Odd-Sector Epsilon Channel

Purpose

The branch/Casimir superselection note repackaged the candidate physical subspace as: \(P_{\mathrm{phys}} = P_{\mathrm{obs}}P_{\mathrm{aux},0} \;+\; P_{j_{\min}}P_{\mathrm{aux},1}.\)

That was already cleaner than the earlier slot language. But the odd-sector projector can be sharpened one step further.

On \(\mathbf 2 \otimes \mathbf 2,\) the weak singlet is not merely the minimal-Casimir channel. It is exactly the unique SU(2)-invariant antisymmetric epsilon channel.

This note records that refinement.

Standard SU(2) decomposition

For the defining weak doublet, \(\mathbf 2 \otimes \mathbf 2 = \mathrm{Sym}^2(\mathbf 2) \oplus \wedge^2(\mathbf 2) = \mathbf 3 \oplus \mathbf 1.\)

So:

  • the triplet is the symmetric part;
  • the singlet is the antisymmetric part.

Equivalently, the singlet is the image of the unique SU(2)-invariant alternating tensor \(\epsilon_{ab}.\)

This means the odd weak-singlet channel can be characterized in three equivalent ways:

  1. as the j_{\mathrm{tot}}=0 channel;
  2. as the antisymmetric channel;
  3. as the \epsilon-contraction channel.

Projector formulas

Let \tau denote the swap operator exchanging the two weak-doublet factors in \(\mathbf 2 \otimes \mathbf 2.\)

Then:

  • \tau = +1 on the symmetric triplet,
  • \tau = -1 on the antisymmetric singlet.

So the singlet projector is exactly \(P_{\epsilon} = \frac12(\mathbf 1 - \tau).\)

Now compare this with the total weak Casimir.

If C_{\mathrm{tot}} is normalized so that:

  • C_{\mathrm{tot}} = 0 on the singlet,
  • C_{\mathrm{tot}} = 2 on the triplet,

then on these two eigenspaces one has \(C_{\mathrm{tot}} = \mathbf 1 + \tau.\)

Therefore \(P_{\epsilon} = \frac12(\mathbf 1 - \tau) = \mathbf 1 - \frac12 C_{\mathrm{tot}}.\)

So the odd-sector projector used earlier, \(P_{\mathrm{odd},0} = \mathbf 1 - \frac12 C_{\mathrm{tot}},\) is exactly the antisymmetric \epsilon-channel projector.

Why this is better than the Casimir-only reading

The Casimir reading said:

keep the minimal total weak-spin channel.

That was already structural, but it still sounded somewhat spectral and abstract.

The \epsilon-channel reading is tighter:

keep the unique SU(2)-invariant antisymmetric scalar pairing between the two doublets.

This matters because it makes the odd-sector choice look less like a minimization convention and more like the standard invariant scalar contraction.

In the present carrier language:

  • the even auxiliary line is governed by observable-branch selection;
  • the odd auxiliary line is governed by invariant antisymmetric scalar pairing.

That is a cleaner split of roles.

Relation to the repo’s antisymmetric-structure theme

The broader repo already treats antisymmetric 2-plane structure as an important hidden motif in other branches of the programme.

This note does not claim that the odd auxiliary selector is already derived from that larger hidden antisymmetric story.

It does show something narrower:

the odd-sector choice in the static carrier is already the unique antisymmetric scalar channel available in ordinary SU(2) representation theory.

So if a later parent derivation privileges antisymmetric pairings or oriented 2-plane structure, it would align naturally with the present odd-sector selector rather than having to replace it.

Rewriting the superselection rule

With this refinement, the candidate physical-subspace projector becomes \(P_{\mathrm{phys}} = P_{\mathrm{obs}}P_{\mathrm{aux},0} \;+\; P_{\epsilon}P_{\mathrm{aux},1},\) where:

  • P_{\mathrm{obs}} is the observable-branch projector on the auxiliary even line,
  • P_{\epsilon} is the antisymmetric SU(2)-invariant scalar-pairing projector on the auxiliary odd line.

This is the cleanest statement so far.

What this changes conceptually

After this refinement, the candidate physical-state rule is no longer best described as:

  • branch selection plus minimal Casimir selection.

It is better described as:

  • observable-branch selection on the even line,
  • invariant antisymmetric scalar-pairing selection on the odd line.

That is a more concrete and more standard representation-theoretic formulation.

What remains open

This note still does not finish the derivation.

The remaining burden is now very localized:

  1. why the even auxiliary line should obey the observable-branch selector P_{\mathrm{obs}};
  2. why the physical odd sector should be built through the invariant \epsilon pairing rather than through the symmetric triplet channel;
  3. whether those two rules arise from one common parent statement.

But the odd part is now much less mysterious than before.

Best current formulation

The strongest safe statement now available is:

the best current candidate for the physical static subspace is selected by observable-branch projection on the auxiliary even line and by the unique antisymmetric SU(2)-invariant scalar-pairing channel on the auxiliary odd line.

That is more precise than the earlier branch/Casimir wording.

What is now established

The following point is now finite:

the odd-sector projector in the current physical-subcarrier candidate is not only the minimal total weak-spin projector; it is exactly the antisymmetric \epsilon-channel projector \(P_{\epsilon} = \frac12(\mathbf 1 - \tau) = \mathbf 1 - \frac12 C_{\mathrm{tot}}\) on \mathbf 2 \otimes \mathbf 2.

This is a genuine structural tightening of the odd-sector selection rule.

What remains open

The next question is:

can the full physical-subspace projector be derived as “observable branch plus invariant antisymmetric pairing” from the parent reduction, or must that rule be taken as part of the reduced state definition?