A little computation shows that this gives the correct interaction with spin.

This Hamiltonian acts on a two component spinor.

We can **extend this concept to use the relativistic energy equation**.
The idea is to replace
with
in the relativistic energy equation.

Instead of an equation which is second order in the time derivative, we can make a first order equation, like the Schrödinger equation, by extending this equation to four components.

Now rewriting in terms of and and ordering it as a matrix equation, we get an equation that can be written as a dot product between 4-vectors.

Define the

With this definition, the relativistic equation can be simplified a great deal

where the gamma matrices are given by

In fact any set of matrices that satisfy the anti-commutation relations would yield equivalent physics results, however, we will work in the above explicit representation of the gamma matrices.

Defining
,

satisfies the equation of a conserved 4-vector current

and also transforms like a 4-vector. The fourth component of the vector shows that the probability density is . This indicates that the normalization of the state includes all four components of the Dirac spinors.

For non-relativistic electrons, the first two components of the Dirac spinor are large while the last two are small.

We use this fact to write an approximate two-component equation derived from the Dirac equation in the non-relativistic limit.

This

For a free particle, each component of the Dirac spinor satisfies the Klein-Gordon equation.

This is consistent with the relativistic energy relation.

The four normalized solutions for a Dirac particle at rest are.

The

The next step is to find the solutions with definite momentum.
The four plane wave solutions to the Dirac equation are

where the four spinors are given by.

is positive for solutions 1 and 2 and negative for solutions 3 and 4. The spinors are orthogonal

and the normalization constants have been set so that the states are properly normalized and the spinors follow the convention given above, with the normalization proportional to energy.

The solutions are not in general eigenstates of any component of spin but are eigenstates of **helicity**,
the component of spin along the direction of the momentum.

Note that with negative, the exponential has the phase velocity, the group velocity and the probability flux all in the opposite direction of the momentum as we have defined it. This clearly doesn't make sense. Solutions 3 and 4 need to be understood in a way for which the non-relativistic operators have not prepared us. Let us simply relabel solutions 3 and 4 such that

so that all the energies are positive and the momenta point in the direction of the velocities. This means we change the signs in solutions 3 and 4 as follows.

We have plane waves of the form

with the plus sign for solutions 1 and 2 and the minus sign for solutions 3 and 4. These sign in the exponential is not very surprising from the point of view of possible solutions to a differential equation. The problem now is that for solutions 3 and 4 the momentum and energy operators must have a minus sign added to them and the phase of the wave function at a fixed position behaves in the opposite way as a function of time than what we expect and from solutions 1 and 2. It is as if solutions 3 and 4 are moving backward in time.

If we change the charge on the electron from to and change the sign of the exponent, the Dirac equation remains the invariant. Thus, we can turn the negative exponent solution (going backward in time) into the conventional positive exponent solution if we change the charge to . We can interpret solutions 3 and 4 as positrons. We will make this switch more carefully when we study the charge conjugation operator.

The Dirac equation should be invariant under Lorentz boosts and under rotations,
both of which are just changes in the definition of an inertial coordinate system.
Under Lorentz boosts,
transforms like a 4-vector but the
matrices are constant.
The Dirac equation is shown to be **invariant under boosts** along the
direction
if we transform the Dirac spinor according to

with .

The
**Dirac equation is invariant under rotations** about the
axis
if we transform the Dirac spinor according to

with is a cyclic permutation.

Another symmetry related to the choice of coordinate system is parity.
Under a **parity inversion operation** the Dirac equation remains invariant if

Since , the third and fourth components of the spinor change sign while the first two don't. Since we could have chosen , all we know is that

From 4 by 4 matrices, we may derive 16 independent components of covariant objects.
We **define the product of all gamma matrices**.

which obviously

For rotations and boosts, commutes with since it commutes with the pair of gamma matrices. For a parity inversion, it anticommutes with .

The simplest set of covariants we can make from Dirac spinors and matrices are tabulated below.

Classification |
Covariant Form |
no. of Components |

Scalar | 1 | |

Pseudoscalar | 1 | |

Vector | 4 | |

Axial Vector | 4 | |

Rank 2 antisymmetric tensor | 6 | |

Total |
16 |

For many purposes, it is useful to write the Dirac equation in the traditional form . To do this, we must separate the space and time derivatives, making the equation less covariant looking.

Thus we can identify the operator below as the Hamiltonian.

The Hamiltonian helps us identify constants of the motion. If an operator commutes with , it represents a conserved quantity.

Its easy to see the
commutes with the Hamiltonian for a free particle so that **momentum will be conserved**.
The components of orbital angular momentum do not commute with
.

The components of spin also do not commute with .

But, from the above, the

The Dirac equation naturally

We can also see that the **helicity**, or spin along the direction of motion does commute.

For any calculation, we need to know the interaction term with the Electromagnetic field.
Based on the interaction of field with a current

and the current we have found for the Dirac equation, the interaction Hamiltonian is.

This is simpler than the non-relativistic case, with no term and only one power of .

The Dirac equation has some unexpected phenomena which we can derive. Velocity eigenvalues for electrons are always along any direction. Thus the only values of velocity that we could measure are .

Localized states, expanded in plane waves, contain all four components of the plane wave solutions.
Mixing components 1 and 2 with components 3 and 4 gives rise to **Zitterbewegung**,
the very rapid oscillation of an electrons velocity and position.

The last sum which contains the cross terms between negative and positive energy represents

It is possible to solve the Dirac equation exactly for Hydrogen in a way very similar to the non-relativistic solution.
One difference is that it is clear from the beginning that the total angular momentum is a constant of the motion and
is used as a basic quantum number.
There is another conserved quantum number related to the component of spin along the direction of
.
With these quantum numbers, the radial equation can be solved in a similar way as for the non-relativistic case
yielding the **energy relation**.

We can identify the standard principle quantum number in this case as . This result gives the same answer as our non-relativistic calculation to order but is also

A calculation of Thomson scattering shows that even simple low energy photon scattering relies on the ``negative energy'' or positron states to get a non-zero answer. If the calculation is done with the two diagrams in which a photon is absorbed then emitted by an electron (and vice-versa) the result is zero at low energy because the interaction Hamiltonian connects the first and second plane wave states with the third and fourth at zero momentum. This is in contradiction to the classical and non-relativistic calculations as well as measurement. There are additional diagrams if we consider the possibility that the photon can create and electron positron pair which annihilates with the initial electron emitting a photon (or with the initial and final photons swapped). These two terms give the right answer. The calculation of Thomson scattering makes it clear that we cannot ignore the new ``negative energy'' or positron states.

The Dirac equation is invariant under charge conjugation, defined as changing electron states into the opposite charged
positron states with the same momentum and spin (and changing the sign of external fields).
To do this the Dirac spinor is transformed according to.

Of course a second charge conjugation operation takes the state back to the original . Applying this to the plane wave solutions gives

which defines new positron spinors and that are charge conjugates of and .

Jim Branson 2013-04-22