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:
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 as used in the popular graduate textbook An Introduction to Quantum Field Theory by Michael E. Peskin and Daniel V. Schroeder.
Construction of Dirac spinor with a given spin direction and charge
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.