Question #efb13

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Vinicius M. G. Silveira
Mar 27, 2016

There are many interesting aspects of the Dirac Equation, but I'd say the three most important points are:

  1. It's a relativistic wave equation
  2. It predicts the existence of antiparticles
  3. It predicts the existence of spin

Explanation:

The Dirac Equation

#[beta mc^2 + c(sum_(n=1)^3 alpha_n hat(p)_n)] psi (vec(r),t) = i ħ (del)/(del t) psi (vec(r),t) ,#

proposed by Paul Dirac in 1928, is considered by many to be one of the most important equations of Physics, since it symbolizes the union of Quantum Mechanics and Special Relativity. In summary, it describes the way a particle that obeys the laws of Quantum Mechanics and behaves as time passes by.

For this reason, in order to have a basic understanding, we need to be acquainted with the basic principles of these two theories. So we must have some basic knowledge of those fields to understand the points above.

To understand the first point, we need to know that a relativistic wave equation is an equation for a wave function (a quantum mechanical description of matter) that obeys the relativistic energy relation #E = +- sqrt((mc^2)^2 + (pc)^2)#.

The first wave equation developed to describe matter in the quantum mechanical level was the Schrödinger equation

# [-1/(2m)(ħ)^2 grad^2 + V(vec(r))]psi(vec(r),t) = i ħ del/(del t) psi (vec(r),t),#

where #i=sqrt(-1)#, #ħ = h/(2 pi)# is the reduced Planck constant, #m# is the mass of the particle discribed by the wave function #psi (vec(r),t)#, #vec(r)# is the position of this particle, #t# is time, #V(vec(r))# is the potential energy of the particle and #grad^2 = (del^2)/(del x^2) + (del^2)/(del y^2) + (del^2)/(del z^2)# is the Laplacian.

One of the basic results of Quantum Mechanics is that a measurement of the momentum of a particle is related (via an eigenvalue equation) to the momentum operator:

#vec(hat(p)) = -i ħ vec(grad)#

Then, if we take a look at the Schrödinger equation, we see that it can be written as:

# [1/(2m) hat(p)^2 + V(vec(r))]psi(vec(r),t) = i ħ del/(del t) psi (vec(r),t),#

Remember that the expression for the kinectic energy in classical mechanics is #T=1/2 m v^2 = 1/(2m) p^2#, wich leaves us with the following expression for the total energy (see also the expression for the Hamiltonian):

#E = T+V = 1/(2m) p^2 + V#

Taking one more look at the Schrödinger equation, we can rewrite it as:

#hat(E) psi(vec(r),t) = i ħ del/(del t) psi (vec(r),t)#

Thus, it makes sense to define the energy operator as:

#hat(E) = i ħ del/(del t)#

Now, we still need to get to the Dirac equation. For simplicity, we'll have to ignore the potential energy term, that is set #V=0# (that was also what Dirac did). If we decide to use the more common form of the relativistic energy relation, #E^2 = (mc^2)^2 + (pc)^2#, to reach a quantum and relativistic equation, we'll arive at the Klein Gordon equation, wich is an adequate description for bosons, but does not account for some of the behaviour of fermions.

So Dirac derived a new form of the relativistic energy relation:

#E = vec(p) cdot vec(v) + sqrt(1-tilde(beta)^2) (mc^2) = vec(p) cdot vec(v) + beta^prime (mc^2),#

where #tilde(beta)^2 = (v/c)^2# and #beta^prime= sqrt(1-tilde(beta)^2)#.
This is just the old relation written in a different way, but for the quantum version of the theory, it has a fundamental difference: it's more similar to the Schrödinger equation since it has an energy term that is linear, and not quadratic (the Klein-Gordon equation has a quadratic term on the energy operator and has some problems with probability densities due to this).

Now we can try to arrive at the Dirac equation. We just need to plug in the energy operator defined above in place of #E#, the momentum operator in place of #vec(p)# and we need to represent #vec(v)# and #beta^prime# in some way.

Dirac's insight was to define matrices #alpha_n#, with #n in {1,2,3}#, and #beta#, and to make the substitutions #v_n to c alpha_n#, #beta^prime to beta#.
We reach this equation:

#[beta mc^2 + c(sum_(n=1)^3 alpha_n hat(p)_n)] psi (vec(r),t) = i ħ (del)/(del t) psi (vec(r),t)#

It turns out there are some requirements that these matrices must abide and the smallest matrices that do are some #4 times 4# ones that are very similar to the Pauli matrices, wich are related to the spin of the particles. Since we have #4 times 4# matrices in the equation, #psi# must be a vector in order for the equation to make sense:

#psi(vec(r),t) = ((psi_1(vec(r),t)), (psi_2(vec(r),t)), (psi_3(vec(r),t)), (psi_4(vec(r),t)))#

To make it easier on notation, we can define #gamma^0 = beta# and #gamma^n = beta alpha_n# and rewrite the equation (in covariant notation) as

#i ħ gamma^(mu) del_(mu) psi = mc psi#

or

#gamma^(mu) hat(p)_(mu) - mc psi = 0#

There is even a beautiful notation (called the Feynman slash notation) that, combined with natural units, that is #ħ = 1 = c#, give us the following neat forms of the Dirac equation:

#(i cancel(del) - m) psi = 0#

#(i cancel(hat(p)) - m) psi = 0#

It turns out that this vector quality of #psi# is profoundly related to the other two points introduced earlier: the existence of antiparticles and spin (matter of fact, we call this kind of wave function a spinor).

If we take a look at the equation #E = +- sqrt((mc^2)^2 + (pc)^2)# we can see that, while #sqrt((mc^2)^2 + (pc)^2)# is always positive, we have positive and negative solutions to #E#. With a lot of (quite hard) calculations, we can see that 2 of the four components of #psi# are associated with positive energy solutions and the other 2 are associated with negative energy solutions.

Dirac's ideia was that all negative energy states were occupied in the vaccuum and to create positive energy states we'd need to supply energy to bump a negative state up to a positive energy level.
This ideia (known as the Dirac Sea) corresponds to pair production and we can relate 2 of the 4 components of #psi# with a particle, while the other 2 correspond to the antiparticle.

What's most remarkable about this is that antimater was first discovered experimentally in 1932 - 4 years after Dirac's prediction!

Finally, we take a look at spin. In Pauli's theory of spin, a wave function is actually a 2 component spinor, in wich the first component corresponds to the up state while the second one corresponds to the down state. Recall that we have 2 components of out 4 component spinor dedicated to particles and 2 dedicated to antiparticles.

As said before, the #gamma# matrices are related to the Pauli matrices. Matter of fact, they are higher dimensional versions of Pauli matrices with positive and negative signs organized in a way that particles and antiparticles have oposite spin relations!

So the Dirac equation not only links Special Relativity and Quantum Mechanics, it also explains a previously known effect (spin) and predicted correctly a previously unknown type of matter (antimatter), making it one of the greatest theoritical successes in Physics - and the reason Dirac won the 1938 Nobel Prize in Physics.

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