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.
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> That would mean that any non-gravitational theory cannot say anything about zero point energies because they are only observable in gravity.
If I remember correctly, "normal ordering" takes care of the zero point energy in conventional qft.
What am I missing ?
Yes it does. However, you are not allowed to do it if you couple to gravity because gravity sees the absolute scale of gravity.
Similarly, I could ask: Why does everybody quantise the harmonic oscillator that it has the spectrum 1/2, 3/2, 5/3, ...
By the same argument, one could also argue that the normal ordered H is the correct quantization of the classical Hamiltonian.
In textbooks, it comes about because you apply the "x and p go to operator rule to
H= 1/2 (p^2+x^2)
but I could classically rewrite H as
H = 1/2 (p+ix)(p-ix)
and then apply this rule but would get a different spectrum without zero point energy.
Yes, one can always add zero to the classical Hamiltonina as
(..)( px - xp ) whcih is not zero for QM.
But what prevents us from "normal ordering" such that the c.c. is zero if we turn gravity on ?
There seems to be more to zero-point energies than meets the eye, see this.
Maybe Husain simply has taken seriously what Witten said at KITP25:
"Maybe *every* quantum mechanical system has a "dual" gravitational interpretation..."
Witten's statement immediately raises the question: what is the gravitational dual of the harmonic oscillator, which Husain seems to have tried to answer.
The example is simply being used to ask what, if anything, can be learned from a duality between a theory with no time structure(such as gravity with no classically fixed asymptotics), and a theory on a fixed background with an unambigious notion of time (ie. a timelike Killing vector field).
One way to get time is to fix a time gauge, find the corresponding reduced Hamiltonian, and then pose a "duality" question. But different gauges give different reduced Hamiltonians, and you need one to talk about vacuum energy.
One could say that a precondition for asking a "duality" question is that notions of time agree on both sides, so one can also compare the spectra of the Hamiltonians. This would be the way to go, but first you have to extract time in some way from a theory with a Hamiltonian constraint.
What is a definition of "duality" if one side is a gravity theory with no apriori notion of time?
Viqar, I think that is a good question. The simple answer is: It is an equivalence of theories, so that one should say "actually, it's the same theory".
Of course that is chickening out, because we have not yet said what the definition of a theory is.
In the case of QFT on fixed backgrounds, there are good definitions, Whiteman axioms being one, Haag-Kaster axioms probably being an even better one.
In the later case, it is quite clear: All theories have the same quasi-local algebra of observables (as a C*-algebra, up to unitary equivalence), what makes the theories different are the inclusion relations (the pre-sheaf structure) and the way the Poincare group acts in terms of automorphisms.
In the case of arbitrary backgrounds, there has been some progress by Brunetti, Fredenhagen and Verch by replacing Poincare transformations with local isomorphisms.
Now, you will say, we don't have all those in theories without time (and other structure). And you propose just to take the algebra of observables. But this hits the problem I mentioned in the blog article:
QM systems with finitely many degrees of freedom (and even some QFT's) typically (for exceptions, see for example our LQG string paper) an infinite dimensional, separable Hilbert space. But as every kid knows, there is again only one. And the observables are some linear operators on that Hilbert space.
(You take unbounded ones, but one could as well take the corresponding Weyl operators. I don't think there is any mileage to be gained from different choices of algebras of unbounded operators. You probably always take some closure of the algebra of polynomials in x and p (or creation and annihilation operators)).
In the bounded case, you will always end up with B(H), the unique algebra of bounded operators. This is shared by all QM systems.
So my whole point is: If your definition of a theory is just the algebra of observables (maybe with a Hilbert space), then there are not too many theories around in the first place and no wonder you find dualities.
In QM, different theorie of course differ by different dynamics (for example encoded in the Hamiltonian). In the case without (and not even without prefered notion of) time, there should be some structure to replace it. Otherewise, your definition of theory is extremely weak.
Robert, I'm using an action to define a theory, not an algebra of observables.
Of course, if one has explicit Hamiltonians, one can compare spectra and see if they match as a criteria for duality. If not, the question is: what is the best one can do?
One answer is to simply say that there is no notion of duality between a timeless theory and a normal one. Another is to ask the question again after a time gauge fixing.
Given one pair (a, a^dagger) and the oscillator H, lets say up to a constant, as a preferred variable, how many different classical actions can one write down?
Yea but if you make your definition of a theory stronger (eg positing Haag Kastler axioms and other constructive field theoretic approaches) you end up with not just a few possible theories, you end up with pretty much no theory (at least most of the interacting sort, open questions on the gauged ones).
You can play around with this by relaxing various assumptions on hilbert spaces and so forth, but it tends to not get you very far.
We're still stuck with the old mathematical ambiguities, and they just get worse (or become undefined) in curved backgrounds.
It really does become a question of 'whats the best we can do' as Viqar asks
Viqar,
ok, you want to define your theories in terms of Lagrangians. But then, to establish a duality, you should show that there is some relation between the Lagrangians and not just between the observables (and this is what happens for example for T-duality).
For the usual notion of AdS/CFT one has for example Witten's prescription of how to compute CFT correlation functions using gravity data. In addition to that one has the information coming from the SO(4,2)xSU(4) symmetry.
The evidence for S-duality is naturally weaker, mainly existence of BPS-states. Obviously, this is a stement about the spectrum of the Hamiltonian.
Anonymous,
the fact that we don't have any examples of higher dimensional interacting theories that fulfil some set of axioms in a strict sense is, I think, of a different sort: That is a lack of our technical abilities to prove these things. But I think most people would agree that say asymptotically free theories exist in a stricter sense and fulfil a suitable set of axioms even if it is far beyond their power to actually prove that.
Hey Robert,
I agree, in general you probably could accomplish this precise isomorphism between theories in the case of an asymptotically free theories, thats the one exception to the rule as thats the one place where the really rigorous treatments are consistent.
I also agree that you don't want to just compare algebras, but full lagrangians as well. You also want to explicitly check that none of your symmetries are anomalous, this can explicitly break a duality.
Anyway, as far as gauge fixing, you might worry that you could potentially miss some dualities, we are after all throwing out a lot of information.
Robert,
In the case where one side is fully background dependent, one has to think outside the box. AdS/CFT is highly special due to the fixed structures/symmetries.
Hi Robert,
I maybe disagree that
"The only way to change this is to turn on gravity because the metric couples to this in form of a cosmological constant.
... That would mean that any non-gravitational theory cannot say anything about zero point energies because they are only observable in gravity."
I guess phi(t) would make an observable, i.e. a geometric phase, in any case where parallel transport with respect to t, or with respect to any other classical external parameters in the Hamiltonian, becomes nontrivial. Hence a "cosmological constant" must not necessarily relate to the parameter t alone, as others may exist to take the role.
Schoene Gruesse von Martin nach Bremen, wusste gar nicht dass Du jetzt an der IUB--witzigerweise bin ich in HB-Grohn zum Teil aufgewachsen. Wie ist die Uni? (Uni HH Physik 1991-1996 :)
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