From Central Extensions to Self-Measurement: Higher Coherence, Non-Linearity, and the End of Divergences in QFT
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From Central Extensions to Self-Measurement: Higher Coherence, Non-Linearity, and the End of Divergences in QFT Andrei T. Patrascu 1 1 FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We propose a new mechanism by which quantum field theory achieves ultraviolet finiteness without recourse to arbitrary cutoffs or external UV completions. Our starting point is the categorical structure underlying algebraic quantum field theory. At the level of triangle coherence diagrams, breakdowns lead only to central extensions classified by 2-cocycles. This structure yields the familiar projective representations and central commutators of quantum mechanics: linearity and unitarity remain intact. In contrast, higher coherence conditions—pentagon and hexagon diagrams—can break in ways that go beyond central extensions. When coherence morphisms cease to be central, purely unitary, or state-independent, the resulting dynamics naturally acquires nonlinear and non-unitary corrections. We interpret these corrections as an intrinsic “self-measurement” of the quantum field. We show that self-measurement dynamics can be represented as local completely positive semigroups acting on the net of algebras in AQFT. These flows act dually on correlation functions by convolution with positive Hadamard kernels, reducing scaling degree and ensuring unique extensions of distributions across coincident points. Thus divergences are dynamically suppressed: time-ordered products become finite without counterterms, and renormalisation is reinterpreted as entropy-increasing self-measurement flow. In this picture, renormalisation is no longer an external prescription but an emergent dynamical process arising from higher categorical coherence breakdown. Finally, we outline experimental scenarios where deviations from perfect unitarity and linearity—such as small Lindblad-like damping in neutrino oscillations, modified braiding phases in condensed matter systems, or anomalous running of couplings—could provide empirical access to this framework. Our approach suggests a radical shift: finiteness and measurement in QFT are two sides of the same categorical phenomenon. I. INTRODUCTION Quantum field theory (QFT) has long been a tale of two triumphs and one persistent unease. On the one hand, renormalisation renders perturbative predictions finite and precise; on the other, the measurement problem leaves the dynamical status of “collapse” conceptually unsettled; and in between, the axiomatic/algebraic programme reveals a remarkably rigid local structure for relativistic quantum physics. We argue here that these three themes are not independent. When the higher categorical coherence of the local theory breaks in specific, controlled ways, the dynamics acquires intrinsic, local completely positive (CP) “self–measurement” corrections. Those corrections dynamically smooth ultraviolet (UV) singularities, thereby turning renormalisation into an entropy–increasing selection principle rather than an external subtraction scheme. In this view, finiteness and measurement are two shadows cast by a single categorical mechanism. 1.1. Historical context and the role of renormalisation From Tomonaga–Schwinger–Feynman–Dyson to BPHZ and the causal Epstein–Glaser (EG) construction, renormalisation has evolved from a calculational art to a mathematically sharp extension problem for distributions: one constructs time–ordered products on configuration space minus the diagonals and then extends them across coincident points subject to locality, covariance and spectrum conditions. This is made completely precise in perturbative algebraic QFT (pAQFT), notably by Brunetti–Fredenhagen and by Hollands–Wald, using microlocal analysis and the Hadamard condition to control singularities and to classify finite renormalisations via scaling degree [1, 2, 4–7, 76]. Concretely, if uis a distribution on Rdaway from the origin, its (Steinmann) scaling degree at 0is sd0(u) = inf δ∈R: lim λ↓0λδu(ϕλ)=0 ∀ϕ∈ D, ϕλ(x) = λ−dϕ(x/λ).(1) Uniqueness of extension holds whenever sd0 ( u ) < d ; in general, the ambiguity consists of a finite sum of derivatives of δ –functions supported on the diagonals. In pAQFT this underlies the classification of counterterms and the notion that different renormalisation prescriptions are “nearby but incompatible distributions”: they agree off the diagonals yet differ by contact terms.
2 Renormalisability of non–Abelian gauge theory—established by ’t Hooft and Veltman using dimensional regularisation—cemented the operational success of this perspective and underwrites the Standard Model [ 8 ]. Wilson’s renormalisation group (RG) reframed UV control as scale–dependent coarse–graining [ 9 ], and functional RG equations (Polchinski, Wetterich) turned the flow into exact evolution equations for effective actions [ 10 , 11 ]. For gravity, asymptotic safety posits a non–Gaussian UV fixed point as a possible completion [ 12 ]. These achievements highlight two conceptual tensions: why UV divergences occur at all (beyond a formal, distribution–theoretic statement), and what dynamical principle selects a specific finite theory without importing an external cutoff or a different UV theory. 1.2. Measurement, unitarity, and the AQFT perspective The algebraic (Haag–Kastler) formulation describes physics by a net O 7→ A ( O )of local C ∗ /von Neumann algebras satisfying isotony and locality, with Poincaré covariance and the spectrum condition. The Reeh–Schlieder property (cyclicity and separating nature of the vacuum for each local algebra) makes manifest the operational richness of local observables and the possibility of local tomography [ 5 , 13 ]. In this setting, unitary dynamics corresponds to inner automorphisms αt ( A ) = U† tA Ut (Heisenberg picture). However, general quantum operations—those which in lab physics model measurement back–action— are described by completely positive, trace–preserving (CPTP) maps and, in the Markovian limit, by GKLS/Lindblad generators, dρ dt =−i[H, ρ] + X jLjρL† j−1 2{L† jLj, ρ},(2) which preserve positivity/complete positivity and monotonically increase entropy under suitable conditions [ 14 – 17 ]. In standard QFT such CP semigroups are effective descriptions of an external apparatus or environment; the underlying Heisenberg evolution remains purely unitary. Our proposal is different: self–measurement is intrinsic—emerging from the failure of higher categorical coherence in the local fusion/braiding data—and thus compatible with AQFT locality and the microlocal spectrum condition. 1.3. Triangle vs. higher coherence: why triangles are “safe” Categorically, the coherence constraints for monoidal/braided structures appear as triangle,pentagon, and hexagon diagrams. Their controlled failures are measured by groupoid– or group–cohomology classes. At the triangle/2–cocycle level, a breakdown yields projective phases α ( g, h ) ∈U (1) and hence central extensions of the symmetry/algebra. Physically, this amounts to multiplying intertwiners by phases that commute with everything: U(g)U(h) = α(g, h)U(gh), δα = 1,[α]∈H2(G, U(1)),(3) so Heisenberg evolutions remain inner, unitarity and linearity are preserved, and commutator deformations are central (e.g. Schwinger terms, Heisenberg [ x, p ] = i~1 ). Equivalently, [ α ]classifies central extensions 1→U(1) −→ b G−→ G→1.(4) On the Lie–algebraic side one sees [Qa, Qb] = ifabcQc+i κab 1, κab ∈R.(5) Thus triangle defects are “safe”: they produce at most central extensions and therefore cannot generate dissipation or nonlinearity. 1.4. Higher coherence can go beyond central extensions At the pentagon/hexagon level one meets associators and braidings. In pointed settings (simple objects labelled by a group G ), the associator ag,h,k is encoded by a 3–cochain ω ( g, h, k )with the pentagon forcing the 3–cocycle condition δω = 1; braiding is governed by a bicharacter b ( g, h )with the hexagon imposing compatibility. In abelian cases, Eilenberg–Mac Lane/Quinn theory identifies abelian 3–cohomology with quadratic forms; Joyal–Street express braided coherence in these terms [18–20]. Crucially, nothing forces higher–coherence defects to remain central or purely unitary:
3 •Non–centrality: the associator aX,Y,Z or braiding cX,Y takes values outside Z(A); •Non–unitarity: polar decomposition a=u eKwith K=K†6= 0; •State dependence: aor cdepends on the local state (e.g. modular data). Transport around pentagon/hexagon loops then implements, on observables, the positive conjugations A7→ eKAe−K , whose infinitesimal generators produce the dissipative part of (2) . If the coherence is state–dependent one obtains controlled Weinberg–type nonlinearities ˙ρ = ··· + Nρ ( ρ ); microcausality plus locality suppress superluminal signalling in the sense of Gisin/Polchinski [23, 105, 106]. Summarising: Triangle defects ⇒ only central extensions (unitary, linear). Higher coherence defects ⇒ either central (safe) or beyond central extensions, yielding intrinsic CP and (possibly) nonlinear corrections. 1.5. Self–measurement as dynamical renormalisation (no cutoff, no external UV completion) We now link the above to renormalisation in EG/pAQFT. Let O 7→ A ( O )be an AQFT net, and let Φ t = etL be a local CP semigroup: Φ tA ( O ) ⊆ A ( O ), commuting with spacelike separated algebras and Poincaré covariance. Such a semigroup can be generated by local Lindblad operators of the form Lj,O = ROd4x gj ( x ) Oj ( x )with smooth test functions gj of compact support, ensuring microcausality is preserved. On the dual level (Wightman/Schwinger distributions), Φ t acts as convolution with positive Hadamard kernels: W(t) n=K⊗n t? Wn,(6) where Kt is a Lorentzian analog of a heat kernel, built from parametrices so that the microlocal spectrum condition is preserved. Two consequences follow: 1. Scaling improvement: sd ( W(t) n ) <sd ( Wn )near the diagonals; for any fixed t > 0one reaches the regime of unique extension (cf. (1) ). In other words, the “nearby but incompatible” EG extensions are dynamically mapped to a single, entropy–preferred extension. 2. UV finiteness: loop integrals are rendered finite by exponential UV damping in momentum space. For the two–point covariance, heuristically e Ct(k)≈1 k2+m2−i0exp−t σ(k), σ(k)∼k2(|k|→∞),(7) so every diagram acquires sufficient fall–off for any t > 0. Hence, the renormalisation ambiguities are not chosen but selected dynamically: EG extensions become unique at t > 0. In the small– t regime one recovers standard RG (with an effective scale Λ ∼t−1/2 ), augmented by definite, testable deformations from the CP part of the flow. No arbitrary cutoff is introduced, and no external UV completion is invoked: finiteness emerges from intrinsic self–measurement. Operationally, this addresses the pAQFT incompatibility issue: distributions that are indistinguishable off the diagonals but differ by local counterterms (“close” in the distributional topology) are driven by Φ t to the same extended distribution. The selection criterion is physical—monotone entropy production under a locality–preserving CP flow—rather than conventional. 1.6. Relation to ’t Hooft’s approaches ’t Hooft’s work pressed two related ideas: (i) the modern renormalisation of gauge theories [ 8 ], and (ii) a deterministic/cellular–automaton view of quantum mechanics aimed at resolving the measurement problem by embedding quantum phenomena in a deeper classical substratum [ 24 ]. Our approach is complementary. We keep the quantum algebraic framework but modify dynamics at the categorical coherence level, turning measurement into a local CP flow intrinsic to the field. In particular, we do not add hidden variables or a new UV theory; instead, we reinterpret renormalisation as dynamical self–regularisation within QFT itself. The gauge–theory successes of dimensional regularisation remain intact; what changes is the conceptual status of finiteness and a new source of small, structured deviations from strict unitarity/linearity with experimental consequences.
4 1.7. Contributions and outlook •Conceptual unification. We show that triangle–level coherence breakdown produces at most central extensions, hence preserves unitary, linear quantum mechanics, while higher coherence breakdown can go beyond central extensions, generating intrinsic CP/nonlinear corrections. •AQFT realisation. We define local CP semigroups on nets compatible with microcausality and the microlocal spectrum condition, and we identify “coherence curvature” (polar positive parts of associators/braidings and state dependence) as the origin of their generators. •Renormalisation without cutoffs. We argue that the Φ t –flow reduces scaling degree and selects unique extensions, rendering time–ordered products finite without counterterms, and connect the t–flow to an intrinsic, entropy–monotone RG. •Phenomenology. We outline tests: Lindblad–like damping in oscillation experiments, modified braiding statistics in condensed–matter platforms, and subtle anomalies in coupling running at high energies (all constrained to respect locality and no signalling). Beyond providing a new angle on the measurement problem (measurement as intrinsic CP dynamics), this programme directly addresses the practical heart of QFT: UV control. It suggests that what we have long treated as a prescription may in fact be the long–time limit of a physical, entropy–producing flow generated by higher categorical structure. The sections that follow give precise definitions (coherence curvature and its polar decomposition), state the microlocal reduction theorem, and present worked examples (e.g. ϕ4 4) alongside constraints ensuring locality, covariance and Ward identities. II. TRIANGLE VS. HIGHER COHERENCE: COHOMOLOGY, CENTRAL EXTENSIONS, AND BEYOND This section develops the precise mathematical backbone behind the slogan: Triangle coherence defects ⇒ only central extensions (unitary, linear quantum mechanics), whereas higher (pentagon/hexagon) coherence defects can either remain central (hence equally safe) or go beyond central extensions, producing intrinsic completely positive (CP) and even state–dependent nonlinear corrections. We proceed in four steps. First we recall cochains, coboundaries, and the group–cohomological classification that already appears at the level of triangle diagrams. Second, we prove that triangle-level noncommutativity is exactly projective and central, hence preserves unitarity and linearity. Third, we formulate the pentagon/hexagon constraints and the pointed classification (including the abelian 3–cohomology/quadratic-form correspondence), highlighting the central and the beyond–central branches. Finally, we give a mechanism showing how non–central or non–unitary higher coherence produces CP (and potentially nonlinear) dynamics, together with a physically transparent example. A. Cochains, coboundaries, and a first reading of coherence Let G be a discrete group and A an abelian G –module; here we take A = U (1) with trivial G –action. Write Cn ( G, A ) = {φ : Gn→A} for inhomogeneous n –cochains. The standard group–cohomology coboundary δ:Cn→Cn+1 is (δφ)(g1, . . . , gn+1) = φ(g2, . . . , gn+1) n Y i=1 φ(g1, . . . , gigi+1, . . . , gn+1)(−1)iφ(g1, . . . , gn)(−1)n+1 .(8) We set Zn= kerδ,Bn= im δ, and Hn(G, A) = Zn/Bn. Two recurring instances are: •H2(G, U(1)) classifies projective multipliers and central extensions. •H3 ( G, U (1)) classifies (pointed) associators (the F –symbols) up to monoidal equivalence; in the abelian case, braided pointed data are governed by abelian 3–cohomology and quadratic forms [18–20].
5 B. Triangle defects are exactly central extensions We now state and prove the precise version of the “triangle is safe” claim. Theorem II.1 (Triangle breakdown ⇔ central extension & unitary linear dynamics) . Let U : G→ U ( H ) be a (strongly continuous) projective representation with multiplier α:G×G→U(1), U(g)U(h) = α(g, h)U(gh).(9) Then: 1. Associativity forces α∈Z2(G, U(1)), i.e. δα = 1. 2. Changing local phases U ( g ) 7→ β ( g ) U ( g )for β : G→U (1) shifts α7→ α·δβ , so inequivalent projective classes are [α]∈H2(G, U(1)). 3. There is a central extension 1 →U (1) →b G→G→ 1with a genuine unitary representation b U:b G→ U(H)s.t. U(g)is the image of any lift of g. 4. If π : A→B ( H )is a (nondegenerate) ∗ –representation of a C ∗ –algebra and U ( g )implements a symmetry by inner automorphisms AdU(g) ( A ) = U ( g ) †A U ( g ), then for any multiplier α the Heisenberg dynamics αt ( A ) = U ( t ) †A U ( t )is a ∗ –automorphism for each t : it is linear, norm– preserving, and (by Wigner’s theorem) unitary on H . In particular, projective defects can only produce central commutator shifts (Schwinger terms) and never nonunitary or nonlinear evolution. Proof. (1) Compute (U(g)U(h))U(k)in two ways: (U(g)U(h))U(k) = α(g, h)α(gh, k)U(ghk), U(g)(U(h)U(k)) = α(h, k)α(g, hk)U(ghk). Equality for all g, h, k yields α ( h, k ) α ( g, hk ) = α ( g, h ) α ( gh, k ), i.e. δα = 1. (2) If U0 ( g ) = β ( g ) U ( g ) then α0 ( g, h ) = β ( g ) β ( h ) β(gh)α ( g, h ) = ( δβ ) ·α . (3) Standard construction: define b G = U (1) ×G with multiplication ( z, g ) · ( w, h ) = ( zwα ( g, h ) , gh ). This is a central extension with b U ( z, g ) = z U ( g ). (4) For any A∈π ( A ), AdU(g) is a ∗ –automorphism (conjugation by a unitary). Projective phases cancel inside Ad , hence AdU(·) is a genuine representation of G by ∗ –automorphisms. Thus dynamics is linear, norm–preserving, and unitary. Central extensions can add central terms to Lie brackets (e.g. [Qa, Qb] = ifabcQc+iκab1) but cannot generate dissipators or nonlinearities. Physics. The appearance of H2 ( G, U (1)) recovers textbook phenomena: Bargmann phases for the Galilei group, spin from double covers, magnetic translations, and central terms in current algebras. All of these are unitary & linear. In AQFT language, implementers acting by Ad are inner ∗ –automorphisms of the local algebras, so the net structure and the modular data are preserved [5]. C. Pentagon and hexagon: pointed classification and two branches We pass to higher coherence. Consider a (unitary) pointed monoidal category Vecω G whose simples are labelled by g∈G and with tensor g⊗h = gh . The associator ag,h,k : ( g⊗h ) ⊗k→g⊗ ( h⊗k )is multiplication by a scalar ag,h,k =ω(g, h, k) idghk, ω :G3→U(1).(10) Mac Lane’s pentagon forces ω∈Z3(G, U(1)): δω(g, h, k, `) = ω(h, k, `)ω(g, hk, `)ω(g, h, k) ω(gh, k, `)ω(g, h, k`)= 1.(11) Changing tensorators by a 2–cochain α sends ω7→ ω·δα , hence pointed monoidal structures are classified by [ω]∈H3(G, U(1)) [19]. If, in addition, the category is braided and pointed, braiding is determined by a bicharacter b : G×G→ U (1), cg,h = b ( g, h ) idgh , subject to the hexagon constraints which couple b to ω . In the abelian case one has the Eilenberg–Mac Lane/Joyal–Street classification H3 ab(G, U(1)) ∼ ={quadratic forms q:G→U(1)}, b(g, h) = q(g+h) q(g)q(h),(12) and ω = ωq is the abelian 3–cocycle determined functorially by q so that both hexagons are satisfied [18, 20].
6 Two branches. At the pentagon/hexagon level there are two qualitatively distinct regimes: 1. Central unitary branch (safe). ω ( g, h, k ) ∈U (1) and cg,h are phases in the center of the relevant endomorphism algebras. All coherence failures are pure phases: dynamics remains unitary/linear. This includes Vecω Gand the abelian data (12). 2. Beyond–central branch (new). Associators and/or braidings acquire a non–central component or a non–unitary positive component, or become state–dependent. Then going around pentagon/hexagon loops no longer yields a mere phase. The induced maps on local algebras need not be inner ∗ –automorphisms; as we show below, they canonically induce CP (and potentially nonlinear) dynamics. D. A cohomological “curvature” and its dynamical meaning Let aX,Y,Z : ( X⊗Y ) ⊗Z→X⊗ ( Y⊗Z )be the associator in a (not necessarily strict) tensor category enriched over Hilbert spaces (or realized by endomorphisms of local algebras in AQFT). For a quadruple (W, X, Y, Z)define the pentagon loop operator ∆(W, X, Y, Z) := aW,X,Y ⊗Z◦aW⊗X,Y,Z ◦(idW⊗aX,Y,Z)◦a−1 W,X⊗Y,Z ◦a−1 W,X,Y .(13) In the strictly unitary central case one has ∆ = id. In general we set ∆ = u eK, u†u=1, K =K†.(14) We call Ω := log ∆ = log u + K (defined via functional calculus on the finite–dimensional intertwiners or by a choice of branch) the coherence curvature. A perfectly coherent theory has Ω=0. From curvature to CP maps. Let A ( O )be a local von Neumann algebra and suppose the associator/braiding implements, by naturality, a rebracketing/exchange map T:A(O)−→ A(O),T(A) := X λ V† λA Vλ,(15) obtained by coarse–graining over the different elementary coherence paths (Kraus operators Vλ ). In the perfectly coherent unitary central case this reduces to T= AdU with U unitary (one Kraus operator), hence a ∗ –automorphism. If ∆ 6 = 1 , a minimal choice is to include two paths—say, the two sides of the pentagon—with Kraus system {V1, V2} given by the partial isometries along the two sides, rescaled to ensure unitality V† 1V1 + V† 2V2 = 1 (Stinespring dilation makes such a choice canonical up to unitary ancillas [ 27 ]). Then Tis CP, unital, and reduces to AdU iff both Vi are isometries with the same range, i.e. iff ∆is (central) unitary. Proposition II.2 (Infinitesimal generator from small curvature) . Assume a one–parameter family of defects with ∆ ε = 1 + εX + O ( ε2 )on each quadruple, where X†6 = −X in general. Define a CP unital map T ε by coarse–graining over the two pentagon paths as above and iterate it n = t/ε times. Then T n ε converges (in the strong operator topology on normal states) to a quantum dynamical semigroup etL with GKLS generator L(A) = i[H, A] + X jL† jALj−1 2{L† jLj, A},(16) where H is determined by the antihermitian part of X and the Lj by its positive part (via a Choi–Kraus decomposition). Moreover, if X is purely antihermitian and central then L = adH is inner (unitary flow). Sketch. Write the two path operators as T1 = 1 + ε ( iH −1 2R ) + O ( ε2 ), T2 = √ε L + O ( ε )with R = L†L chosen so that T† 1T1 + T† 2T2 = 1 + O ( ε2 ). Define T ε ( A ) = T† 1AT1 + T† 2AT2 . A standard Trotter– Kato/Davies weak–coupling argument shows that T t/ε ε→etL with L ( A ) = i [ H, A ] + L†AL −1 2{L†L, A} [ 16 ]. If X is antihermitian and central then L = 0 (no positive part) and H∈Z ; hence the limit is an inner ∗–automorphism group. State dependence and controlled nonlinearity. If the associator/braiding depends on the local state ρ (e.g. via modular data), then the Kraus operators depend on ρ and the coarse–grained map becomes nonlinear: ρ7→ PλVλ [ ρ ] ρ Vλ [ ρ ] † . Under locality/microcausality constraints (operators supported in double cones, spacelike commutation), the resulting nonlinear terms are local and avoid superluminal signalling in the sense of [23, 106]. In the linearised regime one recovers Weinberg–type nonlinear drifts [105].
7 E. DHR/AQFT realisation of the two branches In the DHR framework of superselection sectors, objects are localized endomorphisms ρ of the quasilocal algebra A , fusion is composition, and intertwiners implement charge transport. The (unitary) statistics operator ε ( ρ, σ )gives a braiding, and the associator aρ,σ,τ is constructed from localized intertwiners; the DHR coherence implies the pentagon/hexagon equations [28–30]. •Central unitary branch. If a and ε take values in the center of the relevant morphism algebras (and are unitary), then all pentagon/hexagon loops are central phases. The induced action on A ( O ) is by inner ∗–automorphisms: unitary and linear. •Beyond–central branch. If the polar positive part of a or ε is nontrivial and not central, then pentagon/hexagon loops produce nontrivial ∆with positive part. Coarse–graining over coherence paths yields a CP semigroup on A ( O )as in Prop. II.2. If, furthermore, a or ε depends on the local state (e.g. through modular conjugations associated to Ωand A ( O )), nonlinear corrections appear but remain local. F. Worked examples Example 1 (safe branch: pointed braided, abelian). Let G = ZN = hxi . For k∈Z2N define a quadratic form qk(mx) = expπi k m2 N, bk(mx, nx) = qk((m+n)x) qk(mx)qk(nx)= exp2πi k mn N.(17) The abelian 3–cocycle ωk associated to qk (via Eilenberg–Mac Lane) together with bk solves the hexagon equations. All coherence data are central phases; dynamics is unitary/linear. Different k label distinct topological spins and mutual statistics (Abelian anyons), but no dissipator arises [18, 20]. Example 2 (beyond–central branch: local positive curvature). Let O ⊂ M be a double cone and choose a local selfadjoint field J ( x )(e.g. a conserved current) with smooth compactly supported test function g∈C∞ 0(O). Set Kρ,σ,τ =ZO d4x g(x)J(x), > 0,(18) and deform the associator by aρ,σ,τ 7→ aρ,σ,τ eKρ,σ,τ . Then the pentagon loop has positive part exp ( δK ); coarse–graining over the two pentagon paths produces a CP map with Lindblad operators L=√ROd4xpg(x)J(x)(in the quadratic approximation). The resulting GKLS generator is L(A) = i[H, A] + L†AL −1 2{L†L, A},(19) local and microcausal (since g is compactly supported). For ↓ 0one recovers the unitary limit; for any > 0one obtains intrinsic self–measurement in O. G. Summary and sharp dichotomy The discussion above can be packaged as a clean “if and only if”. Theorem II.3 (Safe vs. beyond–central) . Let C be the (unitary) tensor category of localized sectors/intertwiners in AQFT, with associator a and (when present) braiding c . Consider the induced rebracketing/exchange maps on each local algebra A(O). 1. (Safe branch) The induced dynamics is unitary and linear (inner ∗ –automorphisms) iff all coherence data take values in central unitaries and satisfy the cocycle equations, i.e. the coherence curvature Ωis central antihermitian (pure phases) and δa = δc = 1. Equivalently, the only defects are classes [α]∈H2(G, U(1)) and (pointed) [ω]∈H3(G, U(1)) realised as central phases. 2. (Beyond–central branch) If at least one of the following holds:
8 (i) the polar positive part of a coherence morphism is nontrivial and not central; (ii) the coherence data are state–dependent; (iii) the pentagon/hexagon closure holds only after coarse–graining over multiple paths, then the induced dynamics on A ( O )is generated by a GKLS operator L (possibly augmented by state–dependent nonlinear terms), i.e. a CP, entropy–producing self–measurement flow. Proof. (1) follows from Thm. II.1 and the pointed classification: central phases act by inner ∗ – automorphisms; pentagon/hexagon then commute up to phases. (2) follows from Prop. II.2: any non–central positive part or coarse–graining over inequivalent coherence paths produces a CP map whose continuous limit is GKLS. State dependence yields nonlinearities by dependence of Kraus operators on ρ. Physical upshot. Triangle coherence breakdown can only implement central extensions and hence standard, unitary and linear quantum mechanics. Higher coherence breakdown still includes that safe case (central abelian 3–cocycles and bicharacters), but it also admits intrinsically different behaviours— non–central positive curvature and state dependence—which generate the nonunitary and nonlinear corrections needed for the self–measurement mechanism developed later to drive renormalisation dynamically. III. SELF–MEASUREMENT ON NETS AND DYNAMICAL RENORMALISATION This section realises the programme announced in the Introduction in the precise language of algebraic QFT (AQFT) and microlocal analysis. We construct local completely positive (CP) semigroups on Haag–Kastler nets from higher–coherence “curvature”; we prove that these maps preserve isotony and microcausality; we compute their dual action on n –point distributions and show that they lower the Steinmann scaling degree near the diagonals; and we conclude that time–ordered products admit unique Epstein–Glaser (EG) extensions for any fixed t > 0. Finally, we connect the t –flow to an intrinsic renormalisation group (RG) equation of Polchinski/Wetterich type and illustrate the mechanism on ϕ4 4. Throughout, M is Minkowski spacetime, O 7→ A ( O )is a Haag–Kastler net with vacuum Ω, and fields are understood as (possibly operator–valued) distributions affiliated with the net [ 5 , 7 ]. We use the microlocal notation and results of [4, 32, 33]. A. From higher coherence to local CP semigroups The basic structure behind our construction is the “coherence curvature” introduced in §IID: a pentagon/hexagon loop ∆that, in general, has polar decomposition ∆ = u eK with u unitary and K = K†≥ 0not necessarily central. In a local setting, ∆is built from intertwiners localized in a double cone O, so its positive part Kis an observable affiliated with A(O). Heuristic principle. A nontrivial positive curvature K acts as a local measurement strength that, when one coarse–grains over the inequivalent coherence paths in a pentagon/hexagon, generates a CP instrument on A ( O ); the Markovian limit yields a GKLS generator. This is precisely the content of Prop. II.2 of §II D. We now formulate a canonical class adapted to AQFT. Definition III.1 (Local GKLS generators on a net) . Fix a double cone ObM . Let {Oj}N j=1 be local fields (e.g. Wick polynomials) affiliated with A ( O ), and let Gjk ∈ D0 ( O×O )be a positive kernel in the sense that for all compactly supported test functions f= (f1, . . . , fN) X j,k ZZO×O fj(x)Gjk(x, y)fk(y)d4x d4y≥0, with Gjk smooth away from the diagonal and Hadamard–compatible (defined below). The corresponding local GKLS generator LOacts on A∈ A(O)by LO(A) = i[HO, A] + X j,k ZZO×O Oj(x)AOk(y)−1 2{Ok(y)Oj(x), A}Gjk(x, y)d4x d4y, (20)
9 where HO is a local selfadjoint operator (counterterm Hamiltonian). The associated CP semigroup is ΦO t=etLOon A(O). Remark III.2 (Hadamard–compatibility).Positivity of G ensures complete positivity; Hadamard– compatibility means that Gjk has wavefront set contained in that of a Hadamard bidistribution, so that the product with local fields is well–defined by microlocal Hörmander criteria and preserves the microlocal spectrum condition [4, 76]. Smooth compactly supported Gis included as a trivial example. Theorem III.3 (Isotony and microcausality are preserved).Let LObe as in (20). Then: 1. (Isotony)ΦO tA(O)⊆ A(O)for all t≥0. 2. (Microcausality) If O1⊥ O2 (spacelike) and A∈ A ( O1 ), B∈ A ( O2 ), then [Φ O1 t ( A ) , B ] = 0 for all t≥0. 3. (Covariance) If the kernel family {Gjk} and the set {Oj} transform covariantly, then Φ O t is covariant under the corresponding symmetry group. Proof. (1) By construction, Oj ( x )and HO are affiliated with A ( O ); hence commutators and anticommutators in (20) are in A ( O ), and the generator leaves A ( O )invariant. Trotter–Kato then implies etLO preserves the algebra. (2) If O1⊥ O2 , locality implies [ Oj ( x ) , B ] = 0 for x∈ O1 and B∈ A ( O2 ). From (20) one checks d dt [Φ O1 t ( A ) , B ] = Φ O1 t [ LO1 ( A ) , B ] = 0 with initial condition [ A, B ] = 0, hence [Φ O1 t ( A ) , B ] = 0. (3) Follows from naturality: if U ( g ) Oj ( x ) U ( g ) † = P`Dj` ( g ) O` ( gx )and Gjk transforms as a rank–2 tensor kernel, then U(g)ΦO t(A)U(g)†= ΦgO tU(g)AU(g)†. Origin in higher coherence. When ∆ = u eK is the pentagon/hexagon loop operator in a double cone, the leading nontrivial order in K produces (by the coarse–graining construction of Prop. II.2) a generator of the form (20) with kernels Gjk determined by K (e.g. via its two–point functions in the vacuum). Unitary central curvature K = 0 yields LO = adHO (the “safe” branch of §II C), whereas noncentral positive parts give G6= 0 (the beyond–central branch). B. Quasi–free channels and covariance update To see the effect on correlation functions explicitly, it is convenient to work in the Weyl algebra of a free scalar field, W ( M ), generated by Weyl operators W ( f ) = eiφ(f) with symplectic form σ ( f, g ) = φ ( f ) φ ( g ) −φ ( g ) φ ( f ) = ∆( f, g )(the Pauli–Jordan pairing). A (quasi–free) CP Gaussian channel Φ t is determined by a positive kernel Nt (the noise) and a drift Kt (the dissipation), acting on covariances C as Ct=Kt·C·K∗ t+Nt, Nt≥0.(21) Complete positivity requires Nt≥i 2 ( Kt ∆ K∗ t− ∆) as distributions (the “quantum noise inequality”). In the purely dephasing case we take Kt = 1 (no drift). Then Ct = C + Nt . If Nt is smooth Hadamard–compatible, the Hadamard singularity of Cis preserved [4]. The associated action on n –point Wightman distributions Wn is a positive definite convolution with kernels built from Kt and Nt (Wick calculus). For time–ordered products, the propagator C entering the EG construction is replaced by Ct ; in particular, the singular structure along coincident points is mitigated by Nt . To go beyond a mere smooth addition (which preserves singularities but does not regularise loops enough), one may choose Kt to be a pseudodifferential attenuator of low order (so as not to destroy the Hadamard wavefront set) while Nt compensates the quantum noise relation. A concrete class is discussed below. C. Microlocal effect: lowering scaling degree We now quantify the effect of Φ t on Steinmann scaling degree. Recall that if u∈ D0 ( Rd )is defined away from 0, uniqueness of extension holds whenever sd0 ( u ) < d (see (1) ). In pAQFT, u arises as a linear combination of products of propagators (with derivatives) C ( xi−xj ); the obstruction to extension comes from the large–momentum behaviour of b C (equivalently, the short–distance singularity of C ). The next theorem shows that a broad class of local CP flows improves this behaviour in exactly the way needed for EG uniqueness.
16 V. PHENOMENOLOGY, EXPERIMENTAL PROBES, AND THE QUANTUM–TRAJECTORY PICTURE Having established in Sections II–IV that higher–coherence breakdown beyond central extensions canonically generates a local, symmetry–compatible completely positive (CP) flow Φ t = etL on nets, we now develop its phenomenology. We connect the generator to Schwinger–Keldysh/Feynman–Vernon influence functionals, derive analytic predictions for interferometric platforms (neutrino oscillations, neutral mesons, quantum optics), propose condensed–matter probes (anyon braiding), and extract scheme–independent corrections to running couplings. We also show how the RG picture of §III admits a quantum–trajectory (unravelling) interpretation that links measurement records to effective coarse–graining. A. Influence functionals and Schwinger–Keldysh representation A convenient language for the dual action of Φ t on n –point functions is the closed–time–path (CTP) or Schwinger–Keldysh functional integral. For a (Gaussian) sector of the theory, the effect of a quasi–free CP channel is encoded by an influence functional [41–43]: Ft[φ+, φ−] = expniSSK[φ+, φ−]−1 2ZZd4x d4y(φ+(x)−φ−(x)) Nt(x, y) (φ+(y)−φ−(y))o,(26) where SSK is the usual SK action (with the retarded/advanced structure) and Nt≥ 0is the noise kernel (cf. (21) ). The CP condition for quasi–free maps is precisely Nt≥i 2 ( Kt ∆ K∗ t− ∆) as distributions; detailed balance relative to a KMS state imposes the KMS relation between the dissipative kernel (anti–Hermitian part in SSK) and Nt[38]. Proposition V.1 (Hadamard preservation and microlocal admissibility) . If Nt is of Hadamard type (wavefront set contained in that of a Hadamard bidistribution) and Kt is the pseudodifferential attenuator of Theorem III.4, then the SK two–point correlation matrix transformed by Ft remains Hadamard and satisfies the microlocal spectrum condition. In particular, the flowed Wightman function W(t) equals W plus a smooth positive kernel plus attenuated mid–momentum contributions, and the retarded/advanced functions keep their causal support. Proof. The transformation of the SK covariance by (26) is Gab t = KtGabK∗ t + i Π ab [ Nt ], where a, b ∈ { + ,−} and Π ab inserts Nt on the Keldysh (“+- − ”) leg only. Since Nt is Hadamard and Kt is order–0with mt→ 1at large |k| , the wavefront sets of Gab t are the same as those of Gab ; the only change is a smooth addition and a bounded attenuation in mid–momentum shells. Causality of retarded/advanced parts is preserved by support properties of Ktand Nt. Physics. Eq. (26) exhibits the self–measurement channel as a physical influence functional: the imaginary quadratic form in ( φ+−φ− )is precisely the dephasing noise that drives entropy increase. Unlike ad–hoc regulators, Nt and Kt are fixed by coherence curvature and constrained by locality and symmetry. B. Two–level analytics: closed formulas and complete positivity Many probes reduce effectively to two–level dynamics. Write ρ = 1 2 ( 1 + ~r ·~σ )and consider the general GKLS form ˙ ~r =~ Ω×~r −D ~r +~c, (27) with Hamiltonian 1 2~ Ω·~σ , dissipation matrix D = D>≥ 0, and drift ~c (which vanishes for unital channels). Complete positivity (CP) is equivalent to the positivity of the Kossakowski matrix C≥0in ˙ρ=−i[H, ρ] + 3 X i,j=1 Cijσiρσj−1 2{σjσi, ρ}.(28) For diagonal C = diag ( γx, γy, γz )one has D = diag ( γy + γz, γx + γz, γx + γy )and CP iff γi≥ 0. The exact solution of (27) is ~r(t) = e−Dt R~ Ω(t)~r(0) + Zt 0 e−D(t−s)R~ Ω(t−s)~c ds, (29)
17 with rotation R~ Ω ( t )about ~ Ω . This gives closed formulas for survival probabilities and interferometric visibilities across platforms. C. Neutrino oscillations: intrinsic dephasing and energy scaling For two flavors in vacuum, in the mass basis the Hamiltonian is H = ∆m2 4Eσ3 with frequency ω = ∆ m2/ (2 E )and mixing angle θ . A CP, trace–preserving, unital channel diagonal in the mass basis corresponds to C= diag(γ⊥, γ⊥,0) (pure dephasing). Solving (27) yields the survival probability Pνα→να(L, E) = 1 −sin2(2θ)1 21−e−ΓLcos ∆m2L 2E,Γ := 2γ⊥,(30) where L≃t is the baseline (we set c = 1). The intrinsic dephasing rate Γinherits the locality/curvature origin: for channels whose kernels arise from dimension– d composite densities, dimensional analysis suggests Γ(E)∼ε E d−4up to logarithms (e.g. Γ∝ε E for d= 5 operators) [44, 45]. Proposition V.2 (No–signalling and matter effects) . If the dissipator is local and flavor–diagonal in matter, then in the adiabatic approximation the same formula (30) holds with θ→θm ( E, ne )and ∆ m2→ ∆ m2 m ( E, ne ), and Γreplaced by its matter–dressed value Γ m . Spacelike separated detectors cannot distinguish whether Φtwas applied upstream (Theorem IV.4). Physics. Bounds on Γfrom atmospheric, accelerator, and astrophysical data translate into upper limits on the local curvature parameter ε . Unlike environment–induced decoherence, here locality and covariance tightly constrain the energy dependence Γ( E )and its mixing–angle independence (for mass–basis dephasing) [44, 45]. D. Neutral mesons: K0−¯ K0and B0−¯ B0 Neutral–meson oscillations are another precision interferometer. In the Weisskopf–Wigner approximation, the effective non–Hermitian Hamiltonian Heff = M−i 2 Γacts on the two–dimensional flavor space. A CP Lindbladian correction D [ ρ ] = Pi ( `iρ`† i−1 2{`† i`i, ρ} )adds decoherence parameters ( α, β, γ )in the Bell–Steinberger analysis [ 46 ]. For dephasing in the mass basis, the KS – KL interference term acquires e−Γt damping, and the positivity of the Kossakowski matrix implies ( α, β, γ ) ≥ 0and 2 √αγ ≥β . The CP origin ensures no CPT–violating signal appears unless the curvature breaks the symmetry. Proposition V.3 (Bell–Steinberger with CP channel) . The Bell–Steinberger relation for decay amplitudes remains valid with the replacement hKS|KLi → hKS|KLie−Γt , and CP invariance of the channel implies that |η+−|and φ+−receive only damping corrections at leading order. Physics. Existing bounds on ( α, β, γ )constrain the local curvature strength in the strange and beauty sectors; any observed phase drift beyond damping would point away from the central–unitary branch and toward state–dependent or non–central curvature. E. Quantum optics: Hong–Ou–Mandel (HOM) visibility Consider two single–photon wavepackets impinging on a 50:50 beam–splitter. In the ideal unitary case the coincidence probability Pc ( τ )versus delay τ exhibits the HOM dip Pc ( τ ) = 1 2 (1 −V Λ( τ )), with V the visibility and Λdetermined by spectral overlap. A local CP dephasing channel acting on the temporal mode (e.g. a Gaussian noise kernel with variance σ over a gate region) modifies the two–mode density operator by the map of §V B with γ⊥=γ. The exact result is Pc(τ) = 1 21−e−ΓTgate VΛ(τ),(31) with Γ = 2 γ and Tgate the effective interaction time. Eq. (31) cleanly separates environment–induced loss (which also reduces singles) from intrinsic dephasing (which preserves singles but reduces indistinguishability); the latter is the signature of self–measurement. Microcausality ensures no violation of commutation relations between spacelike separated output fields.
18 F. Anyon interferometry and hexagon curvature In a pointed braided theory, braiding and associator data ( b, ω )are central phases determined by a quadratic form q (safe branch). Suppose a small beyond–central curvature Kg,h corrects the R –move as Rg,h = b ( g, h ) eKg,h with Kg,h = K† h,g ≥ 0supported locally in the interferometer arms. The two paths in a Fabry–Pérot or Mach–Zehnder geometry acquire relative weights A∝b(g, h)e−κg,h ,A∝b(h, g)e−κh,g , with κg,h = Tr(Kg,h)≥0. The interference contrast is then C=2|b(g, h)|e−(κg,h+κh,g)/2 1 + e−(κg,h−κh,g )≈2|b(g, h)|e−¯κ(¯κ=1 2(κg,h +κh,g)1).(32) A reduction of contrast at fixed current and temperature, scaling with the enclosed quasiparticle number as predicted by Kg,h, is the hallmark of hexagon curvature. Physics. Quantum Hall and moiré platforms (e.g. ν = 1 / 3Laughlin states) offer controlled interferometers; (32) predicts a decay of the oscillation amplitude with arm length set by the local channel strength and independent of external noise models [47–49]. G. Running couplings: universal finite shifts We revisit the intrinsic RG correction of §IV E in two gauge contexts. QED vacuum polarization. Let Π µν ( q )denote the photon self–energy. A covariant CP channel constructed from gauge–covariant currents (Prop. IV.6) modifies the electron propagator by S→St = KtSK∗ t+Ntwith transverse Kt. Ward identities enforce qµΠµν t(q)=0. At one loop, Πµν t(q) = Πµν(q) + δΠµν (q), δΠµν (q) = −(gµνq2−qµqν)δΠ(q2),(33) with δΠ(q2) = e2 12π2Zd4k (2π)4|mt(k)|2|mt(k+q)|2−1 (k2−m2)((k+q)2−m2). Thus the QED β –function gets a finite, scheme–independent shift δβ1 ( t )akin to (25) . Gauge invariance is manifest. Non–Abelian case. For Yang–Mills, choose densities Oj in the adjoint representation and Gjk proportional to δjk to preserve color. The one–loop β –function coefficient b0 receives a calculable negative shift δb0 ( t )from the attenuator mt ; asymptotic freedom remains but the slope changes by a definite amount fixed by curvature. H. Quantum trajectories and coarse–grained records Every CP semigroup admits an unravelling into quantum trajectories: between stochastic “jumps” generated by Kraus operators Lj , the state evolves under an effective non–Hermitian Hamiltonian Heff = H−i 2PjL† jLj [ 16 , 50 ]. The (classical) distribution of detection records defines a coarse–graining of quantum histories. In our setting, the jump structure is fixed by higher–coherence curvature; consequently: Proposition V.4 (Trajectory–RG correspondence) . Consider an observable OΛ probing momenta below Λ. Under the trajectory ensemble induced by Φ t with attenuator mt = 1 −χ qt supported on a shell around Λ, the trajectory–averaged expectation E [ OΛ ]evolves according to a Polchinski–type flow with kernel νt = ∂tNt and regulator χ , as in (23) . Thus the RG–like smoothing is the coarse–grained average over self–measurement records. Sketch. Averaging over quantum jumps inserts Lj ( · ) L† j superoperators, which in Gaussian sectors reduce to contractions with Nt ; the no–jump piece implements the attenuator Kt . Restricting to observables with support below Λimplements shell coarse–graining; the resulting generator on expectation values has exactly the functional form of (23).
19 Physics. Prop. V.4 provides an operational picture: the intrinsic RG flow is the shadow of irretrievable information carried away by self–measurement “clicks”—a physical coarse–graining. I. Constraints, benchmarks, and an experimental matrix We summarize signatures and constraints in a compact matrix: Platforms & observables. •Neutrinos: survival probabilities (30) ; look for energy–scaling of Γ( E )compatible with locality; baseline dependence distinct from environmental decoherence. •Kaons/mesons: damping of interference terms with CP–preserving constraints; Bell–Steinberger consistency with only amplitude reduction. •Quantum optics: HOM visibility (31) reduced without loss in singles; dependence on gate region and arm length determined by local channel strength. •Anyons: interferometric contrast (32) ; independence from temperature/edge–disorder when those are separately controlled; scaling with number of enclosed quasiparticles. •High energy: universal finite shifts in β –functions; small deviations in running couplings without new thresholds. Bounds & scaling. •CP & microcausality force Γ≥0and forbid superluminal signalling (Theorem IV.4). •Symmetries restrict allowed Γ( E ); e.g. flavor–diagonal dephasing in mass basis is mixing–angle independent. •Detailed balance (Theorem IV.9) implies entropy production monotonicity; laboratory tests can probe QDB via fluctuation relations on small systems. J. Synthesis The beyond–central branch of higher coherence predicts specific,local, and covariant deviations from exact unitarity/linearity that: 1. manifest as dephasing factors in interferometers (neutrinos, mesons, photons) with energy/length scaling fixed by local operator content; 2. reduce anyon interferometric contrast in a way tied to hexagon curvature rather than environmental noise; 3. produce universal finite shifts of RG data while preserving Ward identities; 4. admit a trajectory picture that explains RG as self–measurement coarse–graining. Because the channels are reconstructed from coherence curvature, the number of free parameters is small and symmetry–constrained—yielding falsifiable predictions. VI. SCATTERING, SPECTRAL REPRESENTATIONS, AND THE OPTICAL THEOREM UNDER SELF–MEASUREMENT Sections III–V established that higher–coherence curvature beyond central extensions generates a local, covariant completely positive (CP) flow Φ t = etL on Haag–Kastler nets, reduces scaling degree, and dynamically selects Epstein–Glaser extensions. We now connect this structure to scattering theory and spectral analysis. Our aims are fourfold: 1. Prove that the self–measurement flow preserves the canonical commutator and Hadamard microlocal spectrum while positively reshaping the Källen–Lehmann spectral density.
20 2. Construct asymptotic dynamics and a CPTP scattering instrument, and derive a modified optical theorem that accounts for dissipative loss channels in a way consistent with complete positivity and locality. 3. Show that cluster decomposition is retained under the flow, ensuring physical separability of distant experiments. 4. Work out explicit one–loop consequences for 2 → 2scattering in ϕ4 4 , tying back to the finite integrals of §III F. Throughout, we assume the covariant, local, Hadamard–compatible GKLS form of §III A and §III B. A. Källen–Lehmann under local CP flow Let φbe a (scalar) Wightman field with two–point function W(x−y) := hΩ, φ(x)φ(y)Ωi=Z∞ 0 dµ2ρ(µ2)∆+(x−y;µ2),(34) where ∆ + is the positive–frequency two–point function and ρ ( µ2 ) ≥ 0is the Källen–Lehmann spectral density [5, 51, 52]. The canonical commutator is [φ(x), φ(y)] = i∆(x−y),(35) with ∆the Pauli–Jordan commutator (a c–number distribution). Consider a quasi–free CP flow with covariance update Wt=KtW K∗ t+Nt,(36) where Kt = Op ( mt ( D )) is an order–0(translation–invariant) pseudodifferential attenuator with symbol mt ( k ) → 1as |k|→∞ , and Nt is a positive, Hadamard–compatible bidistribution supported in a double cone (or covariant family thereof), as in §III B. We assume the commutator preservation condition Kt∆K∗ t= ∆,suppNtspacelike–symmetric, hence Nt−N> t= 0,(37) which is natural for local channels built from densities commuting with the free symplectic form. Theorem VI.1 (Spectral positivity and commutator preservation) . Under (36) – (37) , Wt admits a Källen–Lehmann representation Wt(x−y) = Z∞ 0 dµ2ρt(µ2)∆+(x−y;µ2), ρt(µ2) = |mt(µ)|2ρ(µ2) + ρN(µ2),(38) with ρt ( µ2 ) ≥ 0and ρN≥ 0the (Lorentz–invariant) spectral density of Nt . Moreover, [ φt ( x ) , φt ( y )] = i∆(x−y)and the microlocal spectrum condition (Hadamard property) is preserved. Proof. Fourier transforming, c Wt ( p ) = |mt ( p ) |2c W ( p ) + b Nt ( p ). Since c W ( p ) = (2 π ) θ ( p0 ) ρ ( p2 ) δ ( p2−µ2 ) (integrated over µ2 ), the first term multiplies the spectral measure by |mt|2 . The positivity of Nt implies, by Bochner–Schwartz, that b Nt is a positive measure supported on the forward lightcone (Hadamard compatibility), hence b Nt ( p ) = (2 π ) θ ( p0 ) ρN ( p2 )with ρN≥ 0. This yields the claimed ρt≥ 0. The commutator of the flowed field, computed from the antisymmetric part of Wt, [φt(x), φt(y)] = Wt(x, y)−Wt(y, x) = KtW−W>K∗ t+ (Nt−N> t) = Kti∆K∗ t=i∆, uses (37) and N> t = Nt . Microlocal preservation follows because (i) multiplication by |mt|2 of a distribution that is conormal to the lightcone retains the same wavefront set; (ii) Nt is Hadamard (smooth addition or conormal with the same cone). Physics. Theorem VI.1 shows that the channel does not tamper with causality or canonical commutation, while it does re–weight the physical spectrum by a positive factor and may add positive spectral weight near multi–particle thresholds via ρN . This matches the intuition that self–measurement redistributes spectral weight without violating positivity or microcausality.
21 B. Asymptotic dynamics and a CPTP scattering instrument Scattering in AQFT is built from asymptotic fields (Haag–Ruelle) under unitary dynamics [ 5 ]. In our setting, the Heisenberg evolution is replaced by a contraction semigroup on local algebras generated by L . We construct contractive Møller maps and a CPTP scattering instrument. Let αt be the free unitary dynamics and Φ t = etL the interacting self–measurement flow. For test functions fwith energy–momentum support near a mass shell, define φin/out(f) = lim t→∓∞ Φ∗ tα∗ −t(φ(f))(limit in weak operator topology),(39) whenever the limit exists on a dense domain. The existence can be established by Cook’s method extended to contractions (Kato–Birman theory) under small channel strength and standard mass–gap/regularity assumptions. Proposition VI.2 (Contractive wave operators and scattering instrument) . Suppose the dissipator is infrared–regular and of compact time support in the interaction picture. Then the limits defining φin/out exist, and there are completely positive wave maps W± intertwining the asymptotic free algebra Afree with the physical algebra such that W∗ ±are isometries on asymptotic states. The scattering instrument S(·) := W∗ +◦W−(·)W∗ −◦W+ is completely positive and trace–preserving on the asymptotic (free) Fock space: it maps incoming density matrices to outgoing density matrices. In the unitary limit ( L Hamiltonian) S reduces to the usual S–matrix channel, S(ρ) = SρS†. Sketch. Let Ψ t = Φ ∗ tα∗ −t in the Schrödinger picture. Under infrared regularity, R∞ 0k˙ Ψtψkdt < ∞ for ψ in a dense subspace of asymptotic states (Cook–Kato criterion), ensuring Cauchy convergence of Ψ tψ as t→ + ∞ . This yields W− ; similarly for W+ . Complete positivity follows from that of Φ t and normality of α∗ t . Trace preservation of S follows from that of W± composed appropriately (they are CP contractions with isometric adjoints on asymptotic subspaces). Physics. Instead of a unitary S –matrix, we obtain a CPTP map on asymptotic density matrices. This is the correct physical object when intrinsic loss channels (self–measurement clicks) are present but unobserved. C. A modified optical theorem (inclusive form) Let T denote the scattering transfer superoperator, defined by S = id + T in the interaction picture on asymptotic states. In the unitary case, S†S = 1 implies the standard optical theorem 2 Im Tii = Pf|Tfi|2 . In our CPTP setting, trace preservation and complete positivity impose a generalized optical theorem with a strictly positive loss term arising from the dissipator. Theorem VI.3 (Inclusive optical theorem for CPTP scattering) . Let {|ii} be an orthonormal basis of incoming asymptotic states and Tfi := hf|T (|iihi|)|fi. Then 2Im Tii =X f|Tfi|2+ Σloss(i),Σloss(i)≥0,(40) where Σ loss ( i )is a quadratic functional of the Lindblad operators {Lj} coming from the dissipator along the scattering history of |ii . Equality reduces to the unitary optical theorem when {Lj} = 0 (triangle/central branch). Sketch. Write the Dyson–like expansion for the CPTP channel in the interaction picture (time–ordered cumulants), keeping terms up to second order. The anti–Hermitian part of the effective Hamiltonian is −i 2PjL† jLj, so that T(ρ) = −iZdt [HI(t), ρ] + Zdt X jLj(t)ρLj(t)†−1 2{Lj(t)†Lj(t), ρ}+··· Evaluating hi|·|ii and taking the imaginary part gives 2 Im Tii = Pf|Tfi|2 + Rdt Pjhi|L† j ( t ) Lj ( t ) |ii + ··· , where the first term arises from the norm of the coherent transition amplitudes (via CP positivity) and
22 the second from the anti–Hermitian part (Lindblad loss). Positivity Σ loss ≥ 0follows from L† jLj≥ 0 and CP. The full proof uses CP Schwarz inequalities for superoperators and extends to all orders by resumming the cumulants into S. Physics. The theorem states that inclusive observables (summing over unobserved self–measurement products) obey an optical theorem with an additional non–negative loss probability. In the safe (triangle/central) regime Lj=0 and one recovers unitarity exactly. D. Cluster decomposition under local CP flow The cluster property for the vacuum, lim a→∞hΩ, A αa(B)Ωi=hΩ, AΩihΩ, BΩi,(aspacelike), ensures physical independence of far–separated experiments. We show it persists under Φt. Proposition VI.4 (Cluster property preservation) . Let A∈ A ( O1 )and B∈ A ( O2 ), with O1 and O2 bounded, and let Φtbe local and covariant. Then lim a→∞hΩ,Φt(A)αa(Φt(B))Ωi=hΩ,Φt(A)ΩihΩ,Φt(B)Ωi, for spacelike a translating O2 to infinity. If, moreover, Φ t satisfies detailed balance w.r.t. the vacuum (or a KMS state), the convergence is monotone in t. Proof. Locality and covariance imply αa◦ Φ t = Φ t◦αa . For large spacelike a , microcausality gives [A(O1), αa(A(O2))] = 0, hence the CP Schwarz inequality yields hΩ,Φt(A)αa(Φt(B))Ωi−hΩ,Φt(A)ΩihΩ,Φt(B)Ωi≤ kΦt(A)ΩkkΦt(B)Ω −hΦt(B)iΩk, and the second factor vanishes as a→ ∞ by the cluster property of the vacuum plus continuity of Φ t . QDB implies contractivity of relative entropy along the flow, which strengthens the convergence monotonicity. Physics. The proposition reassures that intrinsic self–measurement does not induce spurious long–range correlations: distant experiments decouple as usual. E. One–loop 2→2amplitude in ϕ4 4revisited We revisit the one–loop correction to 2 → 2scattering in ϕ4 4 under the attenuator mt ( k ) = 1 −χ ( k ) (shell profile) from §III F. The amputated one–loop amplitude in the s–channel reads M(t) s(s) = −iλ + (−iλ)2hI(t)(s) + I(t)(t) + I(t)(u)isym,(41) where, for s= (p1+p2)2, I(t)(s) = Zd4k (2π)4|mt(k)|2|mt(k+p1+p2)|2 (k2+m2−i0)((k+p1+p2)2+m2−i0). As shown in §IIIF, the integral splits as the usual logarithmic piece plus a finite channel correction ∆ t , rendering M(t) sfinite without counterterms. The imaginary part obeys the inclusive optical theorem: 2Im M(t) s(s) = ZdΠ2|M(t) 2→2|2+ Σloss(s), with d Π 2 the two–body phase space measure and Σ loss determined by the Lindblad densities chosen in (20) . At small t , expanding |mt|2 = 1 − 2 χ + χ2 gives analytic control of both the finite part and the loss term, which scales with the shell volume. Physics. The amplitude remains causal and crossing–symmetric; the channel selects definite finite pieces and adds an inclusive loss term consistent with CP. This is exactly what one should feed into phenomenology: look for small, energy–localized deficits (relative to standard logs) in processes dominated by a known loop momentum window.
23 F. Synthesis The self–measurement flow induced by higher–coherence curvature admits a complete scattering–theoretic description consistent with AQFT and positivity: • The Källen–Lehmann representation survives with a positive, physically interpretable reshaping of the spectral density (Theorem VI.1); canonical commutators and Hadamard microlocal structure are preserved. • Asymptotics exist as CP contractions and define a CPTP scattering instrument (Proposition VI.2); the optical theorem acquires a universal, non–negative loss term sourced by the dissipator (Theorem VI.3). • Cluster decomposition remains valid (Proposition VI.4), ensuring the separability of distant experiments. • Explicit loop computations (e.g. ϕ4 4 ) show how the flow renders amplitudes finite and predictive, tying inclusive loss directly to the curvature–fixed channel shape. These results dovetail with the renormalisation picture of §III and the consistency analysis of §IV: finiteness without cutoffs,positivity without unitarity, and predictivity without new UV degrees of freedom are all consequences of one structural input—higher categorical coherence beyond central extensions. VII. FIXED POINTS, ENTROPY MONOTONICITY, AND UNIVERSALITY OF THE SELF–MEASUREMENT RG Sections III–VI established that higher–coherence curvature beyond central extensions produces a local,covariant completely positive (CP) semigroup Φ t = etL on Haag–Kastler nets, which reduces Steinmann scaling degree, dynamically selects Epstein–Glaser (EG) extensions, and yields a consistent scattering framework. We now address the global structure of this flow: fixed points, entropy monotonicity, hypercontractivity, and universality classes. The main results are: 1. The self–measurement flow admits a Dirichlet form and a gradient–flow interpretation with respect to the Bogoliubov–Kubo–Mori (BKM) inner product; the relative entropy to a reference KMS state is a Lyapunov functional that decreases strictly away from the central/unitary (triangle) branch. 2. Under quantum detailed balance (QDB), the flow satisfies a quantum logarithmic Sobolev inequality (qLSI) in a broad class of channels, implying exponential convergence to the stationary state and hypercontractivity of the semigroup. 3. In 1 + 1dimensions, for flows generated by stress–tensor densities, there is a c–theorem–type monotonicity: an entropic c –function decreases along the self–measurement RG, matching Zamolodchikov’s direction at small channel strength. 4. Fixed points are classified into a safe (central/unitary) manifold and a primitive (mixing) manifold; in the latter, the stationary state is unique and faithful, with a spectral gap controlling entropy decay. These structures define universality classes of self–measurement deformation. Throughout we assume the local GKLS form of (20) , Hadamard compatibility, and covariance as in Theorem III.3. We write Φ∗ tfor the Schrödinger–picture dual acting on normal states. A. Dirichlet form, BKM geometry, and gradient–flow structure Fix a region O and a faithful reference state σ on A ( O )(vacuum restriction or a KMS state). Denote by hA, Biσ := Trσ1/2A†σ1/2B the BKM inner product on A ( O )(the GNS Hilbert space of ( A ( O ) , σ )). Assume quantum detailed balance (QDB) with respect to σ (Def. in §IVD), so that L is self–adjoint on this Hilbert space.
24 Definition VII.1 (Dirichlet form and Fisher information) . The Dirichlet form E and the Fisher information Iare E[A] := −hA, L(A)iσ≥0,I(ρkσ) := d dss=0 Sρskσ,(42) where S ( ρkσ ) = Trρ ( log ρ−log σ ) is the Uhlmann relative entropy and ρs is the geodesic in the BKM metric through ρwith tangent given by the modular logarithmic derivative (see, e.g., [37]). Proposition VII.2 (Entropy dissipation identity).Let ρt= (Φt)∗ρ0and assume QDB. Then d dtSρtkσ=−I(ρtkσ) = −EΩt≤0,(43) where Ω t is the σ –centered observable representing the tangent of ρt in the BKM geometry (the “score operator”). Proof. Standard for reversible quantum Markov semigroups: the GKS–Lindblad structure and QDB imply L is self–adjoint in the BKM inner product, and the de Bruijn identity in the noncommutative setting gives d dt S(ρtkσ) = −hΩt,L(Ωt)iσ(see [17, 38]). Physics. The Lyapunov property dtS ( ρtkσ ) ≤ 0expresses entropy production intrinsic to self–measurement. In the triangle (central/unitary) branch, L = adH and E ≡ 0: entropy is conserved, matching standard quantum mechanics. B. Quantum log–Sobolev inequalities and hypercontractivity The quantum logarithmic Sobolev inequality (qLSI) controls the decay of relative entropy by the Dirichlet form: there exists α > 0such that S(ρkσ)≤1 2αEΩfor all ρ. (44) When (44) holds, one obtains exponential convergence S ( ρtkσ ) ≤e−2αtS ( ρ0kσ )and hypercontractivity: Φtmaps noncommutative Lp(σ)to Lq(σ)with q−1 = e2αt(p−1) [58, 59]. We give a sufficient condition tailored to local GKLS generators (20). Theorem VII.3 (qLSI for local elliptic noise) . Consider L of the form (20) with (i) QDB w.r.t. a faithful σ, (ii) a coercive Kossakowski kernel Gjk(x, y)in the sense that there exists c > 0with X j,k ZZ fj(x)Gjk(x, y)fk(y)d4x d4y≥cX jkfjk2 H−s for some s < 2(Sobolev norm), and (iii) local fields Oj whose connected two–point functions in σ satisfy a micro–ellipticity bound compatible with (ii). Then (44) holds with a constant α > 0depending on c , the Sobolev order s, and the micro–ellipticity constants. Sketch. Following the Bakry–Émery Γ–calculus adapted to reversible quantum semigroups [ 59 ], define the carré–du–champ Γ( A ) = L ( A†A ) −A†L ( A ) −L ( A† ) A and its iterate Γ 2 . The coercivity of G plus micro–ellipticity implies Γ 2≥κ Γ(curvature–dimension condition) for some κ > 0. By the quantum Bakry–Émery theorem, this yields (44) with α = κ (see also [ 60 ] for finite–dimensional analogues). Locality and Hadamard bounds ensure that the required products of distributions are well–defined. Corollary VII.4 (Exponential mixing and hypercontractivity) . Under the hypotheses of Theorem VII.3, S ( ρtkσ ) ≤e−2αtS ( ρ0kσ )and k Φ tkp→q≤ 1for q− 1 = e2αt ( p− 1) in Lp ( σ )spaces. In particular, σ is the unique stationary state (primitivity) if the fixed–point algebra is trivial. Physics. Exponential approach to a stationary state (with rate α ) is the quantitative expression of self–measurement equilibration. In the safe triangle regime, α= 0 (no dissipation).
25 C. An entropic c–function and a self–measurement c–theorem in 1+1 In 1 + 1unitary QFT, Zamolodchikov’s c –theorem asserts the existence of a positive function C ( µ )that decreases along RG flows and equals the central charge at fixed points [ 61 ]. Here we show that for flows generated by local stress–tensor–based channels, an entropic c–function decreases along t. Let Abe an interval of length Rand let ρA tbe the reduced state of ρton A(A). Define CA(t) := Rd dR SρA t(1+1 dimensions).(45) For a CFT vacuum, S(ρA) = c 3log(R/)so CA(0) = c 3. Theorem VII.5 (Entropic c –monotonicity under local stress–tensor channel) . In a 1 + 1–dimensional QFT, consider a local CP semigroup generated by densities proportional to components of the stress tensor Tµν with a Hadamard–compatible positive kernel supported in a double cone containing A . Assume QDB w.r.t. the vacuum and boost covariance. Then CA ( t )defined by (45) is nonincreasing in t ; if, moreover, the fixed–point theory is conformal, limt→∞ CA(t) = cIR 3≤cUV 3. Sketch. The first law of entanglement ( δS = δhKAi with KA the modular Hamiltonian of A ) and the Bisognano–Wichmann form of KA imply that, to leading order in the channel strength, d dt S ( ρA t ) = −EA [Ω t ], with EA the Dirichlet form restricted to A . Scale–differentiation commutes with t –evolution under boost covariance. Positivity of EA gives ∂tCA ( t ) ≤ 0. At fixed points, conformal invariance implies S ( ρA ∞ ) = cIR 3log ( R/ ), hence the limit. The argument is the entanglement analogue of Zamolodchikov’s, using relative entropy monotonicity under inclusions in place of the T¯ Ttwo–point function [62]. Physics. The theorem ties self–measurement monotonicity to the irreversibility of RG flows: the channel degrades short–distance entanglement, driving the theory toward lower effective central charge. D. Fixed points and classification We next classify stationary points of Φtand their stability. Definition VII.6 (Fixed–point algebra and primitivity) . The fixed–point algebra is F := {A∈ A ( O ) : L ( A )=0 } . The semigroup is primitive if the only fixed points in the Schrödinger picture are scalar multiples of σ(unique faithful stationary state). Theorem VII.7 (Fixed points: central/unitary vs. primitive) . Let L satisfy QDB and the hypotheses of Theorem VII.3. 1. Central/unitary (triangle) branch: If the Kossakowski kernel vanishes, Gjk ≡ 0, then L = adH is Hamiltonian. The fixed–point algebra is the commutant of H ; entropy is conserved and the flow is unitary. 2. Beyond–central branch: If the representation generated by {Oj} is irreducible on A ( O )and the kernel is coercive, then F = C1 and the semigroup is primitive. There exists a spectral gap λ > 0 such that kρt−σk1,σ ≤e−λt kρ0−σk1,σ (L1–norm weighted by σ). Sketch. (1) Immediate. (2) Irreducibility implies no nontrivial conserved observables; coercivity plus QDB yields a Poincaré inequality hA−hAiσ, A −hAiσiσ≤λ−1E [ A ]and hence a spectral gap λ . Primitivity follows by Frigerio’s structure theorem for quantum Markov semigroups (see [38]). Physics. Fixed points in the beyond–central branch encode self–measurement equilibria: the theory relaxes to a unique, symmetry–respecting stationary state. The central/unitary branch is the standard unitary limit (triangle defects only).
32 Physics. Proposition IX.1 realizes a measurement localized in O1 : the outside algebra is untouched, and the map is CP because we have coarse-grained over (unobserved) Kraus outcomes. It is the axiomatic avatar of the coarse-graining over coherence paths in Sections II D–III A. C. From instruments to semigroups: Hille–Yosida and GKLS Let {Φt}t≥0be a family of local CP maps on A(O2)with Kraus operators in N, such that: •(Semigroup) Φt+s= Φt◦Φs,Φ0= id. • (Strong continuity on bounded sets) For every normal state ω and A∈ A ( O2 ), t7→ ω (Φ t ( A )) is continuous. • (Energy-boundedness) There exists a positive selfadjoint K affiliated with A ( O2 )and constants a, b ≥0such that PαkVα(t)ψk2≤akψk2+bhψ, Kψiuniformly on compact t-sets. Theorem IX.2 (GKLS generator on local algebras) . Under the above assumptions, the generator L = limt↓0 (Φ t−id ) /t exists as a norm-closed densely defined map on A ( O2 ), and has the (local) GKLS form L(A) = i[H, A] + X j∈JL† jALj−1 2{L† jLj, A},(53) where H = H†∈ N and Lj∈ N (countable index set J ) satisfy PjL† jLj convergent in the strong operator topology on the domain of K . Moreover, Φ t = etL is the unique normal CP semigroup with generator L . Sketch. The structural result is a local version of the Evans–Lewis/Arveson–Lindblad theorem [ 73 , 74 ]: strong continuity plus complete positivity and unitality imply the GKLS form on the σ -weakly dense domain generated by products B†AC with A, C ∈ N . Energy-boundedness ensures the series PjL† jLj converges on a core and that L is closable; the Hille–Yosida theorem for CP semigroups gives existence/uniqueness of etL . Localization of H, Lj in N follows because all Kraus/Stinespring operators lie in Nby assumption. Connection to kernels. If Lj are smeared local fields supported in O1 with positive Kossakowski kernel Gjk ( x, y )(Definition III.1), then the series in (53) is the operator-algebraic realization of the integral generator (20) on A(O2). D. Locality, covariance, and detailed balance Proposition IX.3 (No-signalling and covariance revisited) . Let L be of the form (53) with H, Lj∈ N . Then (i) Φ t preserves isotony and microcausality as in Theorem III.3; (ii) if a symmetry group G has an inner implementation on the net and H, Lj transform covariantly in a finite-dimensional representation, then Φtis G-covariant (Prop. IV.6). Proof. (i) Since H, Lj∈ N ⊂ A ( O2 ), L maps A ( O1 )into itself and acts trivially on commuting algebras A(O3)with O3⊥ O2. (ii) Covariance is inherited from the representation on H, Lj. For a KMS state σβ of a stationary spacetime, quantum detailed balance (QDB) with respect to σβ is equivalent to KMS-symmetry of L (Section IVD); for local generators it reduces to a representationtheoretic condition on H, Ljwith respect to the modular group. Theorem IX.4 (Stationarity and mixing of KMS states) . If L satisfies QDB with respect to a faithful σ (e.g. a KMS state restricted to A ( O2 )), then σ◦ Φ t = σ and the entropy S ( ρtkσ )is nonincreasing (Theorem IV.9). If, in addition, the fixed-point algebra of L in A ( O2 )is trivial, then Φ ∗ t is mixing: ρt→σ in the σ -weighted trace norm with an exponential rate governed by the log-Sobolev constant (Theorem VII.3).
33 E. Energy-constrained cb-norm bounds In relativistic QFT the uniform cb-norm is too strong; physically relevant continuity is energy-constrained. For a positive Hamiltonian Hand E > 0, define the energy-constrained cb-norm kΨkcb,E := supk(Ψ⊗idn)(X)k:X∈ B(H⊗Cn),kXk ≤ 1,Tr(ρH)≤Efor all states in the range of X. (54) Theorem IX.5 (Short-time behavior under energy constraints) . Let Φ t = etL be a local CP semigroup with generator (53) such that PjL† jLj≤c01 + c1H (form sense) for some positive H affiliated with A(O2). Then for every E > 0there is C(E)with kΦt−idkcb,E ≤C(E)t(t↓0).(55) Sketch. Dyson expansion and energy bounds yield k (Φ t−id )( A ) k ≤ tk [ H, A ] k + PjkL† jALjk + k{L† jLj, A}k on vectors with energy ≤E . Taking cb-norm and optimizing over ancillary dimension gives the stated bound. Physics. The theorem justifies treating the dissipative corrections as small over short time/length scales at fixed energy density. It underpins perturbative phenomenology (Section V). F. Nuclearity ensures countable Kraus decompositions Theorem IX.6 (Countable Kraus families under nuclearity) . Assume modular nuclearity for A ( O2 )and let Φbe a normal unital CP map localized in O1bO2 . Then Φadmits a Kraus decomposition with a countable index set J and Kraus operators Lj∈ N such that PjL† jLj converges strongly on a dense energy-bounded domain. Sketch. Nuclearity implies that the inclusion map from A ( O1 )to the GNS Hilbert space is nuclear; by a theorem of Davies–Lewis (in the W ∗ -setting) normal CP maps on nuclear operator systems admit countable Kraus decompositions. Localization in Nfollows by Kaplansky density and the split inclusion. Physics. Countable Kraus families are sufficient to define the stochastic quantum trajectories (Section VH) and to give rigorous meaning to the generator series. G. Constructing Ljfrom smeared fields We now connect the abstract Lj to concrete local field densities. Let O ⊂ M be a double cone and let W ( M )be the Weyl algebra of a (free or perturbatively constructed) scalar field. Fix test functions gj∈C∞ 0 ( O )and Wick polynomials Oj ( x )(e.g. ϕk ( x )) such that the microlocal product Oj ( gj )is essentially selfadjoint on the Wightman domain. Proposition IX.7 (Field-smeared Kraus operators) . For small > 0, the operators Lj = √Oj ( gj ) define a completely dissipative quadratic form Q(A) := X jLjψ, ALjψ−1 2ψ, {L† jLj, A}ψ on a common core, which closes to a generator L of the form (53) . Moreover, if the gj transform covariantly, then Lis covariant and local. Sketch. Microlocal spectrum condition ensures that Wick polynomials smeared with gj are closable with energy-bounded domains; Kato’s inequality for quadratic forms then yields complete dissipativity. Localization and covariance are inherited from supports and transformation properties of gj. Connection to kernels. Choosing a positive definite matrix γjk and setting Gjk ( x, y ) = γjk gj ( x ) gk ( y ) recovers the integral representation (20).
34 H. Trotter product and interaction picture For phenomenology and perturbative constructions, one often combines Hamiltonian and dissipative evolutions. Theorem IX.8 (Trotter–Kato product formula for local CP semigroups) . Let αt be the Hamiltonian Heisenberg evolution generated by a selfadjoint H∈ A ( O2 )and let Ψ t = etD be a local CP semigroup with generator Dof the form (53), commuting with αton A(O2). Then lim n→∞ αt/n ◦Ψt/nn=et(adH+D) in the strong operator topology on normal states, uniformly on compact t-sets. Proof. This is a standard Trotter–Kato result for contraction semigroups on operator spaces; complete positivity and locality are preserved by the limit. Commutation on the local algebra ensures Chernoff equivalence of generators. Physics. The theorem legitimizes the intuitive “unitary plus dissipative” splitting used repeatedly in Sections V–VI. I. Curved spacetimes and locally covariant functoriality Let Loc be the category of globally hyperbolic spacetimes and admissible embeddings, and Alg the category of C ∗ -algebras. A locally covariant net is a functor A : Loc →Alg satisfying the time-slice axiom [6]. Theorem IX.9 (Functoriality of local CP flows) . There is a covariant functor Φ t : Loc →CPAlg (CP maps between algebras) such that for each spacetime M and each ObM with split property, Φ M t restricts to a local CP semigroup on AM ( O )with generator of the form (20) , and for each morphism ψ : M→N one has naturality A(ψ)◦ΦM t= ΦN t◦A(ψ). Sketch. Choose Kraus operators from local fields and kernels built functorially from the metric (Hadamard parametrices), ensuring covariance. Split property and nuclearity hold in curved spacetimes for Hadamard states; the construction of Sections IXB–IX C globalizes functorially. Physics. The theorem lifts the self-measurement mechanism to the locally covariant framework: the CP flow is geometric and compatible with general covariance. J. Synthesis We have shown that the algebraic/microlocal self-measurement programme is underpinned by solid operator-algebraic existence and locality results: • The split property and modular nuclearity provide a type I collar enabling localized CP instruments (Prop. IX.1) with Kraus operators supported in the collar. • Under mild energy-boundedness, these instruments exponentiate to strongly continuous local CP semigroups with local GKLS generators (Theorem IX.2), preserving isotony, microcausality, and covariance (Prop. IX.3). • KMS states are stationary and mixing under detailed balance (Theorem IX.4); short-time changes are controlled in energy-constrained cb-norm (Theorem IX.5). • Nuclearity ensures countable Kraus decompositions (Theorem IX.6); concrete Lj arise from smeared local fields (Prop. IX.7). A Trotter product formula holds (Theorem IX.8). • The construction extends functorially to curved spacetimes (Theorem IX.9), consistent with local covariance. Taken together, these results guarantee that the higher-coherence curvature mechanism can always be realized as a bona fide local CP dynamics on physically relevant AQFT nets, closing the conceptual loop between categorical inputs and analytic existence.
35 X. GRAVITY, SEMICLASSICAL BACKREACTION, AND ENTROPY BOUNDS UNDER SELF–MEASUREMENT We now couple the self–measurement dynamics developed in Sections III–VII to gravity in the semiclassical regime. Our goals are: 1. establish that local, covariant CP flows Φ t preserve the Hadamard property and stress–tensor conservation so that the renormalized stress–energy tensor (RSET) hTabit is well–defined and conserved on a curved background (M, g); 2. derive the Einstein–Langevin backreaction equations from the Schwinger–Keldysh influence functional associated to Φ t , identifying the noise kernel and fluctuation–dissipation structure under detailed balance; 3. analyze focusing of null congruences and the Raychaudhuri equation with the flowed RSET, and show how the flow disfavors negative–energy pulses in averaged inequalities; 4. prove a first–order generalized second law (GSL) monotonicity result for stationary horizons when Φ t satisfies KMS detailed balance with respect to the natural modular flow (Rindler/Unruh or Hartle–Hawking); 5. discuss the status of sharp bounds (Bekenstein, QNEC), and outline cosmological implications (de Sitter). Our conventions follow [75–77]. A. Hadamard preservation, RSET, and conservation Let ( M, g )be globally hyperbolic. A state ω on the field algebra is Hadamard if its two–point function W(x, y)has wavefront set WF(W) = {(x, kx;y, −ky)∈T∗(M×M)\{0}: (x, kx)∼(y, ky), kx.0}, with the microlocal spectrum condition ensuring the existence of a local covariant Hadamard parametrix and thus of the RSET hTabiωsatisfying Wald’s axioms [75, 76]. Let Φ t be a local, covariant CP flow as in Theorem IV.10, produced by a pseudodifferential attenuator Ktand a positive Hadamard–compatible bidistribution Nt(cf. Prop. V.1). Set Wt=KtWK∗ t+Nt. Proposition X.1 (Hadamard property and conservation) . If Kt is order–0with principal symbol 1and Ntis Hadamard–compatible and covariantly conserved in the sense that for all test tensors fab, ∇a xNab,cd(x, y) = ∇c yNab,cd(x, y)=0, then Wt is Hadamard and the RSET difference ∆ hTabit := hTabit−hTabi is smooth, locally covariant, and conserved: ∇a∆hTabit= 0. Proof. Hadamard: multiplication by an order–0pseudodifferential operator preserves WF ( W )(principal symbol 1leaves the conormal cone intact); addition of Nt with Hadamard wavefront set preserves the microlocal spectrum condition [ 76 ]. For the RSET, point–splitting subtraction with the Hadamard parametrix Hgives ∆hTab(x)it= lim y→xDab(x, y)(Kt⊗Kt)(W−H)(x, y) + Nt(x, y)−lim y→xDab(x, y)(W−H), so only smooth terms survive and the result is smooth. Conservation follows from the fact that Dab is divergence–free on bi–solutions and from the assumed conservation of Nt together with ( ∇·Kt )=0at principal level; local covariance is inherited from the functoriality of Kt, Nt(Theorem IX.9). Physics. Proposition X.1 ensures that the local self–measurement dynamics does not jeopardize the conservation law required by the semiclassical Einstein equation; the flowed state remains Hadamard, so the renormalized stress tensor is meaningful.
36 B. Backreaction: in–in effective action and Einstein–Langevin We work in the Schwinger–Keldysh/CTP formalism. Let Sm [ g, Φ] be the matter action. The CP flow Φtis encoded by the influence functional Ft[Φ+,Φ−]as in (26). The in–in effective action is Γt[g+, g−] = SEH[g+]−SEH[g−] + Wt[g+, g−], where Wtis obtained by integrating out matter with Ftinserted. Expanding Wtto second order in the metric perturbation hab =g+ab −g−ab yields Wt[g+, g−] = Zd4x√−g1 2hab(x) Πab cd t(x, y)hcd(y)d4y+iNt[h] + O(h3), where Π t is the dissipation kernel (real) and Nt is the noise quadratic form determined by Nt and stress–tensor fluctuations. Variation leads to stochastic equations Gab[g]+Λgab +αHab +βIab = 8πGhTabit[g] + ξab,(56) the Einstein–Langevin equation, where Hab, Iab are the local curvature counterterm tensors, and ξab is a Gaussian conserved stochastic source with hξab(x)i= 0,hξab(x)ξcd(y)i=1 2h{b tab(x),b tcd(y)}iconn t≡Nξ ab cd(x, y).(57) Under KMS detailed balance for Φt,Πtand Nξsatisfy a fluctuation–dissipation relation [77]. Theorem X.2 (Fluctuation–dissipation under QDB) . If Φ t satisfies quantum detailed balance w.r.t. a KMS state σ at inverse temperature β (Theorem IV.9), then in stationary spacetimes the Fourier transforms of Πtand Nξobey c Nξab cd(ω,~ k) = coth βω 2Im b Πab cd(ω,~ k), compatible with causality (Kramers–Kronig). Consequently, the Einstein–Langevin equation (56) is stable and causal at linear order. Sketch. QDB implies KMS symmetry of the Schwinger kernel on the CTP contour. The standard FDT derivation in the in–in formalism identifies Nξ with the Keldysh component and Πwith retarded/advanced components; KMS gives the stated relation (see [77]). Physics. Eq. (56) shows the backreaction of self–measurement: besides the smooth shift in hTabi , there are intrinsic, conserved fluctuations ξab fixed by the channel. Under QDB these fluctuations and the dissipative response are thermodynamically consistent. C. Averaged null energy and focusing Let ka be a null vector field tangent to a complete affinely–parameterized null geodesic γ ( λ ), and let f(λ)≥0be a smooth, compactly supported smearing. Define Et[f] := Zdλ f(λ)hTabkakbitγ(λ). Negative values of Et [ f ]are possible in QFT, but are constrained by quantum energy inequalities (QEIs) [78]. Proposition X.3 (Monotonicity of smeared null energy under local dephasing) . Consider a channel Φ t generated by local densities constructed from the stress tensor (or its covariant smearing along γ ), with positive kernel Gsupported in a tubular neighborhood of γ. Then, to second order in t, d dt Et[f]≥0.
37 Sketch. Let Lbe the generator in the Heisenberg picture. For A=Rf Tkk one has d dt hAit=hL(A)it=X jhL† jALj−1 2{L† jLj, A}it. If Lj are localized near γ and linear in Tkk with positive kernel G , the RHS is the quadratic form of A in the positive Kossakowski matrix evaluated in state ρt , thus nonnegative. More generally, the positivity of the Kossakowski form and microcausality along the tube ensure ˙ Et [ f ] ≥ 0by the CP Schwarz inequality. Physics. Proposition X.3 indicates that local self–measurement suppresses negative energy densities along null congruences in a coarse–grained sense. Via the Raychaudhuri equation, dθ dλ =−1 2θ2−σabσab +ωabωab −8πG hTkki, this disfavors the defocusing induced by negative energy, hence supports entropy–increase statements below. D. Generalized second law at first order Let H be a stationary causal horizon (Rindler, Killing horizon of a stationary black hole). The generalized entropy is Sgen ( t ) = Area(t) 4G~ + Sout ( t ), where Sout ( t )is the von Neumann entropy of quantum fields outside the horizon on a Cauchy slice Σ t . We show that under a local CP flow Φ t satisfying detailed balance with respect to the natural KMS state σ (Unruh/Hartle–Hawking) and supported in the exterior, S0 gen(0) ≥0. Theorem X.4 (First–order GSL under detailed balance) . Assume: (i) the background is stationary with horizon surface gravity κ ; (ii) the reference state σ is the corresponding KMS state; (iii) Φ t is a local CP flow supported outside the horizon, commuting with the modular group of σ (QDB), and preserving Hadamardness; (iv) the linearized semiclassical Einstein equation holds. Then d dtSgen(t)t=0 ≥0. Sketch. Following the relative–entropy proofs of the GSL [ 79 , 80 ], consider the modular Hamiltonian K of the exterior algebra w.r.t. σ (for Rindler, K = 2 πRv Tkk dλ dd−2x⊥ ). For any state ρ , S ( ρkσ ) = ∆hKi−∆Sout ≥0. Detailed balance implies d dt S(ρtkσ)≤0(Theorem IV.9), hence d dt∆Sout(t)≥d dt∆hKit. The linearized area change relates to ∆ hKi by the first law of horizon mechanics and the linearized Einstein equation, yielding (in units ~= 1) d dt Area 4G=d dt∆hKit. Combining gives d dt Sgen ( t ) ≥ 0at t = 0. Technical assumptions ensure differentiability and the commutation of t –variation with modular flow; see [ 79 ] for the underlying structure and Theorem IV.9 for the entropy monotonicity under QDB. Physics. Theorem X.4 shows that the intrinsic self–measurement dynamics is thermodynamically consistent with black–hole mechanics: it increases outside entropy while its stress–tensor response increases area according to the first law.
38 E. Bekenstein bound and QNEC: stability under Φt The Bekenstein bound in flat space can be formulated as a relative–entropy inequality on half–spaces [80]: S(ρkσ)≥2π∆hKi − ∆Sout ≥0. Since (Φ t ) ∗ is CPTP and σ –preserving under QDB, S (Φ ∗ tρkσ ) ≤S ( ρkσ ); thus the bound is stable under Φtand in fact becomes strict unless Φtacts trivially. For the quantum null energy condition (QNEC) 2 πhTkki ≤ S00 out , sharp proofs rely on modular theory and strong subadditivity [ 81 , 82 ]. While a full stability proof is beyond our scope, the ingredients used above (Hadamard preservation, modular covariance under QDB, positivity of the Kossakowski form along null directions) indicate that, at least to leading order in small t , the QNEC is preserved or strengthened (due to S00 increasing faster than hTkkiunder dephasing). F. Cosmology: de Sitter detailed balance and backreaction In de Sitter space (Hubble parameter H ), the Bunch–Davies vacuum is KMS for the static patch. Channels Φ t that satisfy QDB with respect to this KMS state (with β = 2 π/H ) produce a fluctuation–dissipation pair (Π t, Nξ )adapted to the Gibbons–Hawking temperature. At linear order, the Friedmann equation receives a smooth positive correction 3H2= 8πG ρ(t) matter +ρ(t) ξ+ Λren, where ρ(t) matter includes the smooth shift from Kt and ρ(t) ξ is the (small) stochastic energy density associated with ξab . Detailed balance prevents secular runaway. In slow–roll inflation, local CP dephasing of light scalars can reduce super–Hubble phase coherence, potentially modifying non–Gaussianities at a calculable, symmetry–constrained level. G. Synthesis The beyond–central higher–coherence branch extends consistently to semiclassical gravity: •RSET and conservation: Hadamardness and conservation survive under Φ t (Prop. X.1); the semiclassical Einstein equation is meaningful. •Backreaction: The in–in effective action yields an Einstein–Langevin equation with noise and dissipation fixed by the channel; detailed balance ⇒FDT (Theorem X.2). •Focusing and ANE: Local dephasing eliminates negative–energy advantages along null congruences in a smeared sense (Prop. X.3). •Entropy laws: The generalized entropy is nondecreasing at first order for stationary horizons under QDB (Theorem X.4); Bekenstein/QNEC–type bounds are stable under the flow. •Cosmology: de Sitter detailed balance gives controlled backreaction without secular instabilities. Thus, the same self–measurement mechanism that dynamically selects renormalized QFTs also harmonizes with gravitational thermodynamics: renormalisation, measurement, and horizon entropy emerge as facets of a single, local, symmetry–compatible structure. XI. INTEGRABLE QUANTUM FIELD THEORIES: FACTORIZED SCATTERING, FORM FACTORS, AND GENERALIZED HYDRODYNAMICS UNDER SELF–MEASUREMENT The previous sections established that beyond–central higher–coherence defects canonically generate local, covariant CP semigroups Φ t = etL on nets and that these flows are compatible with causality, symmetries, and renormalisation. We now analyze integrable (1 + 1)–dimensional quantum field theories (IQFTs), whose exact structures (factorized scattering, form–factor bootstrap, thermodynamic Bethe ansatz, and generalized hydrodynamics) provide an ideal nonperturbative arena to test the self–measurement paradigm. Our program is fourfold:
39 1. Construct integrability–preserving local GKLS generators built from conserved densities, prove charge conservation, and show that factorized scattering and Yang–Baxter constraints are preserved (“elastic dephasing”). 2. Analyze the effect on exact form factors and finite–temperature correlation functions à la LeClair–Mussardo; prove improved convergence and Hadamard microlocal admissibility. 3. Derive a charge–conserving kinetic equation at Euler scale (generalized hydrodynamics, GHD) with aCP collision integral that increases Yang–Yang entropy while preserving all local charges; prove an H–theorem. 4. Work out explicit examples (free Majorana/Ising, sine–Gordon) and connect to universal phenomenology. We assume standard IQFT structures [83–87]. A. IQFT primer: factorization, bootstrap, and TBA In (1 + 1) dimensions, integrability implies infinitely many local conserved charges Qs (including energy H = Q2 and momentum P = Q1 ) that commute with the S –matrix. Asymptotic states are labelled by rapidities θand internal indices a, |θ1, a1;. . . ;θn, ani, θ1>··· > θn, and scatter elastically with a two–body S –matrix Scd ab ( θ )depending on rapidity differences and solving the Yang–Baxter equation (YBE), unitarity, and crossing: YBE: S12(θ12)S13(θ13)S23(θ23) = S23(θ23)S13(θ13)S12(θ12), Unitarity: Sef ab (θ)Scd ef (−θ) = δc aδd b,Crossing: Scd ab(iπ −θ) = Sc¯ b a¯ d(θ).(58) Local operators O admit exact form factors FO n ( θ1, . . . , θn )satisfying Watson’s equations, kinematic pole axioms, and bound–state residue equations [ 84 ]. Thermodynamics is governed by the thermodynamic Bethe ansatz (TBA): the pseudoenergy ε(θ)at inverse temperature βsatisfies ε(θ) = β E(θ)−φ ? log1 + e−ε(θ), φ(θ) = 1 2πi ∂θlog S(θ),(59) with dressed energy E(θ) = mcosh θfor a single species, and convolution ?. B. Integrability–preserving local GKLS generators Let {Qs}s∈S be the (commuting) tower of local conserved charges with densities j0 s ( x ), i.e. Qs = Rj0 s ( x ) dx and ∂tj0 s + ∂xj1 s = 0. Fix a double cone ObR1,1 and smooth gs∈C∞ 0 ( O ). Consider the Lindblad densities Ls,α =Zdx gs,α(x)j0 s(x), s ∈ S, α ∈Is,(60) with a positive–definite Kossakowski matrix Γ(s,α),(s0,α0). Define the local GKLS generator on the net: L(A) = i[HO, A] + X s,s0,α,α0 Γ(s,α),(s0,α0)L† s,α A Ls0,α0−1 2{L† s0,α0Ls,α, A},(61) with HOlocal and selfadjoint. This is precisely of the form constructed in Section IX. Theorem XI.1 (Charge conservation and elastic dephasing) . For the generator (61) with (60) one has: 1. (All charges conserved)[Qs,L∗]=0, hence d dt hQsit= 0 for all s∈ S. 2. (Factorized scattering preserved) The asymptotic scattering remains elastic and factorized: the LSZ wave operators exist on the Fock space of asymptotic states, and the S –matrix is unchanged. In the Bethe basis, Φ t acts as pure dephasing in rapidity space (damping off–diagonal coherences) without particle production.
40 3. (YBE, crossing, unitarity) The bootstrap axioms (YBE, unitarity, crossing) remain valid for on–shell scattering amplitudes. Proof. (1) Since [ Qs, Ls0,α ] = 0 (they are built from commuting densities), the dissipator commutes with each Qs ; the Hamiltonian part [ HO,· ]is local and also commutes with Qs because HO can be chosen as a local function of the densities. Thus [ Qs,L∗ ]=0. (2) By Haag–Ruelle adapted to contractions (Section VI), the wave operators exist provided the dissipator is infrared supported and does not create particles. Here Ls,α commute with the number of particles asymptotically (being diagonal in the Bethe basis), hence no creation/annihilation occurs; the channel becomes diagonal phases in the rapidity occupation basis, i.e. dephasing. (3) Because the microscopic scattering is unaffected (no change of on–shell S ), the bootstrap axioms continue to hold for the elastic amplitudes. The nonunitarity appears only in off–shell correlators and inclusive loss terms (Section VI). Physics. There is a distinguished integrability–preserving family of channels: they leave the exact S –matrix and thermodynamics (TBA) invariant while introducing local, charge–respecting dephasing. These are the natural nonperturbative avatars of the safe dephasing branch. C. Form–factor expansions with dephasing kernels Equal–time correlators of a local operator O at inverse temperature β admit a LeClair–Mussardo (LM) series [88]: hO(0)O(x)iβ=∞ X n=0 1 n!X a1,...,anZdθ1···dθn (2π)n n Y j=1 1 1 + eε(θj)|FO n(θ1, . . . , θn)|2e−|x|Pjmajcosh θj,(62) with ε solving (59) . The self–measurement channel Φ t acts on correlators by convolution with a dephasing kernel in rapidity space. A general and bootstrap–compatible class is: FO,(t) n(θ1, . . . , θn) := Dt({θij})FO n(θ1, . . . , θn), Dt= exp−tX 1≤i<j≤n χ(θi−θj),(63) with χ even, positive–definite, and analytic in a strip containing Im θ∈ [0 , π ]and satisfying χ ( iπ −θ ) = χ(θ). Proposition XI.2 (Watson’s equations and kinematic poles preserved) . If Dt is symmetric and analytic as above, then the dephased form factors (63) satisfy Watson’s equations and the kinematic residue axiom with the same residues as FO n. Proof. Watson: symmetry under transpositions (i, i+1) requires F(t) n(...,θi, θi+1, . . . ) = S(θi−θi+1)F(t) n(...,θi+1, θi, . . . ). Since Dt is symmetric under permutations, it factors out of the Watson relation, which holds for Fn by assumption. Kinematic poles: near θij →iπ, FO n∼iRes θij −iπ FO n−2+regular. Analyticity and χ ( iπ −θ ) = χ ( θ )imply Dt is regular and equal for the two sets of variables near the pole, so residues are unchanged. Theorem XI.3 (LM series under dephasing: improved convergence) . Let Dt be as in (63) with χ≥ 0 and χ ( θ ) ≥c > 0for |θ| ∈ [ θ0,∞ )(some θ0> 0). Then the LM series (62) with Fn7→ F(t) n converges absolutely and uniformly for all x6 = 0 and any β > 0, and defines a correlation function with improved short–distance (large–momentum) behaviour; in particular, microlocal singularities are not worsened and the Steinmann scaling degree decreases with t. Proof. Each n –particle term acquires a factor exp−tPi<j χ ( θi−θj ) ≤exp ( −ct Npairs ( θ, θ0 )), where Npairs counts pairs with |θi−θj| ≥ θ0 . For large rapidities, many pairs contribute, giving exponential suppression of large–rapidity regions. Combined with the thermal factors and the e−|x|Pmcosh θ , one obtains absolute and uniform convergence by dominated convergence. Improved short–distance follows from the suppression of large momenta (Fourier–transform of θ –space damping) and the microlocal admissibility of Dt(analytic in the crossing strip).
41 Physics. The dephasing factor damps configurations with large rapidity separations—precisely those responsible for subtle resummation issues in finite–temperature correlators. The bootstrap stays intact; only interference is reduced. D. Charge–conserving generalized hydrodynamics with CP collision integral GHD describes Euler–scale dynamics of IQFTs in terms of quasi–particle root densities ρp ( θ ; x, t )and occupation n ( θ ) = ρp/ρs , where ρs is the state density including dressing [ 89 , 90 ]. In the absence of integrability breaking, ∂tρp(θ;x, t) + ∂x veff(θ;x, t)ρp(θ;x, t)= 0,(64) with veff the dressed group velocity. A local CP flow commuting with all charges should appear at Euler scale as a charge–conserving collision integral Ct[ρp]: ∂tρp+∂x(veffρp) = Ct[ρp],Zdθ hdr s(θ)Ct[ρp](θ)=0 ∀s∈ S,(65) where hdr sare dressed one–particle eigenvalues of Qs. Proposition XI.4 (Diffusive CP collision operator and H –theorem) . A canonical CP collision integral consistent with (65) is Ct[ρp] = ∂θDt(θ;ρp)ρs(θ)∂θµ(θ), µ(θ) := log n(θ) 1−n(θ),(66) where Dt ( θ ; ρp ) ≥ 0is a (dressed) rapidity–diffusion coefficient determined by the Kossakowski kernel of the channel. Then: 1. (Charge conservation) For any s , Rdθ hdr sCt [ ρp ] = 0 (integration by parts and the dressing equations). 2. (Yang–Yang entropy production) The Yang–Yang entropy density sY Y = Rdθ ρs−nlog n− (1 − n)log(1 −n)obeys ∂tsY Y +∂xJY Y =Zdθ Dt(θ;ρp)ρs(θ)∂θµ(θ)2≥0.(67) Proof. (1) The dressing map obeys linear integral equations with kernel φ ; using the identity ∂θhdr s = ( φ ? ( n ∂θhdr s ))+ ∂θhs and integrating by parts gives Rhdr s∂θ ( ··· ) = 0 assuming vanishing boundary terms (finite–entropy states). (2) Differentiate sY Y using ∂tn = (1 −n ) n ( Ct/ρs−∂x ( veffρp ) /ρs )and note that the convective term is a divergence; the collisional part gives (67) after substituting (66) and integrating by parts. Positivity follows from Dt≥0. Physics. Equation (66) is the Euler–scale footprint of local dephasing: a rapiditiy–space diffusion increasing Yang–Yang entropy but conserving all charges—a hydrodynamic avatar of the QDB entropy production of Section VII. E. Examples Free Majorana (Ising). The theory has S ( θ ) = − 1, a single species with dispersion E ( θ ) = mcoshθ . Take a translation–invariant pseudodifferential attenuator mt ( k ) = mt ( θ )with |mt| ≤ 1and a smooth Hadamard–compatible noise Nt. Two–point functions are explicitly hψ(x)¯ ψ(0)it=Zdθ 2π|mt(θ)|2f(θ) + δft(θ)e−m|x|cosh θ, where f is the usual spectral weight and δft≥ 0arises from Nt . The Källen–Lehmann representation is modified as in Theorem VI.1 with a positive reshaping of the spectral density; all bootstrap trivialities are preserved. The GHD diffusion coefficient Dt can be computed from the Kossakowski kernel built from ¯ ψψ.
48 Postulate (Self–measurement functor). A small local deviation [Ω] 6 = 0 in degree ≥ 3determines, functorially and uniquely up to equivalence, a local GKLS generator L with Kossakowski kernel fixed by [Ω] and the microlocal/BRST constraints of Sections IX and VIII. The next theorem is the rigorous core of this postulate at the infinitesimal level. B. A categorified Stinespring correspondence Let A be a local net on a globally hyperbolic M , and C its superselection category. Consider a small deformation of the associator a7→ a eκ with κ∈Z3(C, U(1)),1. Theorem XIII.2 (Infinitesimal curvature–to–GKLS map) . Let κ∈Z3 ( C, U (1)) be a localized 3–cocycle (supported in a double cone O) that is trivial on central/trivial sectors and compatible with the braiding. Then there exists a canonical completely dissipative quadratic form Qκon A(O)such that: 1. The closure Lκ of Qκ is a local GKLS generator (as in Theorem IX.2), with Kossakowski kernel Gκ obtained by transgressing κ along the SK contour (cf. §V A) and then projecting to gauge–invariant densities (Prop. IV.6, Theorem VIII.1). 2. If κ = dλ is a 3–coboundary, then Lκ is cohomologically trivial on the observable net (Prop. VIII.2); in particular, it reduces to a Hamiltonian derivation (triangle branch). 3. The map κ7→ Lκ is functorial under inclusions O1bO2 and covariant under spacetime embeddings (Theorem IX.9). Sketch. (1) Use the SK influence functional representation (26) and build a minimal positive noise kernel Nκ by pulling back the phase defect eκ to a quadratic form on field functionals via local Wick monomials obeying BRST covariance (§VIII). Positivity of Nκ follows from the U (1)–valued nature of κ to leading order and Hadamard admissibility by locality of the transgression. The resulting quadratic form is completely dissipative and closes to a GKLS generator by Theorem IX.2. (2) If κ = dλ , the SK transgression reduces to a boundary term that is BRST–exact; by Prop. VIII.2 the dissipator is cohomologically trivial. (3) Naturality is inherited from the functorial construction of Nκ using the locally covariant calculus (Theorem IX.9). Physics. Theorem XIII.2 formalizes the slogan “curvature ⇒ measurement”: higher coherence defects seed precisely the CP channels we have analysed. Coboundaries are gauge artefacts (triangle branch); only nontrivial curvature classes generate irreversible dynamics. C. Measurement problem: comparison with decoherence, GRW/CSL, and nonlinear QM Three families of proposals address the measurement problem: environment–induced decoherence [ 99 – 101 ], dynamical collapse models (GRW/CSL) [ 102 – 104 ], and nonlinear modifications of the Schrödinger equation [105]. Our construction differs in three decisive ways. 1. Intrinsic locality and covariance. The Lindblad densities and kernels are local covariant objects derived from coherence curvature; they preserve causality and Ward identities (Sections VI, VIII). Environment–induced decoherence often relies on additional baths; collapse models introduce explicit stochastic localisation terms that typically require preferred frames. 2. Linearity on states vs. convex–linearity test. Our baseline branch H1 (Section XII) is linear CPTP on density matrices, hence no–signalling is automatic. Weinberg–type nonlinear Schrödinger dynamics leads to superluminal signalling in the presence of entanglement [ 106 ]; our nonlinearity witness (Prop. XII.2) is designed to detect any departure from convex–linearity in a falsifiable way. 3. Renormalisation as measurement. The CP flow reduces scaling degree and selects EG extensions (§III), lending a dynamical origin to finiteness, in contrast with external regulators or ad hoc collapse parameters.
49 Physics. The present framework is conservative on kinematics (local QFT and AQFT axioms) and radical on dynamics (measurement is internal, geometry–tied, and symmetry–constrained). It predicts dephasing without particle loss in integrable sectors (Section XI) and inclusive unitarity deficits otherwise (Theorem VI.3). D. Reeh–Schlieder, Wigner–Araki–Yanase, and apparatus locality The Reeh–Schlieder theorem guarantees that the vacuum is cyclic for local algebras: local operations on Ωgenerate a dense set [ 107 ]. This underlies the in–principle possibility of constructing apparatus–like states within a bounded region. However, the Wigner–Araki–Yanase (WAY) theorem [ 108 ] constrains repeatable measurements of quantities that do not commute with conserved charges. Our channels respect both facts: Proposition XIII.3 (WAY–compatibility) . If the GKLS generator L is built from densities commuting with a global conserved charge Q (Section XI B), then for any A not commuting with Q the Heisenberg–picture instrument Φ ∗ t cannot implement repeatable sharp measurements of A without violating the positivity/commutation constraints; the obstruction is identical to WAY’s. Sketch. The CP Schwarz inequality together with [ Q, L∗ ] = 0 implies that the Naimark dilation of Φ ∗ t preserves the Q –superselection sectors; sharp repeatability of an incompatible A would require cross–sector coherences in the pointer, contradicting the sector–preserving property. Physics. Self–measurement is compatible with the existing algebraic constraints on what is measurable with bounded resources; it does not magically produce forbidden devices. E. Gravity and information: GSL and holographic hints Section X proved first–order generalized second law (GSL) monotonicity under detailed balance (Theorem X.4). Conceptually, this links the microscopic origin of irreversibility to horizon thermodynamics. Two speculative threads emerge: • Coherence curvature as a boundary term. In AdS/CFT–like settings, a boundary 3–cocycle may induce a bulk CP channel that realises a mixed–state analogue of double–trace deformations. The relative–entropy monotonicity then parallels entanglement wedge nesting. • QNEC stability. The stability/strengthening of QNEC under dephasing (Section XE) hints at a deeper equivalence between measurement–induced irreversibility and quantum focusing. These ideas remain conjectural but fit the pattern: curvature–tied dephasing increases appropriate entropies while preserving local constraints. F. Two unification theorems and an equivalence principle We summarise the logical equivalences suggested by our analysis. Theorem XIII.4 (Equivalence I: finiteness, entropy, and locality) . For a local net with Hadamard states, the following are equivalent at leading order in the curvature strength: 1. reduction of the Steinmann scaling degree and unique EG extension for all time–ordered products (§??, §??); 2. existence of a local GKLS generator with QDB such that S ( ρtkσ )is strictly decreasing unless ρt is stationary (§VII A, §VII B); 3. positivity of the reshaped Källen–Lehmann density and preservation of the Pauli–Jordan commutator (§VI A).
50 Sketch. (1) ⇒ (2): Unique EG extension implies well–defined local Wick polynomials and a canonical choice of Nt , hence a QDB–satisfying generator and entropy monotonicity (Prop. VII.2). (2) ⇒ (3): QDB and locality preserve Hadamardness and commutators (Theorem VI.1). (3) ⇒ (1): Positivity and commutator preservation reduce the UV degree of singularities (microlocal smoothing), forcing the canonical extension in EG (Section IIIC). Theorem XIII.5 (Equivalence II: integrability and dephasing) . In (1 + 1)–dimensional integrable QFTs, the following are equivalent for channels supported in a double cone: 1. exact factorized scattering with unchanged S–matrix (Theorem XI.1); 2. GKLS generators built from the full tower of conserved densities (Section XI B); 3. rapidity–space dephasing kernels preserving Watson/kinematic axioms (Prop. XI.2) and inducing entropy–increasing GHD diffusion (Prop. XI.4). Physics. Theorems XIII.4–XIII.5 articulate an equivalence principle for quantum irreversibility: whenever curvature–tied dephasing is present, finiteness, entropy production, and locality co–propagate. G. Open problems and conjectures We list precise questions suggested by our framework. Conjecture XIII.6 (Global curvature–to–GKLS functor) . There exists a functor S from the ( ∞, 2)–category of locally covariant nets with coherence curvature to the category of local CP semigroups such that S is natural, respects disjoint unions (tensoring), and reduces to Theorem XIII.2 infinitesimally. Conjecture XIII.7 (Higher c/a –theorems under self–measurement) . In d≥ 3, there exist monotone functionals (sphere or ball entropies with suitable modular weights) that decrease along curvature–induced flows and reduce to a/F–functions at fixed points; cf. Theorem VII.5 for d= 2. Problem XIII.8 (State–dependent but no–signalling dynamics).Characterise the largest class of state–dependent local CP evolutions Φ t [ ρ ]that preserve no–signalling (in the sense of AQFT locality) and satisfy an entropy monotonicity principle. Does a generalisation of Theorem IV.9 hold in this setting? Problem XIII.9 (QNEC stability).Prove the stability (or strengthening) of QNEC under the local CP flows of Section X, beyond first order in channel strength. Problem XIII.10 (Nonperturbative bootstrap with dephasing).Extend the form–factor bootstrap to include admissible dephasing kernels Dt at the level of axioms and solve explicitly in interacting models; control analyticity domains uniformly in t. H. Synthesis The conceptual upshot is a grand unification: •Triangle vs. higher coherence. Triangle defects ( H2 ) yield only central extensions: unitary, linear QM with projective phases. Higher–degree curvature ( H≥3 ) yields local CP flows: intrinsic measurement compatible with locality. •Renormalisation as measurement. The self–measurement flow smooths short–distance singularities, selects finite EG extensions, and shifts RG coefficients in a universal, scheme–independent way. •Entropy and gravity. Detailed balance makes entropy production a gradient flow and ties it to semiclassical backreaction and horizon thermodynamics (Einstein–Langevin, GSL). •Integrability and hydrodynamics. In exactly solvable sectors, the flow reduces to elastic dephasing and charge–conserving diffusion, providing a bridge to non–equilibrium physics.
51 •Falsifiability. The experimental programme (Section XII) isolates clean signatures: energy–scaling dephasing, inclusive unitarity deficits, slope shifts in running couplings, entropy Lyapunov behaviour, and convex–linearity tests. If borne out empirically, these ideas would reposition the measurement problem from an interpretational puzzle to a structural consequence of quantum locality and categorical coherence—not a bolt–on, but the very mechanism that makes QFT finite, predictive, and thermodynamically consistent. XIV. CONCLUSION: FROM COHERENCE CURVATURE TO FINITE, PREDICTIVE QUANTUM FIELD THEORY This work developed a coherent thread from categorical inputs to concrete, testable physics. The central idea is that higher–categorical coherence—the possibility that pentagon/hexagon (and higher) diagrams fail to commute—is not a peripheral curiosity but a structural datum that, when localized and made covariant, forces a canonical class of completely positive (CP) dynamics on local algebras. This CP dynamics realizes self–measurement, makes renormalisation dynamical, and preserves the fundamental locality and symmetry constraints of quantum field theory (QFT). Main contributions 1. Triangle vs. higher coherence : We gave a cohomological derivation showing that the failure of triangle diagrams (degree–2 cohomology) produces only central extensions, hence unitary, linear quantum mechanics with projective phases. In contrast, higher coherence curvature (degree ≥ 3) cannot, in general, be absorbed into a central extension and therefore induces genuinely dissipative dynamics compatible with locality. 2. Curvature ⇒CP semigroups : We constructed a functorial map from localized higher–coherence classes to local, covariant GKLS generators on Haag–Kastler nets. In the Heisenberg picture, L(A) = i[H, A] + X j,k ZZOj(x)AOk(y)−1 2{Ok(y)Oj(x), A}Gjk(x, y)d4x d4y, (78) with Gjk ≥ 0a Hadamard–admissible kernel fixed (up to equivalence) by the curvature class and symmetry/BRST constraints. 3. Renormalisation as self–measurement : The flow etL reduces the Steinmann scaling degree of time–ordered products and selects a unique Epstein–Glaser (EG) extension, thereby producing finite, predictive correlators without external regulators or UV completions. 4. Consistency with QFT axioms : The flow is local and covariant, preserves the canonical commutator and Hadamard microlocal spectrum, respects Ward/Slavnov–Taylor identities (and the Master Ward identity) in anomaly–free gauge theories, and is compatible with the split property and modular nuclearity (ensuring existence and locality of the semigroup). 5. Scattering and optical theorem : Asymptotic dynamics yields a CPTP scattering instrument. The optical theorem acquires a universal, non–negative inclusive loss term: 2Im Tii =X f|Tfi|2+ Σloss(i),Σloss ≥0,(79) reducing to the unitary equality when the curvature vanishes (triangle branch). 6. Entropy monotonicity and RG fixed points : Under quantum detailed balance (QDB), relative entropy to a faithful reference state is a Lyapunov functional, d dtS(ρtkσ) = −E[Ωt]≤0,(80) and a quantum log–Sobolev inequality implies exponential mixing to a stationary state. In 1 + 1 dimensions we proved an entropic c–theorem–type monotonicity.
52 7. Gravity and thermodynamics : The flowed state remains Hadamard, the renormalized stress tensor is conserved, and the backreaction admits an Einstein–Langevin description with fluctuation– dissipation under QDB. We showed a first–order generalized second law (GSL) for stationary horizons, tying measurement–induced irreversibility to horizon thermodynamics. 8. Integrability and hydrodynamics : In integrable QFTs, channels generated by conserved densities preserve the exact S –matrix and bootstrap axioms and act as elastic dephasing in rapidity space; at Euler scale they induce a charge–conserving, entropy–producing GHD diffusion. 9. Experimental roadmap : We formulated a universal effective parameterisation, convex feasibility for complete positivity and symmetry, Fisher–information design, and platform–specific tests (neutrinos, mesons, quantum optics, anyons, high–energy scattering, quantum simulators). We also proposed a convex–linearity witness as a falsifiable test for genuine state–dependent (nonlinear) dynamics. A unifying equation set The mechanism condenses into a small set of structural relations: Principle Equation Comment Local CP generator (78) Locality, covariance, BRST–compatibility. Källen–Lehmann reshaping ρt ( µ2 ) = |mt ( µ ) |2ρ ( µ2 )+ ρN ( µ2 ) ≥ 0 Positive spectral weight reshaping. Entropy gradient flow (80) Quantum detailed balance ⇒ Lyapunov functional and hypercontractivity. Inclusive optical theorem (79) Scattering as a CPTP instrument; inclusive loss term. EG selection reduced scaling degree ⇒unique finite extension Dynamical resolution of renormalisation ambiguities. What is preserved, what is modified • Preserved: Local commutativity, causality, isotony, covariance, canonical commutation relations, Hadamard spectrum, Ward/Slavnov–Taylor (anomaly–free), cluster decomposition, integrable elastic scattering (for charge–generated channels). • Modified: Off–shell interference (dephasing), spectral weights (positive reshaping), inclusive optical sum rules (nonzero Σ loss ), RG slopes (scheme–independent shifts), entropy (monotone), null–energy averages (suppression of negative pulses under local dephasing), and finite parts of loop amplitudes (dynamically selected). Assumptions and limitations Our constructions rely on: (i) locality and the split property (to implement local instruments); (ii) Hadamard admissibility (to keep the RSET meaningful); (iii) anomaly cancellation for exact gauge consistency; and (iv) small–curvature/weak–channel expansions for some theorems (e.g. perturbative gravity and integrability). We deliberately do not assume new on–shell degrees of freedom or preferred frames. Predictions and falsifiability at a glance 1. Energy–scaling dephasing with symmetry–fixed operator content: rates Γ( E )follow the scaling law extracted from operator dimension, not ad hoc power laws.
53 2. Inclusive unitarity deficits localized in kinematic windows set by the attenuator profile, without new thresholds. 3. Slope shifts in running couplings correlated across processes sharing loop kernels. 4. Entropy Lyapunov behaviour (relative–entropy decay) for channels satisfying QDB. 5. Convex–linearity: violation ⇒ genuine state–dependence; satisfaction alone does not confirm the mechanism but narrows alternatives. Any systematic violation of complete positivity, Ward/Slavnov–Taylor identities (in an anomaly–free sector), or locality would falsify the framework as stated. A practical recipe for using the framework 1. Identify the allowed local densities {Oj} by symmetry/BRST and the physical scale window of interest. 2. Choose an attenuator profile and kernel family Gjk obeying locality and Hadamard admissibility. 3. Compute predictions: (i) scattering with Σ loss ; (ii) RG slope shifts; (iii) equal–time correlators (finite EG–selected pieces); (iv) entropic monotones. 4. Fit the universal curvature parameters ( ε, Λ ,shape )with convex CP/symmetry constraints; deploy null tests (unitarity deficits, convex–linearity, entropy). 5. Cross–validate across platforms that probe the same operator content. Conceptual closing A single structural input—higher coherence curvature—binds together three pillars long treated as disparate: measurement, renormalisation, and thermodynamics. In the flat (triangle/central) case we recover unitary quantum mechanics with projective phases and regulator–dependent renormalisation. With curvature, locality promotes the defect into a self–measurement flow that both explains irreversibility and selects finite QFTs. The same flow preserves causality and gauge structure, dovetails with integrability, and is consistent with gravity’s second law. The path forward is twofold. Mathematically, complete the curvature–to–GKLS functor beyond the infinitesimal regime and extend the entropic monotones to higher dimensions. Experimentally, pursue the targeted signatures identified here, especially those that intertwine independent sectors (interferometry, running couplings, inclusive optical deficits). A convergent positive pattern would reposition the measurement problem from an interpretational quandary to a structural consequence of quantum locality and categorical coherence; a negative pattern would be equally valuable, carving away vast classes of beyond–central dynamics. Either way, the message is crisp: coherence curvature is the missing dynamical datum. When it vanishes, we live in the familiar projective, unitary world; when it is present but local and symmetric, nature self–measures, QFT becomes finite, and entropy acquires a microscopic, geometric origin. XV. STATEMENTS •The authors declare no conflict of interest •No new data has been generated for this manuscript [1] H. Epstein and V. Glaser, The role of locality in perturbation theory, Ann. Inst. H. Poincaré A 19 (1973) 211–295.
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