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 \({C(3,0)}\).

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