Wednesday, December 10, 2008

Symmetry Breaking in Quantum Mechanics

Today, in our Mathematical Quantum Mechanics lecture, I put my foot in my mouth. I claimed that under very general circumstances there cannot be spontaneous symmetry breaking in quantum mechanics. Unfortunately, there is an easy counter example:

Take a nucleus with charge Z and add five electrons. Assume for simplicity that there is no Coulomb interaction between the electrons, only between the electrons and the nucleus (this is not essential, you can instead take the large Z limit as explained in this paper by Frieseke. The only way the electrons see each other is via the Pauli exclusion principle. The Hamiltonian for this system has an obvious SO(3) rotational symmetry. The ground state, however is what chemists would call 1S^2 2S^2 2P^1. That is, there is one electron in a P-orbital and in fact this state is six-fold degenerate (including spin). Of course, there is a symmetric linear combination but in that six dimensional eigen-space of the Hamiltonian there are also linear combinations that are not rotationally invariant. Thus, the SO(3) symmetry here is in general spontaneously broken.

This is in stark contrast to the folk theorem that for spontaneous symmetry breaking you need at least 2+1 non-compact dimensions. This for example is discussed by Witten in lecture 1 of the IAS lectures or here and is even stated in the Wikipedia.

Witten argues using the Stone-von Neumann theorem on the uniqueness of the representation of the Weyl group (the argument is too short for me) and explicitly only discusses the particle on the real line with a potential (the famous double well potential where the statement is true). In 1+1 dimensions, there is the argument due to Coleman, that in the case of symmetry breaking you would have a Goldstone boson. But free bosons do not exist in 1+1d since the 2-point function would be a log which is in conflict with positivity.

I talked to a number of people and they all agreed that they thought that "in low dimensions (QM being 0+1d QFT), quantum fluctuations are so strong they destroy any symmetry breaking". Unfortunately, I could not get hold of Prof. Wagner (of the Mermin-Wagner theorem) but maybe you my dear reader have some insight what the true theorem is?

Monday, December 08, 2008

Spectral Action Part II

After in part I, I have explained how to go back and forth between spaces with metric information and (C*)-algebras it's now time to add some physics: Let us now explore how to formulate an action principle that will lead to equations of motion when extremised.

Let me stress however, that although it may look differently, this is entirely classical physics, there is no quantum physics to be found anywhere here even though having possibly non-commutative algebras might remind you of quantum mechanics. This, however, is only a way of writing strange spaces and has nothing to do with the quantisation of a physical theory. We will even compute divergences of one loop Feynman diagrams, but still this only a technical trick to write the classical Einstein action (integral of the Ricci scalar over the manifold) and has nothing to do with quantum gravity! Our final result will be the classical equations of motion of gravity coupled to a gauge theory and a scalar field (the Higgs)!

The trick goes as follows: Last time, we saw that in order to encode metric information we had to introduce a differentiation operator D so we could formulate the requirement that a function should have a gradient which has length less or equal to 1. One could have taken the gradient directly but there was a slight advantage to take the Dirac operator instead since that maps spinors to spinors instead of the gradient mapping scalars to vector fields.

Another advantage of the Dirac operator is that when it is squared it gives the Laplacian plus the Ricci scalar (which we want to integrate to obtain the Einstein Hilbert action) divided by some number which I vaguely remember to be 12 but which is not essential. In formulas, we have DD = \Delta + R/12. Even better, when we are in d dimensions, taking the d-th power gives us the volume element. Thus, taking these two observations together and using the Clifford algebra, we find that taking d-2 powers of D gives us (\Delta + R/12)dVol. The second term is obviously what we need to integrate to obtain the EH action.

But how to get the integration? Here the important observation is that we can pretend that this operator is the kinetic term for some field. Then, we can compute the one loop divergence of this field. On one hand, we know that this is the functional trace of the log of this operator. On the other hand, we can compute the divergence of this expression either diagrammatically or with slightly advanced technology in terms of the heat-kernel.

I will explain the heat-kernel formalism at some other time. However, the result of that treatment is a series of "heat-kernel coefficients" a_n which are scalars expressions of the curvature of mass dimension 2n. That is a_0 is 1, a_1 is basically the Ricci scalar, a_2 is a linear combination of scalar contractions of the curvature squared and so on. All those can interpreted as the coefficients of a power series in a formal parameter s, i.e. h(s) = a_0 + a_1 s + a_2 s^2 + \ldots.

The important result now is that the effective action is the integral of \int dx\int_0^\infty {ds\over s^{(1+d/2)}}h(s) over s from 0 to infinity and over all space-time points. Because of the negative powers of s, this integral diverges at 0. It turns out, this diverge is nothing but the UV divergence of the one-loop diagrams. Now you can apply your favourite regularisation procedure (dimensinal regularisation, Pauli-Vilars, you name it). Here, for simplicity, we just use a cut-off \Lambda and start our integration at 1/\Lambda^2 instead of 0. Connes and Chamseddine do something very similar. Just instead of a sharp cut-off they use a function f in the integral that decays exponentially when s approaches 0 (NB s has mass-dimension -2 and thus acts as 1/p^2).

For concreteness, take d=4. Then the a_0 term leads to \Lambda^4 times 1 integrated over the space (i.e. a cosmological constant). The term from a_1 gives \Lambda^2 times the Einstein-Hilbert term and a_2 is \log(\Lambda) times an integral of curvature squared. The remaining terms are finite when Lambda is removed. Thus, we find the Einstein action (including a huge cosmological constant) plus a curvature squared correction as the divergence of the one-loop effective action.

But the effective action can also be written as the functional trace of the log of the operator. For this, we don't need any field for which the operator is the kinetic operator of. Imagine we happen to know all the eigenvalues of D^{d-2}. Then we have just found that the Einstein Hilbert action can be written as the divergent part of the sum of the log of all those eigenvalues. And this is the spectral action principle: {\cal S}_{EH} = \tr \log D^{d-2}.

As it happens, Connes and Chamseddine really know the eigenvalues of the Dirac operator on spheres. So, they can really do this sum (which turns out to be expressible in terms of zeta-functions). The spheres are of course compact and thus the eigenvalues are discrete. In our field theory argument above, however, we implicitly used the usual continuous momentum space arguments for the effective action. In the limit of large momentum (which is relevant for the divergence) corresponding to short distances this should not really matter, one can pretend that momentum space is actually continuous. However, with a cut-off, this is not precisely true and the discreteness comes in in sub-leading orders. The difference between the continuous computation and the discrete one is of course tiny for a large cut-off. And it is exactly this difference that lets the two authors find agreement to "astronomical precision" (p. 15).

OK, up to now, we have reformulated the gravitational action in terms of the spectrum of the Dirac operator. But what about gauge interactions. But every physicist should now now how to proceed: Do gravity in higher dimensions and perform a Kaluza-Klein reduction. If the compact space has a symmetry group G then besides some scalars you will find YM-theory with gauge group G.

In the non-commutative setting, one can as well take a non-commutative space for the compact directions. Here, Connes and Chamseddine argue for a minimal example (given some conditions of unclear origin, one might be suspicions that those are tuned to give the correct result). It is minimal in terms of irreducibility. However, the total space being the product (by definition reducible) of this compact space by a classical commutative 4d space time (again reducible) make the naturalness of this requirement a but questionable.

For the concrete specification of the compact space, some Clifford magic is employed (including Bott periodicity) but this is standard material. You end up with a non-commutative description of two points for the two spinor chiralities. The symmetry then determines the gauge group. Here I am not completely sure but it seems to me that they employ the usual representation theory arithmetic from GUT-theories to sort all standard model particles in nice representations.

The non-commutative formulation allows the KK-gauge boson A_mu to also have a leg in the compact direction between the two points. From the 4d perspective this is of course a scalar. This of course will be the Higgs. The two points have a finite distance (see the Landi notes for details) and give a mass term inversely proportional to the distance (that is opposite to a superficially similar D-brane construction as noted by Michael Douglas some years ago).

That's it. We have an algebraic formulation of the classical action for the standard model. Let's recap what went in: The NCG version of a space with metric information in terms of a Dirac operator. Some heat-kernel material to write the gravitational action in terms of eigenvalues, GUT-type representations theory and KK-theory. What kind of 'surprises' were found? GUT type relations are rediscovered, treating the discrete spectrum of D on spheres can well be approximated by continuous momentum space for momenta large compared to the inverse radius. And there is a final observation detailed in appendix A of the paper: For S^3\times S^1 there are some cancellations of unclear origin which make a_3 vanish. This however is a statement about a heat-kernel coefficient and does not have a priori any connection with the non-commutative approach. Furthermore, the physical implications are left in the dark.

Friday, December 05, 2008

KK description of Black Holes?

Still no second part of the spectral action post. But instead a litte puzzle that came up over coffee yesterday: What is the Kaluza-Klein description of a black hole?

To be more explicit: Take pure gravity on R^4xK for compact K and imagine that it is large (some parsec in diameter say). Then you could imagine you have something that looks like a blackhole in this total space-time. What is its four dimenional description in KK theory?

With KK theory I mean the 4d theory with an infinite number of fields. I want to include the whole KK tower. This theory should be equivalent to the higher dimensional one since both are related by a (generalised) Fourier transform on K. One might be worried that a black hole is so singular that this Fourier transform has problems, does not converge or something. But if that is your worry, take a black hole that is not eternal but one that is formed by the collision of graviational waves say. In the past, those waves came in from infinity and if you siufficiently go back in time all fields are weak. This weak field configuration should have no problem being described in KK language and then the evolution is done in the 4d perspective. What would the 4d observer see when the black hole forms in higher dimensions?

The question I would be most interested in is if there is always a black hole in terms of the 4d gravity or if the 4d gravity can remain weak and the action can be entirely in the other fields.

One scenario I could imagine is as follows: The 4d theory has besides the metric some gauge fields and some dilatons. If the black hole is well localised in K then many higher Fourier modes of K will participate. From the 4d perspective, the KK momentum is the charge under the gauge fields and the unit is dependent on the dilaton. So could it be that there is a gauge theory black hole, i.e. a charged configuration that is confined to small region of space time where the coupling is strong with all the causality implications of black holes in gravity?

Tuesday, December 02, 2008

Spectral Actions Imprecisely

A post from Lubos triggered me to write a post on non-commutative geometry models for gravity plus the standard model as for example promoted by Chamseddine and Connes in a paper today.

Before I start I would like to point out that I have not studied this paper in any detail but only read over it quickly. Therefore, there are probably a number of misunderstandings on my side and you should read this as a report of my thoughts when scanning over that paper rather than a fair representation of the work of Chamseddine and Connes.

Most of what I know about Connes' version of non-commutative geometry (rather than the *-product stuff which has only a small overlap with this) I know from the excellent lecture notes by Landi. If you want to know more about non-commutative geometry beyond the *-product this is a must read (except maybe for the parts on POSETs which are a hobby horse of the author and which can be safely ignored).

But enough of these preliminary remarks. Let's try to understand the spectral action principle!

Every child in kindergarden knows that if you have a compact space you get a commutative C*-algebra for free: You just have to take the continuous functions and add and multiply complex conjugate them point-wise. As norm you can take the supremum/maximum norm (here the compactness helps). This is what is presented in every introduction section of a talk on non-commutative geometry, but onfortunately, this is completely trivial.

The non-trivial part (due to Gelfand, Naimark and Segal) is that it works also the other way round: Given a (unital) commutative C*-algebra, one can construct a compact space such that this C*-algebra is the algebra of the functions on it. Furthermore, if one has got the algebra from the functions on a space, the new space is homeomorphic to (i.e. the same as) the original one.

How can this work? Of course, anybody with some knowledge in algebraic geometry (a similar endeavour but there one deals with polynomials rather than continuous functions) knows how: First, we have to find the space as a set of points. Let's assume that we started from a space and we know what the points are. Then for each point x we get a map from the functions on the manifold to the numbers, we simply map f to f(x). A short reflection reveals that this is in fact a representation of the algebra which is one-dimensional and thus irreducible. It turns out that all irreducible representations of the algebra are of this form. Thus, we can identify the points of the space with the irreducible representations of the algebra.

I could have told an equivalent story in therms of maximal ideals which arise as kernels of the above maps, i.e. the ideals of functions that vanish at x.

Next, we have to turn this set into a topological space. One way to do this is to come up with a collection of all open or all closed sets. In this case, however, it is simpler to define the topology in terms of a closure map, that is a map that maps a set of points to the closure of this set. Such a map has to obey a number of obvious properties (for example if I apply the closure a second time it doesn't do anything or a set is always contained in its closure). In order to find this map we have to make use of the fact that if a continuous functions vanishes on a set of points then by continuity it vanishes as well on the limit points of that set, that is on its closure. Therefore, I can define the closure of a set A of points as the vanishing points of all continious functions that vanish on A. This definition then has an obvious reformulation in terms of irreducible representations instead of points. Think about it, as a homework!

Now that we have a topological space, we want to endow it with a metric structure. But instead of giving a second rank symmetric tensor, we specify a measure for the distance between two points x and y as this is closer to our formalism so far. How can we do this in terms of functions?

The trick is now to introduce a derivative. This allows us to restrict our attention to functions which are differentiable and whose gradient is nowhere greater then 1, i.e. the supremum norm of the gradient is bounded by one (this is where the metric enters implicitly since we use it to compute the length of the gradient). Amongst all such functions f we maximize l=|f(x)-f(y)|. Take now the shortest path between x and y (a geodesic). Since the derivative of f along this paths is bounded by one l cannot be bigger than the distance between x and y. And taking the supremum over all possible f we find that l becomes the distance.

Again, I leave it as a homework to reformulate this construction in the algebraic setting in terms of irreducible representations and the distance between them. There is only a slight technical complication: We started with the algebra of scalar functions on the manifold but the gradient maps those to vector fields which is a different set of sections. This makes life a bit more complicated. Connes' solution is here to forget about scalar functions and take spinors (sections of a spinor bundle precisely) instead. If those exist, all the previous constructions work equally well. But now we can use the Dirac operator and this maps spinors to spinors (with another slight complication for Weyl spinors in even dimensions).

In the algebraic setting, the Dirac operator D is just some abstract linear (unbounded) operator D which fulfills a number of properties listed on p2 of the Chamseddine_Connes paper and once that is given by some supernatural being in addition to the algebra, you can actually reconstruct a Riemannian manifold from an abstract commutative C*-algebra and D.

Next, we have to write down actions. Unfortunately, I now have to run to get to a seminar on the status of LHC. This will be continued, so stay tuned!