Thursday, August 21, 2014

Nikolay Bogoliubov: 105th birthday

In Czechia and Slovakia, August 21st is primarily remembered as the anniversary of the 1968 occupation by the "brotherly armies" of the Warsaw Pact that ended the Prague Spring, a period of liberalization of socialism in Czechoslovakia. The invaders' actions 46 years ago look kind of moderate to me today, from the perspective of events in Ukraine and elsewhere, so I won't discuss the year 1968 today.

However, physicists are dying and being born on August 21st, too. In 1836, Claude Louis Navier (of the hydrodynamics fame) died. On August 21st, 1995, he was followed by Subrahmanyan Chandrasekhar.

However, I want to spend more time with Nikolay Bogoliubuv who was born on August 21st, 1909.

He was born in Nizhny Novgorod, the Russian Empire, currently Russia's 5th most populous town but they relocated to the Poltava Governorate, now a part of Ukraine, when he was 10, and to Kiev two years later. You may agree that Nikolay looked like a typical Russian academic. That may be a cultural issue: both parents were teachers (father: of philosophy etc., mother: of music).

Bogoliubov would later work in Moscow, Dubna, and some far East Russia during the evacuation in the early 1940s when Wehrmacht was advancing. He has collected some prizes after he died in 1992, e.g. the Dirac Prize, and tons of medals and prizes before he died, including the Stalin Prize, the Lenin Prize, the Stalin Prize (another one), the USSR State Prize, six Orders of Lenin, Hero of Socialist Labor, Order of the October Revolution, Order of the Red Banner of Labor, Order of the Badge of Honor, the Max Planck medal, and dozens of other medals and prizes. I wanted to enumerate some of the Soviet prizes because their diversity and omnipresent ideological flavor looks funny to me.

He's been an excellent scientist who wrote his first technical paper when he was 15. Later, he would write lots of things about the dynamical systems (related to the Liouville equation) and especially quantum field theory. The latter included results about the analytical continuation of correlation functions.

His most famous result is arguably the Bogoliubov transformation which is a transformation of the (generally multi-dimensional) harmonic oscillator spectrum (and the corresponding raising and lowering operators) according to a harmonic oscillator defined by a Hamiltonian with a different bilinear form. This transformation is essential for us to understand the particle production in accelerating frames and curved spacetimes, the Unruh radiation, the Hawking radiation, and so on.

What is it about? In classical field theory, you may define the vacuum simply as having the fields \(\Phi=0\) and therefore \(\partial \Phi / \partial t = 0\). However, in a quantum theory, these are "coordinates" and "velocities" that can't be simultaneously zero, due to the uncertainty principle. At most, you can make both of them "approximately zero" by considering the ground state \(\ket 0\) of the harmonic oscillator.

The ground state \(\ket 0\) is annihilated by the annihilation operator (or operators)\[

a_i \ket 0 = 0.

\] The annihilation operators are some linear combinations of the coordinates and momenta, schematically\[

a = \frac{x + ip}{\sqrt{2}}.

\] However, the ratio of coefficients in front of \(x\) and \(p\) is chosen according to the Hamiltonian we are interested in. And by considering generators of boosts instead of time translations etc., or by changing the Hamiltonian in a different way, we can modify the Hamiltonian. This also changes the relevant annihilation operator (or operators). The annihilation operator(s) relevant for one Hamiltonian is a linear combination of the annihilation and creation operator(s) relevant for another Hamiltonian; this mixing is what we often call the Bogoliubov transformation – or sometimes "the first steps to derive it".

The ground state of the operator \(10x+i p\) is different than the ground state of \(x+ip\), it is squeezed. The Gaussian wave functions have different widths. A Gaussian of a "wrong width" is of course not the ground state anymore. Just like all other wave functions that are not proportional to the ground state wave function, it describes the system with excitations – e.g. with particles (excitations of quantum fields).

It means that quantum mechanics makes the concept of the "vacuum" relative in one more new way. In classical field theory, the emptiness of the "vacuum" is independent of the acceleration of the observer etc. However, in quantum field theory where the width of the wave functions matters and it is nonzero, the acceleration does change the total number of particles.

Particle physicists often think about the Unruh and Hawking radiation when you tell them about "the Bogoliubov transformation". But the first application of the concept in 1947 was actually concerned with the BCS superconductivity.


  1. Bogoliubov would later work in Moscow, Dubna, and some far East Russia during the evacuation in the early 1940s when Wehrmacht was advancing.
    I am pretty sure that despite all his Lenin and Stalin medals witnessing about his überness and political correctness, he still didn't master supernatural powers.
    That's why he wouldn't presciently evacuate in 1940 because the Wehrmacht started advancing only on 22 June 1941 at 04:00 CET exactly on the same day when Napoleon's Grande Armée entered Russia too.
    Anecdotically both arrived to Moscow and both had to leave it again. The difference being that Napoleon smartly left in 4 months while the Wehrmacht preferred 4 years.
    In any case Bogoliubov was certainly right to prefer the Siberian safety after june 1941.

  2. Tom,

    I think you misread Luboš. He said "early 1940s", not "1940". Also, the tide had turned on the Wehrmacht by a lot earlier than 4 years in. For example, the battle of Kursk was only two years later in July 1943.

    I'm not an historian though.

  3. kashyap vasavadaAug 21, 2014, 5:57:00 PM

    Hi Lubos: As you mentioned, Bogoliubov transformation results in infinite number of field theory vacua.Is there an application to ST where also there are infinite or large number of vacua? Of course the two concepts are probably completely different.

  4. Bogoliubov transformations are included in string theory, too. Of course that string theory also has the Bogoliubov transformation-related "ambiguity" of the word "vacuum".

    But the Bogoliubov transformation is straightforward linear algebra. The mapping of the string landscale, which is also a "diversity of vacua" but of a very different kind, is the most complex thing in mathematics. Obivously they're not the same thing.

  5. kashyap vasavadaAug 21, 2014, 8:05:00 PM

    Thanks. Then If I understand ST vacua are different solutions of ST
    eq not related by Bogoliubov transformations.Right?

  6. His 1947 work would be about superfluidity. Superconductivity was understood only in 1957.