Introduction E9F5FC

Questions FFFFC0

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Objective

Identify the four maps (of Clifford algebras) with the four quantum symmetries (on Hamiltonians).

• Make explicit how the quantum symmetries are defined in terms of Hamiltonians of particles and holes.
• Consider what it means for a matrix representation of a Clifford algebra generator {$e_k$} to satisfy a quantum symmetry.
• Consider what it means for a matrix ({$E$} or {$W$} or {$EW^T$}) to express such a quantum symmetry.
• Show how these matrices ({$E$} and {$W$} and {$EW^T$}) express fundamental maps on Clifford algebras.
• Compare with the expressions of {$C,T,S$} in Stone-Chiu-Roy to understand intuitively the relationship with particles, holes, charge, and the direction of time and the orientation of space.

The four quantum symmetries can be thought of as maps on the identity connected component of the orthogonal group {$O_0(p,q)$} which yield the four connected components of the orthogonal group.

The mappings can be defined as operations on matrices by conjugation by special matrices. These special matrices satisfy quantum symmetries with regard to matrix representations of Clifford algebra generators {$e_k$}.

References

• Cameron Krulewski. K-theory, Bott Periodicity, and Elliptic Operators
  • Given a homomorphism (Bott map) such as {$\beta:K(X)\rightarrow K^{-2}(X)$} or {$\beta:KO(X)\rightarrow KO^{-8}(X)$}, which is defined by multiplication by the Bott class, here {$b=\lambda_\mathbb{C}^*=1-H^*$} and {$u=S^+-1$}, respectively, we want to find a map {$\alpha$} such that {$\alpha\beta=\beta\alpha=1$}. This will establish the isomorphism we seek.
  • 2.2.2. This depends on a "formal trick", considering an involution on {$K_c^{-4}=K_c(\mathbb{R}^4\times X)=K_c(\mathbb{R}^2\times\mathbb{R}^2\times X)$} which switches elements {$\theta(xy)=yx$}.
  • This trick seems to appeal to the antiautomorphisms.
  • Note that the multiplication by the Bott class should relate to multiplication by the projective line for the complex numbers and the octonions, respectively.

Gregory Moore discusses (2013 12.5) notions of a real structure, including:

• C-antilinear involutive anti-automorphism. Thus {$\textrm{deg}(a^∗ ) = \textrm{deg}(a)$} but {$(ab)^∗ = (−1)^{|a||b|} b^∗ a^∗$}. This is the Koszul sign convention. By induction on the number of generators, considering the {$k$}-th generator, we can show that this has fourfold periodicity. Because in {$(e_1\cdots e_{k-1}e_k)^* = \epsilon e_k^*(e_1\cdots e_{k-1})*$}, the sign {$\epsilon$} is {$+1$} if {$k$} is odd and {$-1$} if {$k$} is even, and then we have to multiply by the sign of {$(e_1\cdots e_{k-1})*$}.

Page 142: Now define the algebra automorphism λ : Cℓ−t,s → Cℓ−t,s by dening it on the generators to be λ(ei ) = −ei and extending it to be an algebra automorphism. On homogeneous elements it is just the Z2 -grading.

Page 144: Define φ̄ := λ ◦ β(φ), and the norm function: N (φ) := φφ̄. (17.18) The norm function has some nice properties when restricted to the Cliord group Γ(t, s), namely is the subgroup of Cℓ(t, s)∗ which preserves the vector space Rt,s generated by ei under twisted adjoint action. That is, φ ∈ Γ(t, s) if for all vectors y = y i ei (where y i are real numbers) (17.19) λ(φ) · y · φ−1 ∈ Rt,s

Page 145: One useful application of the norm function is that it gives a neat denition of the groups Pinc and Spinc which are useful in both geometry and physics. To dene these 38 The same argument works for Γ(t, s). – 144 –we work with the complexied Cliord algebras. In the complex case we dene x → x̄ to include complex conjugation. That is, if x is in a real Cliord algebra then (x ⊗ z) = x̄ ⊗ z̄. We can again dene the Cliord group Γc (t, s) ⊂ Cℓ∗d as the group preserving the subspace

Page 180: Finite-dimensional fermionic system. For us the ∗-algebra structure on A := Cli(M, Q) ⊗ C (18.2) is β ⊗ C, where β is the canonical anti-automorphism of Cli(M, Q) and C is complex conjugation on C. Thus ∗ xes M and is an anti-automorphism. (These conditions uniquely determine ∗.) Axioms of quantum mechanics would simply give us some ∗-algebra without extra structure. The fermionic system gives us the extra data (M, Q).

Gregory Moore (2019 page 74)

• Show that quaternion conjugation is an anti-automorphism, that is: {$\bar{q1 q2} = \bar{q2}\bar{q1}$}

Quantum symmetries

Varlamov, 1999:

• The group {$O(p,q)$} consists of four connected components: identity connected component {$O_0 (p, q)$}, and three components corresponding to parity reversal {$P$}, time reversal {$T$}, and the combination of these two {$P T$} , i.e., {$O(p, q) = O_0 (p, q) ∪ P (O_0 (p, q)) ∪ T (O_0 (p, q)) ∪ P T (O_0(p, q))$}. Further, since the four element group (reflection group) {$\{1, P, T, P T \}$} is isomorphic to the finite group {$\mathbb{Z}_2 \otimes \mathbb{Z}_2$} (Gauss-Klein group) [Sal81a, Sal84], then {$O(p, q)$} may be represented by a semidirect product {$O(p, q) \cong O_0 (p, q) \odot (\mathbb{Z}_2 \otimes \mathbb{Z}_2)$}.
• The finite group {$\{1, P, T, P T\}$} is associated with an automorphism group {$\{\textrm{Id}, \star, \widetilde \; ,\widetilde{\star}\}$} of the Clifford algebras {$C\ell_{p,q}$} and {$C_{p+q}$}.

Division algebra automorphisms

For a super division algebra {$A\oplus eA$} we have an automorphism {$a\rightarrow a'$} defined by {$ae=ea'$}. This is a homomorphism in that {$e(a_1a_2)'=a_1a_2e = a_1ea_2'=ea_1'a_2'$}. So it is not clear how that could relate to an antiautomorphism. The antiautomorphism can be defined on {$A$} (if expressed as a Clifford algebra) and likewise on {$eA$} thus on their direct sum. Note that the antiautomorphisms respect degrees and so respect the even and odd components of the super division algebra.

Literature

• Todd Trimble. The Super Brauer Group and Super Division Algebras
• Vadim V. Varlamov. Fundamental Automorphisms of Clifford Algebras and an Extension of Dabrowski Pin Groups