Off-topic but good news: Mathematica 9 is released today and it's big, offering the first useful "predictive interface" with suggestions in the world of software, analysis of social networks, interactive gauges, system-wide support for 4,500 units, systematic addition of legends, support for Markov and random processes, integration of Steve McIntyre's language R, and moreBritish field theorist and string theorist David Olive, Commander of the Most Excellent Order of the British Empire, Fellow of the Learned Society of Wales, and Fellow of the Royal Society died on November 7th, at the age of 75.
If I remember well, I've never met him. But I've experienced lots of his key and beautiful insights.
INSPIRE shows that he co-authored 7 articles above 500 citations and another one above 100 citations. Let's look at them; I apologize that the remaining 81 papers that Olive has written won't be discussed below at all.
Montonen-Olive duality
In 1977, he and Claus Montonen of Finland published the Montonen-Olive duality (Wikipedia). It was a fascinating and utterly modern – according to the 2012 criteria – discovery combining (at that time) rather fresh solutions describing the magnetic monopoles (by 't Hooft and Polyakov) and supersymmetry.
In the spirit we know from some very recent "visionary conjectures", they hypothesized that in the \(\NNN=4\) supersymmetric gauge theory, the magnetic monopole states – that are as heavy as \(C/g^2\) and that we know as quantum states localized around nontrivial classical solutions, regulated versions of the Dirac magnetic monopole – come in degenerate families that have the same structure as the electrically charged states. If it's so, their interactions are likely to mimic the electric interactions as well. In fact, the whole gauge theory is invariant under a strong-weak duality \(g\leftrightarrow 1/g\) that, when combined with the shift of the \(\theta\)-angle, extends to the currently well-known \(SL(2,\ZZ)\) S-duality symmetry.
Note that Maxwell's equations \[
\begin{array}{|c|c|c|}
\hline
\nabla\cdot \vec E =\frac{\rho_E}{\epsilon_0} & \nabla\cdot \vec B = [\rho_M]\\
\hline
\nabla\times \vec B =\mu_0 \vec j_E+\mu_0\epsilon_0\pfrac{\vec E}{t}&
\nabla\times \vec E = [-\frac{\vec j_M}{c^2}] -\pfrac{\vec B}{t}
\\
\hline
\end{array}
\] had always have a symmetry between the electric fields and the magnetic fields (in particular, the electromagnetic induction is mutual and works in "both directions" which is important for the existence of oscillating electromagnetic waves), and if Sheldon Cooper had found genuine magnetic monopoles on the North Pole, the symmetry would extend to the electric and magnetic charges, too. In fact, in the real world, magnetic monopoles indicated by the density \(\rho_M\) and the current \(\vec j_M\) probably exist, too, but the lightest ones are very heavy particles, unlike electrically charged particles such as the electron which are light, so there's no symmetry between the masses of electric and magnetic monopoles. Those particles are so inaccessible that we haven't seen them yet and \(\rho_M,\vec j_M\) is omitted in Maxwell's equations.
However, in some other quantum field theories and/or vacua of string theory, the behavior of the sources, i.e. the electric and magnetic charges, may become symmetric, too. The maximally supersymmetric Yang-Mills theory is currently our simplest example of that. (The complete symmetry between the electric and magnetic sources is violated unless \(g=1\).) The high degree of supersymmetry doesn't make the theory contrived, as anti-supersymmetric crackpots of the Woit type love to say, but they make the theory constrained, more fundamental, and simple to calculate because the strong constraints from supersymmetry guarantee that many easily calculable approximations are actually exact. When we do so, we may verify numerous consistency checks.(The Montonen-Olive results were kind of extended to lower supersymmetry, \(\NNN=2\), by Seiberg and Witten, and to \(\NNN=1\) theories by Seiberg. Lots of new physics and difficulties arise as the less supersymmetric gauge theories are growing less constrained and more general.)
In string theory, the Montonen-Olive duality may be interpreted as a symmetry inherited from some stringy backgrounds, e.g. the type IIB string theory, and it may also be geometrically proved using the (2,0) superconformal field theory compactified on a two-torus. If I see enough readers who want some new fresh perspective on the Montonen-Olive duality and who have a realistic chance to "get it", I will write about it later.
Olive wrote two more famous papers on magnetic monopoles in 1977 and 1978, with Goddard and Nuyts and with Goddard, respectively. And in 1978, he also co-wrote a paper with Edward Witten showing that supersymmetry algebras may include topological (e.g. "magnetic") charges on the right hand side.
In 1985 and 1986, during the (middle and later) first superstring revolution, Olive co-authored three famous papers on Kač-Moody and Virasoro algebras and their representations (with Goddard and, in two cases, also Kent). But let me return to one more key paper he co-authored in his miraculous year 1977.
GSO projection and spacetime SUSY in string theory
Olive's most cited paper (full text) approaching 1,000 citations was written together with Ferdinando Gliozzi and Joel Scherk (who unfortunately tragically died in 1980).
I think that with 50 pages, the paper is incredibly talkative, redundant, and in some aspects confusing (some of the R-symmetry groups seem to be smaller than they are, and so on). To be fair, most of the results were "announced" in an earlier 15-page paper in PLB. And the terminology is surely outdated. If you open the paper, you have to return to the era when string theory was studied by a dozen of heroes in the world, it was called "dual models", and superstring theory (with world sheet fermions) was referred to as the "dual spinor model".
But forget about all the doubts: it was a damn important paper.
It was the first paper that showed that realistic spacetime supersymmetry actually emerges from string theory. For a few years, it's been known from the work by Neveu and Schwarz and the work by Ramond (especially the latter was the first emergence of unbroken supersymmetry in the West – on the two-dimensional world sheet of string theory) that string theories with world sheet fermions may exist. But things had been puzzling.
Neveu and Schwarz talked about closed strings whose fermionic degrees of freedom were antiperiodic:\[
\psi^\mu(\sigma+\pi) = - \psi^\mu(\sigma).
\] But Ramond talked about closed strings with periodic fermions, i.e. the equation above without the minus sign. It wasn't clear which of them was right, whether only one of them was right at all, how to combine them if both of them were right, and what fields in the spacetime actually survive.
GSO clarified those issues. One has to use both the NS antiperiodic boundary conditions as well as the R periodic boundary conditions. But because we "doubled" the number of basis vectors, we have to reduce it to one-half again – in both sectors. That's what the GSO projections are good for. Only states that obey\[
(-1)^{F_L}\ket\psi = (-1)^{F_R}\ket\psi = +\ket\psi
\] are included in the physical spectrum; the remaining 3/4 of states whose eigenvalues of \((-1)^{F_L}\) or \((-1)^{F_R}\) are equal to \(-1\) are labeled as unphysical and filtered out from the spectrum. (Note that the doubling and halving of the basis vectors was done separately both for the left-moving excitations and the right-moving excitations of the closed string).
This is a consistent constraint on the physical spectrum because \((-1)^{F_L}=\pm 1\) which counts whether the excited string state is bosonic or fermionic (even or odd number of fermionic excitations) is a symmetry, and similarly for \((-1)^{F_R}\).
This consistent constraint is actually a natural "flip side" of the fact that we include both the NS and R sectors. To allow the fermions to be periodic or antiperiodic on the closed string (both), we must declare the operator \((-1)^{F_L}\) to be a part of the gauge symmetry group. Note that this operator is what changes the sign of fermions if it acts on them (on these fellow operators) via conjugation (and that's the right way how operators act on other operators)\[
(-1)^{F_L}\psi_L^\mu (-1)^{-F_L} = -\psi_L^\mu
\] which is true because on the left hand side, we count the fermionic excitations, then we add one, and we count them again – so that we get the opposite sign and the excitation itself survives. So the string with the opposite boundary condition is obtained by "twisting" using this operator \((-1)^{F_L}\): the operators on the string are periodic up to a conjugation by this operator.
But we may only say that "it is allowed to twist and conjugate in this way" if this operator \((-1)^{F_L}\) is considered to be "de facto the identity operator", more precisely, if it is treated as an element of the gauge group. But if it is so, the physical states must be invariant under this operator. The latter constraint is the GSO projection.
This GSO projection has lots of desirable consequences. The invariance under the "diagonal" GSO projection \((-1)^{F_L+F_R}\) which follows from the two chiral invariances is nothing else than the condition imposing the correct spin-statistics relationship. It's because you should have been worried that we have world sheet fermions \(\psi_\mu\) that are spacetime Lorentz vectors – vector-like fermions are "bad". And indeed, the "diagonal" GSO projection only allows you to add these spin-statistics-violating excitations in pairs so that the spin-statistics relationship is preserved for the physical states.
However, the GSO projection constraints the left-moving and right-moving excitations separately, as we have said, so it's stronger. (The "superstring" theories with the diagonal GSO projection only are modular-invariant – they obey a certain related consistency condition – but they still predict tachyons so their degree of consistency is pretty much on par with the \(D=26\) bosonic string theory. They're known as type 0A/0B string theories.) It's capable of eliminating the tachyon from the open-string spectrum and the tachyon from the closed-string spectrum, too. It reduces the "Dirac spinor" of states in the Ramond sector to a "Majorana-Weyl spinor" which is also good. And when you look at the remaining bosonic and fermionic states of the superstring, you will find out that their numbers match at each excited or unexcited level due to a nontrivial identity – "aequatio identica satis abstrusa" (a rather obscure formula) – that early string theorist Carl Jacobi discovered in 1829.
In fact, the equal number of states is a symptom of a symmetry – the spacetime supersymmetry – and this supersymmetry holds not only at the level of the free spectrum: it holds nonperturbatively, too. When the "somewhat dull, trivial, and confusing" world sheet supersymmetry was suddenly able to produce this remarkable, realistic, nontrivial, interacting supersymmetry in the spacetime, the physicists had to have a special feeling, indeed.
Note that a few paragraphs above, I mentioned that we declared the fermion-counting parity operators \((-1)^{F_L}\) and \((-1)^{F_R}\) to be elements of the gauge symmetry group so we had to assign all the "tasks" to these operators that elements of gauge groups always have: physical states have to be invariant and we must allow states where the periodicity holds "up to transformations by the gauge group element". There exists an alternative, Feynman-path-integral-based, "more controllable" way to describe why all these tasks have to be fulfilled together. It's called "modular invariance". The torus partition function has to be invariant under the 90-degree rotation of the torus (i.e. on the interpretation which of the two basis cycles of its first homology is which). Equivalent mathematics applies to compactification and orbifolds. In all these cases, we're adding "new sectors" (twisted strings, wound strings...) but we have to project the spectrum so that only the invariant states (invariant under the orbifold group, with quantized momentum in the compact dimensions...) are preserved.
Those things look totally crystal clear today but it had to be very interesting for the first discoverers to find these insights in the shadows of ignorance when the shining light of the truth suddenly emerges. I know this feeling, it's a special one.
RIP, Dr Olive.
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