The Dirac Operator

As the fundamental equation of relativistic quantum mechanics and quantum electrodynamics, the Dirac equation is perhaps the most important equation of modem physics. It is impossible to value the vast range of its applications in a single article and therfore we want to present an introduction to certain aspects only. This chapter has the character of a first overview and introduction to the Dirac operator. It covers material that is largely contained in my book [1], but, as I hope, in a more accessible form. Ref. [1] should be taken as a reference to more details and background information from a more mathematical point of view, and as a guide to the older literature on this subject. 

The aim and scope of this article is to describe the Dirac operator and its associated quantum-mechanical evolution equation. We concentrate on introducing 

The Schrodinger equation with this Hamiltonian is called the square-root Klein-Gordon equation because of its formal similarity to a square-root of the Klein-Gordon equation (Oscar Klein and Walter Gordon in fact had little to do with the square-root equation). Unfortunately, the meaning of the square-root Klein-Gordon equation is obscured by the following points. 

1) The Hamiltonian involves the square-root of a differential operator. It is no problem to define this operator with the help of the Fourier transformation and to investigate its properties, but the resulting operator (2) is non-local. This 

2) For the same reason it is not clear, how to modify the equation for the inclusion of external fields. The principle of minimal coupling p — p — A, E E + V for the (scalar) square-root Klein-Gordon equation was critizised by J. Sucher [4], who states that there are solutions ip x) and electromagnetic potentials, such that the Lorentz transformed solution is not a solution of the equation with the Lorentz-transformed potentials. Moreover, the nonlocal nature of the equation means that the value of the potential at some point influences the wave function at other points and it is not clear at all how one can interpret this. 

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The Dirac operator incorporates relativistic effects for the kinetic energy. In order to describe atomic and molecular systems, the potential energy operator must also be modified. In non-relativistic theory the potential energy is given by the Coulomb operator. 

The Dirac operators a, . ..04, work on the basis vectors ei,. ..04 just as the Pauli spin operators work on the spin eigenvectors a, / . The only other property to be noted is that the Dirac spin-orbitals have two large components and two small components , their ratio being of the order (2moc)" ... 

In a recent work (Brummelhuis et al. 2001) the stability result has been improved up to charge numbers Z 117, which covers the range of all known elements. Again the proof starts by taking the square of the inequality, because the modulus of the Dirac operator with the Coulomb potential is not easy to handle. However, instead... 

Furthermore large basis sets are needed for an accurate description of the region close to the nucleus where relativistic effects become important. Methods based on the replacement of the Dirac operator by approximate bound operators (square of the Dirac operator, its absolute value etc...) have not been very successful as can been understood from the fact that they break the Lorentz invariance for fermions. 

The Dirac operator Ho is interpreted as the operator corresponding to the energy of a free particle, which fits well to its role as the generator of the time evolution. As a consequence of the anti-commutation relations (5), the square of the Dirac Hamiltonian is a diagonal matrix. It is simply given by... 

The operator H is intended to describe the energy of a particle in a given external field. As in nonrelativistic quantum mechanics, the influence of an external field is described by a potential-energy V (x) that is added to the kinetic energy Hq. Hence the Dirac operator with an external field reads... 

This follows from the unitarity of the time-evolution and is in turn a consequence of the self-adjointness of its generator H. The Dirac operator H is indeed self-adjoint for a vast range of external-field situations covering virtually all cases of physical interest (in particular the Coulomb potential up to nuclear charges Z = 137, see below). 

Hence u(p) fpos is an eigenvector of the Dirac operator h(p) in momentum space, belonging to the positive eigenvalue Similarly, if / eg is some... 

We can apply the Dirac operator Hq to this wave packet. 

In Dirac s theory the classical expression for E is replaced by the Dirac operator Hq. Translating the classical expression for the velocity into quantum mechanics would thus lead to the operator... 

The operator F has the remarkable property that each component anticommutes with the Dirac operator Hq,... 

Thus, the negative-energy subspace of H e) is connected via the symmetry transformation C with the positive-energy subspace of the Dirac operator H —e) for a particle with opposite charge, but with the same mass. Hence if ip describes an electron, then Cxp describes a positron. Moreover, we have... 

The Dirac operator with an electromagnetic field reads... 

It has been pointed out before that the self-adjointness of the Dirac operator is a crucial property. Only the self-adjointness guarantees that the time evolution is unitary (which means that the norm is constant in time). Any defect in the self-adjointness of Dirac operators would imply a defect in the unitarity of the time evolution. 

A charge conjugation does not change the sign of the scalar potential relative to the other terms in the Dirac operator. Hence a scalar potential acts in the same way on electrons and positrons. If it is repulsive for electrons, so it is for positrons. 

A physical system is close to the nonrelativistic limit, if all velocities of the system are small compared to the velocity of light. Hence the nonrelativistic limit of a relativistic theory is obtained if we let c, the velocity of light, tend to infinity. In the nonrelativistic theory, there is no limit to the propagation speed of signals. For the Dirac equation, the nonrelativistic limit turns out to be rather singular. If we simply set c = oo, we would just obtain infinity in all matrix elements of the Dirac operator H. We must therefore look for cancellations. 

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