Hopf fibre and invariant channel projectors

The geometric model and its averaging operator

For the conditional Hopf model, write a point of the unit sphere as $z\in\mathbb C^3$, $z^\dagger z=1$. Its common phase defines the map

$$S^1\longrightarrow S^5\longrightarrow\mathbb{CP}^2, \qquad z\longmapsto[z].$$

Complex notation packages six real embedding coordinates; the sphere has five tangent spatial dimensions. The unit radius fixes a mathematical normalization, not a physical length. Applying this model to a particle requires its actual spatial geometry and field bundle. A quotient map also does not by itself identify the quotient with a specified nested spatial submanifold.

For an integer $q$, the fibre-weight projector on scalar functions is

$$(P_qf)(z)=\frac1{2\pi}\int_0^{2\pi} e^{-iq\theta}f(e^{i\theta}z)\,d\theta. \tag{1}$$

With the invariant sphere measure, these are orthogonal projections:

$$P_q^\dagger=P_q,\qquad P_qP_p=\delta_{qp}P_q. \tag{2}$$

Indeed a function with fibre weight $p$ obeys $f(e^{i\theta}z)=e^{ip\theta}f(z)$, so the integral in (1) is the Fourier orthogonality integral. Functions in $\operatorname{Ran}P_0$ descend to ordinary functions on $\mathbb{CP}^2$. They need not be constant there.

A complete quadratic example

For any complex $3\times3$ matrix $A$, define

$$f_A(z)=z^\dagger A z.$$

Every such function has zero fibre weight. Its constant part is $\operatorname{tr}A/3$. Its remaining traceless part is an eight-dimensional complex space of functions transforming under the adjoint action:

$$f_A(Uz)=f_{U^\dagger A U}(z),\qquad U\in SU(3). \tag{3}$$

For example $A=\operatorname{diag}(1,-1,0)$ gives $f_A=|z_1|^2-|z_2|^2$. It survives $P_0$ exactly, yet an $SU(3)$ rotation interchanging the first two axes changes its sign. Thus the quadratic zero-fibre-weight space contains both a constant singlet and a nontrivial eight-dimensional representation.

Full group averaging is a different operator. With normalized Haar measure,

$$\int_{SU(3)}U^\dagger A U\,dU =\frac{\operatorname{tr}A}{3}I, \qquad \int_{SU(3)}f_A(Uz)\,dU=\frac{\operatorname{tr}A}{3}. \tag{4}$$

The first integral commutes with every fundamental group matrix, so it is scalar; its trace fixes the scalar. Equations (1) and (4) therefore remove different field components.

An invariant channel is an operator condition

On a specified finite-dimensional unitary representation $R$ of a compact group $G$, the operator

$$\Pi_G=\int_G R(g)\,dg \tag{5}$$

is the orthogonal projector onto the invariant vectors. Haar invariance gives $R(h)\Pi_G=\Pi_G$, idempotence and self-adjointness. For Hermitian generators $T_a$ of connected $G$, the equivalent test is

$$\ker\left(\sum_aT_a^2\right)=\bigcap_a\ker T_a. \tag{6}$$

The colour singlet criterion derives (6) and distinguishes it from merely vanishing mean colour. Fibre neutrality and the full invariant condition can both be required, but one does not replace the other.

For a physical coupling calculation the action must determine the retained projector and the interaction operator. One then evaluates its action on the retained fields, including any complementary modes required by the dynamics. Declaring an average is not yet a mechanism that performs that average during an encounter. The neutrino channel control specifies the simultaneous electric, colour and surviving weak-response targets.

Verification

geometric_channel_checks.py checks (1) on explicit positive, negative and zero fibre weights, constructs the quadratic functions, and verifies (4) independently with a finite unitary averaging set exact on matrices. It retains a nonzero traceless function after fibre averaging and removes it under full matrix averaging. These are projector identities, not a neutrino interaction-rate calculation.