Friday, November 07, 2014

Tullio Regge (1931-2014)

Tullio Regge died two weeks ago. Click for an obituary.

Regge, a recipient of many awards including the Dirac medal (1996) and a former deputy of the European Parliament (since 1989), was one of the forefathers of string theory. I will discuss Reggeology later but it was not the only approach to quantum gravity that he has pioneered.

Regge calculus

What is the Regge calculus? Well, you don't have to be excessively ambitious to view this framework as an approach to quantum gravity. In fact, you may realistically view it as a very useful method in numerical relativity. What is it?

It sounds pretty simple but someone had to invent it. Regge did it in the early 1960s.

Take a manifold - for example a 4-manifold - and triangulate it in some way so that the faces are flat and they are merged in flat codimension-1 edges which are connected in lower-dimensional edges ... that eventually terminate in vertices. In this setup, the whole curvature (of either sign) is concentrated in the vertices in the form of a deficit/excess angle: recall the methods how to calculate the Euler character.

Regge was able to rewrite Einstein's equations as constraints on the curvature stored in these vertices. This makes numerical calculations in general relativity more "generally covariant" (or "coordinate-independent") than e.g. a metric tensor on the lattice.

I personally find it unlikely that the Regge calculus may be relevant for quantum gravity in any nontrivial way but I surely find it more relevant than various "causal dynamical triangulations" (CDT) etc. because the Regge calculus is at least known to reproduce Einstein's equations when it is dealt with properly. Whenever things like CDT are counted as approaches to quantum gravity, the Regge calculus should be included, too.

Regge has also contributed papers about the stability of black hole singularities, about the solitons, and about many other topics. But for obvious reasons, I will look at:


Reggeology, an unusual yet powerful approach pioneered in 1957 and culminating in the 1960s, is something else than the Regge calculus. It focuses on the analytic properties of scattering amplitudes and interprets them dynamically. The Regge slope is a part of the story.

Let me begin with some general comments. Like many insights, Reggeology will stay with us and many of its results are powerful enough so that they cannot be "forgotten".

They differ from the "routine physics" that has been taught in quantum field theory courses for decades but they are not another example of the "populist armchair physics", cheap ideas that the laymen instantly understand and that the good physicists can instantly see as being on a wrong track. Quite on the contrary: Reggeology is a deep set of methods that can only be appreciated by a sufficiently thoughtful physicist after a sufficiently long effort. And they survive.

In this framework, you write scattering amplitudes as functions of the angular momentum. The angular momentum is normally an integer or a half-integer but you assume that it is a complex number and you analytically continue the scattering amplitude to the whole complex plane.

Such a continuation will have simple poles. You should care about their position. An essential point is that you should forget whether the operations you are doing have a "simple" physical interpretation. Instead, approach these matters mathematically and try to get as much as possible.

You will find out that the amplitudes increase as a power law for large values of \(\cos(\theta)\) that are very close to one. Surprisingly, it is damn important to know how these things behave. It turns out that the exponent in the large-cosine-theta behavior is related to the position of the pole in the angular-momentum plane.

If you want a constructive explanation of a similar relationship, imagine that the scattering amplitude arises from the exchange of a spin-\(j\) particle P in the \(t\)-channel. The interaction with the external scalar particles - that we scatter - must involve \(j\) spacetime derivatives, to contract the Lorentz indices of P in the cubic vertex. That will make the amplitude scale like the \(2j\)-th power of the energy.

This is a powerful observation. I didn't want to introduce the unusual and complicated parametrization in terms of the angles, cosines, and angular momenta, but Regge did so. Quantitatively speaking, the relevant behavior occurs in a physical limit of large \(s\), fixed \(t\) (Mandelstam variables) because
\(2t = -s (1 - \cos \theta),\)
i.e. in the small angle scattering. In this so-called Regge limit, the amplitudes go like
\(A(s,t) \sim s^J\)
where \(J\) is the typical angular momentum, i.e. the location of the pole in the \(t\)-plane. This term in the amplitude should therefore be proportional to \(1/(t-J)\), too, and the position of the pole, \(J\), is a function of \(t\), namely \(\alpha(t)\). You see that the amplitudes increase quite rapidly.

In theories with a better, softer ultraviolet (high-energy, short-distance) behavior, you don't expect a power law increase of the amplitudes, at least not in the ordinary high-energy regime of large (and comparable) \(s,t\), i.e. in the so-called hard scattering limit. But how do these theories get rid of the power law increase? Well, they combine the contributions of intermediate particles of infinitely many values of the spin. In perturbative string theory, the relevant intermediate particles lie on the (simplest choice of the) Regge trajectory,
\(J = \alpha(t) = \alpha' t + \alpha(0)\)
where \(\alpha'\) is the Regge slope - equal to \(1/(2\pi\,{\rm Tension})\) in string theory - and \(\alpha(0)\) is an additive shift, related to the tachyon's squared mass. Such a prescription indicates that there exist infinitely many particle species whose squared mass is a linear function of their spin (a fact that is easy to derive in string theory because every unit of "oscillation" adds a constant to the string's mass squared; but this fact was known long before the "strings" were seen).

When you sum their contributions, you may obtain an expansion of a function that is well-behaved - in fact, exponentially decreasing - in the ordinary hard scattering limit.

This computation was first thought to be relevant for hadrons. Later, it became unpopular and people used to emphasize that the only correct application of this Reggeology was quantum gravity and the previous links to QCD were unjustified.

However, in the last 16 years, the AdS/CFT correspondence has shown that the original motivation behind Reggeology was also justified and many of its ancient results hold because strongly interacting gauge theories are equivalent to quantum gravity, after all. We also know why other results don't hold - some of them have to be modified because of extra dimensions that used to be neglected. In some sense, Maldacena's equivalence has revived and updated some of the most unpopular approaches to nuclear physics and made them really hot and useful. See also:
In the treatment above, we have seen some powers of \(s\) and positions of the poles in the \(t\)-plane. But in the normal Feynman diagrammatic approach, you might also want to add the \(s\)-channel diagrams where the roles of the \(s,t\) invariants are interchanged. Because of a certain deep intuition, the Regge physicists kind of knew that for infinitely many particle species, such a computation would be a sort of double-counting because the \(t\)-channel diagrams already have the capacity to include all the terms you want, as long as you have infinitely many species.

The "dual theories" or more frequently "dual models", later renamed as "string theory", were born. The equivalence between the \(s\)-channel and \(t\)-channel contributions is currently called "world sheet duality" not to confuse it with many other, more modern types of dualities that were later discovered in string theory. The world sheet duality was the most important requirement that led Gabriele Veneziano to write down his amplitude (the Euler Beta function) that turned out to be an exact tree-level open-string scattering amplitude.

The evolving name of string theory

In the early 1970s, the physical stories illuminating all these unexpected relationships between special kinds of amplitudes changed dramatically. For this reason, people also changed the name of the theory they studied. The theory - or theories - used to be called "dual models" in the late 1960s to refer to the world sheet duality. This duality was soon explained as a simple consequence of the stringy picture: there is only one stringy tree level open string diagram (disk) and it can be expanded in two ways as QFT diagrams, either in the \(s\)-channel or the \(t\)-channel way.

Because the previous insights turned out to be merely a "tip of the iceberg" and many more specific insights followed from the structure that could have been derived from strings, the "dual models" were renamed "string theory". In a similar way, we have learned many new things about "the theory" in the 1990s that showed that even the strings themselves were another "tip of the (larger) iceberg" but because people don't want to rename the theory every decade once they learn something new and because our qualitative idea about the theory changed a slightly less dramatically than in the early 1970s, the term "string theory" stuck (even though some people wanted all string theorists to be known as M-theorists).

At any rate, the general and generalized Regge-like approach is something that I believe will be important in the future of theoretical physics, too. The analytical continuations of amplitudes and other objects to unusual regimes and various new relationships between features of seemingly different and unrelated regimes is what tests the depth of our understanding and the rigidity of the theories we use to describe the real world.

Every new extrapolation of amplitudes that makes sense is a good news. And every new relationship between previously unrelated phenomena and quantities is something that makes our description of the Universe more concise and less arbitrary because it removes the number of independent assumptions. Dualities - including the world sheet duality; UV-IR dualities; S-duality; T-duality; U-duality; bulk-boundary duality, and so on - are priceless. Regge has helped to refine this strategy tremendously.

Deja vu.


  1. kashyap vasavadaNov 7, 2014, 5:29:00 PM

    Thanks for
    nice summary of Regge’s remarkable work. May he RIP. It is good to note that
    people who dismiss ST as useless will have to come up with an alternative which
    would explain success of Regge phenomenology and Veneziano model even if you do not mention unification
    and other successes of ST. That will be stumbling block for such people!

  2. The thing is that such "people" either dont know and/or give a damn about the scientific method ...

    Anyway, I like these nice explanations too and I also wish him to rest in very well deserved peace.

  3. Aleksandar MikovicNov 8, 2014, 4:50:00 PM

    Recently I have shown in a series of papers that Regge calculus can be useful for defining a quantum gravity theory, see arxiv:1407.1394, 1407.1124 and 1402.4672. The new ingredients are: (1) application of the effective action formalism (2) the spacetime triangulation is physical, i.e. the spacetime is a PL manifold and not a smooth manifold. By appropriately choosing the path-integral measure one obtains a finite quantum theory of gravity whose classical limit is given by the Regge action when the edge-lengths are much larger than the Planck length (L_P). Furthermore, the effective action for triangulations with a large number of triangles is well approximated by the QFT effective action for GR with a physical cutoff.

    When the matter is coupled, the effective cosmological constant is a sum of the classical CC + quantum gravity CC + matter loops CC, and classical CC + matter loops CC can be set to zero, which leaves the quantum gravity CC of O(l_P^4/L_0^4) << 1. The length L_0 is the parameter of the PI measure, and it is a free parameter with the only restriction L_0 >> L_P. By taking L_0 = 10^{-4} m one recovers the observed CC value.

  4. Hey Lubos,
    some related (and comforting) Italian news: the manslaughter verdict against the geophysicists of the National Committee for the Prevention of Major Risks for "not predicting the L'Aquila earthquake" has been overturned on appeal.

  5. Torsten Asselmeyer-MalugaNov 18, 2014, 1:38:00 PM

    PL manifolds and smooth manifolds are equivalent for dimension smaller than 6.