Parallel transport between completed records

A history carries a comparison, not just an endpoint

Let a history γ lead from record x to record y. A transport map U_γ:V_x→V_y says how an amplitude in the departure carrier is expressed at the arrival carrier. It may describe an internal phase or orientation comparison without specifying a material trajectory, a travel time, or a metric distance. Those additional interpretations require their own operational maps.

The basic rules are U_identity=I, U_γ⁻¹=U_γ⁻¹, and U_{γ₂∘γ₁}=U_γ₂U_γ₁. For norm-preserving complex-linear comparison, the maps are unitary. Tensor preservation means that transport carries each declared tensor at x to the corresponding tensor at y. A common source can motivate the existence of these comparisons, but it does not force all histories between the same endpoints to have the same map. §5 and Corollary 2.4, pp.5 and 9–10.

This history dependence matters for completion. If future tests coherently compare two alternatives, an endpoint description may be insufficient. The predictive-sufficiency guide explains exactly which differences a record is allowed to forget.

Compose the earliest action on the right

For a path x₀→x₁→⋯→x_n, the forward product is

The rightmost factor acts first on the input column. Since each link obeys U′_yx=g_yU_yxg_x⁻¹, every intermediate g⁻¹g cancels and U′_γ=g_xₙU_γg_x₀⁻¹. The rule remains true for any subdivision into shorter histories. Reversing a path reverses the factor order and replaces every factor by its adjoint.

For two routes γ₁,γ₂:x→y, a relative comparison based at x is W=U_γ₂†U_γ₁. It transforms as g_xWg_x⁻¹. This compares γ₁ with the return along γ₂. Its trace, spectrum, and expectation in a consistently transformed input state are independent of local presentation. The construction also fixes orientation: exchanging the two routes replaces W by W†.

Writing matrices in visual path order from left to right would give the wrong action under the column-vector convention. With noncommuting links, that error changes the comparison itself rather than merely its notation. Theorem 5.2 and Proposition 5.3, pp.10–11.

Recover the differential transport equation

Assume a smooth path x(s) in a smooth record chart, with local short links approaching identity. Define A_s=A_μ(x(s))dx^μ/ds through U(s+ds,s)=I−A_sds+O(ds²). Taking the limit of successive comparisons gives

Unitarity of the infinitesimal comparison implies A_s†=−A_s. Consequently d(ψ†ψ)/ds=−ψ†(A_s†+A_s)ψ=0. The solution is U_γ=𝒫exp(−∫_γA), where 𝒫 denotes ordering along the path. Ordinary exponentiation is adequate only when the relevant connection matrices commute or some other special simplification applies.

The standard path-to-connection correspondence also needs coherent composition, reversal, reparametrization invariance, thin-homotopy invariance, and suitable smooth local dependence. A thin deformation sweeps no area; ignoring it is compatible with a curvature two-form. These hypotheses do not declare all ordinary homotopic paths equivalent: a curved connection can distinguish contractible paths that sweep different regions. Corollary 2.4, p.5.

Work through an Abelian interval and a loop

For a one-component carrier choose A=−ia, with a a real one-form. Then U_γ=exp(i∫_γa). Under g=e^iχ, the connection law A′=gAg⁻¹−dg g⁻¹ becomes a′=a+dχ. An open-path integral gains χ(y)−χ(x), exactly the phase required by the endpoint law.

On an interval with coordinate r∈[0,L] and constant a=k dr, the transport is e^ikL. A gauge change χ(r)=−kr sets a′=0 on that interval while changing the endpoint components. The invariant transported inner product is unchanged. Thus an open transport phase alone has meaning only with its endpoint reference information.

On a circle with angular coordinate θ and A=iαdθ, forward transport around the circle is e^−2πiα. Curvature vanishes because the base is one-dimensional. A single-valued allowed gauge transformation can remove the connection globally only when its holonomy is trivial. This is a useful warning: zero local curvature does not imply trivial global transport. The same example prevents a curvature-only proof of total determinant cancellation. No-go theorem 6.6, p.13.

Noncommuting operations retain their order

Let U=e^iεσx and V=e^iεσy act on a two-component carrier, where the Pauli matrices satisfy [σ_x,σ_y]=2iσ_z. Expanding both products through second order gives UV−VU=−2iε²σ_z+O(ε³). The two first-order operations have the same separate sizes; their ordering creates the difference.

This displayed product is an algebraic commutator; its executed column-vector sequence starts with the rightmost V⁻¹. To describe a positively oriented square traversed as +x,+y,−x,−y using constant short forward links e^−Axδx and e^−Ayδy, the actual product is e^Ayδye^Axδxe^−Ayδye^−Axδx. It equals I−[A_x,A_y]δxδy+⋯. Spatially varying connections add the derivative terms in F_xy=∂_xA_y−∂_yA_x+[A_x,A_y].

Rank greater than one permits such noncommutativity but does not guarantee it. All supplied matrices might lie in one Abelian subgroup. A full unitary frame group is therefore different from a realized holonomy group or a controllability result. §13.3 and §15, pp.45–47.

Construct a comparator from a supplied ambient geometry

Suppose u_x,u_y are orthonormal frames for two rank-m subspaces of a common Hilbert space. Projection of u_xψ into the arrival carrier has arrival components M_yxψ, where M_yx=u_y†u_x. Projection may decrease norm, so this overlap is generally not unitary. If it is invertible, define its forward unitary polar factor

If M=LΣR† is a singular-value decomposition, U=LR†. For any unitary V, minimizing ‖M−V‖²_HS is equivalent to maximizing ReTr(V†M). In singular-value coordinates this is Σ_aσ_aReQ_aa≤Σ_aσ_a, with equality only at Q=I when every σ_a>0. Thus the polar factor is the unique nearest unitary. It also minimizes ‖u_x−u_yV‖_HS. Proposition 17.1, pp.51–52.

The overlap transforms as M′_yx=g_yM_yxg_x⁻¹; functional calculus gives the same law for U. Singular overlaps have a principal angle of π/2 and are outside this unique comparison regime. The common ambient geometry and nearest-unitary protocol are additional relational inputs, exactly as the no-natural-comparator theorem requires.

Connect polar comparisons to a protected subspace

Let P(x)=u(x)u(x)† be a smooth rank-m orthogonal projector. Expanding u(x+dx)†u(x) gives I−u†du+O(|dx|²), so the forward polar link has connection A=u†du. Intrinsically, the covariant derivative of a section s=Ps is ∇s=Pds, and its curvature is P(dP∧dP)P. The identity P dP P=0 does not make this derivative vanish: differentiation acts on the varying section, not merely on P. Proposition 13.3, p.44.

A protected spectral cluster can supply such a projector, provided its rank stays constant and its separation from the complement is controlled. A fixed projector with only a changing basis has zero curvature. A moving embedded projector can have nonzero curvature while its protective spectral gap remains fixed.

Geometric transport still need not equal autonomous physical evolution. If the carrier contains unequal energies, slow evolution accumulates internal dynamical phases in addition to the geometric comparison. A degenerate band, an explicit echo, or another declared control can isolate geometry. Forgetting microscopic selector labels does not remove a physical right-carrier Hamiltonian’s splitting. §14.2, p.46. The spectral-bundle guide develops these conditions.

Preserve the declared tensors as well as the norm

For a retained 3+2 splitting with a global parallel product-volume tensor, an allowed connection must obey TrA₃+TrA₂=0 in adapted frames. Raw projected connections need not satisfy this. One supplied compatible correction uses a=TrA₃+TrA₂ and sets A_m^src=A_m−aI_m/5 for m=3,2. The total trace then vanishes. This constructs a compatible comparator; it does not prove that microscopic adiabatic evolution applies the correction. §13.2, p.44.

For forward discrete polar links, let θ=Arg(detU₃detU₂) on a continuously chosen small-link branch. Multiplying both blocks by e^−iθ/5 makes their product determinant one and retains endpoint covariance under adapted frames. Large links and nontrivial bundles require consistent patching; a fifth-root branch cannot be assumed global. §17.1, p.52.

These distinctions locate the physical task. The source must supply carriers, adjacency, comparison resources, and tensor-compatible histories whose admitted tests survive completion. A smooth limit can organize those outputs as gauge geometry. The existence of geometry leaves selection of curvature, electric variables, a field Hamiltonian, and a continuum theory as additional questions.

Continue reading

• Local frames and gauge freedom
• Holonomy and curvature
• When transport descends to records
• Protected spectral carriers