Dirac equation, spin and fine structure Hamiltonian Fernando Chamizo (Dated: May 27, 2019) The Dirac equation is the starting point for relativistic quantum mechanics which evolved into the modern Quantum Field Theory. The purpose of this paper is to introduce it from a historical point of view and focus on two conspicuous applications. The first one is the explanation of the spin of the electron on a theoretical basis. The second is the derivation of the fine structure Hamiltonian that gives the relativistic corrections on the hydrogen atom. I. HISTORICAL INTRODUCTION of Minkowski space-time. In this section one can follow the steps of the proof but the underlying idea remains a Along the following sections we are going to give quite little obscure. Paying a high price, the modern language a number of historical references and comments but we of representations makes this point clearer [2, 4.2]. want to devote this one entirely to the groundbreaking x4. The Hamiltonian for an Arbitrary Field. The paper [1] in which Dirac introduced his new equation. equation for an electron in an electromagnetic field is After 80 years, it is noticeable how close this paper is to the displayed at the beginning and the rest of the section modern reader. Definitively it is not only a fundamental contains manipulations with σ that nowadays could be contribution, it is also a very advisable reading. consider prolix. The paper [1] has six sections and a general introduc- x5. The Angular Momentum Integrals for Motion in a tion that we label as 0. We describe here the contents Central Field. This is an impressive and important part with some comments avoiding deliberately formulas and of the paper. In one sentence, the spin of the electron is ~ technical points. deduced. Namely, it is proved is that L+ 2 σ is a constant of motion under Dirac equation in a central field (and x0. Dirac notices firstly the discrepancy between the ex- L is not). This is an honest theoretical physics answer periments and the quantum theory predictions for atoms. for the philosophical question about the Nature in the After mentioning the contribution of Pauli and Darwin introduction. to solve this problem, he indirectly suggests that they are artificial theories (with the funny sentence \The question x6. The Energy Levels for Motion in a Central Field. remains as to why Nature should have chosen this par- The motivation here is to deduce that the correction ticular model for the electron"). Finally he announces for the hydrogen atom matches the theory of Pauli and in an impersonal way a Hamiltonian in agreement with Darwin. Dirac is sketchy in this section and some inter- relativity and the experiments. esting implications appear in the second part of the paper [3]. Shortly after, Darwin published his study of the fine x1. Previous Relativity Treatments. This is a kind of structure of the hydrogen atom [4]. technical continuation of the introduction. The Klein- Gordon equation (named here as Gordon-Klein) is consid- Along this work we will tackle topics related to the ered as a tentative relativistic Schr¨odingerequation with sections in the paper by Dirac with the only important two difficulties: Its limitation to define probabilities of omission of the Lorentz covariance, which is more tech- dynamical variables and the existence of negative energies. nical in nature. In connection with the last section, we The last paragraph contains an assertion between modest derive the fine structure Hamiltonian which is one of the and prophetic: \we shall be concerned only with the re- aftermaths of it. moval of the first of these two difficulties. The resulting theory is therefore still only an approximation". II. SEVERAL FORMULATIONS x2. The Hamiltonian for No Field. The content of this fundamental section is a derivation of the Dirac equation as a factorization of the Klein-Gordon equation in a way Let us start with some motivation following Dirac's that could be found in any modern book except for slight steps. The idea to create a relativistic Schr¨odingerequa- 2 2 2 2 4 changes in the notation. Something that sounds strange tion is to quantize the relation E = p c + m c promot- from the modern point of view is that Dirac seems to be ing E and p to be the corresponding operators. In this prone to consider the needed extra dimensions (and even way it comes out the Klein-Gordon equation for the free the matrices) as reflecting new \variables". particle (by a slip of the pen there is a wrong sign in [1]) x3. Proof of Invariance under a Lorentz Transformation. 2 2 2 2 2 2 4 − ~ @t Ψ = −~ c r Ψ + m c Ψ: (1) Although the equation has been derived starting from a relativistic expression, it is not clear what Lorentz This is a second order equation in time. It means that covariance means for it. The point to have in mind (that the wave function at a fixed time does not determine its still puzzles students) is that the four coordinates of Ψ evolution in later times. A deeper problem is that it lead (the spinor wave function) are not the four coordinates to negative probabilities (see e.g. [5, x1.2]). 2 The Dirac equation is usually introduced, following the With this notation the Dirac equation becomes original [1, x5], as a \factorization" of Eq. (1). Let us i γµ@ − mcΨ = 0 (8) examine a toy analogy. Consider the classical harmonic ~ µ oscillator ruled by x¨ + !2x = 0 and say that for some where, as usual in relativity, it is assumed µ 2 f0; 1; 2; 3; 4g reason one wants to turn it into a first order linear equa- and the summation convention. This is called the covari- tion αx_ + βx = 0. The first reaction is to claim that ant form of the Dirac equation. In QFT (Quantum Field µ this is blatantly impossible because we need the freedom Theory) very often γ @µ is abbreviated as @= and the to impose two initial conditions, one for position and equation acquires the minimalist form in natural units another for velocity. The key point is that the dimension (i@=− m)Ψ = 0. The explicit form of the γµ corresponding of the solution vector space can be increased keeping the to the choice Eq. (5) is the so-called Dirac representation order promoting x to be a vector and consequently the IO 0 σ coefficients to matrices. In our example x¨ + !2x = 0 is γ0 = ; γj = j : (9) O −I −σ 0 equivalent to the vector equation j In some contexts it is convenient to define γ0 differently, 0 −1 αX_ + βX = O with α = 1 ; β = : (2) namely 2×2 !2 0 OI 0 σ γ0 = ; γj = j : (10) The matrix equation forces X = (x; x_)t with x a solution IO −σj 0 of the scalar equation. This is the cheap and well-known This is called the Weyl representation. It corresponds to a mathematical trick of hiding higher order derivatives in different choice of α and β i.e., to switch the two couples d d coordinates. A calculation shows (α dt − β)(α dt + β) = of columns and to change the sign of the last couple of d2 2 dt2 + ! (Cayley-Hamilton theorem is working here) and rows in Eq. (5). we can say that we have factorized the original equation. The relations Eq. (4) translate into the suggestive com- In the same way, we could write Eq. (1) as pact formula µ ν ν µ µν 2 2 γ γ + γ γ = 2η 14×4 (11) −i~@t +cα·p+βmc i~@t +cα·p+βmc Ψ = 0 (3) where ηµν are the components of the Minkowski tensor. if we impose for i; j 2 f1; 2; 3g, i 6= j, It turns out that under Lorentz transformations γ = (γ0; γ1; γ2; γ3) transforms as a vector and the Lorentz 2 2 αj = β = 1; αiαj + αjαi = 0 and αjβ + βαj = 0: covariance of the Dirac equation means that there is a (4) corresponding representation of the Lorentz group acting Of course this cannot fulfilled with numbers, we need on Ψ. to move to the noncommutative realm of matrices. The Shortly we will examine some of the noteworthy achieve- first two relations for αj are part of those for the Pauli matrices but the third spoils any choice of a 2 × 2 matrix ments of the Dirac equation. Before going on, when one β with β2 = 1. Taking β = 1 would work flipping the sees these 4 × 4 matrices a natural question shows up middle + sign into −. In fact it can be proved [6, XX.7] pointing an inconvenience. The physical meaning of the that the smallest dimension to have a solution of Eq. (4) extra dimensions is unclear. The wave function was orig- with Hermitian matrices is 4. Roughly speaking, flipping inally, as the name suggests, a function representing a signs requires to increase the square roots of 1 doubling wave, afterward it became a vector (a 2-spinor to be pre- the dimension. A possible choice fulfilling Eq. (4) is cise) to encode the two possibilities for the spin of the electron but we now have two more coordinates. The full answer is not easy and it is related to Dirac's self-criticism O σj IO αj = and β = ; (5) in [1, x1]. The equation contains information about two σj O O −I particles, the electron and the positron and they cannot written in blocks of 2 × 2 matrices. It is not the only be separated because they are excitations of the same choice but it can be proved that all possible choices are quantum field [7, 37.4].