From 396c574b2e03aa5094cef78844384a1855f0e5a2 Mon Sep 17 00:00:00 2001 From: Mike Boyle Date: Tue, 10 Dec 2024 13:30:35 -0500 Subject: [PATCH] Sketch an outline --- docs/src/conventions.md | 116 ++++++++++++++++++++++++++++++++++++++++ 1 file changed, 116 insertions(+) diff --git a/docs/src/conventions.md b/docs/src/conventions.md index 9284085..22d9aaf 100644 --- a/docs/src/conventions.md +++ b/docs/src/conventions.md @@ -5,6 +5,104 @@ equation for rotating spherical harmonics. --- +# Outline + +* Three-dimensional Euclidean space + - Cartesian coordinates ``(x, y, z)`` => ℝ³ + - Cartesian basis vectors ``(𝐱, 𝐲, 𝐳,)`` + - Euclidean norm => Euclidean metric + - Spherical coordinates + - Specifically give transformation to/from ``(x, y, z)`` + - Derive metric in these coordinates from transformation + - Integration / measure on two-sphere + - Derive as restriction of full metric, in both coordinate systems +* Four-dimensional Euclidean space + - Eight-dimensional Clifford algebra over the tangent *vector space* ``Tℝ³`` + - Four-dimensional even sub-algebra => ℝ⁴ + - Coordinates ``(W, X, Y, Z)`` + - Basis vectors ``(𝟏, 𝐢, 𝐣, 𝐤)``, but we usually just omit ``𝟏`` + - Show a few essential formulas establishing the product and its conventions + - Unit quaternions are isomorphic to ``\mathbf{Spin}(3) = + \mathbf{SU}(2)``; double covers ``\mathbf{SO}(3)`` + - Be explicit about the mapping between vector in ℝ³ and quaternions + - Show how a unit quaternion can be used to rotate a vector + - Spherical coordinates (hyperspherical / Euler) + - Specifically give transformation to/from ``(W, X, Y, Z)`` + - Derive metric in these coordinates from transformation + - Express unit quaternion in Euler angles + - Integration / measure / Haar measure on three-sphere + - Derive as restriction of full metric, in both coordinate systems +* Angular momentum operators / functional analysis + - Express angular momentum operators in terms of quaternion components + - Express angular momentum operators in terms of Euler angles + - Show for both the three- and two-spheres + - Show how they act on functions on the three-sphere +* Representation theory / harmonic analysis + - Representations show up in Fourier analysis on groups + - Peter-Weyl theorem + - Generalizes Fourier analysis to compact groups + - A basis of functions on the group is given by matrix elements of + group representations + - Representation theory of ``\mathbf{Spin}(3)`` + - Show how the Lie algebra is represented by the angular-momentum operators + - Show how the Lie group is represented by the Wigner D-matrices + - Demonstrate that ``\mathfrak{D}`` is a representation + - Demonstrate its behavior under left and right rotation + - Demonstrate orthonormality + - Representation theory of ``\mathbf{SO}(3)`` + - There are several places in [Folland](@cite Folland_2016) (e.g., + above corollary 5.48) where he mentions that representations of + a quotient group are just representations that are trivial + (evidently meaning mapping everything to the identity matrix) on + the factor. I can't find anywhere that he explains this + explicitly, but it seems easy enough to show. He might do it + using characters. + - For ``\mathbf{Spin}(3)`` and ``\mathbf{SO}(3)``, the factor + group is just ``\{1, -1\}``. Presumably, every representation + acting on ``1`` will give the identity matrix, so that's + trivial. So we just need a criterion for when a representation + is trivial on ``-1``. Noting that ``\exp(\pi \vec{v}) = -1`` + for any ``\vec{v}``, I think we can show that this requires + ``m \in \mathbb{Z}``. + - Basically, the point is that the representations of + ``\mathbf{SO}(3)`` are just the integer representations of + ``\mathbf{Spin}(3)``. + - Restrict to homogeneous space (S³ -> S²) + - The circle group is a closed (normal?) subgroup of + ``\mathbf{Spin}(3)``, which we might implement as initial + multiplication about a particular axis. + - In Eq. (2.47) [Folland (2016)](@cite Folland_2016) defines a + functional taking a function on the group to a function on the + homogeneous space by integrating over the factor (the circle + group). This gives you the spherical harmonics, but *not* the + spin-weighted spherical harmonics — because the spin-weighted + spherical harmonics cannot be defined on the 2-sphere. + - Spin weight comes from Fourier analysis on the subgroup. + - Representation matrices transfer to the homogeneous space, with + sparsity patterns + + + +--- + +Spherical harmonics as functions on homogeneous space. +https://www.youtube.com/watch?v=TnFvOa9v7do gives some nice +discussion; maybe the paper has better references. + +Theorem 2.16 of [Hanson-Yakovlev](@cite HansonYakovlev_2002) says that +an orthonormal basis of a product of ``L^2`` spaces is given by the +product of the orthonormal bases of the individual spaces. +Furthermore, on page 354, they point out that ``\{(1/\sqrt{2\pi}) +e^{im\phi}\}`` is an orthonormal basis of ``L^2(0,2\pi)``, while the +set ``\{1/c_{n,m} P_n^m(\cos\theta)`` is an orthonormal basis of +``L^2(0, \pi)`` in the ``\theta`` coordinate. Therefore, the product +of these two sets is an orthonormal basis of the product space +``L^2\left((0,2\pi) \times (0, \pi)\right)``, which forms a coordinate +space for ``S^2``. I would probably modify this to point out that +``(0,2\pi)`` is really ``S^1``, and then we could extend it to point +out that you can throw on another factor of ``S^1`` to cover ``S^3``, +which happens to give us the Wigner D-matrices. + We first define the rotor that takes ``(\hat{x}, \hat{y}, \hat{z})`` onto ``(\hat{\theta}, \hat{\phi}, \hat{r})``. Then, we can invert that, so that given a rotor that specifies such a rotation exactly, we @@ -169,6 +267,24 @@ Condon-Shortley phase convention.*** ## Angular-momentum operators +* First, a couple points about ``-i\hbar``: + - The finite transformations look like ``\exp[-i \theta L_j]``, but + the factor of ``i`` introduced here just cancels the one in the + ``L_j``, and the sign is just chosen to make the result consistent + with our notion of active or passive transformations. + - Any factors of ``\hbar`` are included *purely* for the sake of + convenience. + - The factor ``i`` comes from plain functional analysis: We need a + self-adjoint operator, and ``\partial_x`` by itself is + anti-self-adjoint (as can be verified by evaluating on ``\langle + x' | x \rangle = \delta(x-x')``, which switches sign based on + which is being differentiated). We want self-adjoint operators so + that we get purely real eigenvalues. [Van Neerven](@cite + vanNeerven_2022) cites this in a more rigorous context in his + Example (10.40) (page 331), with more explanation around Eq. + (15.17) (page 592). The "self-adjoint ``\iff`` real eigenvalues" + condition is item (1) in his Corollary 9.18. + Wigner's $𝔇$ matrices are defined as matrix elements of a rotation in the basis of spherical harmonics. That rotation is defined in terms of the generators of rotation, which are expressed in terms of the