In quantum field theory, the Dirac spinor is the bispinor in the plane-wave solution
Contents
- Derivation from Dirac equation
- Results
- Two spinors
- Pauli matrices
- For particles
- For anti particles
- Completeness relations
- Dirac spinors and the Dirac algebra
- Conventions
- Construction of Dirac spinor with a given spin direction and charge
- References
of the free Dirac equation,
where (in the units
The Dirac spinor for the positive-frequency solution can be written as
where
Derivation from Dirac equation
The Dirac equation has the form
In order to derive the form of the four-spinor
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:
Results
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
Solve the 1st equation for
This solution is useful for showing the relation between anti-particle and particle.
Two-spinors
The most convenient definitions for the two-spinors are:
and
Pauli matrices
The Pauli matrices are
Using these, one can calculate:
For particles
Particles are defined as having positive energy. The normalization for the four-spinor ω is chosen so that
where s = 1 or 2 (spin "up" or "down")
Explicitly,
For anti-particles
Anti-particles having positive energy
Here we choose the
Completeness relations
The completeness relations for the four-spinors u and v are
where
Dirac spinors and the Dirac algebra
The Dirac matrices are a set of four 4×4 matrices that are used as spin and charge operators.
Conventions
There are several choices of signature and representation that are in common use in the physics literature. The Dirac matrices are typically written as
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
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
Note that
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
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