# Spacetime and spinors in geometric algebra

In geometric algebra, the rotor group is the Lie group obtained by restricting $${\textrm{Spin}\left(r,s\right)}$$ to elements whose reverse is their inverse, i.e. elements which satisfy $${U\widetilde{U}=1}$$. For $${n>2}$$ this restriction results in the identity component, i.e the rotor group is just $${\textrm{Spin}^{e}}$$. Thus in this context for $${n>2}$$ we can write $${R_{U}^{e}(v)=Uv\widetilde{U}}$$. This rotation operator can also be applied to any multivector $${A=\Sigma\left\langle A\right\rangle _{k}}$$ to yield the “rotated” multivector $${UA\widetilde{U}}$$. Under this operation, each $${k}$$-blade, consisting of the exterior product of $${k}$$ vectors, is replaced with the exterior product of $${k}$$ rotated vectors.

The vectors in any space-like plane in a vector space can be identified with the complex numbers via the representation of the isomorphism $${C_{0}\left(2,0\right)\cong C\left(0,1\right)\cong\mathbb{C}}$$ effected by $${\hat{e}_{1}\hat{e}_{2}\equiv\Omega\rightarrow i}$$, where $${i}$$ is the unit vector in $${C\left(0,1\right)}$$ identified with the imaginary unit in $${\mathbb{C}}$$. A vector $${v\equiv v^{1}\hat{e}_{1}+v^{2}\hat{e}_{2}\in C\left(2,0\right)}$$ is represented by $${v_{\mathrm{E}}\equiv\hat{e}_{1}v=v^{1}+v^{2}\hat{e}_{1}\hat{e}_{2}}$$ in the even subalgebra $${C_{0}\left(2,0\right)}$$, and therefore by $${v_{\mathbb{C}}=v^{1}+iv^{2}}$$ in $${\mathbb{C}}$$, where the choice of $${\hat{e}_{1}}$$ thus defines the real axis. Complex conjugation is then the reflection across the imaginary axis $${\hat{e}_{1}v_{\mathrm{E}}\hat{e}_{1}=v\hat{e}_{1}=\tilde{v}_{\mathrm{E}}=v^{1}+v^{2}\hat{e}_{2}\hat{e}_{1}=v^{1}-iv^{2}=v_{\mathbb{C}}^{*}}$$, which is also reversion in $${C_{0}\left(2,0\right)}$$. The complex inner product is $${\left\langle v_{\mathbb{C}},w_{\mathbb{C}}\right\rangle _{\mathbb{C}}=v_{\mathbb{C}}^{*}w_{\mathbb{C}}=\tilde{v}_{\mathrm{E}}w_{\mathrm{E}}=\left\langle v,w\right\rangle _{\mathbb{R}}}$$. Note that multiplication by the imaginary unit in $${\mathbb{C}}$$ is represented by right Clifford multiplication by $${\Omega}$$ in $${C_{0}\left(2,0\right)}$$: $${v_{\mathrm{E}}\Omega=\hat{e}_{1}v\Omega=v^{1}\Omega-v^{2}=iv^{1}-v^{2}=iv_{\mathbb{C}}}$$. This means that exponential rotations must also act from the right in $${C_{0}\left(2,0\right)}$$, and since $${\Omega}$$ anti-commutes with vectors, both operations from the left reverse sign: $${e^{\Omega\theta}v_{\mathrm{E}}=e^{\Omega\theta}\left(\hat{e}_{1}v\right)=\hat{e}_{1}\left(e^{-\Omega\theta}v\right)=\left(e^{-\Omega\theta}v\right)_{\mathrm{E}}=e^{-i\theta}v_{\mathbb{C}}}$$.

The representation of the isomorphism $${C_{0}\left(1,3\right)\cong C\left(3,0\right)}$$ effected by $${\hat{e}_{i}\hat{e}_{0}\rightarrow\sigma_{i}}$$ is sometimes called a space-time split in geometric algebra, since the resulting basis of $${C(3,0)}$$ reflects (and depends upon) the particular chosen orthonormal basis $${\hat{e}_{i}}$$ of $${C_{0}(1,3)}$$. An event $${x\in C(1,3)}$$ with spacetime coordinates $${x^{\mu}\hat{e}_{\mu}}$$ is represented by $${x\hat{e}_{0}=x^{0}+x^{i}\sigma_{i}}$$ in $${C(3,0)}$$; such a linear combination of scalar and vector in $${C(3,0)}$$ is then called a paravector (although this term is sometimes used differently). This scheme can be used to treat relativistic physics in a condensed manner. Note that a space-time split “preserves” the scalar and pseudo-scalar basis: $${I\rightarrow I}$$ and $${\Omega\rightarrow\Omega}$$. If spacetime is instead represented by the “mostly pluses” signature algebra $${C(3,1)}$$, $${-x\hat{e}_{0}}$$ can be used as the space-time split with $${\hat{e}_{0}\hat{e}_{i}\rightarrow\sigma_{i}}$$ in order to make the signs come out right.

There is also an interesting alternative to the standard definition of Dirac and Weyl spacetime spinors (as vectors acted on by a faithful complex representation of $${\textrm{Spin}(3,1)^{e}}$$), which instead considers these spinors as elements of the Clifford algebra associated with space. The Dirac spinors are vectors in $${\mathbb{C}^{4}}$$, a complex vector space of dimension 4 that decomposes into two orthogonal 2-dimensional complex subspaces which are each invariant under the action of $${\textrm{Spin}(3,1)^{e}}$$. Now, the even subalgebra $${C_{0}(3,1)\cong C(3,0)\cong\mathbb{C}(2)}$$ can also be viewed as a complex vector space of dimension 4. The action of $${\textrm{Spin}(3,1)^{e}}$$ on $${C_{0}(3,1)}$$ by Clifford multiplication is linear, and $${C_{0}(3,1)}$$ decomposes into two spaces invariant under $${\textrm{Spin}(3,1)^{e}}$$: the bivectors that are real linear combinations of $${e_{0}e_{i}\cong\sigma_{i}}$$ have negative determinant, while linear combinations of the remaining bivectors $${e_{i}e_{j}\cong\sigma_{i}\sigma_{j}=i\sigma_{k}}$$ have positive determinant, as do the scalars. The pseudo-scalars are real multiples of $${\Omega=e_{0}e_{1}e_{2}e_{3}}$$, and so have negative determinant. Thus an element of $${\textrm{Spin}^{e}(3,1)\cong SL(2,\mathbb{C})}$$, having determinant $${+1}$$, leaves invariant these positive and negative determinant subspaces under Clifford multiplication. Note that as $${C(3,0)}$$, these subspaces are exactly the even and odd subspaces $${C_{0}(3,0)}$$ and $${C_{1}(3,0)}$$.

This alternative definition of spinor then readily generalizes to any dimension, i.e. spinors can be defined as elements of the $${2^{n-1}}$$-dimensional vector space $${C_{0}(r,s)\cong C(r,s-1)}$$ acted on by $${\textrm{Spin}(r,s)^{e}}$$ via Clifford multiplication. However, it is important to note that despite the accidental equivalency in signature $${(3,1)}$$, for other signatures and dimensions these definitions are quite distinct. In particular, the above decomposition of $${C_{0}(3,1)\cong\mathbb{C}(2)}$$ as a vector space of spinors under the action of $${\textrm{Spin}(3,1)^{e}}$$ is completely unrelated to the chiral decomposition of the Dirac rep $${C\mathbb{^{C}}(3,1)\cong\mathbb{C}(4)}$$ when restricted to $${C_{0}(3,1)}$$. This is underscored by the fact that while the Dirac rep decomposes as a rep of any part of the even subalgebra, the decomposition of the spinor space $${C_{0}(3,1)}$$ only occurs under the action of the identity component $${\textrm{Spin}(3,1)^{e}}$$: the determinant is not preserved under the action of $${\textrm{Spin}(3,1)\cong SL^{\pm}(2,\mathbb{C})}$$, and so there is no decomposition in this case. Lastly, note that this gives us a new characterization of $${\textrm{Spin}\left(3,1\right)^{e}}$$ as plus or minus the exponentials of the Lie algebra of vectors and bivectors in $${C(3,0)}$$.