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 4dimensional complex vector space considered as a (½,0)⊕(0,½) representation of the Lorentz group.^{[1]} Bispinors are, for example, used to describe relativistic spin½ wave functions.
In the Weyl basis, a bispinor
consists of two (twocomponent) Weyl spinors and which transform, correspondingly, under (½,0) and (0,½) representations of the group (the Lorentz group without parity transformations). 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 bispinorsEdit
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 (for spin 1/2) of the Lorentz group.
In the Weyl basis, explicit transformation matrices for a boost and for a rotation are the following:^{[2]}
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.
PropertiesEdit
A bilinear form of bispinors can be reduced to five irreducible (under the Lorentz group) objects:
 scalar, ;
 pseudoscalar, ;
 vector, ;
 pseudovector, ;
 antisymmetric tensor, ,
where and are the gamma matrices.
A suitable Lagrangian for the relativistic spin½ 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 representationEdit
IntroductionEdit
This outline describes one type of bispinors as elements of a particular representation space of the (½,0)⊕ (0,½) representation of the Lorentz group. This representation space is related to, but not identical to, the (½,0)⊕ (0,½) 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 D1 below. The basis elements of so(3;1) are labeled M^{μν}.
A representation of the Lie algebra so(3;1) of the Lorentz group O(3;1) will emerge among matrices that will be chosen as a basis (as a vector space) of the complex Clifford algebra over spacetime. These 4×4 matrices are then exponentiated yielding a representation of SO(3;1)^{+}. This representation, that turns out to be a (1/2,0)⊕(0,1/2) representation, will act on an arbitrary 4dimensional complex vector space, which will simply be taken as C^{4}, and its elements will be bispinors.
For reference, the commutation relations of so(3;1) are

(M1)
with the spacetime metric η = diag(−1,1,1,1).
The gamma matricesEdit
Let γ^{μ} denote a set of four 4dimensional gamma matrices, here called the Dirac matrices. The Dirac matrices satisfy

^{[3]}
(D1)
where {, } is the anticommutator, I_{4} is a 4×4 unit matrix, and η^{μν} is the spacetime metric with signature (+,,,). This is the defining condition for a generating set of a Clifford algebra. Further basis elements σ^{μν} of the Clifford algebra are given by

^{[4]}
(C1)
Only six of the matrices σ^{μν} are linearly independent. This follows directly from their definition since σ^{μν} =−σ^{νμ}. They act on the subspace V_{γ} the γ^{μ} span in the passive sense, according to

^{[5]}
(C2)
In (C2), the second equality follows from property (D1) of the Clifford algebra.
Lie algebra embedding of so(3;1) in Cℓ_{4}(C)Edit
Now define an action of so(3;1) on the σ^{μν}, and the linear subspace V_{σ} ⊂ Cℓ_{4}(C) they span in Cℓ_{4}(C) ≈ M^{n}_{C}, given by

.
(C4)
The last equality in (C4), which follows from (C2) and the property (D1) of the gamma matrices, shows that the σ^{μν} constitute a representation of so(3;1) since the commutation relations in (C4) are exactly those of so(3;1). The action of π(M^{μν}) can either be thought of as sixdimensional matrices Σ^{μν} multiplying the basis vectors σ^{μν}, since the space in M_{n}(C) spanned by the σ^{μν} is sixdimensional, or be thought of as the action by commutation on the σ^{ρσ}. In the following, π(M^{μν}) = σ^{μν}
The γ^{μ} and the σ^{μν} are both (disjoint) subsets of the basis elements of Cℓ_{4}(C), generated by the fourdimensional Dirac matrices γ^{μ} in four spacetime dimensions. The Lie algebra of so(3;1) is thus embedded in Cℓ_{4}(C) by π as the real subspace of Cℓ_{4}(C) spanned by the σ^{μν}. For a full description of the remaining basis elements other than γ^{μ} and σ^{μν} of the Clifford algebra, please see the article Dirac algebra.
Bispinors introducedEdit
Now introduce any 4dimensional complex vector space U where the γ^{μ} act by matrix multiplication. Here U = C^{4} will do nicely. Let Λ = e^{ωμνMμν} be a Lorentz transformation and define the action of the Lorentz group on U to be
Since the σ^{μν} according to (C4) constitute a representation of so(3;1), the induced map

(C5)
according to general theory either is a representation or a projective representation of SO(3;1)^{+}. 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 matricesEdit
It remains to choose a set of Dirac matrices γ^{μ} in order to obtain the spin representation S. One such choice, appropriate for the ultrarelativistic limit, is

^{[6]}
(E1)
where the σ_{i} are the Pauli matrices. In this representation of the Clifford algebra generators, the σ^{μν} become

^{[7]}
(E23)
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 4dimensional, the only possibility is that it is a (1/2,0)⊕(0,1/2) representation, i.e. a bispinor representation. Now using the recipe of exponentiation of the Lie algebra representation to obtain a representation of SO(3;1)^{+},

(E3)
a projective 2valued representation is obtained. Here φ is a vector of rotation parameters with 0 ≤ φ^{i} ≤2π, and χ is a vector of boost parameters. With the conventions used here one may write

(E4)
for a bispinor field. Here, the upper component corresponds to a right Weyl spinor. To include space parity inversion in this formalism, one sets

^{[8]}
(E5)
as representative for P = diag(1,−1,−1,−1). It is seen that the representation is irreducible when space parity inversion is included.
An exampleEdit
Let X=2πM^{12} so that X generates a rotation around the zaxis by an angle of 2π. Then Λ = e^{iX} = I ∈ SO(3;1)^{+} but e^{iπ(X)} = I ∈ GL(U). Here, I denotes the identity element. If X = 0 is chosen instead, then still Λ = e^{iX} = I ∈ SO(3;1)^{+}, but now e^{iπ(X)} = I ∈ GL(U).
This illustrates the double valued nature of a spin representation. The identity in SO(3;1)^{+} gets mapped into either I ∈ GL(U) or I ∈ GL(U) depending on the choice of Lie algebra element to represent it. In the first case, one can speculate that a rotation of an angle 2π will turn a bispinor into minus itself, and that it requires a 4π rotation to rotate a bispinor back into itself. What really happens is that the identity in SO(3;1)^{+} is mapped to I in GL(U) with an unfortunate choice of X.
It is impossible to continuously choose X for all g ∈ SO(3;1)^{+} so that S is a continuous representation. Suppose that one defines S along a loop in SO(3;1) such that X(t)=2πtM^{12}, 0 ≤ t ≤ 1. This is a closed loop in SO(3;1), i.e. rotations ranging from 0 to 2π around the zaxis under the exponential mapping, but it is only "half"" a loop in GL(U), ending at I. In addition, the value of I ∈ SO(3;1) is ambiguous, since t = 0 and t = 2π gives different values for I ∈ SO(3;1).
The Dirac algebraEdit
The representation S on bispinors will induce a representation of SO(3;1)^{+} on End(U), 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 SO(3;1)^{+} representations, is described in the article on Dirac algebra. One of the consequences is the decomposition of the bilinear forms on U×U. This decomposition hints how to couple any bispinor field with other fields in a Lagrangian to yield Lorentz scalars.
See alsoEdit
 Dirac spinor
 Spin(3,1), the double cover of SO(3,1) by a spin group
NotesEdit
 ^ Caban & Rembieliński 2005, p. 2
 ^ David Tong, Lectures on Quantum Field Theory (2012), Lecture 4
 ^ Weinberg 2002, Equation 5.4.5
 ^ Weinberg 2002, Equation 5.4.6
 ^ Weinberg 2002, Equation 5.4.7
 ^ Weinberg 2002, Equations (5.4.17)
 ^ Weinberg 2002, Equations (5.4.19) and (5.4.20)
 ^ Weinberg 2002, Equation (5.4.13)
ReferencesEdit
 Caban, Paweł; Rembieliński, Jakub (5 July 2005). "Lorentzcovariant reduced spin density matrix and EinsteinPodolskyRosen–Bohm correlations". Physical Review A. American Physical Society (APS). 72 (1): 012103. arXiv:quantph/0507056v1. doi:10.1103/physreva.72.012103.CS1 maint: ref=harv (link)
 Weinberg, S (2002), The Quantum Theory of Fields, vol I, ISBN 0521550017.