Bispinor

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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 (½,0)⊕(0,½) representation of the Lorentz group.[1] Bispinors are, for example, used to describe relativistic spin-½ wave functions.

In the Weyl basis, a bispinor

consists of two (two-component) 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 bispinors

A bispinor field transforms according to a rule

where is a Lorentz transformation. Here the coordinates of physical points are multiplied on the left by , so the point with coordinate before the transformation has coordinate after the transformation.

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.

Properties

A bilinear form of bispinors can be reduced to five irreducible (under the Lorentz group) 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 (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 Template:EquationNote below. The basis elements of so(3;1) are labeled Mμν.

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 ({{ safesubst:#invoke:Unsubst||$B=1/2}},0)⊕(0,{{ safesubst:#invoke:Unsubst||$B=1/2}}) representation, will act on an arbitrary 4-dimensional complex vector space, which will simply be taken as C4, and its elements will be bispinors.

For reference, the commutation relations of so(3;1) are

Template:NumBlk

with the spacetime metric η = diag(−1,1,1,1).

The Gamma Matrices

Let γμ denote a set of four 4-dimensional Gamma matrices, here called the Dirac matrices. The Dirac matrices satisfy

Template:NumBlk

where {, } is the anticommutator, I4 is a 4×4 unit matrix, and ημν is the spacetime metric with signature (-,+,+,+). This is the defining condition for a generating set of a Clifford algebra. Further basis elements σμν of the Clifford algebra are given by

Template:NumBlk

Only six of the matrices σμν are linearly independent. This follows directly from their definition since σμν =−σνμ. They act on the subspace Vγ the γμ span in the passive sense, according to

Template:NumBlk

In Template:EquationNote, the second equality follows from property Template:EquationNote of the Clifford algebra.

Lie algebra embedding of so(3;1) in C4(C)

Now define an action of so(3;1) on the σμν, and the linear subspace VσC4(C) they span in C4(C) ≈ MnC, given by

Template:NumBlk

The last equality in (C4), which follows from (C2) and the property (D1) of the gamma matrices, shows that the σμν constitute a representation of so(3;1) since the commutation relations in Template:EquationNote are exactly those of so(3;1). The action of π(Mμν) can be either be thought of as 6-dimensional matrices Σμν multiplying the basis vectors σμν, since the space in Mn(C) spanned by the σμν is 6-dimensional, or it can be thought of as the action by commutation on the σρσ. In the following, π(Mμν) = σμν

The γμ and the σμν are both (disjoint) subsets of the basis elements of C4(C), generated by the 4-dimensional Dirac matrices γμ in 4 spacetime dimensions. The Lie algebra of so(3;1) is thus embedded in C4(C) by π as the real subspace of C4(C) spanned by the σμν. For a full description of the remaining basis elements other than γμ and σμν 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 U = C4 will do nicely. Let Λ = eωμνMμν be a Lorentz transformation and define the action of the Lorentz group on U to be

Since the σμν according to Template:EquationNote constitute a representation of so(3;1), the induced map

Template:NumBlk

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 matrices

It remains to choose a set of Dirac matrices γμ in order to obtain the spin representation Template:Mvar. One such choice, appropriate for the ultrarelativistic limit, is

Template:NumBlk

where the σi are the Pauli matrices. In this representation of the Clifford algebra generators, the σμν become

Template:NumBlk

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 ({{ safesubst:#invoke:Unsubst||$B=1/2}},0)⊕(0,{{ safesubst:#invoke:Unsubst||$B=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)+,

Template:NumBlk

a projective 2-valued representation is obtained. Here φ is a vector of rotation parameters with 0 ≤ φi ≤2π, and χ is a vector of boost parameters. With the conventions used here one may write

Template:NumBlk

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

as representative for P = diag(1,−1,−1,−1). It seen that the representation is irreducible when space parity inversion included.

An example

Let X=2πM12 so that Template:Mvar generates a rotation around the z-axis by an angle of . Then Λ = eiX = I ∈ SO(3;1)+ but eiπ(X) = -I ∈ GL(U). Here, Template:Mvar denotes the identity element. If X = 0 is chosen instead, then still Λ = eiX = I ∈ SO(3;1)+, but now eiπ(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 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 SO(3;1)+ is mapped to -I in GL(U) with an unfortunate choice of Template:Mvar.

It is impossible to continuously choose Template:Mvar for all g ∈ SO(3;1)+ so that Template:Mvar is a continuous representation. Suppose that one defines Template:Mvar along a loop in SO(3;1) such that X(t)=2πtM12, 0 ≤ t ≤ 1. This is a closed loop in SO(3;1), i.e. rotations ranging from 0 to around the z-axis 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 algebra

The representation Template:Mvar 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 also

Notes

  1. Caban and Rembielinski 2005, p. 2.
  2. David Tong, Lectures on Quantum Field Theory (2012), Lecture 4

References

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