Dirac equation

The Dirac equation in quantum mechanics describes the properties and behavior of a fundamental fermion with spin 1 /2 ( for example, electron, quark ). It was developed in 1928 by Paul Dirac and meets the requirements of the special theory of relativity.

The Dirac equation is a partial differential equation of the same order in space and time, in accordance with the information required by the theory of relativity invariance under Lorentz transformations. In the nonrelativistic limit () it goes to the Pauli equation. Each solution of the Dirac equation corresponds to a possible state of that particle, with the particularity that the representation of this state four spatial wave functions are needed (see Dirac spinor ), instead of two in the nonrelativistic theory or one in the case of spinless particles. For the particles described by the Dirac equation yields:

• For a free particle, the relativistic energy-momentum relation is satisfied.

• For a particle in an electrostatic field of a point charge, the hydrogen spectrum results with its fine structure.

• The particle has an intrinsic angular momentum (spin ), which has the quantum number 1/2 and therefore can not be due to the rotation of a mass.

• Supporting the particles an electrical charge associated with the spin and a magnetic dipole. In comparison with the magnetic dipole, which would cause the particles by a rotational movement in the same size of angular momentum, which is connected with the spinning moment the double thickness (see Abnormal magnetic moment of the electron).

• For the particle exists an antiparticle with the same mass and the same spin but with opposite charge and magnetic moment.

All of these properties correspond to the experimental findings. At the time of discovery of the Dirac equation, 1928, the first four were already known, but not their common ground. The latter property has been predicted by the Dirac equation, and the first evidence of a antiparticle succeeded in 1932 Carl David Anderson (see positron).

The occurring in the Dirac equation differential operator plays an important role ( Dirac operator ) in mathematics ( differential geometry).

Dirac equation of an uncharged particle

The Dirac equation is a system of four coupled partial differential equations for the four components of the Dirac spinor functions. The variable which stands for the upper index zero time and the indices 1-3 indicate the location coordinates.

In natural units with down the Dirac equation for an uncharged particle of mass

The expression in square brackets is a Dirac operator.

The constant gamma or Dirac matrices and act in the space of four components of the spinor and link them together. The products of two gamma matrices have the following properties:

Thus, they form a Clifford or Dirac algebra.

If the Dirac operator applied to both sides of the Dirac equation, entkoppen the four differential equations is obtained for each component of the Klein-Gordon equation:

The repeated application of a Dirac operator thus leads to the Klein-Gordon equation, which is why the Dirac equation is also regarded as the " root " of the Klein-Gordon equation. For a particle in a pulse eigenstate of the Klein-Gordon equation ( in the order of their terms ), ie the relativistic energy - momentum relation of a particle of mass

Every irreducible representation of the Dirac algebra consists of matrices. In the standard Dirac representation or they have the following form. Disappearing matrix elements with value zero are not written

In this illustration, the two go lower components of the spinor in the nonrelativistic limit as tends to zero. Thus it is particularly suitable for the treatment of slowly moving electrons. In this mathematically and physically equivalent Weyl spinor representation of the transformation behavior is simply under Lorentz transformations, which also equivalent Majorana representation of the Dirac equation is a real equation system. Other representations are obtained by equivalence transformations.

The four gamma matrices can be used in symbolic form to the contravariant 4- vector

Summarized. Then, the first term of the Dirac equation is in the form of an inner product of the vectors and. However, this is not invariant under Lorentz transformation, because remains constant. The Lorentz invariance of the Dirac theory arises only by the fact that the Dirac operator acting on a spinor, whose four components are transformed as appropriate. The end result is thus a solution of the Dirac equation by the Lorentz transformation into a solution of the corresponding transformed Dirac equation.

Momentum space and slash notation

In addition to the above described form in coordinate space, the Dirac equation can be written in momentum space. It is then

Where for brevity the Einstein summation convention has been used. (This means that summing over repeated indices. ) In the further simplified Feynman slash notation the scalar product is expressed with the gamma matrices with a slash. There are in position space:

And is in momentum space:

Gauge invariance and electromagnetic interaction

If the Dirac equation solves, then also solves multiplied by a phase spinor, the Dirac equation. Since all physically measurable quantities also included with each factor a complex conjugate factor, they are and the Dirac equation is invariant under this phase transformation of the Dirac spinor.

The invariance also calls under all phase transformations that depend continuously differentiable of time and place,

Have to the partial derivatives in the Dirac equation by a so-called covariant derivative

Be replaced. The occurring here four functions are called in physics four-potential or gauge field. Mathematically, it is a Konnexion or a relationship. If we define the transformed gauge field by

Then solves the Dirac equation with the gauge field

Or in slash notation

If and only if the transformed Dirac spinor, the Dirac equation with the transformed gauge field met. Transformations whose parameters depend on the time and place as here the phase arbitrarily, hot in physics local gauge transformations.

In the gauge field is the scalar potential Φ and the vector potential of electrodynamics,

When they are transformed as indicated, the electric and magnetic field strength remain

And all other measurable quantities unchanged.

The Dirac equation with covariant derivative and electrodynamics are invariant under arbitrary time-and position-dependent transformations of the phase of the Dirac spinor. The parameters in the covariant derivative determines the strength of the coupling of the electromagnetic potentials at the Dirac spinor. It corresponds exactly to the electric charge of the particle.

The replacement of the partial derivatives in the Dirac equation with a covariant derivative couples the electromagnetic potentials at the Dirac spinor. One speaks of so-called minimal coupling as opposed to a coupling term such as " magnetic field strength times the Dirac spinor ", which would also gauge invariant, but is not necessary to complete a derivative to a covariant derivative.

Schrödinger form

After multiplication with you can because in the Dirac equation for the time derivative dissolve and bring the Dirac equation in the form of a Schrödinger equation,

The occurring here 4 × 4 matrices can be compactly write two matrices using the Pauli matrices with blocks of 2 ×:

The differential operator on the right side of the Schrödinger equation, the Dirac equation corresponding to the Hamiltonian, the potential energy of the particle are eigenvalues ​​of this Hamiltonian.

The mathematical analysis shows, in the case of an uncharged particle ( ) that the spectrum contains both positive and negative values ​​, as well as one of the energy-momentum relation of the Klein-Gordon equation ( in natural units ), the positive and negative energy values receives.

Since particles with negative energy have never been observed and there a world would be unstable with particles whose energies upwards and downwards is unlimited, Dirac postulated that the vacuum is a Dirac sea, in which every conceivable state of negative energy is already occupied so that more electrons could accept only positive energies. Add to this Dirac sea enough energy, at least the rest energy of two electrons are added, could be a lake - electron impart positive energy and the resulting hole would be different as a state with the rest, also positive energy and the lack of, opposite charge. So Dirac predicted the existence of antiparticles and the pair creation of electron -positron pairs that were observed a year later.

The notion of a Dirac sea applies, however today is untenable and is replaced by the Feynman - Stückelberg interpretation. It implies the Dirac equation as an equation for a quantum field which is mathematically an operator that created or destroyed in the quantum mechanical states of particles or antiparticles. The creation and annihilation of particles during the interaction of the electron with the proton results in quantum electrodynamics to a small shift of energies of various states of the hydrogen atom, which would have the same energy without the creation and annihilation processes. The calculated size of the Lamb shift is consistent within the measurement accuracy of six digits with the measured value.

The creation and annihilation of particles during the interaction of the electron with a magnetic field changes the Dirac value of the gyromagnetic factor. It causes a so-called anomalous magnetic moment, they talk as g -2 anomaly. The calculated in quantum electrodynamics value agrees with the measured value to 10 decimal places.

Derivation of the gyromagnetic factor

We start from the Schrödinger form of the Dirac equation for a particle in an electromagnetic field and split the spinor in two Zweierspinoren on. We use the summation convention and write not from the string associated with an index pair sum

We assume that the particle moves only slowly, so that its energy is only slightly larger than the rest energy. That is, after removal of the fast development time, resulting from the rest energy

The time derivatives of Zweierspinoren and are small,

In the second line, the time derivative is small by assumption and the kinetic energies and the electrostatic energy small compared to the rest energy therefore is against small and approximately equal

Inserted in the first line is the

For the product of the Pauli matrices obtained

Therefore, the spinor satisfies the Pauli equation with the non-classical value

Here, the components of the staple operator.

In the homogeneous magnetic field is considered, and with

If one neglects terms which are quadratic in. Then says the Pauli equation

Consequently, the magnetic field coupled not only to the angular momentum and not only contributes to energy at. The factor is the magneton of the particle. In angular momentum eigenstates is an integer multiple of the magnetic field strength. However, it follows a half-integral multiple which is an integer only after multiplication by g.

Realizations in high energy and condensed matter physics

The Dirac equation is ( after quantization of the corresponding classical field ) the basis of relativistic quantum field theories of high energy physics. Only in recent years, we know that even in the non-relativistic energies realizations exist, namely graphene, which are layer systems associated with graphite. And that it takes only the vanishing threshold mass ( the so-called chiral limit ) m = 0 to be considered, and it is also the speed of light c by the limit speed Vf of the electron system to replace the so-called Fermi velocity. As a consequence of energy E and momentum p are proportional to each other in this system, E ~ p, while otherwise ~ is in non- relativistic electrons E p2. In addition, numerous other features arise.