Example. The Lorentz Spin action is Hermitian matrix congruence [fcap-0040]
✍️source

In the Pauli model Example [fcap-002L], an odd vector becomes an even element after multiplication by $e_0$. Encode $v=te_0+x_1e_1+x_2e_2+x_3e_3$ by \[X(v)=\Psi (ve_0)=t1-\sum _{j=1}^3x_j\sigma _j =\begin {pmatrix}t-x_3&-x_1+i x_2\\-x_1-i x_2&t+x_3\end {pmatrix}.\] A matrix is Hermitian when $Z^*=Z$, where $*$ denotes conjugate transpose. Every Hermitian two-by-two matrix has the form $Z=\begin {pmatrix}a&z\\\overline z&d\end {pmatrix}$ with $a,d\in \mathbb R$. The inverse coordinates are \[X^{-1}(Z)=\left (\frac {a+d}{2},-\operatorname {Re}z, \operatorname {Im}z,\frac {d-a}{2}\right ).\] Thus $X$ is a real linear equivalence, and direct expansion gives \[\det X(v)=t^2-x_1^2-x_2^2-x_3^2=Q_{1,3}(v).\]

For $s\in \operatorname {Spin}(Q_{1,3})$, put $M=\Psi (s)\in SL_2(\mathbb C)$. Then the vector action is \[X(svs^{-1})=M X(v)M^*.\] To prove it, conjugation by $e_0$ defines an automorphism $\gamma (x)=e_0xe_0$ of the even algebra. It negates each $A_j=e_0e_j$ and the volume element $I=e_0e_1e_2e_3$. On the matrix side, $M\mapsto \operatorname {adj}(M)^*$ has the same effect on $\sigma _j$ and $-i1$. It is a real algebra automorphism: both adjugation and conjugate transpose reverse products, and in size two adjugation is linear. Checking the even generators therefore gives \[\Psi (\gamma (x))=\operatorname {adj}(\Psi (x))^*.\] Now multiply the Clifford action by $e_0$: \[(svs^{-1})e_0=s(ve_0)\gamma (s^{-1}).\] Since $\det M=1$, we have $\operatorname {adj}(M^{-1})=M$. Applying $\Psi $ proves the congruence formula. It preserves Hermitian matrices, and its determinant is unchanged because $\det M=\det M^*=1$.

The square makes the coordinate meaning of the action explicit:

$\text{[math]}$ For example, let $M=\operatorname {diag}(e^{r/2},e^{-r/2})$. Its congruence multiplies $t-x_3$ by $e^r$, multiplies $t+x_3$ by $e^{-r}$, and fixes the off-diagonal entries. Hence \[\begin {aligned} t'&=\cosh (r)t-\sinh (r)x_3,& x_1'&=x_1,\\ x_3'&=-\sinh (r)t+\cosh (r)x_3,& x_2'&=x_2. \end {aligned}\] This is the boost with the sign selected by $X=t1-x\cdot \sigma $. The null cone is exactly the Hermitian matrices of rank at most one.

For comparison with the formal opposite-signature model, use $T$ and $E$ from Lemma [fcap-003Z]. TauCeti's vector equivalence has coordinates \[H(y)=\begin {pmatrix}y_3+y_2&-y_1-i y_0\\ -y_1+i y_0&y_3-y_2\end {pmatrix},\qquad \det H(y)=-Q_{3,1}(y)\] TauCeti.coe_realCliffordThreeOneVectorEquivHermitian_apply TauCeti.realCliffordThreeOneVectorEquivHermitian_det. Substitution gives $H(Tv)=X(v)$. In the even algebra this is the same identity as $E(ve_0)=-T(v)f_3$, since $H(y)=\Phi (-yf_3)$. The group coercion formula TauCeti.coe_realSpinThreeOneEquivSpecialLinear_apply identifies the matrix of $E(s)$ with $\Phi (E(s))=\Psi (s)=M$. Consequently the action dictionary in Lemma [fcap-003Z] translates TauCeti's exact Hermitian congruence theorem TauCeti.realSpinThreeOneEquivSpecialLinear_action into the displayed Pauli action. This checks both the signature sign and the matrix axes, not just the abstract group type.

Multiplication by a fixed nonisotropic vector is the even-algebra action mechanism of [porteous1995clifford, Chapter 16, Proposition 16.9, pp. 143--144]. His Lorentz group calculation uses adjugation and the norm-one carrier [porteous1995clifford, Chapter 17, Proposition 17.5, pp. 151--152]. The coordinates and conjugate-transpose action above are derived in our chosen axes. The orthogonal image retains the spinor-norm condition of Theorem [fcap-0023]; a topological component identification is separate.

Example. The Lorentz Spin action is Hermitian matrix congruence [fcap-0040]
✍️source

Reference. Clifford algebras and the classical groups [porteous1995clifford]

Hyperbolic recurrence, real signatures and quaternary centers [fcap-0010]

Convention. Real signature and Clifford signs [chevalley1954algebraic, II.2.9, pp. 65--66] [fcap-0011]

realCliffordForm realCliffordForm_apply
L∃∀N

Theorem. Witt extension and the regular-subspace interface [fcap-002A]

Proposition. A Lorentz frame can be prescribed [fcap-002B]

Proof.

Remark. The radical blocks an unqualified extension theorem [fcap-002C]

Definition. The hyperbolic plane [lawson2016spin, I.4, Theorem 4.1 and (4.3), pp. 25--26] [fcap-0012]

realCliffordForm_one_one_apply realBottSplitIsometry
L∃∀N

Theorem. The hyperbolic matrix recurrence [lawson2016spin, I.4, Theorem 4.1, (4.3), pp. 25--26] [fcap-0013]

hyperbolicEquivTensor realCliffordBottEquiv_ι
L∃∀N

Lemma. Finite hyperbolic reduction [chevalley1954algebraic, II.2.9, pp. 65--66] [fcap-0014]

realCliffordBottIterEquiv realCliffordSignatureReductionEquiv
L∃∀N

Theorem. The signature-switch recurrence [lawson2016spin, I.4, Theorem 4.1 and (4.1), pp. 25--26] [fcap-001X]

realCliffordSignatureSwitchRecurrenceEquiv
L∃∀N

Remark. Why algebraic periodicity is a separate branch [chevalley1954algebraic, II.2.1, II.2.5, and II.2.9, pp. 42--46, 65--66]; [lawson2016spin, I.4, Theorems 4.1 and 4.3, pp. 25--29] [fcap-001Y]

Example. Four real Clifford base entries [lawson2016spin, I.4, Theorem 4.3 and Tables I--II, pp. 27--29] [fcap-0015]

realCliffordOneZeroEquivProd realCliffordZeroOneEquivComplex realCliffordZeroTwoEquivQuaternion realCliffordOneOneEquivMatrix
L∃∀N

Lemma. Reverse norm one is the Spin carrier in real dimension three [fcap-002G]

Example. The compact three-dimensional Spin group is the unit quaternions [fcap-002H]

Example. The compact three-dimensional Spin action is quaternion conjugation [fcap-003J]

Example. The split three-dimensional Spin group is \(SL_2(\mathbb R)\) [fcap-002I]

Example. The split three-dimensional Spin action is symmetric matrix congruence [fcap-003X]

Lemma. Reverse norm one is the Spin carrier in real dimension four [fcap-002J]

Example. The compact four-dimensional Spin group is a pair of unit quaternions [fcap-002K]

Example. The compact four-dimensional Spin action is left and right quaternion multiplication [fcap-003K]

Example. The split four-dimensional Spin action is left and inverse-right multiplication [fcap-003Y]

Example. The Lorentzian four-dimensional Spin group is \(SL_2(\mathbb C)\) [fcap-002L]

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

Example. The Lorentz Spin action is Hermitian matrix congruence [fcap-0040]
✍️source

Proposition. The discriminant center of a four-dimensional Spin group [fcap-0043]

Proof.

Lemma. The volume term does not obstruct Spin in dimension five [fcap-003H]

Example. A norm-one even unit can leave the vector space in dimension six [fcap-003I]

Remark. From real recurrences to the residue table [fcap-0016]

Example. The real Clifford algebra of signature (1,3) [fcap-0026]

realClifford_classification_of_residue_eq_two
L∃∀N

Proposition. The real algebra generated by the Dirac gamma matrices [fcap-002U]

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

Definition. Spin group [fulton2013representation] [spin-000O]

Example. The Lorentzian four-dimensional Spin group is \(SL_2(\mathbb C)\) [fcap-002L]

Theorem. The Spin image is the spinor-norm kernel [fcap-0023]

range_spinToSpecialOrthogonal_eq_ker_spinorNorm
L∃∀N

Proof.

Definition. grade reversion [wieser2022formalizing] [ca-000J]
CliffordAlgebra.reverse
L∃∀N