April 23, 1999

This Week's Finds in Mathematical Physics (Week 133)

John Baez

I'd like to start with a long quote from a paper by Ashtekar:

1) Abhay Ashtekar, Quantum Mechanics of Geometry, preprint available as gr-qc/9901023.

During his Goettingen inaugural address in 1854, Riemann suggested that the geometry of space may be more than just a fiducial, mathematical entity serving as a passive stage for physical phenomena, and may in fact have direct physical meaning in its own right. General relativity provided a brilliant confirmation of this vision: curvature of space now encodes the physical gravitational field. This shift is profound. To bring out the contrast, let me recall the situation in Newtonian physics. There, space forms an inert arena on which the dynamics of physical systems - such as the solar system - unfolds. It is like a stage, an unchanging backdrop for all of physics. In general relativity, by contrast, the situation is very different. Einstein's equations tell us that matter curves space. Geometry is no longer immune to change. It reacts to matter. It is dynamical. It has "physical degrees of freedom" in its own right. In general relativity, the stage disappears and joins the troupe of actors! Geometry is a physical entity, very much like matter.

Now, the physics of this century has shown us that matter has constituents and the 3-dimensional objects we perceive as solids are in fact made of atoms. The continuum description of matter is an approximation which succeeds brilliantly in the macroscopic regime but fails hopelessly at the atomic scale. It is therefore natural to ask: Is the same true of geometry? If so, what is the analog of the `atomic scale?' We know that a quantum theory of geometry should contain three fundamental constants of Nature, c, G, ħ, the speed of light, Newton's gravitational constant and Planck's constant. Now, as Planck pointed out in his celebrated paper that marks the beginning of quantum mechanics, there is a unique combination,

L = sqrt(ħ G/c^3),

of these constants which has dimension of length. (L ~ 10^{-33} cm.) It is now called the Planck length. Experience has taught us that the presence of a distinguished scale in a physical theory often marks a potential transition; physics below the scale can be very different from that above the scale. Now, all of our well-tested physics occurs at length scales much bigger than L. In this regime, the continuum picture works well. A key question then is: Will it break down at the Planck length? Does geometry have constituents at this scale? If so, what are its atoms? Its elementary excitations? Is the space-time continuum only a `coarse-grained' approximation? Is geometry quantized? If so, what is the nature of its quanta?

To probe such issues, it is natural to look for hints in the procedures that have been successful in describing matter. Let us begin by asking what we mean by quantization of physical quantities. Take a simple example - the hydrogen atom. In this case, the answer is clear: while the basic observables - energy and angular momentum - take on a continuous range of values classically, in quantum mechanics their eigenvalues are discrete; they are quantized. So, we can ask if the same is true of geometry. Classical geometrical quantities such as lengths, areas and volumes can take on continuous values on the phase space of general relativity. Are the eigenvalues of corresponding quantum operators discrete? If so, we would say that geometry is quantized and the precise eigenvalues and eigenvectors of geometric operators would reveal its detailed microscopic properties.

Thus, it is rather easy to pose the basic questions in a precise fashion. Indeed, they could have been formulated soon after the advent of quantum mechanics. Answering them, on the other hand, has proved to be surprisingly difficult. The main reason, I believe, is the inadequacy of standard techniques. More precisely, to examine the microscopic structure of geometry, we must treat Einstein gravity quantum mechanically, i.e., construct at least the basics of a quantum theory of the gravitational field. Now, in the traditional approaches to quantum field theory, one begins with a continuum, background geometry. To probe the nature of quantum geometry, on the other hand, we should not begin by assuming the validity of this picture. We must let quantum gravity decide whether this picture is adequate; the theory itself should lead us to the correct microscopic model of geometry.

With this general philosophy, in this article I will summarize the picture of quantum geometry that has emerged from a specific approach to quantum gravity. This approach is non-perturbative. In perturbative approaches, one generally begins by assuming that space-time geometry is flat and incorporates gravity - and hence curvature - step by step by adding up small corrections. Discreteness is then hard to unravel.

[Footnote: The situation can be illustrated by a harmonic oscillator: While the exact energy levels of the oscillator are discrete, it would be very difficult to "see" this discreteness if one began with a free particle whose energy levels are continuous and then tried to incorporate the effects of the oscillator potential step by step via perturbation theory.]

In the non-perturbative approach, by contrast, there is no background metric at all. All we have is a bare manifold to start with. All fields - matter as well as gravity/geometry - are treated as dynamical from the beginning. Consequently, the description can not refer to a background metric. Technically this means that the full diffeomorphism group of the manifold is respected; the theory is generally covariant.

As we will see, this fact leads one to Hilbert spaces of quantum states which are quite different from the familiar Fock spaces of particle physics. Now gravitons - the three dimensional wavy undulations on a flat metric - do not represent fundamental excitations. Rather, the fundamental excitations are one dimensional. Microscopically, geometry is rather like a polymer. Recall that, although polymers are intrinsically one dimensional, when densely packed in suitable configurations they can exhibit properties of a three dimensional system. Similarly, the familiar continuum picture of geometry arises as an approximation: one can regard the fundamental excitations as `quantum threads' with which one can `weave' continuum geometries. That is, the continuum picture arises upon coarse-graining of the semi-classical `weave states'. Gravitons are no longer the fundamental mediators of the gravitational interaction. They now arise only as approximate notions. They represent perturbations of weave states and mediate the gravitational force only in the semi-classical approximation. Because the non-perturbative states are polymer-like, geometrical observables turn out to have discrete spectra. They provide a rather detailed picture of quantum geometry from which physical predictions can be made.

The article is divided into two parts. In the first, I will indicate how one can reformulate general relativity so that it resembles gauge theories. This formulation provides the starting point for the quantum theory. In particular, the one-dimensional excitations of geometry arise as the analogs of "Wilson loops" which are themselves analogs of the line integrals exp(i integral A.dl) of electromagnetism. In the second part, I will indicate how this description leads us to a quantum theory of geometry. I will focus on area operators and show how the detailed information about the eigenvalues of these operators has interesting physical consequences, e.g., to the process of Hawking evaporation of black holes.

I feel like quoting more, but I'll resist. It's a nice semi-technical introduction to loop quantum gravity - a very good place to start if you know some math and physics but are just getting started on the quantum gravity business.

Next, here are some papers by younger folks working on loop quantum gravity:

2) Fotini Markopoulou, The internal description of a causal set: What the universe looks like from the inside, preprint available as gr-qc/9811053.

Fotini Markopoulou, Quantum causal histories, preprint available as hep-th/9904009.

Fotini Markopoulou is perhaps the first person to take the issue of causality really seriously in loop quantum gravity. In her earlier work with Lee Smolin (see "week99" and "week114") she proposed a way to equip an evolving spin network (or what I'd call a spin foam) with a partial order on its vertices, representing a causal structure. In these papers she is further developing these ideas. The first one uses topos theory! It's good to see brave young physicists who aren't scared of using a little category theory here and there to make their ideas precise. Personally I feel confused about causality in loop quantum gravity - I think we'll have to muck around and try different things before we find out what works. But Markopoulou's work is the main reason I'm even daring to think about these issues....

3) Seth A. Major, Embedded graph invariants in Chern-Simons theory, preprint available as hep-th/9810071.

In This Week's Finds I've already mentioned Seth Major has worked with Lee Smolin on q-deformed spin networks in quantum gravity (see "week72"). There is a fair amount of evidence, though as yet no firm proof, that q-deforming your spin networks corresponds to introducing a nonzero cosmological constant. The main technical problem with q-deformed spin networks is that they require a "framing" of the underlying graph. Here Major tackles that problem....

And now for something completely different, arising from a thread on sci.physics.research started by Garrett Lisi. What's the gauge group of the Standard Model? Everyone will tell you it's U(1) x SU(2) x SU(3), but as Marc Bellon pointed out, this is perhaps not the most accurate answer. Let me explain why and figure out a better answer.

Every particle in the Standard Model transforms according to some representation of U(1) x SU(2) x SU(3), but some elements of this group act trivially on all these representations. Thus we can find a smaller group which can equally well be used as the gauge group of the Standard Model: the quotient of U(1) x SU(2) x SU(3) by the subgroup of elements that act trivially.

Let's figure out this subgroup! To do so we need to go through all the particles and figure out which elements of U(1) x SU(2) x SU(3) act trivially on all of them.

Start with the gauge bosons. In any gauge theory, the gauge bosons transform in the adjoint representation, so the elements of the gauge group that act trivially are precisely those in the center of the group. U(1) is abelian so its center is all of U(1). Elements of SU(n) that lie in the center must be diagonal. The n x n diagonal unitary matrices with determinant 1 are all of the form exp(2 π i / n), and these form a subgroup isomorphic to Z/n. It follows that the center of U(1) x SU(2) x SU(3) is U(1) x Z/2 x Z/3.

Next let's look at the other particles. If you forget how these work, see "week119". For the fermions, it suffices to look at those of the first generation, since the other two generations transform exactly the same way. First of all, we have the left-handed electron and neutrino:

(ν_L, e_L)

These form a 2-dimensional representation. This representation is the tensor product of the irreducible rep of U(1) with hypercharge -1, the isospin-1/2 rep of SU(2), and the trivial rep of SU(3).

A word about notation! People usually describe irreducible reps of U(1) by integers. For historical reasons, hypercharge comes in integral multiples of 1/3. Thus to get the appropriate integer we need to multiply the hypercharge by 3. Also, the group SU(2) here is associated, not to spin in the sense of angular momentum, but to something called "weak isospin". That's why we say "isopin-1/2 rep" above. Mathematically, though, this is just the usual spin-1/2 representation of SU(2).

Next we have the left-handed up and down quarks, which come in 3 colors each:

(u_L, u_L, u_L, d_L, d_L, d_L)

This 6-dimensional representation is the tensor product of the irreducible rep of U(1) with hypercharge 1/3, the isospin-1/2 rep of SU(2), and the fundamental rep of SU(3).

That's all the left-handed fermions. Note that they all transform transform according to the isospin-1/2 rep of SU(2) - we call them "isospin doublets". The right-handed fermions all transform according to the isospin-0 rep of SU(2) - they're "isospin singlets". First we have the right-handed electron:


This is the tensor product of the irreducible rep of U(1) with hypercharge -2, the isospin-0 rep of SU(2), and the trivial rep of SU(3). Then there are the right-handed up quarks:

(u_R, u_R, u_R)

which form the tensor product of the irreducible rep of U(1) with hypercharge 4/3, the isospin-0 rep of SU(2), and the fundamental rep of SU(3). And then there are the right-handed down quarks:

(d_R, d_R, d_R)

which form the tensor product of the irreducible rep of U(1) with hypercharge 2/3, the isospin-0 rep of SU(2), and the 3-dimensional fundamental rep of SU(3).

Finally, besides the fermions, there is the - so far unseen - Higgs boson:

(H_+, H_0)

This transforms according to the tensor product of the irreducible rep of U(1) with hypercharge 1, the isospin-1/2 rep of SU(2), and the 1-dimensional trivial rep of SU(3).

Okay, let's see which elements of U(1) x Z/2 x Z/3 act trivially on all these representations! Note first that the generator of Z/2 acts as multiplication by 1 on the isospin singlets and -1 on the isospin doublets. Similarly, the generator of Z/3 acts as multiplication by 1 on the leptons and exp(2 π i / 3) on the quarks. Thus everything in Z/2 x Z/3 acts as multiplication by some sixth root of unity. So to find elements of U(1) x Z/2 x Z/3 that act trivially, we only need to consider guys in U(1) that are sixth roots of unity.

To see what's going on, we make a little table using the information I've described:

         ACTION OF            ACTION OF             ACTION OF
       exp(π i / 3)            -1               exp(2 π i / 3)
          IN U(1)             IN SU(2)               IN SU(3)

e_L         -1                  -1                     1
ν_L        -1                  -1                     1
u_L     exp(π i / 3)           -1               exp(2 π i / 3)
d_L     exp(π i / 3)           -1               exp(2 π i / 3)

e_R          1                   1                     1
u_R     exp(4 π i / 3)          1               exp(2 π i / 3)
d_R     exp(4 π i / 3)          1               exp(2 π i / 3)

 H          -1                  -1                     1

And we look for patterns!

See any?

The most important one for our purposes is that if we multiply all three numbers in each row, we get 1.

This means that the element (exp(π i / 3), -1, exp(2 π i / 3)) in U(1) x SU(2) x SU(3) acts trivially on all particles. This element generates a subgroup isomorphic to Z/6. If you think a bit harder you'll see there are no other patterns that would make any more elements of U(1) x SU(2) x SU(3) act trivially. And if you think about the relation between charge and hypercharge, you'll see this pattern has a lot to do with the fact that quark charges in multiples of 1/3, while leptons have integral charge. There's more to it than that, though....

Anyway, the "true" gauge group of the Standard Model - i.e., the smallest possible one - is not U(1) x SU(2) x SU(3), but the quotient of this by the particular Z/6 subgroup we've just found. Let's call this group G.

There are two reasons why this might be important. First, Marc Bellon pointed out a nice way to think about G: it's the subgroup of U(2) x U(3) consisting of elements (g,h) with

(det g)(det h) = 1.

If we embed U(2) x U(3) in U(5) in the obvious way, then this subgroup G actually lies in SU(5), thanks to the above equation. And this is what people do in the SU(5) grand unified theory. They don't actually stuff all of U(1) x SU(2) x SU(3) into SU(5), just the group G! For more details, see "week119". Better yet, try this book that Brett McInnes recommended to me:

4) Lochlainn O'Raifeartaigh, Group structure of gauge theories, Cambridge University Press, Cambridge, 1986.

Second, this magical group G has a nice action on a 7-dimensional manifold which we can use as the fiber for a 11-dimensional Kaluza-Klein theory that mimics the Standard Model in the low-energy limit. The way to get this manifold is to take S^3 x S^5 sitting inside C^2 x C^3 and mod out by the action of U(1) as multiplication by phases. The group G acts on C^2 x C^3 in an obvious way, and using this it's easy to see that it acts on (C^2 x C^3)/U(1).

I'm not sure where to read more about this, but you might try:

5) Edward Witten, Search for a realistic Kaluza-Klein theory, Nucl. Phys. B186 (1981), 412-428.

Edward Witten, Fermion quantum numbers in Kaluza-Klein theory, Shelter Island II, Proceedings: Quantum Field Theory and the Fundamental Problems of Physics, ed. T. Appelquist et al, MIT Press, 1985, pp. 227-277.

6) Thomas Appelquist, Alan Chodos and Peter G.O. Freund, editors, Modern Kaluza-Klein Theories, Addison-Wesley, Menlo Park, California, 1987.

© 1999 John Baez