1.1 Dirac Equation

1.1.1 Attempts for a Relativistic Theory

From the Schrödinger equation in non-relativistic quantum mechanics, you have known that there is a formal connection between the Schrödinger equ-ation and the classical energy-momentum relation. For a non-relativistic free particle whose kinetic energy is the total energy, its energy-momentum relation reads E=. The Schrödinger equation can be written down with the help of the following correspondence

The matter-wave equation

is the well known Schrödinger equation for a non-relativistic free particle.

In Einstein's special relativity, the relativistic energy-momentum rela-tion reads E=√. In order to generalize the non-relativistic quantum mechanics to relativistic case, a naive idea is to consider the following wave equation

But this left a completely wrong direction in the old days due to insur-mountable difficulties in the attempt for describing the interactions be-tween electrons and electromagnetic fields. What is the right clue that you may think of? In fact, there have been two right routes known to physicists now: one leads to the so-called Klein-Gordon equation and the other leads to the so-called Dirac equation.

Klein-Gordon equation

Referring to the quadratic form of energy-momentum relation E2=c2p2+m2c4, one can write down a partial differential equation

which is called Klein-Gordon equation. Obviously, the above Klein-Gordon equation solves the following plane-wave solution

withħ2ω2=ħ2k2c2+m2c4, which means ω=±√. Then the de Bröglie relation gives rise to E=ħω=±. You may have realized one puzzle that the Klein-Gordon equation possesses solutions with negative energy even for free particles.

Multiplying(1.3)by Ψ∗on the left, and the complex conjugate of(1.3) by Ψ on the right; then subtracting these two equations, you obtain

This implies a continuity equation

with

According to the experience when you learnt non-relativistic quantum me-chanics, you might regard ρ here as the probability density and j the probability-current density(or probability flux called in some text books). However, from the first equation of (1.7), you realize another puzzle at once that the ρ is not positively definite. Then how to comprehend a neg-ative probability density? For there were two puzzles in the interpretation of wavefunctions for Klein-Gordon equation, this equation used to be laid aside for seven years since it was written out. Do you have any ideas to settle down those two puzzles? In 1937, Pauli and Weisskopf reinterpreted the physics meanings of the aforementioned ρ and j. With a new inter-pretation, they defined eρ and ej as electric charge density and electric charge-current density, respectively, rather than simply consider ρ and j. Consequently, Eq. (1.6) multiplied with elementary electric charge e be-comes the continuity equation for the conservation of electric charge, and thus it is reasonable because the density of electric charge is not necessarily to be positively definite. Nowadays, the Klein-Gordon equation is the well known equation for scalar fields that characterize the feature of spinless particles. It is a task in the course of quantum field theory to investigate in detail.

Dirac equation

Mathematically, one can easily recognize that the emergence of negative probability arises from the fact that the time derivative in Klein-Gordon equation is of second order. In special relativity E=, clearly, → mc2when p → 0, while → pc when m→0. This arouses your attempt to assume

where αi(i=1,2,3) and β might be something new that we do not understand yet.

Taking square of the above equation, you obtain

where the repeated subscripts imply the Einstein's abbreviation of sum-mation. Here you must carefully keep the relative order in taking the multiplication because they should not be the traditional number. The compatibility of(1.9)gives rise to

Actually, those relations given in (1.10) define the called Clifford algebra in mathematician's terminology. The above creative clue was first realized by Dirac. Now you are in the position to write out the Dirac equation as follows

where α=(α1, α2, α3). Here, of course, you must have α†=α and β†=β so that the Hamiltonian in (1.11) is a Hermitian operator. After some simple calculation

where the Dirac equation(1.11)has been used in deriving the second line of the above equation, you can get

and realize that the ρ and j can be defined as the probability density and probability-current density, respectively. Clearly, they are positively defi-nite and there are no puzzles as ever appeared in Klein-Gordon equation.

As to the Dirac equation for a free particle, you can separate the vari-ables t and r to look for plane-wave solutions

Substituting it into the Dirac equation(1.11), you get the stationary Dirac equation

The operators αiand β introduced by Dirac can be represented in matrix form(they must be 4 by 4 matrices at least)

where σidenotes the conventional Pauli matrices

The expression(1.16)is called the Pauli-Dirac representation of the Clifford algebra (1.10). Of course, the other set of U αiU-1and U βU-1for any unitary matrix U†U=1 also fulfils the relation of Clifford algebra(1.10), which gives an equivalent Dirac equation. Since those 4 by 4 matrices (1.16)appear to be block forms of 2 by 2 submatrices, you can conveniently consider

where both φ and χ are two-component wavefunctions. Using(1.16), you can write the stationary Dirac equation(1.15)as

which is actually a set of two matrix equations

Till now, you may worry about the Lorentz covariance of the Dirac equation for it is not exhibited explicitly. Actually, if multiplying (1.11) from left by β, you can rewrite the Dirac equation in 4-dimensional form

with xµand γµ(µ=1,2,3,4) being defined by (x, x4)=(r, ict) and (γ, γ4)=(-iβα, β). Thus the Lorentz covariance is guaranteed.