T-Duality and the Uncertainty Relations

I have just come back from Stockholm, where I spent part of the workshop Dualities, Emergence and Realism at the Department of Philosophy of Stockholm University. T-duality came up often enough in those days that I started thinking about it.

String theorists often say that their theory contains a shortest length: below a certain scale, the string length $\ell_s$, distances are said to stop making sense. The claim is usually traced to a property of the theory called T-duality. This note explains T-duality with as little machinery as possible and then asks what it shows about length. My suggestion is that it does not show that small distances are missing. It shows that, at these scales, the size of space depends on what one measures it with.

Suppose that one of the spatial directions is compact, i.e. it closes on itself as a circle of radius $R$, while the remaining directions extend indefinitely and particles and strings move along them as usual.

A closed string has two options to move along that compact direction.

• The first is to travel around the circle, like a bead sliding along a ring. In quantum mechanics a particle confined to a circle behaves like a wave that must fit a whole number of times around it. The smaller the circle, the shorter the wave, and the shorter the wave, the more energy it carries. The energy of the bead therefore grows as the circle shrinks: it goes like $1/R$.
• The second option is not available to a point particle. A closed string can wrap around the circle, like a rubber band stretched around a tin can. A stretched string has a tension, and its energy is tension times length. The larger the circle, the longer the band, and the more energy it carries. The energy of the band therefore grows with the circle: it goes like $R$.

Physicists call the first kind of state a momentum state and the second a winding state. Each is labelled by a number: $n$ counts how many wavelengths fit around the circle, $w$ how many times the string wraps around it.

Taking both contributions into account, the mass of a string state on the circle is

$$M^2 = \left(\frac{n}{R}\right)^2 + \left(\frac{w R}{\ell_s^2}\right)^2 + \text{(vibrations of the string)}.$$

Now exchange $n$ and $w$, so that every bead becomes a band and every band a bead, and at the same time replace $R$ with $\ell_s^2/R$. The list of possible masses does not change. A large circle and a small circle, with beads and bands exchanged, give exactly the same particles, and the ways in which the strings interact also coincide.¹

This is T-duality. A string theory on a circle of radius $R$ is physically indistinguishable from a string theory on a circle of radius $\ell_s^2/R$. The two descriptions coincide when $R = \ell_s$, the self-dual radius.

The standard argument runs as follows. Try to shrink the circle below $\ell_s$: the result is a theory identical to one with a circle larger than $\ell_s$. Shrinking further only takes one back up. There is therefore no physically distinct situation corresponding to a circle smaller than the string length, and $\ell_s$ is the shortest length.

What follows about space has been read in three ways, and each of them has a close precedent in the debate on the uncertainty relations between position and momentum in quantum mechanics. Indeed, the structure of the two problems is the same: a theory fixes a lower bound on what can be resolved (there the product of the uncertainties in position and momentum, here the radius of a compact dimension), and one has to decide whether the bound concerns our access to the world, the meaning of our terms, or the world itself.

On an epistemic reading, small circles exist, but we cannot find out about them, because the probes needed to resolve them are too heavy to produce.² This is the position of the disturbance account of the uncertainty relations, which Heisenberg himself offered in 1927 with the thought experiment of the γ-ray microscope: an electron has a definite position and a definite momentum at all times, and the uncertainty relations express the fact that any measurement of one disturbs the other in an uncontrollable way.³ The view keeps the classical picture of space, and its difficulty is the same in both cases: the theory gives no role to the values it says we cannot reach. Indeed, nothing in the dynamics of a closed string refers to a radius below $\ell_s$ that differs from its dual, as nothing in standard quantum mechanics refers to the exact position and momentum that the microscope would disturb.

On an eliminative reading, small distances are simply not there; this is how Witten put it.⁴ The precedent here is the verificationist strand of the Copenhagen tradition, on which a quantity that cannot be measured even in principle has no physical meaning. Heisenberg’s 1927 paper already goes in this direction, with the claim that the path of a particle comes into being only when we observe it, and Bridgman’s operationalism turned the same thought into a general criterion for physical concepts.⁵ Applied to T-duality, the argument runs as follows: no experiment can distinguish a circle of radius $R$ from a circle of radius $\ell_s^2/R$, so the question of which one we are in has no content, and distances below $\ell_s$ go together with it. The difficulty is the familiar one for verificationism, i.e. the criterion of meaning does more work than the physics. In particular, the elimination of the small radius depends on a convention, the choice of $R \geq \ell_s$ as the range in which radii are described, which the duality itself does not supply.

On an indeterminist reading, the two descriptions disagree about the radius while agreeing on everything physical, so there is no fact about what the radius is, and the definite space we observe must emerge from something that is not spatial in the ordinary sense.⁶ Its precedent is the reading of quantum mechanics on which a system that is not in an eigenstate of position has no definite position at all, a reading supported by the Kochen–Specker theorem, which excludes the assignment of definite values to all observables independently of the context of measurement.⁷ Recent work treats this as indeterminacy in the world, not in our descriptions of it.⁸ Indeterminacy is invoked where the theory, read in another way, gives definite answers to well-posed questions, such as the radius registered by a given kind of probe.

The debate in quantum mechanics did not end with these three options. Bohr’s reply to Heisenberg was that position and momentum are each well defined relative to an experimental arrangement, and that two arrangements which cannot be realised together give complementary descriptions of the same system.⁹ On this view the bound in the uncertainty relations is neither a limit on knowledge nor a limit on meaning, and it does not make the world indeterminate: it records that the relevant quantities are defined relative to the context of measurement.

The same move could be available for T-duality. A momentum state registers a circle of radius $R$, a winding state registers a circle of radius $\ell_s^2/R$, and the duality exchanges the two probes together with the circles they register.¹⁰ On this reading the radius is defined relative to the probe, as position and momentum are defined relative to the experimental arrangement in Bohr’s account, and the string length marks the point at which the lightest probe changes from one kind to the other.

Footnotes

1. J. Polchinski, String Theory, vol. 1, Cambridge University Press, 1998, ch. 8; T. Ortín, Gravity and Strings, Cambridge University Press, 2004, §14.3. ↩

2. B. Greene, The Elegant Universe, Norton, 1999, ch. 10. ↩

3. W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik”, Zeitschrift für Physik 43 (1927), 172–198. ↩

4. E. Witten, “Reflections on the Fate of Spacetime”, Physics Today 49 (1996), 24–30. ↩

5. Heisenberg, “Über den anschaulichen Inhalt”, cit.; P. W. Bridgman, The Logic of Modern Physics, Macmillan, 1927. ↩

6. N. Huggett, “Target space ≠ space”, Studies in History and Philosophy of Modern Physics 59 (2017), 81–88; N. Huggett and C. Wüthrich, Out of Nowhere, ch. 9, arXiv:2005.10943. ↩

7. S. Kochen and E. P. Specker, “The problem of hidden variables in quantum mechanics”, Journal of Mathematics and Mechanics 17 (1967), 59–87. ↩

8. C. Calosi and J. Wilson, “Quantum metaphysical indeterminacy”, Philosophical Studies 176 (2019), 2599–2627. ↩

9. N. Bohr, “The quantum postulate and the recent development of atomic theory”, Nature 121 (1928), 580–590. ↩

10. R. Brandenberger and C. Vafa, “Superstrings in the early universe”, Nuclear Physics B 316 (1989), 391–410. ↩