7.1 The spinor representation

7.1 The spinor representation

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · adjoint representation · irreducible representation

In chapter 6 we examined in great detail various ways of looking at a particular three dimensional irreducible real representation of the groups $SO(3),SU(2)$ and $Sp(1)$ . This was the adjoint representation for those three groups, and isomorphic to the vector representation for $SO(3)$ . In the $SU(2)$ and $Sp(1)$ cases, there is an even simpler non-trivial irreducible representation than the adjoint: the representation of 2 by 2 complex matrices in $SU(2)$ on column vectors ${\mathbf{C}}^2$ by matrix multiplication or the representation of unit quaternions in $Sp(1)$ on $H$ by scalar multiplication. Choosing an identification ${\mathbf{C}}^2=\mathbf{H}$ these are isomorphic representations on ${\mathbf{C}}^2$ of isomorphic groups, and for calculational convenience we will use $SU(2)$ and its complex matrices rather than dealing with quaternions. We thus have:

Definition (Spinor representation)

The spinor representation of $Spin(3)=$ $SU(2)$ is the representation on ${\mathbf{C}}^2$ given by

$$g\in SU(2)\to {\pi }_{spinor}(g)=g$$

Elements of the representation space ${\mathbf{C}}^2$ are called “spinors”.

The spin representation of $SU(2)$ is not a representation of $SO(3)$ . The double cover map $\Phi :SU(2)SO(3)$ is a homomorphism, so given a representation $(\pi ,V)$ of $SO(3)$ one gets a representation $(\pi \circ \Phi ,V)$ of $SU(2)$ by composition. One cannot ${}_{\mathrm{go}}$ in the other direction: there is no homomorphism $SO(3)SU(2)$ that would allow one to make the spin representation of $SU(2)$ on ${\mathbf{C}}^2$ into an $SO(3)$ representation.

One could try and define a representation of $SO(3)$ by

$$g\in SO(3)\to \pi (g)={\pi }_{spinor}(\tilde{g})\in SU(2)$$

where $\tilde{g}$ is some choice of one of the two elements $\tilde{g}\in SU(2)$ satisfying $\Phi (\tilde{g})=g$ The problem with this is that it won’t quite give a homomorphism. Changing the choice of $\tilde{g}$ will introduce a minus sign, so $\pi$ will only be a homomorphism up to sign

$$\pi (g_1)\pi (g_2)=\pm \pi (g_1{g}_2)$$

The nontrivial nature of the double covering map $\Phi$ implies that there is no way to completely eliminate all minus signs, no matter how one chooses $\tilde{g}$ (since a continuous choice of $\tilde{g}$ is not possible for all $g$ in a non-contractible loop of elements of $SO(3))$ . Examples like this, which satisfy the representation property only one up to a sign ambiguity, are known as “projective representations”. $\mathrm{So}$ , the spinor representation of $SU(2)=Spin(3)$ can be used to construct a projective representation of $SO(3)$ , but not a true representation of $SO(3)$

Quantum mechanics texts sometimes deal with this phenomenon by noting that there is an ambiguity in how one specifies physical states in $\mathcal{H},$ since multiplying a vector in $H$ by a scalar doesn’t change the eigenvalues of operators or the relative probabilities of observing these eigenvalues. As a result, the sign ambiguity noted above has no physical efect since arguably one should be working with states modulo the scalar ambiguity. It seems more straightforward though to not try and work with projective representations, but just use the larger group $Spin(3)$ , accepting that this is the correct group reflecting the action of rotations on three dimensional quantum systems.

The spin representation is more fundamental than the vector representation, in the sense that the spin representation cannot be found only knowing the vector representation, but the vector representation of $SO(3)$ can be constructed knowing the spin representation of $SU(2)$ . We have seen this using the identification of ${\mathbf{R}}^3$ with 2 by 2 complex matrices, with equation 6.5 showing that rotations of ${\mathbf{R}}^3$ correspond to conjugation by spin representation matrices. Another way of seeing this uses the tensor product, and is explained in section 9.4.3. Note that taking spinors as fundamental entails abandoning the description of three dimensional geometry purely in terms of real numbers. While the vector representation is a real representation of $SO(3)$ or $Spin(3)$ , the spinor representation is a complex representation.

Chapter contents · Next section: 7.2 The spin 1/2 particle in a magnetic field · 中文版本

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