Bispinor


In physics, a bispinor is an object with four complex components which transform in a specific way under Lorentz transformations: specifically, a bispinor is an element of a 4-dimensional complex vector space considered as a ⊕ representation of the Lorentz group. Bispinors are, for example, used to describe relativistic spin-½ wave functions.
In the Weyl basis, a bispinor
consists of two Weyl spinors and which transform, correspondingly, under and representations of the group. Under parity transformation the Weyl spinors transform into each other.
The Dirac bispinor is connected with the Weyl bispinor by a unitary transformation to the Dirac basis,
The Dirac basis is the one most widely used in the literature.

Expressions for Lorentz transformations of bispinors

A bispinor field transforms according to the rule
where is a Lorentz transformation. Here the coordinates of physical points are transformed according to, while, a matrix, is an element of the spinor representation of the Lorentz group.
In the Weyl basis, explicit transformation matrices for a boost and for a rotation are the following:
Here is the boost parameter, and represents rotation around the axis. are the Pauli matrices. The exponential is the exponential map, in this case the matrix exponential defined by putting the matrix into the usual power series for the exponential function.

Properties

A bilinear form of bispinors can be reduced to five irreducible objects:
  1. scalar, ;
  2. pseudo-scalar, ;
  3. vector, ;
  4. pseudo-vector, ;
  5. antisymmetric tensor, ,
where and are the gamma matrices.
A suitable Lagrangian for the relativistic spin-½ field can be built out of these, and is given as
The Dirac equation can be derived from this Lagrangian by using the Euler–Lagrange equation.

Derivation of a bispinor representation

Introduction

This outline describes one type of bispinors as elements of a particular representation space of the ⊕ representation of the Lorentz group. This representation space is related to, but not identical to, the ⊕ representation space contained in the Clifford algebra over Minkowski spacetime as described in the article Spinors. Language and terminology is used as in Representation theory of the Lorentz group. The only property of Clifford algebras that is essential for the presentation is the defining property given in below. The basis elements of are labeled.
A representation of the Lie algebra of the Lorentz group will emerge among matrices that will be chosen as a basis of the complex Clifford algebra over spacetime. These matrices are then exponentiated yielding a representation of. This representation, that turns out to be a representation, will act on an arbitrary 4-dimensional complex vector space, which will simply be taken as, and its elements will be bispinors.
For reference, the commutation relations of are
with the spacetime metric.

The gamma matrices

Let γμ denote a set of four 4-dimensional gamma matrices, here called the Dirac matrices. The Dirac matrices satisfy
where is the anticommutator, is a unit matrix, and is the spacetime metric with signature. This is the defining condition for a generating set of a Clifford algebra. Further basis elements of the Clifford algebra are given by
Only six of the matrices are linearly independent. This follows directly from their definition since. They act on the subspace the span in the passive sense, according to
In, the second equality follows from property of the Clifford algebra.

Lie algebra embedding of so(3;1) in ''C''ℓ''4''(C)

Now define an action of on the, and the linear subspace they span in, given by
The last equality in, which follows from and the property of the gamma matrices, shows that the constitute a representation of since the commutation relations in are exactly those of. The action of can either be thought of as six-dimensional matrices multiplying the basis vectors, since the space in spanned by the is six-dimensional, or be thought of as the action by commutation on the. In the following,
The and the are both subsets of the basis elements of Cℓ4, generated by the four-dimensional Dirac matrices in four spacetime dimensions. The Lie algebra of is thus embedded in Cℓ4 by as the real subspace of Cℓ4 spanned by the. For a full description of the remaining basis elements other than and of the Clifford algebra, please see the article Dirac algebra.

Bispinors introduced

Now introduce any 4-dimensional complex vector space U where the γμ act by matrix multiplication. Here will do nicely. Let be a Lorentz transformation and define the action of the Lorentz group on U to be
Since the according to constitute a representation of, the induced map
according to general theory either is a representation or a projective representation of. It will turn out to be a projective representation. The elements of U, when endowed with the transformation rule given by S, are called bispinors or simply spinors.

A choice of Dirac matrices

It remains to choose a set of Dirac matrices in order to obtain the spin representation. One such choice, appropriate for the ultrarelativistic limit, is
where the are the Pauli matrices. In this representation of the Clifford algebra generators, the become
This representation is manifestly not irreducible, since the matrices are all block diagonal. But by irreducibility of the Pauli matrices, the representation cannot be further reduced. Since it is a 4-dimensional, the only possibility is that it is a representation, i.e. a bispinor representation. Now using the recipe of exponentiation of the Lie algebra representation to obtain a representation of,
a projective 2-valued representation is obtained. Here is a vector of rotation parameters with, and is a vector of boost parameters. With the conventions used here one may write
for a bispinor field. Here, the upper component corresponds to a right Weyl spinor. To include space parity inversion in this formalism, one sets
as representative for. It is seen that the representation is irreducible when space parity inversion is included.

An example

Let so that generates a rotation around the z-axis by an angle of. Then but. Here, denotes the identity element. If is chosen instead, then still, but now.
This illustrates the double valued nature of a spin representation. The identity in gets mapped into either or depending on the choice of Lie algebra element to represent it. In the first case, one can speculate that a rotation of an angle will turn a bispinor into minus itself, and that it requires a rotation to rotate a bispinor back into itself. What really happens is that the identity in is mapped to in with an unfortunate choice of.
It is impossible to continuously choose for all so that is a continuous representation. Suppose that one defines along a loop in such that. This is a closed loop in, i.e. rotations ranging from 0 to around the z-axis under the exponential mapping, but it is only "half"" a loop in, ending at. In addition, the value of is ambiguous, since and gives different values for.

The Dirac algebra

The representation on bispinors will induce a representation of on, the set of linear operators on U. This space corresponds to the Clifford algebra itself so that all linear operators on U are elements of the latter. This representation, and how it decomposes as a direct sum of irreducible representations, is described in the article on Dirac algebra. One of the consequences is the decomposition of the bilinear forms on. This decomposition hints how to couple any bispinor field with other fields in a Lagrangian to yield Lorentz scalars.