Spinor representation

The spinor representation of the spin group and the half spinor representation of the spin group are used in physics to describe the spin of a particle. ${\displaystyle Spin(2k)}$ ${\displaystyle Spin(2k+1)}$

Derivation of the representation

In the following we denote with the Clifford algebra of the vector space with the square form . ${\displaystyle \mathbb {C} l_{n}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle q(z_{1},\ldots ,z_{n})=z_{1}^{2}+\ldots +z_{n}^{2}}$

The Clifford algebra is isomorphic to and in particular has two- dimensional representations. The Clifford algebra is by definition generated by with the relations and . On the other hand, as -vector space has the basis ${\displaystyle \mathbb {C} l_{1}}$ ${\displaystyle \mathbb {C} \oplus \mathbb {C} }$ ${\displaystyle 1}$ ${\displaystyle \mathbb {C} l_{2}}$ ${\displaystyle e_{1},e_{2}}$ ${\displaystyle e_{1}^{2}=-1=e_{2}^{2}}$ ${\displaystyle e_{1}e_{2}+e_{2}e_{1}=0}$ ${\displaystyle Mat(2,\mathbb {C} )}$ ${\displaystyle \mathbb {C} }$

${\displaystyle 1=\left({\begin{array}{cc}1&0\\0&1\end{array}}\right),\ g_{1}=\left({\begin{array}{cc}i&0\\0&-i\end{array}}\right),\ g_{2}=\left({\begin{array}{cc}0&i\\i&0\end{array}}\right),\ g_{1}g_{2}=\left({\begin{array}{cc}0&-1\\1&0\end{array}}\right)}$

with the relations and . So you have an isomorphism ${\displaystyle g_{1}^{2}=-1=g_{2}^{2}}$ ${\displaystyle g_{1}g_{2}+g_{2}g_{1}=0}$

${\displaystyle \mathbb {C} l_{2}\cong Mat(2,\mathbb {C} )}$

and in particular a -dimensional representation of . ${\displaystyle 2}$ ${\displaystyle \mathbb {C} l_{2}}$

By

${\displaystyle e_{1}\to 1\otimes g_{1},\ e_{2}\to 1\otimes g_{2},\ e_{j}\to (ie_{j-2}\otimes g_{1}g_{2}){\mbox{ für }}3\leq j\leq n+2}$

one obtains an isomorphism

${\displaystyle \mathbb {C} l_{n+2}\to \mathbb {C} l_{n}\otimes M_{2}(\mathbb {C} )}$.

For an even number it follows by induction ${\displaystyle n=2k}$

${\displaystyle \mathbb {C} l_{2k}\cong M_{2}(\mathbb {C} )\otimes \ldots \otimes M_{2}(\mathbb {C} )\cong M_{2^{k}}(\mathbb {C} )}$,

in particular, a representation of on a -dimensional vector space is obtained . ${\displaystyle \mathbb {C} l_{2k}}$ ${\displaystyle 2^{k}}$ ${\displaystyle \Delta _{n}}$

For an odd number one obtains by complete induction ${\displaystyle n=2k+1}$

${\displaystyle \mathbb {C} l_{2k+1}\cong (\mathbb {C} \oplus \mathbb {C} )\otimes M_{2}(\mathbb {C} )\otimes \ldots \otimes M_{2}(\mathbb {C} )\cong (\mathbb {C} \oplus \mathbb {C} )\otimes M_{2^{k}}(\mathbb {C} )\cong M_{2^{k}}(\mathbb {C} )\oplus M_{2^{k}}(\mathbb {C} )}$,

In particular, one obtains two representations of on -dimensional vector spaces. ${\displaystyle \mathbb {C} l_{2k+1}}$ ${\displaystyle 2^{k}}$

In each case one has a complex vector space for or ${\displaystyle n=2k}$ ${\displaystyle n=2k+1}$

${\displaystyle \Delta _{n}\cong \mathbb {C} ^{2^{k}}}$,

so that

${\displaystyle {\begin{array}{cc}\mathbb {C} l_{n}=End(\Delta _{n})&n=2k{\mbox{ gerade}}\\\mathbb {C} l_{n}=End(\Delta _{n})\oplus End(\Delta _{n})&n=2k+1{\mbox{ ungerade}}\end{array}}}$.

The spinor representation of the spin group is the restriction of the representation to . ${\displaystyle Spin(n)}$ ${\displaystyle \Delta _{n}}$ ${\displaystyle Spin(n)\subset \mathbb {C} l_{n}}$

More generally one can consider for the square shape on the associated spin group . This is also included and thus are respectively representations of . In physics, the elements of are called Dirac spinors . ${\displaystyle n=p+q}$ ${\displaystyle q(z_{1},\ldots ,z_{n})=z_{1}^{2}+\ldots +z_{p}^{2}-z_{p+1}^{2}-\ldots -z_{p+q}^{2}}$ ${\displaystyle \mathbb {R} ^{p+q}}$ ${\displaystyle Spin(p,q)}$ ${\displaystyle \mathbb {C} l_{n}}$ ${\displaystyle \Delta _{n}}$ ${\displaystyle \Delta _{n}^{\pm }}$ ${\displaystyle Spin(p,q)}$ ${\displaystyle \Delta _{n}}$

properties

• The spinor and half spinor representations are faithful representations .
• For all the picture has in or the determinant . ${\displaystyle g\in Spin(n)}$ ${\displaystyle GL(\Delta _{n})}$ ${\displaystyle GL(\Delta _{n}^{\pm })}$ ${\displaystyle 1}$
• On or there is an -invariant Hermitian scalar product. The images of the spinor and half-spinor representations are therefore in and . ${\displaystyle \Delta _{n}}$ ${\displaystyle \Delta _{n}^{\pm }}$ ${\displaystyle Spin(n)}$ ${\displaystyle SU(\Delta _{n})}$ ${\displaystyle SU(\Delta _{n}^{\pm })}$

Half spinor representations

For odd, the spinor representation is an irreducible representation of . In contrast, for the spinor representation, the direct sum of two irreducible representations, which are referred to as half- spinor representations . ${\displaystyle n=2k+1}$ ${\displaystyle \Delta _{n}}$ ${\displaystyle Spin(n)}$ ${\displaystyle n=2k}$ ${\displaystyle \Delta _{2k}^{+}\oplus \Delta _{2k}^{-}}$

These subspaces are obtained as eigenspaces of the effect of to the eigenvalues and . In physics, the elements of these two subspaces are called positive and negative Weyl spinors . ${\displaystyle e_{1}\ldots e_{2k}}$ ${\displaystyle (-i)^{k}}$ ${\displaystyle -(-i)^{k}}$

literature

• Blaine Lawson , Marie-Louise Michelsohn : Spin Geometry. Princeton University Press, ISBN 978-0-691-08542-5 .
• John Roe : Elliptic Operators, Topology, and Asymptotic Methods. Second edition. Chapman & Hall, CRC Research Notes in Mathematics Series, ISBN 978-0-582-32502-9 .
• Thomas Friedrich : Dirac operators in Riemannian geometry. Vieweg Verlag, ISBN 978-3-528-06926-1 .