This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts.
Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin
is not a proper member of the main set of four; it is used for separating nominal left and right chiral representations.
The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation where the curly brackets
This defining property is more fundamental than the numerical values used in the specific representation of the gamma matrices.
Note that the other sign convention for the metric, (− + + +) necessitates either a change in the defining equation: or a multiplication of all gamma matrices by
Under the alternative sign convention for the metric the covariant gamma matrices are then defined by The Clifford algebra
over spacetime V can be regarded as the set of real linear operators from V to itself, End(V), or more generally, when complexified to
The bispinor fields Ψ of the Dirac equations, evaluated at any point x in spacetime, are elements of Ux (see below).
The last identity can be recognized as the defining relationship for matrices belonging to an indefinite orthogonal group, which is
It also means that indices can be raised and lowered on the γ using the metric ημν as with any 4 vector.
When these matrices, and linear combinations of them, are exponentiated, they are bispinor representations of the Lorentz group, e.g., the S(Λ) of above are of this form.
Switching to Feynman notation, the Dirac equation is It is useful to define a product of the four gamma matrices as
For example, a Dirac field can be projected onto its left-handed and right-handed components by: Some properties are: In fact,
[3]: 68 Thus, one can employ a bit of a trick to repurpose i γ 5 as one of the generators of the Clifford algebra in five dimensions.
that says: So lets use these two facts to prove this identity for the first non-trivial case: the trace of three gamma matrices.
This means that we anticommute it an odd number of times and pick up a minus sign.
Anticommute three times instead of once, picking up three minus signs, and cycle using the cyclic property of the trace.
The anticommutation rules imply that interchanging two of the indices changes the sign of the trace, so
Now, The gamma matrices can be chosen with extra hermiticity conditions which are restricted by the above anticommutation relations however.
We can impose and for the other gamma matrices (for k = 1, 2, 3) One checks immediately that these hermiticity relations hold for the Dirac representation.
takes is dependent on the specific representation chosen for the gamma matrices, up to an arbitrary phase factor.
Given a representation of gamma matrices, the arbitrary phase factor for the charge conjugation operator can also be chosen such that
The reason for making all gamma matrices imaginary is solely to obtain the particle physics metric (+, −, −, −), in which squared masses are positive.
However, the transformation required to bring the bilinear form to the complex canonical form is not a Lorentz transformation and hence not "permissible" (at the very least impractical) since all physics is tightly knit to the Lorentz symmetry and it is preferable to keep it manifest.
They argue that it is generally possible (and usually enlightening) to identify the presence of an imaginary unit in a physical equation.
Some of these proponents also question whether it is necessary or even useful to introduce an additional imaginary unit in the context of the Dirac equation.
The geometric point of this is that it disentangles the real spinor, which is covariant under Lorentz transformations, from the
The bispinor, insofar as it has linearly independent left and right components, can interact with the electromagnetic field.
In Euclidean space, there are two commonly used representations of Dirac matrices: Notice that the factors of