Lemma. Opposite Lorentz signatures share an even-algebra action dictionary [fcap-003Z]

✍️sourceAGENTDRAFTED

Keep the Pauli convention of Example [fcap-002L]: \(Q_{1,3}(v)=t^2-x_1^2-x_2^2-x_3^2\), \(A_j=e_0e_j\), and \(\Psi (A_j)=\sigma _j\). For \(Q_{3,1}(y)=y_0^2+y_1^2+y_2^2-y_3^2\), write \(f_0,\ldots ,f_3\) for the coordinate generators. The real linear bijection \[T(t,x_1,x_2,x_3)=(-x_2,x_1,-x_3,t)\] satisfies \(Q_{3,1}(Tv)=-Q_{1,3}(v)\). There is a real algebra equivalence \[E:\mathcal {C}\kern -2pt\ell ^+(Q_{1,3})\longrightarrow \mathcal {C}\kern -2pt\ell ^+(Q_{3,1}), \qquad E(vw)=-T(v)T(w).\] It preserves reversion and restricts to an equivalence of the actual Spin groups. Their vector actions satisfy \[T(svs^{-1})=E(s)T(v)E(s)^{-1}.\]

For the construction, complexify and send \(v\mapsto iT(v)\). Its square is \(Q_{1,3}(v)\), so the Clifford universal property gives an algebra map. On an even word of length \(2k\) its value is \((-1)^kT(v_1)\cdots T(v_{2k})\), which is real. This defines \(E\) by scalar extension as in Theorem [fcap-003V]. The reverse construction sends \(w\mapsto iT^{-1}(w)\); the two composites negate generators and therefore fix every even word. Reversing an even word commutes with \(E\). The dimension-four carrier lemma Lemma [fcap-002J] now identifies the reverse-norm-one even units on both sides, so \(E\) and its inverse restrict to the stated actual groups. Applying the complex map to \(svs^{-1}\) and cancelling \(i\) proves the action identity. This is the sign-transfer mechanism used for compact forms in Proposition [fcap-0039], with the additional coordinate change \(T\).

The algebraic construction is also the composite of sign scaling and the isometry \(T:(-Q_{1,3})\to Q_{3,1}\). TauCeti's general scaling formula at the unit \(-1\) TauCeti.CliffordAlgebra.evenEquivEvenSMul_ι, its reversion compatibility TauCeti.CliffordAlgebra.evenEquivEvenSMul_reverseEven, and the isometry formula CliffordAlgebra.evenEquivOfIsometry_ι provide these three component statements. The displayed \(T\) and its Spin-action dictionary are the explicit specialization proved here.

Let \(\Phi :\mathcal {C}\kern -2pt\ell ^+(Q_{3,1})\to M_2(\mathbb C)\) be TauCeti's concrete matrix equivalence. Its bilinear-generator formula TauCeti.realCliffordThreeOneEvenEquivComplexMatrix_ι gives \[\begin {aligned} \Phi (f_0f_3)&=-\sigma _2,&\Phi (f_1f_3)&=\sigma _1,& \Phi (f_2f_3)&=-\sigma _3,\\ \Phi (f_0f_1)&=i\sigma _3,&\Phi (f_0f_2)&=i\sigma _1,& \Phi (f_1f_2)&=i\sigma _2. \end {aligned}\] Since \(T(e_0)=f_3\), \(T(e_1)=f_1\), \(T(e_2)=-f_0\), and \(T(e_3)=-f_2\), these formulas give \(\Phi (E(A_j))=\sigma _j\). The \(A_j\) generate the even algebra, hence \(\Phi \circ E=\Psi \). Also \(E(e_0e_1e_2e_3)=f_0f_1f_2f_3\), whose matrix is \(-i1\). Thus the Pauli group equivalence agrees, through \(E\), with TauCeti.realSpinThreeOneEquivSpecialLinear. This dictionary concerns the even algebras and their Spin actions; the full real Clifford algebras retain their different signature types.