Beauty of Dirac equation and its extensions

Many people consider the Dirac equation - and sometimes Einstein's equations of general relativity - to be the most beautiful equation of physics.

I understand where they're coming from. When I was 15 years, there would be lots of Einstein's equations written in beautiful fonts at many places of my notebooks. ;-) When I appreciated the Dirac equation a year or two later, I was already too old for mindlessly writing equations just to appreciate their beauty. :-)




The Dirac equation is beautiful but many people, e.g. commenter Gunn at the US LHC blogs, conclude that the equation is beautiful because its extensions and completions are ugly. These people are completely wrong.

In this text, I will show that the Dirac equation is just the first step towards many other important structures in theoretical physics. With a little bit rational, predictive, and comprehensive sense of beauty, the extensions are at least as beautiful as the Dirac equation itself. Let us return to the 1920s and begin.

The first equations of quantum mechanics

In the 1920s, people understood that particles such as electrons exhibited some wave properties. Louis de Broglie has postulated a wave associated with every particle and Erwin Schrödinger wrote the differential equation that this wave could satisfy in the presence of external forces - or potentials.

In fact, Schrödinger first discovered the relativistic equation

( d²/dt² - Δ + m² ) Ψ = 0.

He could immediately see that it couldn't describe the right levels of the Hydrogen atom so he abandoned it. It was rediscovered by Klein, Gordon, and Fock, and became known as the Klein-Gordon equation in the West.

Inspired by this failure, Schrödinger temporarily abandoned special relativity and wrote his conventional non-relativistic equation

[ -id/dt - Δ/2m + V(x,y,z) ] Ψ = 0.

which became the cornerstone explaining all (at least) qualitative features of atoms, molecules, chemistry, biology, and material science. Planck's reduced constant hbar is set to one. De Broglie and Schrödinger themselves have never understood the probabilistic character of quantum mechanics; so although they were fathers of wave mechanics, they shouldn't be counted among the founding fathers of quantum mechanics.

After his proof of the equivalence between different formulations of non-relativistic quantum mechanics, Paul Dirac returned to the problem of a relativistic quantum equation. He realized that the Klein-Gordon equation admitted both positive- and negative-energy solutions because "E^2-p^2=m^2" - the non-differential version of the Klein-Gordon equation - can be solved by both signs of "E".

He needed a Hamiltonian that looked positively definite. Single-component wave functions couldn't do the job. So he eventually found out that the wave function had to have several components. With several components, the part of the Hamiltonian with the spatial derivatives (the kinetic part) may be written as the product of two vectors

H = i gamma⁰ ( gamma^j ∂_j - m )

The energy "H" is no longer squared - it is a first-order equation rather than a second-order equation - so there was suddenly a hope that the new equation may resemble the non-relativistic Schrödinger equation and its solutions with different signs of the energy could be separated, a point that will have to be clarified later. With the right commutation relations between the Dirac gamma matrices, one may also demonstrate that the equation is Lorentz-invariant.

The equation implies the existence of spin of the electron and, as we will discuss later, antiparticles. In the non-relativistic limit, it reduces to the non-relativistic Schrödinger equation with spin, also known as the Pauli equation. It adds some relativistic corrections - that are very small for the Hydrogen atom because the speed of light is very large - and they were successfully verified.

However, the Dirac equation leads to many conclusions and extensions that will are going to discuss in some detail:

• need to second-quantize the Dirac field; antiparticles; quantum field theory;
• non-Abelian charges of the fermionic particle; QCD
• mass term coming from the Higgs mechanism instead of an explicit number
• Weyl fermions; higher-dimensional generalizations; general understanding of spinors and other representations of Lie groups in maths
• supersymmetric extensions
• similar equations with a higher spin, e.g. the spin-3/2 equation for gravitinos
• superstring theory as the final extension of the Dirac equation

Ladies and Gentlemen, let us begin.

Dirac sea, antiparticles

Dirac wrote a first-order equation to get rid of the "squaring" that was present in the Klein-Gordon equation. In that way, he could hope that the negative-energy solutions would be eliminated. However, his final product, while being a first-order equation (first derivative with respect to time only), still predicted negative-energy solutions.

The Dirac wave function has four components. Two of them may correspond to positive-energy solutions with the two allowed values of the electron's spin, up and down, but the other two inevitably allowed negative-energy solutions, too.

Dirac knew what Nature would do to save energy - and Nature usually tries to save energy, especially when it looks for the right ground state (vacuum). The right way for Her to save is to actually occupy all the negative-energy "boxes" or solutions. According to the Pauli exclusion principle that was already known at that time, only one electron could be inserted to each "box" with a given frequency, direction, and spin.

All the negative-energy "boxes" are therefore filled in the vacuum - by a continuum of electrons known as the Dirac sea. On the contrary, if an electron is missing in this supposed-to-be-full Dirac sea, it will look like minus one particle, or a hole, i.e. a particle with the opposite energy and charges than the particle that's missing. Because the missing particle had a negative charge (an electron) and a negative energy (as everyone in the Dirac sea), the hole will look like a positive-energy, positive-charge particle: a physical anti-electron or a "positron", as it is normally called.

Paul Dirac was able to predict the existence of antimatter - before it was observed - which is what he officially received his Nobel prize for.

Using a modern pragmatic vocabulary, this visual exercise with the Dirac sea is only meant to revert the convention of the creation and annihilation operators for the negative-energy solutions relatively to the positive-energy solutions. In this generalized form, it may also be applied to bosons even though bosons don't obey the Pauli exclusion principle and you can never quite fill a "Dirac sea" of bosons.

Quantum field theory

Implicitly, Dirac was already thinking about arbitrarily many electrons, so he already needed a quantum field theory. The single-particle Dirac equation is nice but you won't be able to make sense out of its negative-energy solutions. The only way to solve these puzzles is to promote the Dirac wave function to a quantum field. Once you second-quantize it, you will see that it may create or annihilate an arbitrary number of particles - and their antiparticles.

Although it was confusing for many physicists even in the 1930s, this second quantization applied to the fermions - obeying the Dirac equation - is the same procedure as what we have to do with bosonic fields - such as the electromagnetic field - to see that their energy is also quantized (the energy comes in packets known as photons), as required by the insights of Max Planck, Albert Einstein, Arthur Compton, and others.

So there is no doubt that in the real world, the Dirac field - much like the electromagnetic field - has to have a hat above it. It is an operator that acts on the Hilbert space of states. The states may describe an arbitrary number of particles and antiparticles with the allowed values of momenta and spin. The Dirac equation makes it inevitable to switch from single-particle quantum mechanics to variably multi-particle quantum mechanics - quantum field theory.

Note that this transition was forced upon us by looking for a relativistic version of the Schrödinger equation. In this sense, special relativity itself makes it unavoidable for the number of particles to be variable. Special relativity implies that antiparticles have to exist and that particle-antiparticle pairs must be allowed to be created and annihilated.

Coupling to force fields, QCD

The Dirac equation may be written down for a free particle. But what people really wanted was to study the motion of the electrons in atoms. So it was necessary to understand how these Dirac particles reacted to the external electromagnetic field.

The electromagnetic field is included simply by replacing the partial derivative by the covariant derivative

∂_μ → ∂_μ + ieA_μ

One finds out that the phase of the Dirac field can be changed at each point - as long as a corresponding "canceling" transformation is performed on the electromagnetic potential "A". This operation may be interpreted as a local U(1) symmetry.

A beautiful generalization exists; we can combine several Dirac fields into a multi-dimensional representation of a non-Abelian group such as SU(2) or SU(3) or SU(5) or spin(10) or E_6 or something else. The electromagnetic field "A" is promoted to a field transforming in the adjoint representation of the gauge group. The local transformations of the Dirac fields can change not only the overall phase now; they may mix the different components of the representations by pretty general matrices. These transformations are still symmetries if the more complicated "A" transforms accordingly.

Although the Standard Model Lagrangian doesn't fit on your T-shirt so easily, it's clearly "at least as beautiful" as the original Dirac equation. Adding color fields and isospin fields and their GUT generalizations is as natural as adding an electromagnetic field to the equation.

Not to employ this possible generalization would be just like saying that the number 4 is pretty but numbers greater than 4 are taboo, ugly, and forbidden. It is just like saying that no pretty painting could be created after Mona Lisa. Of course that Nature doesn't respect these unjustifiable restrictions on beauty so it uses the gauge symmetries based on groups that are more complicated than U(1), too.

The electroweak theory, the Quantum Chromodynamics, and various Grand Unified Theories (and their generalizations that are sometimes not meant to describe the actual observed phenomena but are studied for conceptual reasons) use Dirac fields transforming under additional internal groups and including the corresponding extra gauge fields in the covariant derivatives. Nature would be dumb not to use this possibility.

Higgs mechanism as a mass donor

The Dirac equation contains an explicit mass term, "m.psi", that gives the electron or another Dirac particle its rest mass. However, this can't be the end of the story because of the existence of the weak nuclear force.

For example, a neutron may decay into a proton, electron, and an antineutrino. At the level of quarks, it's a down-quark (inside the neutron) that gets transformed into an up-quark (inside the proton), electron, and an antineutrino. Such a transformation is mediated by a massive gauge boson - namely a W-boson.

The gauge bosons must be amassive counterparts of the photons and gluons discussed in the previous section: they must result from a gauge group. But because the weak force is a short-range force - two distant objects can't measurably attract because of the "beta-decay force" - the gauge bosons responsible for the weak force must be massive (so that they can't travel too far as virtual particles).

However, massive gauge bosons would lead to many unpleasant consequences. At too high energies, around 1 TeV or so, the probabilities predicted from their interactions would jump out of the allowed interval between 0% and 100%. The only good enough fix is to give these gauge bosons masses by a Higgs mechanism. The exchange of a virtual Higgs boson returns the probability between the appropriate mantinels.

However, for this fix to work, the theory must have a full electroweak gauge symmetry. This symmetry, SU(2) x U(1), predicts new gauge bosons - neutral Z-bosons - besides the W-bosons that are responsible for the beta-decay. It must be a symmetry of all the laws of physics, including the laws governing the Dirac fermions. This condition bans the explicit fermion masses.

However, the masses may be obtained from the interactions with the Higgs field. That's true for the W-bosons and Z-bosons - that interact with the Higgs field because the Higgs field is charged (transforms) under the symmetries corresponding to the W-bosons and Z-bosons. But it's also true for the fermions; they interact with the Higgs field via the Yukawa interactions.

Now, giving masses to the electron and other fermions by the Higgs mechanism is more complex and the required formulae won't be printed on your T-shirt that nicely. The text won't be too readable if you try. But Nature isn't really controlled by the marketing of the T-shirts. She has higher and less commercials criteria for Her beauty.

According to these criteria, the mass generated by the Higgs mechanism is at least as pretty as the explicit mass that was inserted to the Dirac equation in the 1920s. It has the same consequences at low energies; this Higgs-generated mass gives the electron the same inertia and the same gravitational field as the explicit mass term written down by Dirac (although the US LHC Blogs commenter thinks it doesn't).

But at higher energies, it becomes clear that this mass is just a manifestation of the interactions between the fermion and another field, the Higgs field, that inevitably exhibits many new physical phenomena as well. The full theory including the Higgs field is more contrived when written on your T-shirt; but from all the more aristocratic perspectives, it is a more well-behaved theory than a theory in which the mass terms are written down explicitly.

Spinors, Lie groups, and higher dimensions

Aside from the Dirac fermion which has 4 complex components, one may construct various "reduced" versions of the equation, namely an equation for a Majorana fermion or a Weyl fermion. In four dimensions, these two types of fermions are pretty much equivalent. What are they?

A Majorana fermion is the "real part" of a Dirac fermion; a Weyl fermion is the "left-handed part" of a Dirac fermion. In 3+1 dimensions, these two are described by "equivalent collections of numbers" because in both cases, we may rewrite the 4-component Dirac fermion as a pair of two 2-component fermions. The Majorana or Weyl conditions tell us that these two 2-component fermions are complex conjugates of one another.

In the Majorana case, this condition comes from the requirement that the "imaginary part of the Dirac fermion" has to be zero. The "imaginary part" is the difference between one 2-component fermion and the complex conjugate of the other 2-component fermion. In the Weyl case, we want to eliminate one of the 2-component fermions altogether because it is "right-handed". However, we may still imagine that it survives - it's just given by the complex conjugation of the left-handed fermion. The two theories, Majorana and Weyl, are totally equivalent in 3+1 dimensions (and in (8k+3)+1 dimensions, too).

The Majorana picture is appropriate for "totally neutral" particles. We usually imagine that unlike the electron, a particle described by the Majorana equation carries no conserved charges. Consequently, it may be coupled to itself and be given a "Majorana mass term". This mass term allows to create two copies of the particle out of pure energy (none of them is a distinct antiparticle, the two particles are totally identical!). Neutrinos could perhaps have such Majorana mass terms in which case the lepton number is not conserved.

The Weyl fermion interpretation is preferred if we want to preserve the symmetry that rotates the phases of the left-handed and right-handed fermions differently; such a symmetry prohibits the Majorana mass terms.

When we talk about these different types of 2-component spinors etc., it becomes clear that the true "engine" that makes the Dirac equation possible is the existence of a previously unknown representation of the Lorentz (and rotational) groups, the spinors. The cute properties of the Dirac matrices that make the Dirac equation so pretty are just reflections of the existence of the spinor representation.

If you think rationally about the beauty of the Dirac equation, and you're not constrained by taboos - e.g. a ban on the question where the beauty comes from - you will figure out that the beauty really boils down to the properties of spinor representations of the Lorentz group. Once you know that, you want to know all kinds of similar "beauties". In other words, you should comprehensively learn and analyze the theory of representations of Lie groups and Lie algebras.

Once you do so, you will also be led to the representations of groups different from SO(3,1), for example SO(9,1). You will understand that the spinors exist in arbitrary dimensions. The Dirac equation becomes possible in any dimension you like but it has new properties that deserve to be studied (some of the qualitative ones are periodic functions of the dimension but the periodicity may be as high as eight). Clearly, the beauty of the Dirac equation in 9+1 dimensions is not smaller than the beauty of its 3+1-dimensional counterpart. On the contrary, there is "more" of the same beauty.

A better knowledge of the Lie groups and Lie algebras - and their representations - also allows you to construct and understand more complicated gauge groups and their representations, and understand various theorems that the two types of the groups - internal groups (including gauge groups) and spacetime groups (Lorentz and Poincaré symmetries) - can't non-trivially mix.

Supersymmetry

There is a major exception to the rule above. The rule is known as the Coleman-Mandula theorem and the exception is supersymmetry. When we talk about the supersymmetry and the beauty of the Dirac equation, we can't omit the following fact: supersymmetry makes the existence of the Dirac equation unavoidable if your theory has at least one Klein-Gordon field or at least one "vector field" (such as the electromagnetic one).

It's because supersymmetry generators themselves transform in a very similar way as the Dirac field itself - namely as (Weyl) spinors. So if these generators act on a Klein-Gordon field, they inevitably transform it into its superpartner which must carry Lorentz spinor indices. Obviously, if the Dirac equation were unknown until the 1970s and someone discovered the bizarre new kind of symmetry known as supersymmetry from mathematical, group-theoretical arguments - which is exactly what the Soviets have done - they would discover the Dirac equation later, anyway.

In this sense, supersymmetry is a "more fundamental" explanation of the beauty of the Dirac equation. It is an explanation why the spinors are used and can be used by the laws of physics. Supersymmetry is important as a calculational tool in the "most symmetric" field theories as well as vacua of string theory. However, it is also compatible with the observational physics. The supersymmetric extensions of the Standard Model are extremely promising models to describe the observations at the LHC that many of us expect to be revealed in 2011.

Higher spins and strings

Once you become familiar with the inner workings and mathematical reasons that make the beauty of the Dirac equation possible, you will look at the equation more rationally - like at a pretty woman you have spent years with. ;-)

With this perspective, you know that there also exist other beautiful structures. I don't advise you to do the same thing with the women, but in the case of the equations, you should try to get in touch with every conceivable equation at any spin. You will learn that just like the bosonic equations include the Klein-Gordon, Maxwell, Proca, and Einstein equations - among a few less important friends - the fermionic equations include the Dirac equation, its Majorana and Weyl reductions, but also the Rarita-Schwinger equation that describes a spin-3/2 particle, a gravitino, which is a superpartner of the graviton.

With the knowledge of possible free equations, the list of allowed gauge symmetries, and possible interactions compatible with these symmetries, one has a pretty extensive menu of conceivable quantum field theories. (Strongly coupled field theories that are not easily obtained from a non-interacting limit also exist but I won't discuss this generalization here.) If you ask why there are many fields - even though you can't relate all of them by supersymmetry - you will see that the particles must have an internal structure and string/M-theory is the only framework in which the internal structure can explain the common ancestry of fields with different charges and spins.

In theoretical physics, the commmon ancestry of different (particle) species is known as unification. At the level of field theory, unification is only possible for particles of the same spin; (minimal) supersymmetry allows the spin to deviate by 1/2 in the "evolutionary siblings" but you must go beyond field theory - to string theory - to see more "distant" parts of the tree of life.

Looking at the internal structure of the particles becomes most transparent in the perturbative limits of string theory where the particles are vibrating loops of strings, described by 1+1-dimensional conformal field theories, and the quantized vibrational patterns can be mapped to the particle species. Perturbative string theory recycles many insights and beautiful structures that we have known from the 3+1-dimensional and higher-dimensional spacetimes; they reuse the wisdom in the 1+1-dimensional world sheet. These beauties and clever ideas include supersymmetry and others.

Beyond perturbative expansions, lots of insights are known about string/M-theory, too. But without a perturbative limit, one often doesn't have a "constructive starting point": a general situation in string/M-theory cannot be described as a collection of "free objects" with some particular "interactions" added on top of that. That's why perturbative string theory remains very important in string theory (much like perturbative Feynman diagrams remain very important in field theory), despite the huge amount of insights about non-perturbative string/M-theory that have been gathered since 1995.

The Dirac equation in 1+1 dimensions also plays a role in perturbative string theory and at the very end, you may obtain a deeper perspective on the Dirac field in four dimensions, other fields, possible symmetries between them, their mutual interactions and other processes, as well as the actual profound relationships between all these structures.

Just like string/M-theory, the most complete description of physics and all of its internal wisdom, may be thought of as the most sophisticated conceivable generalization of the concept of gravity or geometry, it may also be thought of as the final achievement of a journey attempting to uncover the source of the beauty of the Dirac equation - and to reveal all similar treasures that are hiding in the mountains. After all, all roads lead to string theory.

This task has been partly realized; lots of the extra treasures have been found. However, the amount of precious material that may still be hiding may be both smaller and greater than what we have already learned. Only the future generations will know the answer.

And that's the memo.