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Clifford algebras

Sources

• Michael Stone, Ching-Kai Chiu, Abhishek Roy. Symmetries, Dimensions, and Topological Insulators: the mechanism behind the face of the Bott clock.
• John Baez. The Tenfold Way. Video
• Gregory W. Moore. Quantum Symmetries and Compatible Hamiltonians.
  • Greg Moore. Quantum Symmetries and the 10-Fold Ways (1 of 2), (2 of 2), Greg Moore

CPT Symmetry

Consider a function {$f:V\rightarrow W$} between two complex vector spaces. Suppose it is additive: {$f(x+y)=f(x)+f(y)$} for all {$x,y\in V$}. Then we say it is a

• linear map, if {$f(sx)=sf(x)$}
• antilinear map, if {$f(sx)=\bar{s}f(x)$}

for all {$s\in\mathbb{C}, x\in V$}.

Wigner's theorem says that if there is a symmetry transformation of a ray space {$P(H)$}, then it is compatible with a unitary or antiunitary transformation on {$H$}.

The Altland-Zirnbauer classes are characterized by the presence or absence of the following three symmetries. Given a Hamiltonian {$H$}:

• Parity {$\mathcal{P}$} is a linear map that anticommutes with {$H$}, so that {$\mathcal{P}H=-H\mathcal{P}$}.
• Charge conjugation {$\mathcal{C}$} is an antilinear map that anticommutes with {$H$}, so that {$\mathcal{C}H=-H\mathcal{C}$}, and furthermore {$\mathcal{C}^2=\pm\mathbb{I}$}.
• Time reversal {$\mathcal{T}$} is an antilinear map that commutes with {$H$}, so that {$\mathcal{T}H=-H\mathcal{T}$}, and furthermore {$\mathcal{T}^2=\pm\mathbb{I}$}.

When an operator squares to {$+1$}, then it can be used to define a real structure. When it square to {$-1$}, it can be used to define a quaternionic structure.

• In the case of {$SU(2)$}, half-integer spin {$T$} defines quaternionic structure (for fermions), whereas integer spin defines real structure (bosons).

Given the parity, the unitary operator {$S=CT$}, we have two cases, {$S^2=1$} and {$S^2=-1$}. When {$S^2=1$}, we can divide up our Hilbert space into the direct sum of two subspaces, the particles (vectors for which {$Sv=v$}) and the holes (vectors for which {$Sv=-v$}).

• Then {$C$} maps particles to holes, and holes to particles. {$C$} is odd.
• Whereas {$T$} maps particles to particles, and holes to holes. {$T$} is even.

The metaphysics of parity, charge conjugation and time reversal

• Moore: Physics takes place in space and time, and time evolution is described, in quantum mechanics, by unitary evolution of states. There should be a family of unitary operators {$U(t_1, t_2)$}, strongly continuous in both variables and satisfying composition laws {$U(t_1,t_3) = U(t_1,t_2)U(t_2,t_3)$} so that {$ρ(t_1) = U(t_1,t_2)ρ(t_2)U(t_2,t_1)$}
• Moore: We can assume our spacetime is time-orientable. Any physical symmetry group {$G$} must be equipped with a homomorphism {$τ:G\rightarrow\mathbb{Z}_2$} telling us whether the symmetry operations preserve or reverse the orientation of time. That is {$τ(g) = +1$} are symmetries which preserve the orientation of time while {$τ(g) = −1$} are symmetries which reverse it.

Relation with linear complex structures

Compare

• {$J_1J_4J_5$} splits {$V_+$} into subspaces with {$W_-=J_2W_+$}.
• {$J_2J_4J_6$} splits {$W_+$} into subspaces with {$X_-=J_1X_+$}.
• {$J_1J_6J_7$} splits {$X_+$} into subspaces where subsequently {$Y_-=J_7J_8Y_+$}.

Connectivity and homotopy

Symmetry of the space depends on {$G_i$} being transitive. The orbit is connected. Homotopy describes the other parts.

Interpretation

A perspective is a mutually anti-commuting orthogonal complex structure, which can be thought of as a generator (squaring to -1) in a Clifford algebra. A division of everything is the collection, the assembly of such orthogonal complex structures that reconstructs the overall space from the remaining space after the complex structures have been imposed. The shifts in perspective are given by the order in which the perspectives are determined. Compare that also with the external view in terms of representations of the Clifford algebra.

We consider the matrices {$M$} that commute with a {$J_k$}. They are the change of bases that do not affect {$J_k$}, for {$MJ_kM^{-1}=J_k$}, thus they are symmetries with respect to {$J_k$}. We get an ever more restricted group of symmetries, which get distinguished from nonsymmetries. As regards perspectives, we distinguish what is invariant from what is not.

Try to describe this as an adjunction, for the category of symmetries, where one functor forgets a symmetry by breaking it, making it more specific, and the adjoint functor recovers it by constructing it, yielding the perspectives structured as a division of everything.

In the case of the three-cycle, starting from {$Sp(2)\times Sp(2)$}, I imagine that there are three ways to start, and any one of them will work, and they yield the same string, but in principle, that makes for a three-cycle, and observing this from the outside, we get {$\mathbb{Z}$}, an infinite cycle. And the threefold choice, much like choosing between three minds, is the basis for our free will.

Anti-commuting makes the perspectives distinct. Squaring to negative one makes them fermionic, in that there can't be multiple copies of the same perspective.

A perspective seems related to the imposition of a complex structure, and also, the imposition of a {$2\times 2$} matrix upon the existing structure. Thus a perspective relates the freedom in what is hidden (the ambiguity of {$i$} and {$\bar{i}$} in the imaginary dimension) and the definitess in what is manifest (the real dimension).

A perspective separates the self and the world, dividing the Hilbert space into two subspaces, accordingly. Thus we have a chain of perspectives upon each other. And each perspective allows us to step in or step out, until we arrive at the seventh perspective, a logical system, where we are stepped out, and if we step in, then we get a contradiction, and the system collapses.

The separation of the self and the world is very much in the spirit of John's projection of quantum physics from classical physics.

A perspective is a subspace that results from halving a space. This yields a sequence of perspectives, with the most recent perspective halved again. It becomes a perspective upon a perspective. Thus a perspective is related to the creation of a {$2\times 2$} matrix, the division of a Hilbert space into {$2^k$} subspaces.

It seems that we keep adding more structure, an additional perspective, an additional symmetry.

Consider how each generator contributes to carving up a matrix into submatrices. How are those subspaces related? What is their effect on the representations, which Morita equivalence cares about?