So what constitutes a duality? Roughly speaking it means that there is a single theory (defined in an operational sense, the theory is the collection of what one could measure) that has at least two different looking descriptions. For example, there is one theory that can either be described as type IIB strings on an AdSxS5 background or as N=4 strongly coupled large N gauge theory. Husain gives a more precise definition when he claims:

Two [...] theories [...] are equivalent at the quantum level. "Equivalent" means that there is a precise correspondence between operators and quantum states in the dual theories, and a relation between their coupling constants, at least in some limits.

Then he goes on to show that there is a one to one map between the observables in some topological theories and the observables of the harmonic oscillator. Unfortunately, such a map is not enough for a duality in the usual sense. Otherwise, all quantum mechanical theories with a finite number of degrees of freedom would be dual to each other. All have equivalent Hilbert spaces and thus operators acting on one Hilbert space can also be interpreted as operators acting in the other Hilbert space. But this is only kinematics. What is different between the harmonic oscillator and the hydrogen atom say is the dynamics. They have different Hamiltonians. By the above argument, the oscillator Hamiltonian also acts in the hydrogen atom Hilbert space but it does not generate the dynamics.

So what does Husain do concretely? He focusses on BF theory on space-times of the globally hyperbolic form R x Sigma for some Euclidean compact 3-manifold Sigma. There are two fields, a 2-form B and a (abelian for simplicity) 1-form A with field strength F=dA. The Lagrangian is just B wedge F. This theory does not need a metric and is therefore topological.

Classically, the equations of motion are dB=0 and F=0. For quantization, Husain performs a canonical analysis. From now on, indices a,b,c run over 1,2,3. He finds that epsilon_abc B_bc is the canonical momentum for A_a and that there are first class constraints setting F_ab=0 and the spatial dB=0.

Observables come in two classes O1(gamma) and O2(S) where gamma is a closed path in Sigma and S is a closed 2-surface in Sigma. O1(gamma) is given by the integral of A over gamma, while O2(S) is the integral of B over S. Because of the constraints, these observables are invariant under deformations of S and gamma and thus only depend on homotopy classes of gamma and S. Thus one can think of O1 as living in H^1(Sigma) and O2 as living in H^2(Sigma).

Next, one computes the Poisson brackets of the observables and finds that two O1's or two O2's Poisson commute while {O1(gamma),O2(S)} is given in terms of the intersection number of gamma and S.

As the theory is diffeomorphism invariant, the Hamiltonian vanishes and the dynamics are trivial.

Basically, that's all one could (should) say about this theory. However Husain goes on: First, he specialises to Sigma = S1 x S2. This means (up to equivalence) there is only one non-trivial gamma (winding around S1) and one S (winding around the S2). Their intersection is 1. Thus, in the quantum theory, O1(gamma) and O2(S) form a canonical pair of operators having the same commutation relations as x and p. Another example is Sigma=T3 where H^1 = H^2 = R^3 so this is like 3d quantum mechanics.

Husain chooses to form combinations of these operators like for creation and annihilation operators for the harmonic oscillator. According to the above definition of "duality" this constitutes a duality between the BF-theory and the harmonic oscillator: We have found a one to one map between the algebras of observables.

What he misses is that there is a similar one to one map to any other quantum mechanical system: One could directly identify x and p and use that for any composite observables (for example for the particle in any complicated potential). Alternatively, one could take any orthogonal generating system e1, e2,... of a (separable) Hilbert space and define latter operators a+ mapping e(i) to e(i+1) and a acting in the opposite direction. Big deal. This map lifts to a map for all operators acting on that Hilbert space to the observables of the BF-theory. So, for the above definition of "duality" all systems with a finite number of degrees of freedom are dual to each other.

What is missing of course (and I should not hesitate to say that Husain realises that) is that this is only kinematical. A system is not only given by its algebra of observables but also by the dynamics or time evolution or Hamiltonian: On has to single out one of the operators in the algebra as the Hamiltonian of the system (leaving issues of convergence aside, strictly one only needs time evolution as an automorphism of the algebra and can later ask if there is actually an operator that generates it. This is important in the story of the LQG string but not here).

For BF-theory, this operator is H_BF=0 while for the harmonic oscillator it is H_o= a^+ a + 1/2. So the dynamics of the two theories have no relation at all. Still, Husain makes a big deal out of this by claiming that the harmonic oscillator Hamiltonian is dual to the occupation number operator in the topological theory. So what? The occupation number operator is just another operator with no special meaning in that system. But even more, he stresses the significance of the 1/2: The occupation number doesn't have that and if for some (unclear) reason one would take that operator as a generator of something, there would not be any zero point energy. And this might have a relevance for the cosmological constant problem.

What is that? There is one (as it happens background independent) theory that has a Hamiltonian. But if one takes a different, random operator as the Hamiltonian, that has its smallest eigenvalue at 0. What has that to say about the cosmological constant? Maybe one should tell these people that there are other dualities that not only identify the structure of the observable algebra (without dynamics). But, dear reader, be warned that in the near future we will read or hear that background independent theories have solved the cosmological constant problem.

Let me end with a question that I would really like to understand (and probably, there is a textbook answer to it): If I quantise a system the way we have done it for the LQG string, one does the following: One singles out special observables say x and p (or their exponentials) and promotes them to elements of the abstract quantum algebra (the Weyl algebra in the free case). Then there are automorphisms of the classical algebra that get promoted to automorphisms of the quantum algebra in a straight forward way. For the string, those were the diffeomorphisms, but take simply the time evolution. Then one uses the GNS construction to construct a Hilbert space and tries to find operators in that Hilbert space that implement those automorphisms: Be a_t the automorphism in the algebra sending observable O to a_t(O) and p the representation map that sends algebra elements to operators on the Hilbert space. Then one looks for unitary operators U(t) (or their hermitian generators) such that

p( a_t(O) ) = U(t)^-1 p(O) U(t)

In the case of time evolution, this yields the quantum Hamilton operator.

However, there is an ambiguity in the above procedure: If U(t) fulfils the above requirement, so does e^(i phi(t)) U(t) for any real number phi(t). Usually, there is an additional requirement as t comes from a group (R in the case of time translations but Diff(S^1) in the case of the string) and one could require that U(t1) U(t2) = U(t1 + t2) where + is the group law. This does not leave much room for the t-dependence of phi(t). In fact, in general it is not possible to find phi(t) such that this relation is always satisfied. In that case we have an anomaly and this is exactly the way the central charge appears in the LQG string case.

Assume now, that there is no anomaly. Then it is still possible to shift phi by a constant times t (in case of a one dimensional group of automorphisms, read: time translation). This does not effect any of the relations about the implementation of the automorphisms a_t or the group representation property. But in terms of the Hamiltonian, this is nothing but a shift of the zero point of energy. So, it seems to me that none of the physics is affected by this. The only way to change this is to turn on gravity because the metric couples to this in form of a cosmological constant.

Am I right? That would mean that any non-gravitational theory cannot say anything about zero point energies because they are only observable in gravity. So if you are a studying any theory that does not contain gravity you cannot make any sensible statements about zero point energies or the cosmological constant.