Wednesday, May 08, 2019

Dualities rule out "realism"

Edwin has observed that an interviewer sometimes speaks like a left-wing ideologue who (intensely or primarily) cares about feelings, appearances, and the duty of science to evolve in an ideologically predetermined direction of progress. Well, science doesn't have the duty.

Also, Edwin has correctly connected two other misunderstandings of the interviewer – who is both a "realist" meaning that he believes that quantum mechanics must ultimately be replaced with a classical theory again; and who seems to have a problem with the AdS/CFT correspondence. These two misunderstandings aren't independent, Edwin pointed out.

I think that this connection that Edwin emphasized is another argument in favor of both of the following statements, often advocated on this website:
  • String theory teaches us important conceptual things about physics
  • Realism in the sense of an opposition to the foundations of quantum mechanics as clarified in Copenhagen conflicts with most of the progress in 20th century physics
Well, as I wrote in the first point, string theory teaches us lots of things – including the second point. ;-)

The holographic AdS/CFT correspondence may be considered an example of a stringy duality. That interviewer seems bothered by this duality – and probably other dualities – and there's a good reason for that. He's a "realist" i.e. a critic of the basic framework of quantum mechanics. And dualities actually show that the basic, non-realist underpinnings of quantum mechanics are unavoidable. There are many additional proofs – like the locality combined with Bell-like inequalities; or the low heat capacity of materials – but string dualities may be added to this list.

Why do dualities make the observer-dependent, non-realist quantum mechanics unavoidable? Because they are intrinsically quantum mechanical equivalences, equivalences that couldn't exist if the world were completely classical at every level.

Consider T-duality in string theory. Totally analogous observations may be made about every duality. T-duality is an equivalence between string theory on a spacetime with a circular dimension of radius \(R\); and another string theory with a circular dimension of radius \(R'\); the momentum and winding modes of the two spacetimes get interchanged:\[

R' = \frac{\alpha'}{R}, \quad n=w', \quad w=-n'

\] The spectrum of excited strings may be shown to be invariant under this operation. It's because the squared mass \(m^2\) of a string has quadratic contributions proportional to \(n^2\), the squared momentum (from kinetic energy), as well as \(w^2\), the squared winding number (from stretching of the string and its tension), and these two terms get interchanged by the T-duality operation.

But T-duality extends to the full spectrum and the interactions – and as we know e.g. from Matrix theory, it holds non-perturbatively, too. Why does T-duality work for all perturbative interactions?

The answer is actually rather simple and crisp. The world sheet dynamics contains scalar fields \(X^\mu(\sigma,\tau)\) that remember where the appropriate point of the world sheet is located within the target space, i.e. the spacetime. These fields are basically Klein-Gordon fields on the world sheet and the solution to the wave equation – massless Klein-Gordon equation – is a sum of the left-moving and right-moving waves.

So the general solution is of the form\[

X^\mu (\sigma,\tau) = X_L(\sigma+\tau) + X_R(\sigma-\tau)

\] Sorry if my signs are reversed relatively to your convention. The funny aspect of the world sheet is that the left-moving and right-moving parts of all fields are pretty much segregated. And it has consequences. It's clear that the theory – e.g. bosonic string theory – has a spacetime reflection symmetry \(X^{25}\to -X^{25}\). Superstrings may be chiral in the spacetime so this mirroring symmetry may be broken, or you need to add the action on fermions, or you may always get away with reflecting two coordinates because that's equivalent to a rotation by 180 degrees.

Because we have separated \(X^{25}\) to the left-movers and right-movers (the superscript is twenty-five because it's the highest possible value in a 26-dimensional spacetime if time is the zeroth coordinate), \(X^{25}_L\) and \(X^{25}_R\), we may do a wonderful thing: we may spacetime-reflect \(X^{25}_L\) only, while keeping \(X^{25}_R\) unchanged:\[

X^{25}_L(\sigma,\tau)\to - X^{25}_L(\sigma,\tau),\quad
X^{25}_R(\sigma,\tau)\to + X^{25}_R(\sigma,\tau)

\] And that's nothing else than T-duality. Why is it equivalent to the exchange of the momentum and winding? Well, it's because \(\partial_L X_L\) is basically the density of "momentum \(\partial_\tau X^{25}\) minus winding \(\partial_\sigma X^{25}\)" on the given point of the string, while \(\partial_R X_R\) is the density of "momentum plus winding". The full momentum and winding is obtained as the integral of \(X_L\pm X_R\) over \(\sigma\), with some appropriate numerical coefficients.

You may see that the sign flip of \(\partial_L X_L\) i.e. of \(X_L\) is equivalent to the exchange of the momentum density and the winding density! The sum of the two densities, \(\partial_R X_R\), remains unchanged. Here \(\partial_L,\partial_R\) represented \(\partial_\pm\), meaning the derivative with respect to \(\tau\pm \sigma\).

Alternatively, the selective sign flip of \(X_L\) may be interpreted as the "Hodge dualization" of the 1-form \(\partial_\alpha X^{25}\):\[

\partial_\alpha X^{25} \to \epsilon_{\alpha\beta} \partial^\beta X^{25}

\] Great. So on the world sheet, the T-duality is some "left-moving-only" truncation of the spacetime reflection symmetry; or, equivalently, the electromagnetic-like duality exchanging the components of the world sheet derivatives of \(X^{25}\) in a way analogous to \(F\leftrightarrow *F\) which exchanges Maxwell's electric and magnetic fields in four dimensions. It's not hard to believe that this is a rather natural symmetry that may be extended to world sheets of all topologies, assuming that the boundary conditions obey some consistency criteria.

Note that \(\tilde X^{25}\), the T-dualized coordinate, is written as \(-X_L+X_R\) instead of the sum with plus signs. A problem is that you can't easily write a formula for \(\tilde X^{25}(\sigma,\tau)\) in terms of \(X^{25}(\sigma,\tau)\). Why? Because there's no simple way to write \(X_L,X_R\) as a function of \(X^{25}(\sigma,\tau)\). A general solution to the wave equation may be decomposed to left-moving and right-moving waves, up to the undetermined constant mode that may be hidden in both parts, but the general formula isn't local.

The formula for \(\partial_\alpha \tilde X^{25}\) does exist – it's simply obtained from the \(\partial_\pm\) components of \(\partial_\beta X^{25}\) where you add the minus sign for one derivative but not the other. But it's only possible for the world sheet derivatives of \(\tilde X^{25}\), not \(X^{25}\) itself. To calculate \(\tilde X^{25}\) at a given world sheet point, you would need to integrate the world sheet derivative over \(\sigma\). Because of the \(\sigma\)-integral, the T-duality operation may be said to be non-local on the world sheet. It's non-local in the spacetime, too. Also, one may check that \(X^{25}(\sigma,\tau)\) and \(\tilde X^{25}(\sigma,\tau)\) have a nonzero commutator – one incorporates some integral of the momentum of the other.

What does it have to do with realism?

Let us consider two basic "realist theories" claimed to compete with quantum mechanics:
  • Bohmian mechanics with some "beables" on top of the wave function
  • "Many Worlds Interpretation" with some notion of "parts of the wave function that become separate and behave as separated worlds"
We may see that both of them are incompatible with T-duality – and analogously all other dualities. If there were beables but you would still consider both mutually T-dual descriptions to be equally good, it would be necessary for the list of beables to include both some quantities and their T-duals. But it's not possible because fields like \(X^{25}(\sigma,\tau)\) and \(\tilde X^{25}(\sigma,\tau)\) totally refuse to commute with each other.

If you included both to the list of your beables, it would be just like Bohm-visualizing the position and the momentum \(x,p\) simultaneously. It would be a problem because Bohmian mechanics wants to claim that the detection of the particle is ultimately determined by the real, "beable" position. But here you would have two equally good – by the assumption of T-duality – "beables" that would fight which of them determines what the detector actually sees. You would literally have a particle that has a precise position and a precise momentum at the same moment. It can't work.

Similarly, the "Many Worlds Interpretation" splitting of the wave function to many worlds can't work. The many worlds people want to imagine that the wave function evolves unitarily but when it becomes a superposition of "two distant wave packets", they behave as independent "two worlds" where people have separate feelings. However, for that to work, you surely need some notion of a "distance" between two wave packets that add to a superposition.

However, the distance between two "position eigenstates" of a string shape calculated e.g. as\[

\int d\sigma\,(X_{A} - X_{B})^2

\] is completely different – and refusing to commute – with the analogous T-dual distance\[

\int d\sigma\,(\tilde X_{A} - \tilde X_{B})^2

\] which is why the two mutually T-dual notions of the "distance between two wave packets" sharply contradict one another. There is no universal notion of "locality" that could tell you when it becomes natural for a wave function to be interpreted as a "sum of two or more wave packets representing different worlds among many worlds".

So T-duality, like all other dualities, are really operations that are "non-diagonal" in any basis which is why they disrespect any "classical structure" that you could invent within the theory. The unavoidable conclusion is that because dualities are so rock-solid, natural, established, and beautiful, things that conflict with them must be wrong. And the choice of a preferred basis of the Hilbert space, preferred "beables", or preferred "separation between some position-like eigenstates" conflict with the T-duality operation, so they just can't be correct.

As an undergraduate freshman, I was already exposed to basic texts on string theory – a librarian generously xeroxed a textbook by Green, Schwarz, Witten for me (for free), she probably risked copyright infringements as well – and I knew something about T-duality soon afterwords (other dualities were only discovered in the mid 1990s and made the picture thicker). So even if I had some tendencies for "realism" at that time, I had to choose: T-duality or the realist prejudices? Clearly, I would have chosen T-dualities because they're beautiful. They are really local symmetries of string theory and a symmetry is always a great improvement of a theory that makes the theory more likely. There exists absolutely no reason why the symmetry should be fundamentally broken so it's almost certainly not broken. Arguments based on symmetries and other kinds of mathematical beauty always trump purely metaphysical prejudices – everyone who has at least some physical intuition agrees.

You need to allow Nature as She wants to be – quantum mechanical, without preferred bases, and prepared to arbitrary measurements – given by any linear Hermitian operator – that are decided by the observer. The right quantities to measure can't be determined a priori; and they can't be determined objectively by any separation of a wave function into "mutually distant" packets, either.

In practice, an observer will be capable of distinguishing outcomes that are "separated" in some geometric sense but it will be because the locality respected by his measurement apparatus, not because of intrinsic properties of the measured object plus universal rules to choose the "right" observables. There can't be any universal criterion to determine "right" observables. The most relevant observables are always determined a posteriori, by dynamics, while the details about the observer and his apparatus always matter.

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