understand why the mathematical formalism of quantum theory is precisely what it is. Research in quantum logic has done a wonderful job of understanding the field of candidates from which the particular formalism we use has been chosen. But what is so special about this particular choice? Why, for example, do we use complex Hilbert spaces rather than real or quaternionic ones? Is this decision made solely to fit the experimental data, or is there a deeper reason? Since questions like this do not yet have clear answers, I shall summarize the physical insight behind ħ by saying simply that a good theory of the physical universe should be a quantum theory - leaving open the possibility of eventually saying something more illuminating.

Having attempted to extract the ideas lying behind the constants c, G, and ħ, we are in a better position to understand the task of constructing a theory of quantum gravity. General relativity acknowledges the importance of c and G but idealizes reality by treating ħ as negligibly small. From our discussion above, we see that this is because general relativity is a background-free classical theory with local degrees of freedom propagating causally. On the other hand, quantum field theory as normally practiced acknowledges c and ħ but treats G as negligible, because it is a background-dependent quantum theory with local degrees of freedom propagating causally.

The most conservative approach to quantum gravity is to seek a theory that combines the best features of general relativity and quantum field theory. To do this, we must try to find a *background-free quantum theory with local degrees of freedom propagating causally*. While this approach may not succeed, it is definitely worth pursuing. Given the lack of experimental evidence that would point us towards fundamentally new principles, we should do our best to understand the full implications of the principles we already have!

From my description of the goal one can perhaps see some of the difficulties. Since quantum gravity should be background-free, the geometrical structures defining the causal structure of spacetime should themselves be local degrees of freedom propagating causally. This much is already true in general relativity. But because quantum gravity should be a quantum theory, these degrees of freedom should be treated quantum-mechanically. So at the very least, we should develop a quantum theory of some sort of geometrical structure that can define a causal structure on spacetime.

Then I talk about topological quantum field theories, which are background-free quantum theories without local degrees of freedom, and what we have learned from them. Basically what we've learned is that there's a deep analogy between the mathematics of spacetime (e.g. differential topology) and the mathematics of quantum theory. This is interesting because in background-free quantum theories we expect that spacetime, instead of serving as a "stage" on which events play out, actually becomes part of the play of events itself - and must itself be described using quantum theory. So it's very interesting to try to connect the concepts of spacetime and quantum theory. The analogy goes like this:

     DIFFERENTIAL TOPOLOGY                  QUANTUM THEORY            

  (n-1)-dimensional manifold                Hilbert space             
          (space)                             (states)                  

cobordism between (n-1)-dimensional           operator    
          manifolds                           (process)                 

  composition of cobordisms            composition of operators  

      identity cobordism                  identity operator         

And if you know a little category theory, you'll see what we have here are two categories: the category of cobordisms and the category of Hilbert spaces. A topological quantum field theory is a functor from the first to the second....

Okay, now for some other papers:

2) Geraldine Brady and Todd H. Trimble. A string diagram calculus for predicate logic, and C. S. Peirce's system Beta, available at http://people.cs.uchicago.edu/~ brady

Geraldine Brady and Todd H. Trimble, A categorical interpretation of Peirce's propositional logic Alpha, Jour. Pure and Appl. Alg. 149 (2000), 213-239.

Geraldine Brady and Todd H. Trimble, The topology of relational calculus.

Charles Peirce is a famously underappreciated American philosopher who worked in the late 1800s. Among other things, like being the father of pragmatism, he is also one of the fathers of higher-dimensional algebra. As you surely know if you've read me often enough, part of the point of higher-dimensional algebra is to break out of "linear thinking". By "linear thinking" I mean the tendency to do mathematics in ways that are easily expressed in terms of 1-dimensional strings of symbols. In his work on logic, Peirce burst free into higher dimensions. He developed a way of reasoning using diagrams that he called "existential graphs". Unfortunately this work by Peirce was never published! One reason is that existential graphs were difficult and expensive to print. As a result, his ideas languished in obscurity.

By now it's clear that higher-dimensional algebra is useful in physics: examples include Feynman diagrams and the spin networks of Penrose. The theory of n-categories is beginning to provide a systematic language for all these techniques. So it's worth re-evaluating Peirce's work and seeing how it fits into the picture. And this is what the papers by Brady and Trimble do!

3) J. Scott Carter, Louis H. Kauffman, and Masahico Saito, Structures and diagrammatics of four dimensional topological lattice field theories, to appear in Adv. Math., preprint available as math.GT/9806023.

We can get 3-dimensional topological quantum field theories from certain Hopf algebras. As I described in "week38", Crane and Frenkel made the suggestion that by categorifying this construction we should get 4-dimensional TQFTs from certain Hopf categories. This paper makes the suggestion precise in a certain class of examples! Basically these are categorified versions of the Dijkgraaf-Witten theory.

4) J. Scott Carter, Daniel Jelsovsky, Selichi Kamada, Laurel Langford and Masahico Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, preprint available as math.GT/9903135.

Yet another attack on higher dimensions! This one gets invariants of 2-links - surfaces embedded in R^4 - from the cohomology groups of "quandles". I don't really understand how this fits into the overall scheme of higher-dimensional algebra yet. They show their invariant distinguishes between the 2-twist spun trefoil (a certain sphere knotted in R^4) and the same sphere with the reversed orientation.

5) Tom Leinster, Structures in higher-dimensional category theory, preprint available at http://www.dpmms.cam.ac.uk/~leinster

This is a nice tour of ideas in higher-dimensional algebra. Right now one big problem with the subject is that there are lots of approaches and not a clear enough picture of how they fit together. Leinster's paper is an attempt to start seeing how things fit together.

6) Claudio Hermida, Higher-dimensional multicategories, slides of a lecture given in 1997, available at http://www.math.mcgill.ca/~hermida

This talk presents some of the work by Makkai, Power and Hermida on their definition of n-categories. For more on their work see "week107".

7) Carlos Simpson, On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani's weak n-categories, preprint available as math.CT/9810058.

For quite a while now James Dolan and I have been talking about something we call the "stabilization hypothesis". I gave an explanation of this in "week121", but briefly, it says that the nth column of the following chart (which extends to infinity in both directions) stabilizes after 2n+2 rows:

                   k-tuply monoidal n-categories 

              n = 0           n = 1             n = 2

k = 0         sets          categories         2-categories

k = 1        monoids         monoidal           monoidal
                            categories        2-categories

k = 2       commutative      braided            braided
             monoids         monoidal           monoidal
                            categories        2-categories 

k = 3         " "           symmetric            weakly
                             monoidal          involutory
                            categories          monoidal

k = 4         " "             " "               strongly 

k = 5         " "             " "                "  "

Carlos Simpson has now made this hypothesis precise and proved it using Tamsamani's definition of n-categories! And he did it using the same techniques that Graeme Segal used to study k-fold loop spaces... exploiting the relation between n-categories and homotopy theory. This makes me really happy.

8) Mark Hovey, Model Categories, American Mathematical Society Mathematical Surveys and Monographs, vol 63., Providence, Rhode Island, 1999. Preprint available as http://www.math.uiuc.edu/K-theory/0278/index.html

Speaking of that kind of thing, the technique of model categories is really important for homotopy theory and n-categories, and this book is a really great place to learn about it.

9) Frank Quinn, Group-categories, preprint available as math.GT/9811047.

This one is about the algebra behind certain topological quantum field theories. I'll just quote the abstract:

A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible simple objects. This paper gives a detailed exploration of "topological quantum field theories" for group-categories, in hopes of finding clues to a better understanding of the general situation. Group-categories are classified in several ways extending results of Frohlich and Kerler. Topological field theories based on homology and cohomology are constructed, and these are shown to include theories obtained from group-categories by Reshetikhin-Turaev constructions. Braided-commutative categories most naturally give theories on 4-manifold thickenings of 2-complexes; the usual 3-manifold theories are obtained from these by normalizing them (using results of Kirby) to depend mostly on the boundary of the thickening. This is worked out for group-categories, and in particular we determine when the normalization is possible and when it is not.

10) Sjoerd Crans, A tensor product for Gray-categories, Theory and Applications of Categories, Vol. 5, 1999, No. 2, pp 12-69, available at http://www.tac.mta.ca/tac/volumes/1999/n2/abstract.html

A Gray-category is what some people call a semistrict 3-category: not as general as a weak 3-category, but general enough. Technically, Gray-categories are defined as categories enriched over the category of 2-categories equipped with a tensor product invented by John Gray. To define semistrict 4-categories one might similarly try to equip Gray-categories with a suitable tensor product. And this is what Crans is studying. Let me quote the abstract:

In this paper I extend Gray's tensor product of 2-categories to a new tensor product of Gray-categories. I give a description in terms of generators and relations, one of the relations being an ``interchange'' relation, and a description similar to Gray's description of his tensor product of 2-categories. I show that this tensor product of Gray-categories satisfies a universal property with respect to quasi-functors of two variables, which are defined in terms of lax-natural transformations between Gray-categories. The main result is that this tensor product is part of a monoidal structure on Gray-Cat, the proof requiring interchange in an essential way. However, this does not give a monoidal {(bi)closed} structure, precisely because of interchange And although I define composition of lax-natural transformations, this composite need not be a lax-natural transformation again, making Gray-Cat only a partial Gray-Cat-cateegory.

© 1999 John Baez