31.5 An example: spinors for SO(4)

31.5 An example: spinors for SO(4)

Concept links · terms present in this machine draft; source roles are unverified: complex inner product · Lie algebra · Lie algebra representation · unitary representation · Lie bracket · complexification

We saw in chapter 6 that the spin group $Spin(4)$ was isomorphic to $Sp(1)\times$ $Sp(1)=SU(2)\times SU(2)$ . Its action on ${\mathbf{R}}^4$ was then given by identifying ${\mathbf{R}}^4=\mathbf{H}$ and acting by unit quaternions on the left and the right (thus the two copies of $Sp(1))$ ). While this constructs the representation of $Spin(4)$ on ${\mathbf{R}}^4$ , it does not provide the spin representation of $Spin(4)$

A conventional way of defining the spin representation is to choose an explicit matrix representation of the Cliford algebra (in this case $\mathrm{Cliff}(4,0,\mathbf{R}))$ , for instance

$${\gamma }_0=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\gamma }_1=-i\left(\begin{array}{cc}0& {\sigma }_1\\ -{\sigma }_1& 0\end{array}\right),{\gamma }_2=-i\left(\begin{array}{cc}0& {\sigma }_2\\ -{\sigma }_2& 0\end{array}\right),{\gamma }_3=-i\left(\begin{array}{cc}0& {\sigma }_3\\ -{\sigma }_3& 0\end{array}\right)$$

where we have written the matrices in 2 by $2$ block form, and are indexing the four dimensions from 0 to $3.$ . One can easily check that these satisfy the Cliford algebra relations: they anticommute with each other and

$${\gamma }_0^2={\gamma }_1^2={\gamma }_2^2={\gamma }_3^2=1$$

The quadratic Cliford algebra elements $-\frac{1}{2}{\gamma }_j{\gamma }_k$ for $j<k$ satisfy the commutation relations of $\mathfrak{so}(4)=\mathfrak{spin}(4)$ . These are explicitly

$$\begin{array}{ll}-\frac{1}{2}{\gamma }_0{\gamma }_1=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_1& 0\\ 0& -{\sigma }_1\end{array}\right),& -\frac{1}{2}{\gamma }_2{\gamma }_3=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_1& 0\\ 0& {\sigma }_1\end{array}\right)\\ -\frac{1}{2}{\gamma }_0{\gamma }_2=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_2& 0\\ 0& -{\sigma }_2\end{array}\right),& -\frac{1}{2}{\gamma }_1{\gamma }_3=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_2& 0\\ 0& {\sigma }_2\end{array}\right)\\ -\frac{1}{2}{\gamma }_0{\gamma }_3=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_3& 0\\ 0& -{\sigma }_3\end{array}\right),& -\frac{1}{2}{\gamma }_1{\gamma }_2=-\frac{i}{2}\left(\begin{array}{cc}{\sigma }_3& 0\\ 0& {\sigma }_3\end{array}\right)\end{array}$$

The Lie algebra spin representation is just matrix multiplication on $S={\mathbf{C}}^4$ and it is obviously a reducible representation on two copies of ${\mathbf{C}}^2$ (the upper and lower two components). One can also see that the Lie algebra spin $(4)=$ su $\mathfrak{i}(2)\oplus \mathfrak{su}(2)$ , with the two su(2) Lie algebras having bases

$$-\frac{1}{4}({\gamma }_0{\gamma }_1+{\gamma }_2{\gamma }_3),-\frac{1}{4}({\gamma }_0{\gamma }_2+{\gamma }_1{\gamma }_3),-\frac{1}{4}({\gamma }_0{\gamma }_3+{\gamma }_1{\gamma }_2)$$

and

$$-\frac{1}{4}({\gamma }_0{\gamma }_1-{\gamma }_2{\gamma }_3),-\frac{1}{4}({\gamma }_0{\gamma }_2-{\gamma }_1{\gamma }_3),-\frac{1}{4}({\gamma }_0{\gamma }_3-{\gamma }_1{\gamma }_2)$$

The irreducible spin representations of $Spin(4)$ are just the tensor product of spin $\frac{1}{2}$ representations of the two copies of $SU(2)$ (with each copy acting on a diferent factor of the tensor product).

In the fermionic oscillator construction, we have

$$S=S^{+}+S^{-},S^{+}=\mathrm{span}\{1,{\theta }_1{\theta }_2\},S^{-}=\mathrm{span}\{{\theta }_1,{\theta }_2\}$$

and the Cliford algebra action on $S$ is given for the generators as (now indexing dimensions from 1 to 4)

$${\gamma }_1=\frac{\partial }{\partial {\theta }_1}+{\theta }_1,{\gamma }_2=i\left(\frac{\partial }{\partial {\theta }_1}-{\theta }_1\right)$$ $${\gamma }_3=\frac{\partial }{\partial {\theta }_2}+{\theta }_2,{\gamma }_4=i\left(\frac{\partial }{\partial {\theta }_2}-{\theta }_2\right)$$

Note that in this construction there is a choice of complex structure $J=J_0$ This gives a distinguished vector $|0⟩=1\in S^{+}$ , as well as a distinguished sub-Lie algebra $\mathfrak{u}(2)\subset \mathfrak{so}(4)$ of transformations that act trivially on |0⟩, given by linear combinations of 2 2

$${\theta }_1\frac{\partial }{\partial {\theta }_1},{\theta }_2\frac{\partial }{\partial {\theta }_2},{\theta }_1\frac{\partial }{\partial {\theta }_2},{\theta }_2\frac{\partial }{\partial {\theta }_1},$$

There is also a distinguished sub-Lie algebra $\mathfrak{u}(1)\subset \mathfrak{u}(2)$ that has zero Lie bracket with the rest, with basis element

$${\theta }_1\frac{\partial }{\partial {\theta }_1}+{\theta }_2\frac{\partial }{\partial {\theta }_2}$$

Spin(4) elements that act by unitary (for the Hermitian inner product 31.3) transformations on the spinor state space, but change |0⟩ and correspond to a change in complex structure, are given by exponentiating the Lie algebra representation operators

$$i(a_{F_1}^{†}{a}_{F_2}^{†}+a_{F_2}{a}_{F_1}),a_{F_1}^{†}{a}_{F_2}^{†}-a_{F_2}{a}_{F_1}$$

The possible choices of complex structure are parametrized by $SO(4)/U(2)$ which can be identified with the complex projective sphere ${\mathbf{CP}}^{\tilde{1}}=S^2$

The construction in terms of matrices is well-suited to calculations, but it is inherently dependent on a choice of coordinates. The fermionic version of Bargmann-Fock is given here in terms of a choice of basis, but, like the closely analogous bosonic construction, only actually depends on a choice of inner product and a choice of compatible complex structure $J,$ producing a representation on the coordinate-independent object ${\mathcal{F}}_d^{+}={\Lambda }^{\ast }{V}_J^{+}$

In chapter 41 we will consider explicit matrix representations of the Cliford algebra for the case of $Spin(3,1)$ . The fermionic oscillator construction could also be used, complexifying to get a representation of

$$\mathfrak{so}(4)\otimes \mathbf{C}=\mathfrak{sl}(2,\mathbf{C})\oplus \mathfrak{sl}(2,\mathbf{C})$$

and then restricting to the subalgebra

$$\mathfrak{so}(3,1)\subset \mathfrak{so}(3,1)\otimes \mathbf{C}=\mathfrak{so}(4)\otimes \mathbf{C}$$

This will give a representation of $Spin(3,1)$ in terms of quadratic combinations of Cliford algebra generators, but unlike the case of $Spin(4)$ , it will not be unitary. The lack of positivity for the inner product causes the same sort of wrong-sign problems with the CAR that were found in the bosonic case for the CCR when $J$ and Ω gave a non-positive symmetric bilinear form. In the fermion case the wrong-sign problem does not stop one from constructing a representation, but it will not be a unitary representation.

Previous section: 31.4 Complex structures, U(d) SO(2 d) and the spinor representation · Next section: 31.6 For further reading · 中文版本

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