## Monday, February 27, 2017

Giotis brought us the bad news that Ludvig Faddějev died yesterday. To theoretical physicists, he is primarily known as a co-father of some ghosts (namely Faddějev-Popov $$bc$$ ghosts, along with Victor Popov, which are the "good ghosts" in the core of the modern covariant, BRST quantization of theories with local symmetries) and as a forefather of quantum groups.

He was born in Leningrad in 1934 (go polar bears: do you agree that the janitor looks like Faddějev? OK, Faddějev was in between the janitor and Arnold Schwarzenegger). His father was a well-known algebraist, his mother was doing numerical linear algebra. Not a bad pedigree. But he wanted to revolt and chose an occupation that was entirely different from his parents'. So he went to theoretical physics instead of mathematics even though he received a very good background in mathematics, partly thanks to Fock and Smirnov.

Obviously, my comment about the "very different field" was meant as a joke. His work remained very close to mathematical physics and like Dirac, he has always considered the mathematical beauty to be the key principle in the search for the laws of physics. As he said in the interview embedded below, that's why he considered himself a "mathematical", not "theoretical", physicist and why he was thinking differently than Landau's school that focused on the "physical sense". I am probably using the terms "mathematical physics" and "physical/mathematical sense" in a different way than he did so it's hard for me to agree with this logic.

He has received numerous prizes including the Dirac Prize, the Poincaré Prize, and the Orders of Lenin, of Friendship of Peoples, and of Merit for the Fatherland LOL. He led or co-founded various institutes including one he named after Euler.

The work that helped quantum groups to be born was his and his students' quantum inverse scattering method.

When Faddějev and Popov invented their Faddějev-Popov $$bc$$ ghosts in the 1960s, they showed how tightly integrated the Soviet theoretical physics was within the global science at that time. But as we mention below the video, it wasn't all or "theoretical" Russian physicists who were working hard in quantum field theory – only the "mathematical physicists" (that's also the Russian community that co-discovered supersymmetry in the 1970s). I think that Feynman technically proposed the FP ghosts before Faddějev and Popov did (and to make his power more shocking, his first example where he introduced them was quantum gravity – the graviton scattering) but they were probably the first ones to write a comprehensive paper on it.

A 2015 interview. It also says that 50 years ago, Landau considered quantum field theory a dead end. Oops. And in the video above, Faddějev said the same thing about string theory. Oops. I've never realized such things. That was quite a blunder from the Russian physics genius. Veltman was visiting Russia and was very helpful for the publication of the Faddějev-Popov paper in a journal (that required no payments). Why was he helpful? Because unsurprisingly, he didn't understand a word in the paper but the names of the authors sounded good enough for him. (That's a similar contribution that has also allowed Veltman to earn his Nobel prize winner for 't Hooft's work.)

In Yang-Mills theories, the contribution from loops of these $$bc$$ ghosts was needed to subtract some unwanted contributions to the amplitude that may be identified as artifacts of gauge-fixing. These days, the extra $$bc$$ ghosts are usually taught as extra fields needed in the BRST quantization. In theories with gauge symmetries, one may construct the fermionic charge $$Q$$, the BRST operator, that obeys $$Q^2=0$$ and physical states are cohomologies of $$Q$$.

It means that in the space of all possible states, we basically labeled "non-closed states" i.e. those obeying$Q \ket\psi \neq 0$ to be unphysical and removed by brute force, while states obeying$\ket\psi = Q\ket\lambda$ are also considered unphysical, but in a different sense: they are pure gauge. So physical states are only those that are annihilated by $$Q$$ but it mustn't be for the simple reason that they are $$Q$$ acting on something themselves (it's trivial because $$Q^2\ket\lambda=0$$). Their norm is zero – they are null – so their addition to physical states don't modify any probabilities. With this setup, the $$D$$ polarizations of the Yang-Mills field $$A_\mu$$, to mention one of the most familiar examples, along with the two fermionic polarizations obtained from the $$b,c$$ excitations, are reduced from $$D+2$$ to $$D-2$$ physical, transverse polarizations.

An equally important basic example of the FP ghosts appears in string theory – the conformal symmetry is the relevant local symmetry that is being treated in this way. In bosonic string theory, the central charge of the $$bc$$ system turns out to be $$1-3k^2=-26$$ because $$k=3$$, and that's why they must be accompanied by 26 fields $$X^\mu$$ parameterizing a 26-dimensional target spacetime. That's an easy way to derive the critical dimension of bosonic string theory; the calculation of $$D=10$$ in superstring theory is analogous but involves new technicalities with superghosts and physical fermionic fields.

It's an elegant method and cohomology may be said to be "fundamental" and one is tempted to generalize the method and ideas. On the other hand, I still think it's important that it may also be said to be just a clever trick. In principle, one could derive all the predictions – including the loop corrections – without ever talking about the FP ghosts.