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克利福德代数

读物

• 维基百科: Spinor
• 维基百科: Spin group
• 维基百科: Spin representation Defines spinors in terms of spin group representations and Clifford algebras.
• Higher-dimensional gamma matrices
• Crystal-Ann McKenzie. An Interpretation of Relativistic Spin Entanglement Using Geometric Algebra explains spinors as minimal left ideals, considers entanglement of frames
• David Hestenes. Spinor Particle Mechanics
• Jose Figueroa-O'Farrill. Majorana Spinors. Understands real and quaternionic representations as complex representations with extra structure. The extra structure will manifest itself in the existence of nondegenerate invariant complex-bilinear forms. Manifests Bott periodicity. Used by Chris.
• Fulton and Harris – Representation Theory: A First Course
• Jean Gallier. Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups. 2012. Includes a proof of Bott periodicity.
• The lecture notes on Clifford algebras by Lundholm and Svensson.
• Varadarajan - Supersymmetry for mathematicians
• Max Horn, Ralf Köhl. Generalized spin representations. Part 2: Cartan-Bott periodicity for the split real En series
• Cameron Krulewski. K-Theory, Bott periodicity, and Elliptic Operators relates real spinor representations and Bott periodicity, summarizes Atiyah's paper
• Bott Periodicity and the Index of Elliptic Operators
• Theo Johnson-Freyd. Bott periodicity via quantum Hamiltonian reduction
• What is Spin? A Geometric explanation Discusses negative probability as physically the same as positive probability but mathematically distinct.
• Richard Delanghe. Clifford Analysis: History and Perspective.

Breakdown

Given real or complex vector space {$V$} and quadratic form {$Q$}.

• Understand the spin group {$\textrm{Spin}(V,Q)$}, which is the double cover of {$SO(V,Q)$}.
• Understand the Lie algebra {$\frak{so}$}{$(V,Q)$} which they share.
• For real {$V$}, understand how {$\frak{so}$}{$(V_C,Q_C)$} is the complexification of {$\frak{so}$}{$(V,Q)$}.

A complex spin representation of {$\textrm{Spin}(n,\mathbb{C})$}

• This is a complex vector space {$S$} and a group homomorphism {$\rho:\textrm{Spin}(n,\mathbb{C})\rightarrow GL(S)$} such that {$\rho(-1)\neq 1$}.

Break down {$V=\mathbb{C}^n$} into maximal totally isotropic subspaces {$W$} and {$W^*$}

• The inner product is the symmetric bilinear form on {$V$} associated to Q by polarization.
• In a complex vector space, if {$Q(e_j)=1$} and {$Q(e_k)=1$}, we have that {$Q(e_j-ie_k)=0$} and {$Q(e_j+ie_k)=0$}. The latter elements are isotropic and if we multiply them by scalars we can get one-dimensional totally isotropic subspaces. If we are careful to pair generators {$e_j$} and {$e_k$} and not mix them, then we can get larger isotropic subspaces. Indeed, we can get a pair of subspaces of equal dimension.
• Then {$n=2m$} or {$n=2m+1$} but either way there are maximal totally isotropic subspaces {$W$} and {$W^*$}, each of dimension {$m$}, which are also dual vector spaces. If {$n=2m$}, then {$V=W\oplus W^*$}. If {$n=2m+1$}, then {$U$} is the one-dimensional subspace orthogonal to both {$W$} and {$W^*$}, and {$V=W\oplus U\oplus W^*$}.

Build {$\frak{so}$}{$(n,C)$} out of matrices acting on` {$W$} and {$W^*$}

• Any {$m\times m$} matrix {$A$} induces an endomorphism of {$W$} and likewise {$A^T$} induces an endomorphism of {$W^*$}. We have {$<Aw,w^*>=<w,A^Tw^*>$}.
• We can define endomorphism {$\rho_A$} of {$V$} which equals {$A$} on {$W$}, {$-A^T$} on {$W^*$} and {$0$} on {$U$} (if {$n$} is odd).
• Note that {$\rho_A$} is skew. {$<\rho_A u,v>=-<u,\rho_Av>$} for all {$u,v\in V$}. Thus {$\rho_A\in so(n,C)\subset \textrm{End}(V)$}.
• The diagonal matrices {$\rho_D$} define a Cartan subalgebra {$\mathbf{h}$} of {$\frak{so}$}{$(n,C)$}.
• We can write out {$\frak{so}$}{$(n,C)$} explicitly, including its root system.

The representation space {$S$} is the exterior algebra of {$W$}

• Consider the exterior algebra {$S=\wedge\bullet W$}. This is the space of Dirac spinors.
• Similarly, consider {$S'=\wedge\bullet W^*$}.

{$V$} acts on {$S$} by the geometric product

• Consider the action of {$V$} on {$S$}, where for {$v=w+w^*\in W\oplus W^*, \psi\in S$}, we have {$v\dot \psi=2^{\frac{1}{2}}(w\wedge\psi+\iota(w^*)\psi)$} where the second terms is interior multiplication.
• Understand geometric algebra and the geometric product.
• This action respects the Clifford relations {$v^2 = Q(v)1$}.
• This induces a homomorphism from {$Cl_n C$} of {$V$} to {$\textrm{End}(S)$}. This defines {$S$} as a Clifford module.
• Similarly, {$S'$} is a Clifford module. {$S$} and {$S'$} are equivalent representations.

A real spin representation of {$\textrm{Spin}(p,q)$}

• We have a real vector space {$S$} and a real spin representation {$\rho:\textrm{Spin}(p,q)\rightarrow GL(S)$}, such that {$\rho(-1)\neq 1$}.
• Consider its complexification {$S_C$}.
• {$S$} is a real representation of {$\frak{so}$}{$(p,q)$} and it therefore extends to a complex representation of {$\frak{so}$}{$(n,\mathbb{C}$}.
• Thus start with complex spin representations of {$\textrm{Spin}(n,\mathbb{C})$} and {$\frak{so}$}{$(n,\mathbb{C}$}.
• Restrict these to complex spin representations of {$\textrm{Spin}(p,q)$} and {$\frak{so}$}{$(p,q)$}.
• Reduce these to real spin representations.

Ideas

• Symmetry breaking distinguishes vectors and spinors.
• Think of Clifford algebra generators that square to {$-1$} as spinors, and those that square to {$+1$} as vectors. Each generator can be thought of as yielding a turn of {$\pi=180^\circ$}. Does that make sense?
• Cameron Krulewski. K-Theory, Bott periodicity, and elliptic operators Explains Attiyah's proof which uses the index of elliptic operators. The real case is the spinor case and mentions Dirac operators.

Spinor

• Low dimension spinor representations have 8fold periodicity https://en.m.wikipedia.org/wiki/Spinor

Harvey, F. Reese (1990), "Chapter 2: The Eight Types of Inner Product Spaces", Spinors and calibrations, Academic Press, pp. 19–40, ISBN 0-12-329650-1

MacMahon Master Theorem

• The coefficient of {$x_1^{k_1}\cdots x_m^{k_m}$} in {$\frac{1}{\det (I_m - TA)}$} equals its coefficient in {$\prod_{i=1}^m \bigl(a_{i1}x_1 + \dots + a_{im}x_m \bigl)^{k_i}$}
• My thesis has combinatorial interpretations for the generating function {$\frac{1}{\det (I - A)} = \sum_{n=0}^{\infty}x^nh_n(\xi_1,...,\xi_n)$}
• Julian Schwinger: It is the MacMahon Master Theorem that unifies the angular momentum properties of composite systems in the binary build-up of such systems from more elementary constituents. (James D Louck. Unitary symmetry and combinatorics. 2008.)
• AI: The theorem is referenced in the study of "spinor spherical harmonics" or "Pauli central field spinors," which are mathematical tools used in quantum mechanics to describe particles with spin (like electrons) in a central potential.
• AI: In its standard mathematical form, the MMT relates the permanent of a matrix to the coefficient of a formal power series involving a determinant. In physics, determinants of operators (like Laplacians) on tensors and spinors are used in certain quantum field theories and string theories. The MMT provides a powerful combinatorial identity that can be used to prove other related identities in these fields.
• In the Standard Model, fermions are not their own antiparticles, but in some theories they can be. Among other things, this involves the question of whether the relevant spinor representations of the groups Spin(p,q) are complex, real (‘Majorana spinors’) or quaternionic (‘pseudo-Majorana spinors’). The options are well-understood, and follow a nice pattern depending on the dimension and signature of spacetime modulo 8.

Twistors

https://en.wikipedia.org/wiki/Twistor_theory

Peter Woit. Topics in Representation Theory

• Cultural Background
• Lie Groups, Lie Algebras and the Exponential Map
• The Adjoint Representation
• More About the Exponential Map
• Maximal Tori and the Weyl Group
• Roots and Weights
• Roots and Complex Structures
• SU(n), Weyl Chambers and the Diagram of a Group
• The Killing Form, Reflections and Classification of Root Systems
• SU(2) Representations and Their Applications
• Fundamental Representations and Highest Weight Theory
• The Weyl Integral and Character Formulas
• Homogeneous Vector Bundles and Induced Representations
• Decomposition of the Induced Representation
• Borel Subgroups and Flag Manifolds
• The Borel-Weil Theorem
• Clifford Algebras
• Spin Groups
• The Spinor Representation
• The Heisenberg Algebra
• The Metaplectic Representation
• Hamiltonian Mechanics and Symplectic Geometry
• The Moment Map and the Orbit Method
• Other Topics