Closed-loop conjugacy data and endpoint-contracted comparisons are gauge invariant. Coherent averaging selects invariant amplitudes, while averaging density matrices can leave a charged carrier with an invariant mixed state.
Describe what an experiment can compare
Gauge-dependent components can help calculate a process, but its recorded outcome must have the same meaning in every allowed local frame. This motivates invariant observables: quantities built from carriers, links, and reference states so that their frame changes cancel. Invariance is a consistency requirement on the description.
SMM/DTT adds an operational question: does the actual completion instrument retain that quantity? A formal invariant may refer to a sector that the apparatus cannot prepare or read. A source-generated observable therefore needs both an invariant definition and a concrete route from the inner operations to the completed record.
An open path can enter an invariant comparison
An open transport matrix Uwv transforms at its two endpoints, so its entries alone are not invariant. With endpoint vectors transforming as ψ′v=hvψv, the contracted scalar ψw†Uwvψv is invariant. The endpoint states act as relational references.
A measured real signal might use its real part or an interference probability involving it. The instrument decides which part is accessible. Keeping endpoint labels without their coherent phase relation can lose this information even if every label remains perfectly readable.
Loop observables retain conjugacy information
For a closed loop based at v, the transport W transforms by hvWhv−1. Its character in a chosen representation, such as tr W, is invariant. For an SU(2) matrix with eigenvalues exp(±iθ), tr W=2cosθ. Reversing the path replaces W by W† and conjugates the trace.
Different observables retain different distinctions. The real trace does not distinguish an Abelian phase Φ from −Φ. A trace in one representation also need not separate every conjugacy class of a larger group. A reconstruction should specify a family of invariant tests and calculate the quotient they define, rather than assume that one convenient scalar captures all field information.
An invariant density matrix need not be neutral
Let R(g) be a finite-dimensional irreducible unitary representation of a compact group. Averaging a density matrix over frame changes gives T(ρ)=∫R(g)ρR(g)†dg, with normalized invariant measure. The result commutes with every R(g). Irreducibility and its unit trace then give T(ρ)=I/m, where m is the carrier dimension.
The mixed state is invariant, but it still lives in the same representation space. It has not become an invariant amplitude or acquired zero charge merely because its density matrix commutes with the group. For U(1), even a pure state in a one-dimensional nonzero-charge sector has an invariant density matrix, since the phase cancels between the vector and its adjoint.
Coherent amplitude averaging is a different operation
Coherent averaging Pinv=∫R(g)dg acts on vectors. It is the orthogonal projector onto invariant amplitudes. A nontrivial irreducible representation has no such vector, so that projector vanishes there. To construct a neutral amplitude, supply a carrier V and a co-located dual V*.
(R(g)⊗R(g)*)s=s.
For irreducible V, the invariant subspace of V⊗V* is one-dimensional, so coherent averaging on the pair yields |s⟩⟨s|. This conclusion uses the supplied dual and the coherent comparison rule. Density-matrix twirling and sharp singlet acceptance are physically different operations.
A one-shot neutral-return test
Prepare the normalized singlet s, transport only its mobile factor around a loop with holonomy W, and test the return projector |s⟩⟨s|. The neutral return amplitude is tr W/m, giving:
For SU(2), this is cos²θ. In particular W=I and W=−I both give probability one. Their singlet amplitudes differ by a sign, but this yes-or-no probability discards that phase. A coherent reference can recover additional information. Thus neutral acceptance is not the same observable as Re tr W, and it should not be substituted for a loop-energy response without analyzing the actual instrument.
Make invariance part of the completion design
A source model can now state a precise completion task: create a carrier and its stationary dual, preserve the required coherence during transport, apply an invariant acceptance instrument, and retain both success and failure. The six nested inner roles could organize these prerequisites, with outer completion publishing the resulting record.
The next useful result would derive the instrument from the same source dynamics that generates the transport. It should also identify which invariant data are lost and which remain sufficient for subsequent predictions. This connects gauge freedom with observation rather than treating symmetry of a Hamiltonian as proof that a desired neutral state has been prepared.
Continue with the articles
- Relational transport and gauge freedom
Explain why independently unframed record fibers require a rule for comparison. Introduce gauge transport as a conditional reconstruction idea, distinct from the draft’s exact Hamiltonian results.
- An exact singlet-loop response
Introduce the four-position mobile carrier and stationary dual used in the solvable model. Explain how the holonomy-dependent resolvent determines the ground-state response under the supplied Hamiltonian.
- Completion without biasing an unknown input
Explain the construction of a loaded completion channel with input-independent success probability. Clarify why this condition matters when the matter state is unknown.
- Invisible holonomy in inaccessible sectors
Explain the draft’s classification of residual parallel monodromy invisible to the prepared interface. Nontrivial structure can survive outside the behavior an experiment can access.