of the free Dirac equation,
where (in the units )
- is a relativistic spin-1/2 field,
- is the Dirac spinor related to a plane-wave with wave-vector ,
- is the four-wave-vector of the plane wave, where is arbitrary,
- are the four-coordinates in a given inertial frame of reference.
The Dirac spinor for the positive-frequency solution can be written as
- is an arbitrary two-spinor,
- are the Pauli matrices,
- is the positive square root
Derivation from Dirac equationEdit
The Dirac equation has the form
In order to derive the form of the four-spinor we have to first note the value of the matrices α and β:
These two 4×4 matrices are related to the Dirac gamma matrices. Note that 0 and I are 2×2 matrices here.
The next step is to look for solutions of the form
while at the same time splitting ω into two two-spinors:
Using all of the above information to plug into the Dirac equation results in
This matrix equation is really two coupled equations:
Solve the 2nd equation for and one obtains
Note that this solution needs to have in order for the solution to be valid in a frame where the particle has .
We consider the potentially problematic term .
- If , clearly as .
- On the other hand, let , with a unit vector, and let .
Hence the negative solution clearly has to be omitted, and .
Alternatively, solve the 1st equation for and one finds
In this case one needs to enforce that for this solution to be valid in a frame where the particle has . This can be shown analogously to the previous case.
This solution is useful for showing the relation between anti-particle and particle.
The most convenient definitions for the two-spinors are:
The Pauli matrices are
Using these, one can calculate:
Particles are defined as having positive energy. The normalization for the four-spinor ω is chosen so that the total probability is invariant under Lorentz transformation. The total probability is:
where is the volume of integration. Under Lorentz transformation, the volume scales as the inverse of Lorentz factor: . This implies that the probability density must be normalized proportional to so the total probability is Lorentz invariant. The usual convention is to choose . Hence the spinors, denoted as u are:
where s = 1 or 2 (spin "up" or "down")
Anti-particles having positive energy are defined as particles having negative energy and propagating backward in time. Hence changing the sign of and in the four-spinor for particles will give the four-spinor for anti-particles:
Here we choose the solutions. Explicitly,
Note that these solutions are readily obtained by substituting the ansatz into the Dirac equation.
The completeness relations for the four-spinors u and v are
- (see Feynman slash notation)
Dirac spinors and the Dirac algebraEdit
There are several choices of signature and representation that are in common use in the physics literature. The Dirac matrices are typically written as where runs from 0 to 3. In this notation, 0 corresponds to time, and 1 through 3 correspond to x, y, and z.
The + − − − signature is sometimes called the west coast metric, while the − + + + is the east coast metric. At this time the + − − − signature is in more common use, and our example will use this signature. To switch from one example to the other, multiply all by .
After choosing the signature, there are many ways of constructing a representation in the 4×4 matrices, and many are in common use. In order to make this example as general as possible we will not specify a representation until the final step. At that time we will substitute in the "chiral" or "Weyl" representation.
Construction of Dirac spinor with a given spin direction and chargeEdit
First we choose a spin direction for our electron or positron. As with the example of the Pauli algebra discussed above, the spin direction is defined by a unit vector in 3 dimensions, (a, b, c). Following the convention of Peskin & Schroeder, the spin operator for spin in the (a, b, c) direction is defined as the dot product of (a, b, c) with the vector
Note that the above is a root of unity, that is, it squares to 1. Consequently, we can make a projection operator from it that projects out the sub-algebra of the Dirac algebra that has spin oriented in the (a, b, c) direction:
Now we must choose a charge, +1 (positron) or −1 (electron). Following the conventions of Peskin & Schroeder, the operator for charge is , that is, electron states will take an eigenvalue of −1 with respect to this operator while positron states will take an eigenvalue of +1.
Note that is also a square root of unity. Furthermore, commutes with . They form a complete set of commuting operators for the Dirac algebra. Continuing with our example, we look for a representation of an electron with spin in the (a, b, c) direction. Turning into a projection operator for charge = −1, we have
The projection operator for the spinor we seek is therefore the product of the two projection operators we've found:
The above projection operator, when applied to any spinor, will give that part of the spinor that corresponds to the electron state we seek. So we can apply it to a spinor with the value 1 in one of its components, and 0 in the others, which gives a column of the matrix. Continuing the example, we put (a, b, c) = (0, 0, 1) and have
and so our desired projection operator is
The 4×4 gamma matrices used in the Weyl representation are
for k = 1, 2, 3 and where are the usual 2×2 Pauli matrices. Substituting these in for P gives
Our answer is any non-zero column of the above matrix. The division by two is just a normalization. The first and third columns give the same result:
More generally, for electrons and positrons with spin oriented in the (a, b, c) direction, the projection operator is
where the upper signs are for the electron and the lower signs are for the positron. The corresponding spinor can be taken as any non zero column. Since the different columns are multiples of the same spinor. The representation of the resulting spinor in the Dirac basis can be obtained using the rule given in the bispinor article.