## Saturday, August 08, 2009

### Paul Dirac: a birthday

Paul Adrien Maurice Dirac was born in Bristol, England, on August 8th, 1902, i.e. 107 years ago. He died in 1984 in Tallahassee, Florida.

His mother was English while his father was an immigrant from Switzerland and an obnoxious family Hitler who taught French and who forced his family to speak French at their English home. Paul's elder brother committed suicide in 1925. In Bristol, Paul gratefully avoided the classics which was the dominant subject taught to his contemporaries, and studied electrical engineering instead.

By the way, at his elementary school, he was asked to solve a famous problem with the three fishermen which may have helped him with the antiparticles 15 years later. I will discuss the episode later in my essay.

After some time spent with electrical engineering, he managed to move to Cambridge to meet his true love, mathematical physics. His PhD in 1926 was based on the work that described the analogy between the commutators on the quantum side (more precisely, in Heisenberg's matrix mechanics) and the Poisson brackets on the classical side. Note that Dirac was 24.

Later, he would study this map in detail. So he was also predestined to figure out how to quantize systems with second-class constraints (using Dirac brackets). Also, Dirac was convinced that the action from classical physics should be relevant for a formulation of quantum mechanics, and he sketched some very rough arguments that Richard Feynman was able to streamline when he invented the path integrals.

Dirac equation and Fermi-Dirac statistics

His following greatest discoveries were very natural in the context of the quantum revolution, too. In 1928, he built on Wolfgang Pauli's non-relativistic spinor equation and constructed his relativistic spinorial Schrödinger-like wave function, the Dirac equation, which is often praised as one of the most beautiful equations in physics.

Obviously, this equation had solutions with both positive and negative energies, because "E^2-p^2 = m^2" is satisfied by values of "E" of both signs. Pairs of positive and negative energy electrons could be created, destabilizing the naive vacuum.

Dirac realized that the true vacuum in Nature has to have the minimum allowed energy. If Nature can fill states with negative energy, such a decision of Hers actually reduces the energy of the state. So Nature wants to fill all of the negative-energy electron states: the union of these filled states is referred to as the Dirac sea. It fills the space. Dirac had to assume that only one electron can be put in each state: he was able to determine other aspects of the Fermi-Dirac statistics, too (he called it "Fermi statistics", and for the sake of "symmetry", used the term "Einstein statistics" for bosons).

On the contrary, individual negative-energy particles may be removed from the Dirac sea, which creates holes, or antiparticles: in this case, they're the positrons. Carl Anderson observed them in 1932, earning Dirac a Nobel prize as early as in 1933.

Bosons and fermions used to be treated differently for quite some time: there was a lot of confusion about the question whether bosons were just "waves" while fermions were just "particles". In Weinberg's Prague talk yesterday, it was argued that even Richard Feynman bought into this confused "dualistic" viewpoint. Wolfgang Pauli was the main person to advance the modern "unitary" viewpoint: both bosons and fermions are described by quantum fields and may behave as waves or particles.

Dirac found the relativistic equation for the electron, the lightest and most important charged particle in Nature. Because it's charged, its interactions with the electromagnetic field are important. Meanwhile, the electromagnetic field became quantized, too. Dirac realized that one should study the field theory of interacting charged matter fields (such as the electron's Dirac field) and the quantized electromagnetic field.

Quantum Electrodynamics

We may argue whether he was the true father of Quantum Electrodynamics but one thing is clear: he gave it the name. ;-) QED remains the most accurately verified theory in all of science (especially thanks to the electron's anomalous magnetic moment). This fact is possible because of the small value of the fine-structure constant that makes the perturbative expansions immensely precise.

Dirac's 1930 textbook on Principles of Quantum Mechanics remains a golden standard. We may say that the book was not just about a presentation of known facts: in fact, in this book, he presented his bra-ket formalism that implies the equivalence between Schrödinger's wave mechanics and Heisenberg's matrix mechanics. Dirac's unified treatment of both approaches was the real beginning of full-fledged quantum mechanics.

The inner products of the bra-kets naturally led him to discover the continuous bases and the Dirac delta function, another object named after him (much like the related Dirac comb).

When I mentioned that QED remains the most accurate theory, you shouldn't think that Dirac should be credited for this precision. Not at all. ;-) All the precise calculations involve loop diagrams which require renormalization, or careful removal of infinities. Dirac has never understood it (or "has never accepted it", to be polite). In his opinion, one could only neglect small things, not infinite things! He insisted on this crazy opinion even in the mid 1970s.

Renormalization is needed

Well, there are several ways to explain why Dirac's argument is bogus. With counterterms, the infinities are properly canceled and one doesn't neglect anything. And these terms have to exist. In terms of the bare constants, they must simply be considered "infinite" for the observed results to be finite, and there's nothing wrong about unobservable parameters' including infinite parts.

Dirac was the main champion of mathematical beauty of physical equations. In Moscow, he wrote that "physical laws should have mathematical beauty" somewhere on the door and they were quite proud about the sentence. But it is important that he was talking about "mathematical beauty" and not some generic "beauty", e.g. one found in poetry: Dirac proved that poetry - attracting physicists such as Oppenheimer - was an unscientific enterprise. Poetry and science are incompatible because the goal of science is to make difficult things comprehensible in a simple way; the goal of poetry is to articulate simple things in an incomprehensible way. ;-)

Women et al.

Paul Dirac's relationship to the social activities and women was pretty much identical to that of Sheldon Cooper; some sources talk about Dirac's autism. For example, he investigated why Heisenberg was dancing. He was told that it was a pleasure when there were nice girls. However, Dirac didn't stop his penetrating intellect: "But, Heisenberg, how do you know beforehand that the girls are nice?" :-)

Once upon a time, a visitor was shocked to see an attractive woman in Dirac's house - Dirac's wife. Dirac realized he was in trouble so he said that the woman was Wigner's sister. Of course, Dirac was right! ;-) They married in 1937 and had 2 children of hers from a previous relationship plus 2 children of theirs.

Religion

While Dirac was convinced that God used beautiful mathematics in creating the world, he didn't accept any religious "myths" because of many things, starting with their mutual inconsistency. ;-)

At the 1927 Solvay Conference, Dirac presented his anti-religious arguments. Heisenberg was open-minded while Pauli summarized Dirac's monologue: "Dirac also has a religion whose first commandment says: there is no God and Paul Dirac is His prophet." Everyone exploded in laughter, including Dirac.

Other things named after him

We have talked about the Dirac equation. It is an equation with the Dirac operator acting on a wave function that transforms as a Dirac spinor (and describes a Dirac fermion). The operator is defined in terms of the Dirac matrices which satisfy the Dirac algebra. Many things may be done with the Dirac spinors. For example, one can calculate the Dirac adjoint which is the reshuffled Hermitian conjugate spinor (psi-dagger gamma_0) whose multiplication with the original spinor gives us Lorentz-invariant scalars.

The analysis of charged particles interacting with the electromagnetic field led him to calculate e.g. the Abraham-Lorentz-Dirac force, the relativistic completion of the Abraham-Lorentz force (which is proportional to the jerk, i.e. the time derivative of the acceleration).

The bra-ket notation naturally led him to include the Dirac delta function among generalized functions: this "function" is really a distribution and defines the Dirac measure. The Fermi-Dirac statistics allows Nature to fill the Dirac sea and sometimes forces the physicists to calculate the Fermi-Dirac integral.

But there are many other ideas and concepts that Dirac invented or discovered that have affected physics, including physics of string theory.

For example, Dirac formulated the Dirac large number hypothesis. Unexpectedly large (or tiny) dimensionless ratios in Nature were conjectured to evolve in time, and be increasing (or decreasing) functions of the age of the Universe (which is getting larger as we're getting older). Most of these hypotheses that would naturally "explain" something have already been falsified because the constants don't change much, as observations indicate.

Dirac's idea could have always been considered to be an example of numerology, and not a very accurate one. But similar mechanisms appear in the theory of quintessence and even Louise Riofrio's famous theories ;-) are simplified versions of Dirac's old incorrect proposals.

Minus two fish

At the beginning, I mentioned the famous school problem he was solving when he was a kid.

Three fishermen catch N fish. In the morning, one of them wakes up. He fairly wants to take 1/3 of them and leave. But the number is not a multiple of three, so he throws one fish back to the pond and takes 1/3 of the rest.

The second fisherman doesn't know that one of his colleagues is already gone. Once he wakes up, he also wants to take 1/3, but the fish ensemble he sees is not a multiple of three, so he throws one fish away and takes 1/3 of the rest.

The third fisherman doesn't know that they're gone, so he throws one fish and takes 1/3 of the rest, too. What's the minimum number of fish N that they had to catch?

The clever classmates of Dirac found the conventional answer, 25. (25-1 = 24, 24-8 = 16, 16-1 = 15, 15-5=10, 10-1=9, 9-3 = 6). However, Dirac was better because he found a smaller solution. Of course, the fisherman caught N=-2 fish (-2-1 = -3, -3-(-1) = -2, and so on). The advantage of N=-2 is that it is a fixed point. (Other solutions differ by multiples of 27.)

I can imagine that this natural way of thinking about negative numbers helped him to deal with the Dirac sea and the negative-energy solutions to the Dirac equation later. In this sense, he may have been predetermined to theoretically discover antimatter.

In fact, Paul Dirac was much more radical when it came to negative numbers. He was convinced that negative probabilities should have been considered seriously, much like the minus two fish. Well, I am OK if they're "quasi-probabilities", some intermediate results (like in Wigner's distribution), but I don't think that they can appear as final predictions for probabilities because the number of events we see can't be negative.

By the way, Cumrun Vafa is among the physicists who are most intrigued by the negative probabilities. He believes that quantum mechanics may be formulated in a new way in which negative probabilities will become much more important.

Dirac monopole

Dirac was also the first guy who wrote the solution for the magnetic monopole. He realized that the vector potential couldn't be defined globally because the divergence of "B" is no longer quite zero (but a Dirac delta function). But one could define the vector potential almost globally - in the whole space with a semi-infinite "Dirac string" removed.

The Dirac string is a thin solenoid that connects the magnetic monopole with the opposite monopole at infinity. Together, they make up an ordinary magnet with both poles. However, we only want the monopole itself, so the rest must be undetectable.

The other monopole at infinity is undetectable because of the distance but the solenoid is only invisible if the Aharonov-Bohm effect produces no phase for charged particles orbiting the solenoid. That led Dirac to realize the Dirac quantization rule - namely the fact that the magnetic monopoles' charges must be multiples of "2.pi/e" where "e" is the minimum elementary electric charge. I think it was many years before the Aharonov-Bohm effect was understood in the normal physical context (where we're interested in the situation in which the phase is nontrivial rather than a multiple of 2.pi).

The Dirac monopole trick has been generalized to arbitrary numbers of dimensions etc. Whenever you count charges and their quantization in string theory, you deal with a lot of numbers and arguments that generalize Dirac's quantities and arguments.

Similar comments apply to the Dirac-Born-Infeld action: these guys were able to write down the effective action for a D-brane in string theory, decades before string theory was born.

At any rate, Dirac was a giant. He had an excellent intuition about beauty and many other things, he was stubborn about most questions, and much like other geniuses, he was sometimes wrong - especially when it came to renormalization. I guess that he would be much more satisfied with our current, stringy picture of the ultraviolet physics where the divergences are literally absent - and maybe he would finally understand why it doesn't hurt  to have intermediate divergent quantities, either.

In Lev Landau's logarithmic classification, Dirac has made it to the top 5 physicists of the 20th century. While Einstein got "0.5", close to Newton's "0.0", Dirac shared the "1.0" grade with Bohr, Heisenberg, and Schrödinger.

1. Also from Dirac we have the bra ket notation lbracket p|H|q rbracket where p and q are vectors and H the Hamiltonian. This is obviously the generalised inner product. But bras and kets provide easy methods of descrbing the phase of vectors, and not their norms lbracket p| is a bra and |q rbracket is a ket.

Note that if there is but one Dirac magnetic monople all charge is quantised, which is to say, the quantised field itself is the divergence of he B field of that monopole.

That led people to search (rather in vain) for that one or how many of them there are magnetic monopoles; about the most we can say right now is, the density of magnetic monopoles in the Universe is lower than about one per several cubic parsec.

lbracket and rbracket are the HTML delineators and won't print!

So you can't do quantum mechanics in HTML!

2. So what would A and B be in the equation for the fishermen problem?

3. Dude, what the hell did you do to my comment :(

4. There is no universal "A, B for the fishermen problem". What you probably wanted to ask is what are A,B in the sentence

Other solutions differ by multiples of 27.

In that sentence, A is a solution, B is -2 (quoted in the previous sentence, a solution), and A-B = 27k. I simply said that the other solutions have the form

A = -2 + 27k