Gauge observables, coherent comparisons, and Gauss sewing

An invariant formula still needs an experiment

A local coordinate change acts simultaneously on every mathematical object that represents the same experimental arrangement. A quantity is gauge invariant when its value is unchanged by this re-expression. That is a consistency requirement. To call it an observable of a proposed source model, one must also identify its preparation, physical coupling, and measurement procedure.

This distinction matters in the completion program because records are defined relative to future tests. One family may contain only local adjoint observables; another may contain coherent paths with calibrated phase; a third may include charged boundary references. These families can distinguish different amounts of transport information. There is no universal record quotient independent of the experiments it must predict. §2 and §5.2, pp.3–5 and 10–11.

The full defining carrier and the algebra of observables acting on it also play different roles. A scalar phase acts trivially by conjugation on M_m, but can affect an interference experiment comparing two histories. Removing that phase from a carrier transport model is justified only if every admitted experiment is insensitive to it. A declaration of “gauge freedom” cannot replace that check.

Open transport is measured with boundary data

An open link U_yx changes under two endpoint frames, so its matrix alone is not invariant. Supply departure and arrival amplitudes and form c=ψ_y†U_yxψ_x. With ψ′_v=g_vψ_v and U′_yx=g_yU_yxg_x⁻¹, both endpoint changes cancel. This is a relational measurement involving the link and the two boundary states.

For a mixed input ρ at x, another invariant comparison uses Tr[O_yU_yxρU_yx†], with O_y the arrival observable. This protocol is insensitive to a scalar phase multiplying U because that phase cancels between U and U†. A coherent controlled alternative need not have this insensitivity: the phase can occur relative to a branch in which the transport was not applied.

The relevant boundary data are therefore part of the observable. Fixing an external detector frame is not the same experiment as transforming the detector with a passive coordinate change. Likewise, a prepared cross-record reference must remain in the completed data if the channel fixes the full local carrier algebras. Remark 4.5, Proposition 4.7, and Theorem 5.2, pp.7–10.

Closed loops produce class functions and correlated tests

A closed product W based at x transforms by conjugation. TrW, detW, and TrW^k are therefore invariant; its eigenvalues can be regarded as a basis-independent list. For a unitary matrix, the complete spectrum determines its conjugacy class. A single real trace generally supplies less information. A family of loop measurements should be chosen to match the desired predictive precision.

On a graph with several cycles, the cycles are simultaneously conjugated at a common root. Separate loop spectra can miss the relative orientation of their matrices. Products such as Tr(W₁W₂), reversed products involving W†, and commutator loops can reveal that orientation. The two-loop example shows configurations with identical separate traces and different product traces.

The positive real trace deficit has a direct metric identity:

This is useful for a comparison cost because it vanishes exactly when the loop map is identity. Its use as an energy requires a coefficient and a microscopic energy interpretation. Gauge invariance alone also permits other class functions and combinations of traces. §17.3–17.5, pp.53–55.

A coherent loop probe resolves relative phase

Prepare a two-path control in |+〉=(|0〉+|1〉)/√2 and a carrier in ρ. Apply identity on one control path and W on the other. The control coherence contains Tr(ρW)/2. Measuring the control in the plus/minus basis gives

A controlled reference phase φ replaces ReTr(ρW) by Re[e^iφTr(ρW)], allowing both quadratures of the complex signal to be sampled. If the input is invariant under every defining-block frame, it must be ρ=I_m/m. Then p₋=(m−ReTrW)/(2m). The uniqueness of the invariant mixed input follows from irreducibility of the defining SU(m) action and unit trace. Proposition 17.3, p.53.

For a central holonomy W=e^iθI, p₋=sin²(θ/2). The carrier by itself returns to the same ray, but its phase relative to the reference path changes the control statistics. The control preparation and phase calibration are physical reference resources. They are not created by renaming the carrier’s frame.

Forced singlet return is a different measurement

Prepare the normalized carrier–dual singlet |s〉=m^−1/2Σ_a|a〉⊗|ā〉. Transport only its carrier factor once around a prescribed loop and project onto the original singlet. Its return amplitude is 〈s|(W⊗I)|s〉=TrW/m, so its probability is

This is one for every central W=e^iθI. A prescribed circuit and final singlet test therefore lose the phase that the controlled-path experiment resolves. The two results are consistent because their reference arrangements differ. Retaining a faithful carrier does not imply that every chosen detector extracts all of its possible transport information. §5.2, pp.10–11.

The four-position singlet Hamiltonian supplies another protocol: the carrier evolves coherently through waiting, backtracking, and circulation alternatives before the final completion measurement. Its survival probability can distinguish central holonomy at fourth order in elapsed time. Frequent intermediate projections or a forced single circuit change that dynamics and can introduce a measurement-induced limit. The exact static response cannot be transferred to such a changed protocol without recalculation. Theorem 20.4 and equation (80), pp.61–62.

Twirling removes a frame label while retaining charged sectors

For a compact group representation R(g), density-matrix twirling is T(ρ)=∫R(g)ρR(g)†dg with normalized Haar measure. It makes the output invariant under conjugation. It does not require every vector in the output’s support to be invariant.

A nontrivial irreducible representation of dimension d gives the simplest counterexample: I_d/d is fixed by twirling, although the representation contains no singlet vector. More generally, decompose the Hilbert space as ⊕_λ(V_λ⊗M_λ). Twirling removes coherences between inequivalent irreducible sectors and replaces the representation part in each block by I_Vλ/dimV_λ, while preserving the block’s multiplicity-state data. It does not delete all λ≠trivial blocks. Proposition 7.3, p.16.

Even vanishing mean generators would be insufficient. In the normalized spin-½ SU(2) mixed state, the three generator expectations vanish, but the quadratic Casimir is nonzero and no invariant vector exists. A neutral Gauss sector is a restriction on the representation content of physical amplitudes, not merely a collection of zero mean charges or a lack of knowledge of the frame.

Coherent invariant sewing selects the Gauss sector

The source therefore states a separate internal composition rule. At an internal record v, let R_v(g) act on all incident link and matter indices. Sew the boundary amplitude coherently by the Haar average

Haar invariance gives P_v²=P_v, P_v†=P_v, and R_v(h)P_v=P_v. Hence P_v is the orthogonal projector onto invariant amplitudes. For a connected group, invariance is equivalent to G_v^aψ=0 for every Lie-algebra generator on the common domain. Link endpoint actions are left and right translations and commute; with the stated tensor-product matter structure, vertex projectors can therefore impose all internal constraints together. Axiom 7.1 and Theorem 7.2, pp.15–16.

The neutral initial internal sector is part of the rule. External boundaries may instead carry specified charges. A Hamiltonian commuting with every R_v(g) preserves whichever sector is selected, but its symmetry alone does not select that sector. For S(U(3)×U(2)), there are 8+3+1=12 Lie-algebra generators per internal vertex; the linked block centers do not supply a thirteenth independent Abelian Gauss constraint.

From observable comparison to an interaction claim

The controlled-loop probability, a singlet survival probability, a chosen rejection energy, and an exact ground-energy response are different quantities. The first two are dimensionless and depend on a measurement protocol. A rejection energy introduces a calibrated penalty. A ground-energy response follows from a specified Hamiltonian and has energy units. They can share the same small-angle trace structure without being interchangeable.

For example, an independent-block quadratic comparison cost has central coefficient κ₁=12κ₃+18κ₂ in the primitive charges (−2,3). Gauge invariance also allows an additional central quadratic invariant, so the relation characterizes a restricted model class. The exact singlet Hamiltonian preserves the relation at the static Hessian level through all hopping orders, under its stated assumptions. Physical gauge couplings require additional normalization, electric dynamics, and matching. Corollary 2.4, p.11; §§2.4–2.5, pp.17–18.

A stronger completion theory must derive its test family, reference preparation, coherent internal sewing, and dynamics from the same independently motivated rule. The present constructions make the consequences of each choice calculable. They do not establish the Standard Model or a quantum continuum solely from a statement of local frame freedom.

Continue reading

• Closed-loop information
• Choose the right record quotient
• What completion preserves
• A coherent singlet Hamiltonian
• Wilson-loop tomography: which measurements determine the transport?
• The full SU(3) holonomy domain: alcoves, traces and exact response
• Unitary return protocols: coherent histories, exact amplitudes and measurement timing