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 (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

\psi=\left(\begin{array}{c}\psi_L\\ \psi_R\end{array}\right)

consists of two (two-component) Weyl spinors $\psi_L$ and $\psi_R$ which transform, correspondingly, under (½,0) and (0,½) representations of the $SO(1,3)$ 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,

\psi\rightarrow{1\over\sqrt2}\left[ \begin{array}{cc}1&1\\-1&1 \end{array} \right]\psi= {1\over\sqrt2}\left(\begin{array}{c}\psi_R+\psi_L\\ \psi_R-\psi_L \end{array}\right) .

The Dirac basis is the one most widely used in the literature.

Expressions for Lorentz transformations of bispinors

A bispinor field $\psi(x)$ transforms according to the rule

\psi^a(x) \to {\psi^\prime}^a(x^\prime) = S[\Lambda]^a_b \psi^b(\Lambda^{-1}x^\prime) = S[\Lambda]^a_b \psi^b(x)

where $\Lambda$ is a Lorentz transformation. Here the coordinates of physical points are transformed according to $x^\prime = \Lambda x$ , while $S$ , 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 $\Lambda_{boost}$ and for a rotation $\Lambda_{rot}$ are the following:^[2]

S[\Lambda_{boost}] = \left( \begin{array}{cc} e^{+\chi\cdot\sigma / 2}&0\\ 0&e^{-\chi\cdot\sigma / 2} \end{array} \right)

S[\Lambda_{rot}] = \left( \begin{array}{cc} e^{+i\phi\cdot\sigma / 2}&0\\ 0&e^{+i\phi\cdot\sigma / 2} \end{array} \right)

Here $\chi$ is the boost parameter, and $\phi^i$ represents rotation around the $x^i$ axis. $\sigma_i$ 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:

scalar, $\bar{\psi}\psi$ ;
pseudo-scalar, $\bar{\psi}\gamma^5\psi$ ;
vector, $\bar{\psi}\gamma^\mu\psi$ ;
pseudo-vector, $\bar{\psi}\gamma^\mu\gamma^5\psi$ ;
antisymmetric tensor, $\bar{\psi}(\gamma^\mu\gamma^\nu-\gamma^\nu\gamma^\mu)\psi$ ,

where $\bar{\psi}\equiv\psi^\dagger\gamma^0$ and $\{\gamma^\mu,\gamma^5\}$ are the gamma matrices.

A suitable Lagrangian for the relativistic spin-½ field can be built out of these, and is given as

\mathcal{L}={i\over2}\left( \bar{\psi}\gamma^\mu\partial_\mu\psi-\partial_\mu\bar{\psi}\gamma^\mu\psi\right)-m\bar{\psi}\psi\;.

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 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 $4\times4$ matrices are then exponentiated yielding a representation of $SO (3;1) +$ . This representation, that turns out to be a $(1 / 2,0)\oplus(0, 1 / 2)$ representation, will act on an arbitrary 4-dimensional 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

$[M^{\mu\nu},M^{\rho\sigma}] = i(\eta^{\sigma\mu}M^{\rho\nu} + \eta^{\nu\sigma}M^{\mu\rho} - \eta^{\rho\mu}M^{\sigma\nu} -\eta^{\nu\rho}M^{\mu\sigma})$

(M1)

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

The Gamma Matrices

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

$\{\gamma^\mu,\gamma^\nu \} = 2\eta^{\mu\nu}I_4,$ ^[3]

(D1)

where ${,$ } is the anticommutator, $I 4$ is a $4\times4$ 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

$\sigma^{\mu\nu} = -\frac{i}{4}[\gamma^{\mu}\gamma^{\nu} - \gamma^{\nu}\gamma^{\mu}].$ ^[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

$[\sigma^{\mu\nu}, \gamma^{\rho}] = -i\gamma^{\mu}\eta^{\nu\rho} + i\gamma^{\nu}\eta^{\mu\rho}.$ ^[5]

(C2)

In (C2), the second equality follows from property (D1) of the Clifford algebra.

Lie algebra embedding of so(3;1) in Cℓ₄(C)

Now define an action of $so (3;1)$ on the $σ μν$ , and the linear subspace $V σ \subset C ℓ 4 (C)$ they span in $C ℓ 4 (C) \approx M n C$ , given by

$\pi(M^{\mu\nu})(\sigma^{\rho\sigma}) = [\sigma^{\mu\nu}, \sigma^{\rho\sigma}] = i(\eta^{\sigma\mu}\sigma^{\rho\nu} + \eta^{\sigma\nu}\sigma^{\rho\mu} - \eta^{\mu\rho}\sigma^{\nu\sigma} - \eta^{\nu\rho}\sigma^{\mu\sigma}),$ .

(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 be either be thought of as 6-dimensional matrices $Σ μν$ multiplying the basis vectors $σ μν$ , since the space in $M n (C)$ spanned by the $σ μν$ is 6-dimensional, or it can 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 Cℓ₄(C), generated by the 4-dimensional Dirac matrices $γ μ$ in 4 spacetime dimensions. The Lie algebra of $so (3;1)$ is thus embedded in Cℓ₄(C) by $π$ as the real subspace of Cℓ₄(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 introduced

Now introduce any 4-dimensional complex vector space U where the γ^μ act by matrix multiplication. Here $U = C 4$ will do nicely. Let $Λ = e ω μν M μν$ be a Lorentz transformation and define the action of the Lorentz group on U to be

u \rightarrow S(\Lambda)u = e^{i\pi(\omega_{\mu\nu}M^{\mu\nu})}u;\quad u^\alpha \rightarrow [e^{\omega_{\mu\nu}\sigma^{{\mu\nu}}}]^\alpha{}_\beta u^\beta.

Since the $σ μν$ according to (C4) constitute a representation of $so (3;1)$ , the induced map

$S:SO(3;1)^+ \rightarrow \mathrm{GL}(U);\quad \Lambda \rightarrow e^{i\pi(X)};\quad e^{iX} = \Lambda, \quad X = \omega_{\mu\nu}M^{\mu\nu}\in \mathfrak{so}(3;1)$

(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 matrices

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

$\begin{align} \gamma^{0} &= -i\biggl(\begin{matrix} 0 & I\\ I & 0\\ \end{matrix}\biggr),\\ \gamma^{i} &= -i \biggl(\begin{matrix} 0 & \sigma_i\\ -\sigma_i & 0\\ \end{matrix}\biggr), \quad i = 1,2,3\\ \end{align}.$ ^[6]

(E1)

where the $σ i$ are the Pauli matrices. In this representation of the Clifford algebra generators, the $σ μν$ become

$\begin{align} \sigma^{i0} &= \frac{i}{2}\biggl(\begin{matrix} \sigma_i & 0 \\ 0 & -\sigma_i \\ \end{matrix}\biggr),\\ \sigma^{ij} &= \frac{i}{2}\epsilon_{ijk} \biggl(\begin{matrix} \sigma_i & 0 \\ 0 & \sigma_i \\ \end{matrix}\biggr)\\ \end{align}.$ ^[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 4-dimensional, the only possibility is that it is a $(1 / 2,0)\oplus(0, 1 / 2)$ representation, i.e. a bispinor representation. Now using the recipe of exponentiation of the Lie algebra representation to obtain a representation of $SO(3;1) +$ ,

$\begin{align} S(\Lambda_{B}) &= e^{i\pi(\phi \cdot \mathbf{J})} = \biggl(\begin{matrix} e^{-\frac{1}{2}\chi\cdot\sigma} & 0 \\ 0 & e^{\frac{1}{2}\chi\cdot\sigma} \\ \end{matrix}\biggr),\\ S(\Lambda_{R}) &= e^{i\pi(\chi \cdot \mathbf{K})} = \biggl(\begin{matrix} e^{\frac{i}{2}\phi\cdot\sigma} & 0 \\ 0 & e^{\frac{i}{2}\phi\cdot\sigma} \\ \end{matrix}\biggr)\\ \end{align},$

(E3)

a projective 2-valued representation is obtained. Here $φ$ is a vector of rotation parameters with $0 \leq φ i \leq2π$ , and $χ$ is a vector of boost parameters. With the conventions used here one may write

$\psi=\begin{pmatrix} \psi_R \\\psi_L \end{pmatrix},$

(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

$\beta = i\gamma^0 = \biggl(\begin{matrix} 0 & I\\ I & 0\\ \end{matrix}\biggr),$ ^[8]

(E5)

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

An example

Let $X=2πM 12$ so that $X$ generates a rotation around the z-axis by an angle of $2π$ . Then $Λ = e iX = I \in SO(3;1) +$ but $e iπ(X) = -I \in GL(U)$ . Here, $I$ denotes the identity element. If $X = 0$ is chosen instead, then still $Λ = e iX = I \in SO(3;1) +$ , but now $e iπ(X) = I \in GL(U)$ .

This illustrates the double valued nature of a spin representation. The identity in $SO(3;1) +$ gets mapped into either $-I \in GL(U)$ or $I \in GL(U)$ 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 $g \in SO(3;1) +$ so that $S$ is a continuous representation. Suppose that one defines $S$ along a loop in $SO(3;1)$ such that $X(t)=2πtM 12, 0 \leq t \leq 1$ . This is a closed loop in $SO(3;1)$ , i.e. rotations ranging from 0 to $2π$ 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 $I \in SO(3;1)$ is ambiguous, since $t = 0$ and $t = 2π$ gives different values for $I \in SO(3;1)$ .

The Dirac algebra

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 $U \times U$ . This decomposition hints how to couple any bispinor field with other fields in a Lagrangian to yield Lorentz scalars.

Notes

↑ Caban and Rembielinski 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)

References

P. Caban and J. Rembielinski, http://arxiv.org/abs/quant-ph/0507056v1 [Phys. Rev. A 72, 012103 (2005)]
Weinberg, S (2002), The Quantum Theory of Fields, vol I, ISBN 0-521-55001-7 .

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