Are dualities not just an expression of equivalence in physical theories?

[S.Daedalus]

By 'equivalence', I mean of the computational kind -- i.e. in the same way any universal computer can emulate any other.

First of all, hi there, I'm not sure I put this question in exactly the right forum, but it seems to me that most dualities currently being discussed fall under the 'beyond the standard model', and most often concretely the string theory, category, so I thought I'd try here first.

To continue, basically, it seems to me that one can essentially view a physical theory as a Turing machine generating observational data -- the quality of the theory is then judged by how well this data agrees with experiment. So if one thinks back to how any universal computer can emulate any other, certain kinds of dualities ought not to be surprising; yet, they are generally treated as something of a big deal. (That's not to say that they're not capable of providing deep insights, or at least being greatly useful, quite to the contrary.)

So basically, it seems to me that a theory dual to some other theory looks a lot like a Turing machine emulating another; is this totally off base?

Of course, this view to a certain extent implies that physical theories have little, if anything, to say about the ontology of our world -- their fundamental constituents, be they strings or quantum foams or whatever else, are perhaps more an artefact of the programming language, rather than some deep insight into fundamental reality. In a word, even if string theory is right, it might not be the case that there are tiny vibrating strings in actuality.

But consider the following thought experiment: let's say we've finally thought up a honest-to-goodness theory of everything, and now, we're extracting predictions from it. Chances are we're going to do this with a computer, through simulation. What this means, basically, is that there's a mapping between states of the computer and states of the fundamental theory. However, a computer is little more than an evolving electromagnetic field configuration -- there's a mapping between states of the electromagnetic field and states of the computer. So in the end, if the mappings are reasonable enough, we have maps between states of the electromagnetic field and states of the final theory -- meaning we could just as well have used our favourite theory of electromagnetism to arrive at the predictions made by the TOE.

Of course, this isn't surprising if both theories are computationally universal. But I've never heard this expressed anywhere, so I figure I'm probably just mistaken about some very basic concepts; hence, having followed some quality discussions on here in the past from the shadows (hope nobody minds), I thought I'd just boldly step forwards and ask...

 

[suprised]

By 'equivalence', I mean of the computational kind -- i.e. in the same way any universal computer can emulate any other.

The answer to the question in the title is yes, but in a physical sense. Equivalence is defined in physics by saying that there is no possible measurement to distinguish the "dual" entities.

This has nothing to do with computation - there is no sign that the physical world would be an emulating computer (there are attempts at that, though). There are notions of universality and emergence, but this is something else than duality.

 

[S.Daedalus]

This has nothing to do with computation - there is no sign that the physical world would be an emulating computer (there are attempts at that, though). There are notions of universality and emergence, but this is something else than duality.

I'm not sure I understand where you draw the difference. Could you elaborate?

Also note that I was merely talking about physical theories, not actual physical systems. (Although an equivalent argument could be made there: if you can use some system to build a universal computer, and the system itself can be simulated on a universal computer, it's computationally universal (and nothing more) in itself; I know that there are arguments that physical reality isn't computable, such as Feynman's observation that if spacetime is continuous, there's really no digital machine that can approximate it arbitrarily well -- though I've been wondering whether that can't be challenged on the basis of the Shannon-Nyquist theorem, which asserts that for some suitable sampling condition, the total information about a continuous function over some interval can be stored in a finite set of data points -- but personally, I'll believe in hypercomputation when I see it -- nevermind that proving a device capable of hypercomputation seems quite a challenge in itself.)

In the end, what's a Turing machine? A partial recursive function from inputs to outputs. That seems also a reasonable description of physical theories -- from initial states, the evolution of some system is computed (think, as an analogy, of computing $$|out\rangle$$-states from $$|in\rangle$$-states in the S-matrix formalism).

So how about those dualities? Two universal Turing machines U and U' are equivalent when there exists some partial recursive function f such that for every input x, $$U(x) = U'(f(x))$$, IIRC. Think for instance of f as a function that sends x to a code for U that first establishes a simulation of U' on U, then passes x to that simulation.

Now as an example let's look at the T-duality between string theories: if I'm not mistaken (don't wish to pretend any expertise), you can transform a state of, say, IIA string theory into a state of IIB string theory by replacing the radius of the compactified dimension R with its inverse times the string tension (somehow, preview seems to be wonking out on me, and I can't get the formula to look right, so it's words instead; it seems that anything I put between tex-tags gets replaced by things I put between such tags earlier, but then deleted). So if f is then the application of this substitution, this looks a lot like the above formula for Turing equivalence. And while I know that looks can be deceiving, this seems kinda significant to me, in particular with respect to the notions of ontic content and correctness of physical theories -- as I've already mentioned, it would seem that string theory's correctness not necessarily implies the existence of actual tiny little vibrating strings, and more to the point, it would not imply that there are no other, equally correct pictures of fundamental physics, based on completely distinct notions.

Of course, the notion of quantities in physical theories not necessarily having a definite fundamental existence is familiar at least since the discovery of the gauge freedom of the electromagnetic potential, but I think there's potentially an even deeper lesson to be learned here...

 

[suprised]

... it would seem that string theory's correctness not necessarily implies the existence of actual tiny little vibrating strings, and more to the point, it would not imply that there are no other, equally correct pictures of fundamental physics, based on completely distinct notions.

Of course, the notion of quantities in physical theories not necessarily having a definite fundamental existence is familiar at least since the discovery of the gauge freedom of the electromagnetic potential, but I think there's potentially an even deeper lesson to be learned here...

This is exactly the modern way of thinking about strings etc. There seems to be no absolute meaning of fundamental degrees of freedom, of the geometry of extra dimensions and so on; many "naively completely different" theories are simply different dual descriptions, or parametrizations, of one and the same physical model. That is the lesson we learned some 15 years ago.

Unfortunately this important message isn't appropriately reflected in popular books and movies, and therefore a lot of confusion about eg the absolute meaning of extra dimensions, Calabi-Yau manifolds, etc, persists.

 

[S.Daedalus]

This is exactly the modern way of thinking about strings etc. There seems to be no absolute meaning of fundamental degrees of freedom, of the geometry of extra dimensions and so on; many "naively completely different" theories are simply different dual descriptions, or parametrizations, of one and the same physical model. That is the lesson we learned some 15 years ago.

Well, I'm usually more behind the times than that, so I'll consider this a personal success.

Out of curiosity, is there any reason you cite this specific time frame? Are you referring to the discovery of AdS/CFT?

 

[suprised]

Out of curiosity, is there any reason you cite this specific time frame? Are you referring to the discovery of AdS/CFT?

Not really, that was a bit later. I refer to the "duality revolution" in 1994/5 when it was discovered that all string theories (as well as 11d M-theory) are just different parametrizations of the same thing.

 

[Orbb]

But the idea equally applies to AdS/CFT and its relatives (if they exist). If, for example, a working bulk theory for QCD is developed, which talks about gravity and strings in one more dimension and is entirely equivalent, there is no reason to believe that one description has more implications for ontology than the other. And I agree that given such a situation, one should mostly refrain from ontological statements (eventhough I rather dislike that conclusion ;-) ) and I think it's really important to keep in mind. On the positive side, it gives you more conceptual freedom and you're less likely to get stuck with an interpretation the mathematical formulation at hand is suggesting. Consider, for example, the interpretation of extra dimensions in String Theory.

 

[tom.stoer]

Let's study a very simpleexample, namely the one-dim. qm harmonic oscillator. There are two dual descriptions, one living in L² Hilbert space of square-integrable functions, one living in l² Hilbert space of square-summable sequences. In addition the L² description is self-dual in the sense that the Fourier transform maps the L² space onto itself (the interpretation is that one L² space is the position space whereas the other one is momentum space).

This is exactly the modern way of thinking about strings etc. ... many "naively completely different" theories are simply different dual descriptions, or parametrizations, of one and the same physical model.

Yes, this is what string theorists have in mind, but unfortunately it was not possible to provide a strict mathematical proof for all dualities. Some of them are known exist in certain limits of the different string theories but need not be valid on the full theory space (of course there are indications that the dualities are valid even beyond the regime where they can be established rigorously)

For the harmonic oscillator is different: here the duality is rigorously established.

I think it's worth to stress this difference: If there is one unique underlying description of string theory, then the dual dessriptions are more or less calculational tools only. If there is no unique description, then one description may have more physical relevance in the sense that is exactly this description which applies to the world we live in (whereas other descriptions may apply to other worlds which may exist somewhere else in the landscape).

Think about QCD: we know its fundamental degrees of freedom (quarks and gluons), but we also know a low-energy regime where chiral perturbation theory as an effective low-energy theory is appropriate. I don't think that photo-pion production can be described using the QCD fundamental degrees of freedom, whereas chiral perturbation theory is able to do the job. So in some sense both quarks and gluons and mesons have physical relevance.

 

[suprised]

The probably first non-trivial example for such a duality is the Ising model in d=2, whose physical degrees of freedom can be written as fermions or bosons. One set can be seen as solitons in terms of the other, and the unique S-Matrix can be constructed from either set.

So dualities per se are by no means new, what was new in the mid-90s was a huge web of dualities, relating all sorts of string-, gauge- and M-theories in various dimensions to each other.

…. there is no reason to believe that one description has more implications for ontology than the other.

Unfortunately this is often not realized and ppl believe in the unambiguous existence of branes, higher dimensions, etc. Popular books and articles do not make this point sufficiently clear, and the public confusion about these matters is gigantic.

Certainly a particular description or parametrization may have specific technical advantages over another, for a concrete problem. For another problem, another description may be better suited. Typically one chooses a description of the model at hand for which the degrees of freedom are weekly coupled, so perturbation theory can be applied.

 

[tom.stoer]

Again: compare the situation in string theory with my examples
A) one-dim. harmonic oscillator with dual L² and l² description
B) QCD with low-energy effective chiral perturbation theory

Now what are the string theories we have constructed so far? Are they something like specific "harmonic oscillator descriptions" with an exact duality beween Hilbert spaces in the sense of A)? Or are they effective theories in the sense of B) waiting for the completion in terms of "fundamental QCD degrees of freedom"?

If you vote for A) then eventually one has to provide a sound mathematical proof. If you vote for B) then unfortunately the duality is not valid at the fundametal level (as is the case for QCD: chiral perturbation theory is valid only in a rather special regime). So one has to provide a fundamental formulation of the theory (named M-theory which is still not known so far).

 

[suprised]

Again: compare the situation in string theory with my examples
A) one-dim. harmonic oscillator with dual L² and l² description
B) QCD with low-energy effective chiral perturbation theory

Now what are the string theories we have constructed so far? Are they something like specific "harmonic oscillator descriptions" with an exact duality beween Hilbert spaces in the sense of A)? Or are they effective theories in the sense of B) waiting for the completion in terms of "fundamental QCD degrees of freedom"?
.

There are all sorts of dualities in string theory, of both kinds A) and B).

A) refers to an exact duality of the complete Hilbert space, so this is the kind of dualities I was referring to: dualities in the sense of different representations of one and the same theory. An example is eg., the equivalence of the heterotic string on T4 and type IIA strings on K3.

B) refers to a duality in a limit, most often the low-energy limit. This is not a strict duality, but an emergent one. An example are Seiberg dualities between seemingly different gauge theories in the IR limit.

As for rigorous proofs, certainly the physicists' point of view is to get the idea first and collect evidence, and support it by further computations. Maybe later, perhaps a long time, a rigorous proofs follows. Many problems are so complex that they probably can never be rigorously proven, but are correct beyond any doubt. An example is the AdS/CFT correspondence, for which hard evidence has been amassed in thousands of papers.

The situation in string dualities is that certain computations can be done exactly, even non-perturbatively. That usually requires supersymmetry as helping hand, in the form of BPS properties. There are zillions of test computations, eg of certain correlation functions, that give the correct results, often given by very non-trivial functions. The point is that these computations _never ever have failed_ ! This is extremely non-trivial and a strong argument for the validity of the duality conjectures.

At a logical level one might have the objection that these results, which are related to the BPS subsector of sypersymmetric theories, do not extend to the full theories, ie, that there are other correlation functions which we cannot compute, but which would be different if we could compute them. That cannot be excluded. But we are physicists and not mathematicians, and as said, the amount of evidence that the dual theories are really the same, is so enormous, that it is a safe bet to accept the validity as a working hypothesis.

 

[tom.stoer]

There are all sorts of dualities in string theory, of both kinds A) and B).

...

that it is a safe bet to accept the validity as a working hypothesis.

Thanks for the explanation.

So iff this working hypothesis is really true, we are mostly in the situation A) w/o being able to prove it. But you are right, evidence as expected for physical theories may be sufficient.

That means that many different versions of string theory (or different theories so to speak) are only computational tools which - in the very end - generate exactly the same physical predictions - which is what really matters. And it means that difficulties due to interpretation may go away just because if we do not like a certain interpretation we can simply switch to a different picture which provides a different explanation - or maybe no explanation at all as it is on a formal level only.

 

[suprised]

What you write here pretty much expresses how many ppl think about it.

In some sense, one has to distinguish the physical content from the language, or picture, in which it is represented. This is a bit similar to choosing coordinates in mathematics, where one also shouldn't confuse abstract mathematical properties with properties that pertain to the choice of a specific coordinate system (like coordinate singularities). Many people who criticize string theory because they don't like extra dimensions or Calabi-Yau spaces don't realize this; these are convenient tools to parametrize the theory, and often one can refomulate things where these notions do not appear explicitly.

See for example again the Ising model. We can write it in terms of a free fermion psi. But we can bosonize the fermion by writing it as a soliton:

psi= exp(i phi)

where phi is a free boson (which is periodic: phi ~ phi + 2 Pi). We can view this periodic boson as a space-time coordinate compactified on a circle, S1. So in this formulation we could talk about an extra dimension. In the fermionic formulation, we don't see that.

The extra-dimension-haters are cordially invited to stick with the latter, and everybody can be happy ;-)

 

[tom.stoer]

I think this is a simple misunderstanding.

I understand that extra dimensions need not be physical extra dimensions like x,y,z,z',z'',... That's what I tried to express: they could be (perhaps in a certain limit only) nothing else but coordinates, parameters or whatever.

But the key point is that if you compare it to the case B) with quarks and mesons, they may very well be physically relevant.

We do not understand enough about string theory to exclude this possibility. Before the discovery of the quark model and of QCD there was no good reason why mesons shouldn't be physically real particles. Even today the "are particles", even so we know that they are not the fundamental degrees of freedom.

In order to decide which picture applies to the extra dimensions or other concepts of string theory we need to understand better what the theory really is. Perhaps we will come to the conclusion that extra dimensions are artifacts of some limit which does not apply to the whole theory space. In that sense they may be (become) irrelevant physically. But the mesons are not valid in the wholetheory space of QCD either, nevertheless in the regime where they exist they really do exist as physical particles.

It is hard to exclude something that looks "as if" from the physical world if this "as if" is very convincing.

 

[suprised]

I was not saying that all parametrizations are on equal footing. Whether a particular geometrical picture or parametrization (if any at all) is meaningful, depends on the regime on parameter/energy region one is looking at. Certainly a string compactification, where the string coupling and the curvatures are small, can be well approximated by a sigma model on the relevant geometric manifold, and one may perform scattering measurements and determine the shape of the manifold etc. The higher dimensional nature could be revealed by discovering a higher dimensional Lorenz invariance, KK and winding modes at high energies; in this sense, the higher dimensional geometry can be very real (though not necessarily unique… in general there are still various different geometrical interpretations of the same set of scattering amplitudes).

On the other hand, in a regime were the coupling is strong, geometrical notions are blurred out and in general there would not be a useful way to give a higher dimensional geometric interpretation of the theory.

Thus, for “dualities” of type B), all boils down to identify the relevant, preferably weakly coupled degrees of freedom in a given regime (similar to choosing quarks or mesons at high/low energies). Sometimes weakly coupled degrees of freedom don’t even exist, and then it is difficult write down something concrete like a lagrangian.

PS: Actually upon re-reading I realize that the discussion turns into a confusing direction, mainly because several different notions of duality were mixed. Let’s just define dualities in this way:

Type A) Dualities on the nose, between theories at the “same” point in the parameter/energy region. Here the duality is literally a reparametrization of the same thing. An example is the fermion/boson duality in the Ising model.

Type B) Dualities in a broader sense: as relationships between theories at different parameter or energy regions. These are not equivalences of theories, rather than maps between different parametrizations of the same theory at different parameter or energy scales. An example is AdS/CFT. QCD/Mesons would fit here too but I am somewhat hesitant to call it a duality.

 

[tom.stoer]

I was not saying that all parametrizations are on equal footing. Whether a particular geometrical picture or parametrization (if any at all) is meaningful, depends on the regime on parameter/energy region one is looking at. Certainly a string compactification, where the string coupling and the curvatures are small, can be well approximated by a sigma model on the relevant geometric manifold, and one may perform scattering measurements and determine the shape of the manifold etc. The higher dimensional nature could be revealed by discovering a higher dimensional Lorenz invariance, KK and winding modes at high energies; in this sense, the higher dimensional geometry can be very real (though not necessarily unique… in general there are still various different geometrical interpretations of the same set of scattering amplitudes).

On the other hand, in a regime were the coupling is strong, geometrical notions are blurred out and in general there would not be a useful way to give a higher dimensional geometric interpretation of the theory.

Thus, for “dualities” of type B), all boils down to identify the relevant, preferably weakly coupled degrees of freedom in a given regime (similar to choosing quarks or mesons at high/low energies). Sometimes weakly coupled degrees of freedom don’t even exist, and then it is difficult write down something concrete like a lagrangian.

Fine!

PS: Actually upon re-reading I realize that the discussion turns into a confusing direction, mainly because several different notions of duality were mixed. Let’s just define dualities in this way:

Type A) Dualities on the nose, between theories at the “same” point in the parameter/energy region. Here the duality is literally a reparametrization of the same thing. An example is the fermion/boson duality in the Ising model.

Type B) Dualities in a broader sense: as relationships between theories at different parameter or energy regions. These are not equivalences of theories, rather than maps between different parametrizations of the same theory at different parameter or energy scales. An example is AdS/CFT. QCD/Mesons would fit here too but I am somewhat hesitant to call it a duality.

I agree.

Regarding QCD/mesons: it's like M-theory/strings iff M-theory is really the unique underlying theory cobvering the full theory space. If this is not the case (and I know that you doubt that this is possible) then the mesons are not part of a theory which is dual to QCD but which is a low-energy approx. only.

The main point is that I still do want to believe in this unique, fundamental description of string theory. That's why this description could be something like QCD is for mesons / chiral perturbation theory / heavy quark effective theories, current algebra / non-relativistic quark model / chiral bag / ...

But I think you classification of dualities is correct and convincing even if we do not agree on the objective :-)

 

[S.Daedalus]

I'm glad to see this has sparked some debate, and I'm really happy with the replies so far. I share the reservations voiced by Orbb, to a certain extent; however, I likewise also see the appreciation of such a view of dualities as a chance to throw off the last remaining shackles of intuition that may guide research down blind (or at least, needlessly winding) alleys.

I'd like to get back to the little thought experiment in the OP, though. To me, this seems to imply that we ought to expect being in situation A in general (though there doubtlessly are examples of B-type dualities (thanks for this classification, btw.) as well). The reason for that being simply that it should be possible in every theory capable of supporting universal computation -- every Turing-complete theory (which encompasses even basic Newtonian mechanics, I believe) -- to find an emulation of every other theory (that's not hypercomputational); i.e. in principle, one should be able to calculate, for instance, electrodynamics within Newtonian mechanics, by setting up a mechanical system that acts as a computer, and encoding the 'program' that calculates a solution of the electrodynamic field equations in the initial conditions. You then compute the behaviour of the mechanical system (in general, I should imagine, through numerical simulation, as an analytic solution for a universal system would entail a solution for the halting problem), and in the end, when the system reaches a pre-defined output state, may read of some electromagnetic field configuration.

So in this sense, it seems to me, that all universal theories should be expected to be 'dual' to each other, and perhaps not only in the clunky way of hacking up a computer within one to simulate the other (which only serves to establish the in-principle possibility), but with more subtle translation functions.