Phase decomposition and Wightman reconstruction

Phase decomposition and Wightman reconstruction

From positive readings to operators on a common domain. Reflection-positive pairings determine a seminormed word space. Zero-norm words are quotiented before Hilbert completion. Equality of the entire reflected Gram matrices under readout therefore gives an isometry on the quotients and a unitary identification of their completions. It does not turn every insertion into a bounded operator. The reconstructed fields act on the specified common invariant span of smeared field words applied to the vacuum.

The following slab, heat and boundary arguments identify that domain, the attachment map and the tube boundary distributions. Character evaluation is applied to whole compressed words, with the common phase measure retained. Full support and continuity have a specific role in upgrading certain almost-everywhere statements to every character; they are not a universal permission to remove exceptional sets. Likewise the one-dimensional invariant vacuum follows from the heat and phase arguments. No positive lower bound on the rest of the energy spectrum follows from vacuum uniqueness. These distinctions are preserved when the same Hilbert-space readings are transported to the realized family.

19.1 Bounded slab insertions and their norm derivatives

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Fix an axis a, width d > 0, and word w = ((xi, vi))n
i=1 of distinct points inside 0 < xa
i < d. A
buffer 0 < r ≤1 requires pairwise maximum-coordinate distance at least r and r ≤xa
i ≤d −r. For
degrees at most D and coefficient norms at most R, we construct

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∥Ud(w)∥≤Cn(n/r)An,
(BS1)

with depending only on. The empty word has bound one. On a future plant the prescribed action inserts before that plant translated by. Boundedness precedes extension to the Hilbert completion.

The centered one-cell estimate gives a threshold, exponent, and 0, such that one insertion at level has price . Include all signed rotations and reflected conjugates by enlarging the coefficient ball by their finitely many operator norms on the finitely many raw degree spaces. These constants consequently work for every reflected copy.

The decreasing schedule satisfies At the first below32, we have ; decrease 0 to include. Choose fixed sufficiently small 0 and the first below. The same first-crossing argument gives

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ak ≥c(r/n)4,
20nak < am,
k ≥m.
(BS2)

Indeed 20, and strict decrease gives. Exterior insertions are paid at still finer scales, chosen after this main scale; their prices may depend on the exterior word.

First let be an integer multiple of. Alternating insertion of and its reflection produces, in the squared norm after repetitions, 4 main copies and two exterior copies. The main positions repeat the original 2 position patterns modulo. Thus cell packing needs only 20 independently of. The buffer separates the main train from the translated exterior. The rooted product estimate and the fixed finite-word lattice limit give

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∥plant of the repeated word∥2 ≤Bp[Ja−P
k
]4nq,
(BS3)

with independent of. This is the train estimate in Next/PHASE/Trains.travel_price and Seats.replicas. By (BS2), its main price is

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Q = [Ja−P
k
]n ≤[Jc−P ]n(n/r)4Pn.

We may use the latter expression for

To remove the exterior prefactor, let send algebraic future words to the reflection Hilbert space, insert the main word, andinsert its opposite reflected word. Reflection and translation invariance give. Put . Taking square roots in (BS3) and finite linear combinations gives. Therefore = supis finite. Moving powers of across the pairing yields

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(Q−2q∥XT qu∥)2 ≤∥Xu∥su.

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u ≤∥Xu∥su, hence su ≤∥Xu∥. Finally ∥XV u∥2 = ⟨Xu, XTu⟩≤Q2∥Xu∥2.
Taking suprema gives s2
Insertion thus annihilates null vectors, descends to the quotient, and extends uniquely with norm
at most Q. This is OS/Train.producer.
For general d, use d′ = am⌊d/am⌋. Since am < r/32, the same word lies in the narrower slab
with buffer r/2. Compose its operator on the right with the heat contraction of width d −d′. On
plants this is exactly the required insertion at width d, and its norm does not increase. The empty
word is the heat contraction. All intermediate choices give the same operator because they agree
on the dense plants. Replacing r by r/2 only enlarges C, proving (BS1) for one common family of
operators. This is the padding and uniqueness argument of OS/Halves and Work/B01.

We next prove operator-norm regularity. Fix labels and width, and write the open set of distinct slab configurations. For a slot-axis index), replace its label by a representative of its physical-derivative and call the resulting placement). Set

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B(x)h =
X
b
hbFb(x).
(BS4)

Near a fixed configuration there is one positive buffer for the original word and these finitely many differentiated words. Equation (BS1) makes locally bounded in operator norm. Between two plants, the placement is a separated correlation with a fixed exterior word. The scalar slot-derivative law gives its derivative as the matrix element of), also for finite linear combinations of plants. On a convex position ball suppose. The scalar mean-value estimate gives

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|⟨u, [F(y) −F(x) −B0(y −x)]v⟩| ≤η∥u∥∥v∥∥y −x∥.

Density gives the same estimate in operator norm. First set = 0: local boundedness makes every placement locally Lipschitz. Apply this also to each differentiated label array; all, hence , are norm continuous. Then set) and shrink the ball. Its oscillation tends to zero, proving) in norm. Induction with differentiated labels gives smoothness of every order on the same position domain. This is the argument of Work/B02/Analysis.norm_remainder and Work/B02/Slab.smooth_place; it establishes the norm regularity used in the waves and compression-valued moments below.

Source–readout formulation. Insertion of a word is first defined on the algebraic word space (the source side); Proposition 19.1 is the congruence statement that allows it to be read on the reflection quotient: it maps the null space to itself and is bounded. The bound (BS1) is uniform in the exterior word, which is what separates an operator on the recovered Hilbert space from a family of matrix-element estimates. The derivative) is computed with the quotient derivative of Theorem 18.3, so the operator-valued map) is differentiated entirely on the readout side, with no reference to a representative. By Theorem C.1 the lifted slab operators are the unitary images of these and satisfy the same bounds.

Proposition 19.1 (Removal of the exterior prefactor). Let send algebraic future words to the reflection Hilbert space, insert the main word andits opposite reflected word, so that for and all, then Hence insertion annihilates null vectors, descends to the quotient and extends uniquely with norm at most

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Proof. su = supq Q−2q∥XT qu∥is finite. Moving powers of T across the pairing and using Cauchy–
Schwarz, ∥XT qu∥2 = ⟨Xu, XT 2qu⟩≤∥Xu∥∥XT 2qu∥, so (Q−2q∥XT qu∥)2 ≤∥Xu∥su; taking the
supremum, s2
u ≤∥Xu∥su and su ≤∥Xu∥.
Finally ∥XV u∥2 = ⟨Xu, XTu⟩≤∥Xu∥· Q2su ≤
Q2∥Xu∥2.

19.2 Surjectivity of the reflection attachment

The boundary attachment is an isometry from a complete Hilbert space, so its range is closed. For regularized words integrated against Fourier transforms of compact smooth tests, the boundary tilt identifies the vector with an attached boundary distribution on the chilled test. Explicitly the joint pairing gives; regularized approximants to and partial-boundary convergence give. Thus = 0. The strict-time and independently heated constructions below supply these pairing and tilt inputs. Compact cutoffs approximate Fourier Schwartz tests. The word bound and closed range extend membership to all integrated regularized words. Pairing a pointwise word with a vector orthogonal to the range gives zero smeared pairings; continuity makes the pairing zero everywhere. Double orthogonal complement and closedness place pointwise words in the range. Strictly ordered positive-time Euclidean words are regularized words: choose regularization below the first time and all gaps, then recursively apply local fields, translations and heat operators. This reproduces the reflection plant with its insertion representatives; quotient representatives agree against every future word and give the same plant. For arbitrary future words sort times by permutation symmetry, then add to theth time. For small positive epsilon the configurations remain separated with strict positive time order and converge back. Expanding plant squared-distance into three table pairings proves norm continuity from smooth separated correlations. Since the range is closed, the unperturbed plant also lies in it. Future plants span densely by GNS, so is onto. Here GNS denotes the Gelfand–Naimark–Segal positive-functional construction [8, 9]; the domain and quotient recovery are the ones specified above. The lift retains boundary word tests, chilled tests, representatives and heat/translation recursion. Recovery preserves the pairing identities and plant norm limits, transporting this surjectivity to the lifted completed space with its actual word domain.

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Attachment recovery after Hilbert completion.
Let UB : HB →HZ
B and UR : HR →HZ
R
be the section-induced isometries onto the completed source boundary and reflection spaces. They
descend through the exact null preimages before completion. Define

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JZ = URJU−1
B ,
U−1
R JZUB = J,
Ran JZ = UR Ran J.

Thus is isometric with closed range. For the source’s regularized vectors, its two pairing equalities give on both carriers

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∥URA −URB∥2 = ∥A∥2 + ∥B∥2 −2 Re⟨A, B⟩= 0.

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Boundedness of UR commutes with the vector-valued test integrals and their L1 approximation.
The orthogonal-complement reader is recovered by U−1
R , preserving the passage from zero tested

pairings to zero continuous pointwise pairings. For future plants the same map preserves the squared-distance table formula and hence the norm limit. Therefore closedness and the dense span of future plants recover surjectivity on the completed source reflection space as well. This construction preserves the supplied tilt, partial-boundary and joint pairing dependencies and uses their actual heat/translation word recursion and insertion representatives.

19.3 Explicit tested boundary formula

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For word v smear its wave as Ψ(y, f) against a joint Schwartz test. Set ai = i and D = −i∂dawn(a).
The tested Cauchy–Riemann identity reads d
dtΨ(y + ta, Dkf) = Ψ(y + ta, Dk+1f). Assume |W(x +
iy)| ≤Ck−m for gaps at least k, 0 < k ≤1. For nondecreasing y, define

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m
X
(−t)k
k!
Ψ(y + ta, Dkf).
Pm(y, f; t) =
k=0

Adjacent derivative terms cancel, leaving!, bounded for . Thus

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m
X
(−1)k
k!
Ψ(y + a, Dkf)
B(y, f) =
k=0
−
Z 1
0
(−t)m
m!
Ψ(y + ta, Dm+1f) dt

is defined on all nondecreasing heights. For strict heights the fundamental theorem gives Interior continuity of the finite terms and dominated convergence of the remainder prove continuity at every nondecreasing. Hence every strict-height approach has limit), with unrestricted gap ratios. Moreover

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m
X
k=0
∥Dkf∥L1/k! + C
m!∥Dm+1f∥L1.
|B(y, f)| ≤C

This continuous Schwartz seminorm, linearity and label multilinearity give a tempered multilinear boundary distribution. Uniqueness makes admissible choices of agree. Character-independent give boundary character continuity and passage of barycenters to the boundary. The construction of the waves and the tested Cauchy–Riemann and polynomial gap estimates used here are given in (SW1)–(SW7) and (CR1)–(CR2) below. The lift retains the joint test, differential operator and cancellation polynomial. Recovery commutes with finite differentiation and the dominated integral, preserving the boundary distribution and its unrestricted strict-height limit.

19.4 Character integration and heat-ground identification

For an arity- word, regularized factor bounds at8 give, for time gaps at least

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 Y
a
Qa8da
!
k−P
a da,
|Wχ(v; x + iy)| ≤

uniformly in character and real positions. Phase-to-unselected factor-norm comparison supplies the same constants: the factors are constructed in (SW1)–(SW3), and the phase norm comparison is (ST1)–(ST4), using the bounded slab construction (BS1)–(BS4). Strict phase values are continuous

and barycenter to the unselected value. For joint Schwartz is integrable; dominated convergence proves smeared character continuity, and probability normalization plus Fubini gives the strict smeared barycenter. The same bound, combined with the probability normalization of the phase measure, makes the double integral absolutely integrable. The common finite Taylor-ray boundary representation passes continuity through its finite strict terms and dominated remainder. Its polynomial part uses finitely many strict evaluations; its remainder has a common integrable bound on the ray parameter after enough test derivatives. Its uniform boundary bound permits dominated character integration, giving the Wightman barycenter with the same gap-order and derivative inputs. The tested boundary formula and strict-time wave estimates below provide this Taylor representation and its common polynomial-order bound.

For positive reflection heat contractions with strong limit Π, the joint momentum measure gives

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⟨x, Esx⟩=
Z
e−sE(p) dνx(p).

Nonnegative energy makes the integrand bounded by one and tending to. Dominated convergence and strong convergence give equal quadratic forms for Π and); polarization proves equality. For attachment has boundary Boltzmann diagonal measures. Its zero-energy intertwining and give strong limit). No rank-one premise enters this step; locality and character arguments establish vacuum rank later. Separate surjectivity realizes all reflection heat evolution on the boundary space. The lift retains whole-word character readings, common factor bounds and joint spectral measures. Recovery commutes with dominated test/character integration and spectral quadratic forms, preserving the barycenter and zero-energy projection identification without strengthening its rank claim.

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Tested boundary and spectral recovery maps.
Use the section-induced Hilbert identifica-
tions already constructed to lift each word before taking its scalar matrix element. For a fixed
word, write ΨZ
χ for that scalar source reader. Unitary recovery gives ΨZ
χ(y, f) = Ψχ(y, f) at strict
heights, for the same joint test and pinned labels. Since D acts on the test alone, for every k

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ΨZ
χ(y, Dkf) = Ψχ(y, Dkf),
P Z
m,χ(y, f; t) = Pm,χ(y, f; t).

In particular the cancellation in the differentiated finite sum recovers exactly the same remainder integrand

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RZ
m,χ(y, f; t) = (−t)m
m!
Ψχ(y + ta, Dm+1f),
|RZ
m,χ(y, f; t)| ≤C
m!∥Dm+1f∥L1.

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The constant is independent of χ, 0 < t ≤1, and nondecreasing y in the specified wave-bound
regime. Finite-sum recovery and this dominated integral define the same Bχ(y, f) on the closed
height chamber. Its Schwartz seminorm, multilinearity, uniqueness across admissible m and all
strict-height approaches are therefore preserved. For the normalized character probability measure,
the strict Fubini identity and this common majorant give
Z
Bχ(y, f) dP(χ) = B(y, f).

The finite strict terms and dominated remainder also recover continuity in. Thus the boundary barycenter and character continuity use the same factor-norm bound and tested Cauchy–Riemann inputs as the source.

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For the reflection-space unitary section UR, retain its joint spectral measure by P Z(A) =
URP(A)U−1
R
and heat evolution by EZ
s = UREsU−1
R . The exact measure reader is

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⟨URx, P Z(A)URx⟩= ⟨x, P(A)x⟩,
ΠZ = URΠU−1
R
= P Z({E = 0}).

Strong limits commute with these bounded identifications. With the boundary section and retained attachment, one also has

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(JZ)∗EZ
s JZ = UB(J∗EsJ)U−1
B
strong
−−−−→
s→∞
UBPboundary({E = 0})U−1
B .

This recovers the zero-energy projection with precisely its supplied rank scope. Subsequent vacuumrank arguments remain separate dependencies.

19.5 Weak irreducibility from heat evolution

Let fields Φ act on their common word domain, with cyclic unit vacuum Ωand translations Suppose bounded satisfies

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⟨Φ(a, f)∗u, Bv⟩= ⟨u, BΦ(a, f)v⟩

for domain vectors, with the actual adjoint restricted there. No domain preservation by is assumed. Iteration moves words across the pairing. ForΩ, the heat attachment gives bounded continuous upper-half-plane functions, holomorphic inside, with boundary values ΩandΩ. Covariance, vacuum invariance and weak word commutation give). Glue) above to) below. Continuity and rectangular Cauchy integrals with cancelling axis boundaries prove entire holomorphy. Boundedness and Liouville make it constant; thus ΩΩ

The constructed strong heat limit to the vacuum projection givesΩΩ word density. Moving words once more yieldsΩΩ. Density and boundedness imply. Scalars satisfy the identity by the field adjoint relation, so the weak commutant is precisely scalar. Heat contraction, holomorphic-edge and vacuum-projection results remain its stated inputs. The lift retains the common domain and actual adjoint pairings. Unitary recovery transports the bounded operator and word vectors, preserving this weak identity and proof without an extra domain-invariance assumption.

Recovery of the weak commutant on its actual domain. Fix the completed-space section unitary. Set Φ on, and . Unitary equivalence of the actual adjoint gives (Φ Φ with domain Dom Φ. Consequently the precise weak pairing is recovered by

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⟨(ΦZ)∗V u, BZV v⟩= ⟨Φ∗u, Bv⟩,
⟨V u, BZΦZV v⟩= ⟨u, BΦv⟩,
u, v ∈D.

The two weak commutants are therefore in bijection under bounded-operator conjugation; no image of under a field is introduced. Finite word pairings recover in the same order, withΩ = Ω. The heat and translation readers give the identical scalar above, so the glued entire function and its constant value are unchanged. WritingΩΩ, the final density step reads

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BZΩZ = aΩZ,
⟨V u, BZwZΩZ⟩= a⟨V u, wZΩZ⟩,
BZ = aI.

Inverse conjugation recovers. This transports the supplied vacuum-projection argument as a dependency of weak irreducibility, without changing the common word domain or the weak hypothesis.

19.6 Low-energy shells and uniform heat decay

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Let positive forward-cone spectral measures μ, ν of a compact word combination and its adjoint
have total mass sum at most M. Their commutator difference A(x, t) = bμ(x, t)−bν(−x, −t) vanishes
for |x| > R, |t| ≤1, uniformly in character by locality. Set

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D(x) =
Z 1
2i
dt,
ρ(E) =
Z 1
0
sin(Et) dt.
0
A(x, t) −A(x, −t)

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Here ρ = (1 −cos E)/E ≥0 for E > 0, ρ(0) = 0, |ρ| ≤1 and ρ(E) ≥E/π on [0, 1]. Fubini makes
D the spatial Fourier transform of the sum of positive ρμ and spatially reflected ρν measures. It
is bounded by M and supported in radius R. For 0 < u ≤1, choose smooth bump Bu equal to
one on momentum radius u, supported within 2u, between zero and one. Its Fourier transform is
bounded in absolute value by the volume of the radius-2u ball, hence by C3u3. Fourier inversion
and positivity imply
Z
ρ(E)Bu(pspatial) dμ ≤K(R, M)u3,
μ{u/2 < E ≤u} ≤2πK(R, M)u2.

The second uses cone support and) on the shell. Dyadic decomposition of (0] sums and yields a constant times. Compact-character word-norm continuity supplies uniform masses. For any such measure with, Tonelli gives

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Z
E>0
e−sE dν = s
Z ∞
0
e−stν{0 < E ≤t} dt ≤2Cs−2.

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Substitution and two integrations by parts evaluate the last integral. Increase C to total mass to
extend an initially small-u bound. This bounds ⟨x, (e−sH −P0)x⟩, keeping the zero atom separate.
Character continuity of translated word pairings gives characteristic- function continuity. Fourier
inversion with majorant M|bg| gives continuity for compact smooth momentum tests. Approximate
heat multipliers by cutoffs equal to one in radius 2N and supported in 3N; cone support bounds
the omitted tail by Me−sN uniformly in character. Heat pairings are thus continuous uniform
limits. These arguments supply the heat continuity and shell-to-heat steps used in the removal
of exceptional characters. Fubini applies since both measures are finite and the integrands are
bounded. The lift retains word measures, locality radius, positive time averaging and spectral cone.
Recovery preserves these Fourier/Tonelli integrals and uniform majorants, transporting shell and
heat bounds while retaining the zero-energy atom.

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Positive spectral averaging and heat-decay recovery.
For each retained phase section Vχ,
the scalar spectral measure of a compact word vector is unchanged: if xZ
χ = Vχxχ and P Z
χ (A) =
VχPχ(A)V −1
χ , then
νZ
χ (A) = ⟨xZ
χ, P Z
χ (A)xZ
χ⟩= νχ(A).

The same identity holds for the adjoint word combination. Thus their commutator reader gives and the time-average reader gives . In particular locality preserves the same

radius, while positivity of the two weighted measures and reflected survives unchanged. The source recovers the exact dyadic decomposition

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∞
X
j=0
μ{u2−j−1 < E ≤u2−j},
0 < u ≤1.
μZ{0 < E ≤u} =

The geometric shell bill and the total-mass extension therefore give the same bound. For the heat reader recovers

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⟨xZ
χ, (e−sHZ
χ −P Z
0,χ)xZ
χ⟩=
Z
E>0
e−sE dνχ ≤2Cs−2.

Here the zero-energy atom is removed before the Tonelli calculation. The common mass bound and cone support also preserve the cutofferror for heat pairings. Hence their character continuity follows from the same compact-test Fourier readers and uniform cutofflimit, with no extra continuity premise on the ground projection.

19.7 Every-character vacuum uniqueness

Full support of the compact character probability means every nonempty open set has positive measure. A closed almost-everywhere set is therefore all characters. For compact-word sum vectors , the almost-everywhere unique heat ground and inverse-square estimate give, with common

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0 ≤⟨xχ, e−sHχxχ⟩−|⟨Ωχ, xχ⟩|2 ≤Ms−2,
s ≥1.

Attachment identifies zero energy with the one-dimensional heat ground in those phases; normalization fixes its vacuum coefficient. Heat pairings and overlaps are continuous, so the displayed inequality defines a closed set for fixed. Full support extends it to every character without assuming ground-projection continuity. For any fixed character, the spectral strong limit and the inequality give

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∥Pχxχ −⟨Ωχ, xχ⟩Ωχ∥2 = ⟨xχ, Pχxχ⟩−|⟨Ωχ, xχ⟩|2 = 0.

Compact test cutoffs approach Schwartz words by joint-word continuity. Bounded projections then agree on Schwartz words and, by cyclicity, on the whole Hilbert space. Thus zero energy has rank one in every character, supplying the weak-commutant heat limit. The lift retains full-support measure, continuous word pairings and attachment. Recovery preserves the closed-set extension and spectral limit, transporting vacuum uniqueness with the stated low-energy and cyclicity inputs.

19.8 Propagation of analytic limits on a complex slice

Let connected open carry eventually holomorphic. At each point assume a ball with common tail bound, the tail allowed to depend on the ball. Suppose holomorphic is the limit on a set accumulating inside. Choose = min( = max(1). For ), the closed disc of radius about lies in. Cauchy derivative estimates on interior radius- discs give local tail bounds

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|F ′
a(z)| ≤D,
|Fa(y) −Fa(x)| ≤D|y −x|.

Any everywhere pointwise limit inherits this Lipschitz bound. Choose) and a tail with center error below4; triangle inequality gives local uniform convergence and holomorphy