Phase decomposition and Wightman reconstruction · Part 6

19.33 Spacetime representation, cone support and attachment

Transform every letter by the finite-dimensional label action and inverse placement of its test. Boundary covariance and compatibility with the letter adjoint preserve

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U(g)[b] = [g · b]

extends to an isometry with inverse) and satisfies the group law. Strong continuity requires no topology on the full label union: pin the finitely many labels in one finite field space and expand in a fixed basis. Each moved expectation is a finite sum of continuous label coordinates times continuous fixed-label pairings with moved tests (BC4). Mixed word pairings are continuous; the squared-norm identity gives norm continuity on words and finite sums. For an arbitrary vector, approximation within3 and isometry control both errors uniformly in. The vacuum is fixed, is invariant, and on (FR4)

The attachment is isometric by FR1, sends the vacuum to Ω, and intertwines translations.

For completeness retain the group-generation calculation. For a strictly future timelike vector with time 0, a coordinate boost of rapidityartanh() sets to zero, since sinh cosh = 0. It preserves the other spatial components and the strict future cone. Three such boosts send any future unit timelike vector to. Given proper orthochronous, choose their product with Preservation of the Lorentz pairing makes preserve time, and preservation of the quadratic form then makes it an orthogonal spatial turn with the original determinant. Hence) is generated by the coordinate boosts and spatial turns used in the boundary-covariance argument.

The absolute-frequency cone is the intersection of inverse images, under all, of the FT3 time cone. Pulling a test disjoint from one such inverse image through, applying time-cone support for the transformed labels and transforming back proves support in each inverse image. Its intersection has the DC1 suffix description: a weakly increasing step height, zero before a suffix and one on it, forces nonnegative suffix time momentum in every Lorentz frame. A vector with that property is future causal: negative time is excluded by the identity, and spatial norm larger than nonnegative time is excluded by a boost toward its spatial direction. Constant positive and negative heights force total time momentum zero; the total suffix is future causal, hence also has zero spatial part. Conversely, zero total momentum and future-causal proper suffixes imply every time-cone inequality by telescoping, in every frame. This uses absolute frequencies without switching to relative coordinates. Let be the joint translation projection-valued measure, with

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⟨v, U(a)v⟩=
Z
ei⟨p,a⟩dμv(p),
μv(B) = ⟨v, P(B)v⟩.

The isometric translation attachment makes these characteristic functions equal to the chamber ones at. This follows by conjugating translations and comparing characteristic functions, using preservation of Hilbert inner products. Uniqueness of Fourier transforms of finite measures identifies their diagonal measures; the chamber measures have nonnegative time support by the following semigroup construction. Lorentz conjugation of translations gives

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μU(r)v = (r∨)∗μv.
(SP1)

Ifsupp, every therefore has nonnegative time component. The preceding characterization puts in. Every diagonal measure has this support, and polarization yields

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P(V c
+) = 0.
(SP2)

This concerns the actual field translation representation in addition to the hierarchy’s distributional support. In the lifted representation, FR4 and the intertwining isometry retain the field domain and all translation matrix elements. Fourier uniqueness recovers their finite spectral measures; SP1 transports those measures and SP2 recovers the spectral projection statement. No surjectivity onto the entire chamber is required for this transfer.

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Spacetime and spectral projection recovery.
For the completed word section VW , define
UZ(g) = VW U(g)V −1
W . On each word class this is exactly the lifted letterwise motion, so its group
law, vacuum and invariant domain recover those of the source.
Strong continuity is preserved
quantitatively by
∥UZ(g)VW v −UZ(g0)VW v∥= ∥U(g)v −U(g0)v∥.

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For the source’s dense-word approximation p to v, the latter is bounded by 2∥v −p∥+ ∥U(g)p −
U(g0)p∥. This keeps the finite-label/Schwartz proof on the dense span and its extension to arbitrary
vectors. Conjugating the actual domain identity (FR4) recovers covariance on DZ.
Let VR identify the chamber source and define its attachment by IZ = VRIV −1
W . It remains an
isometry and reads
IZVW [b] = VRηb,
IZUZ(a) = UZ
R(a)IZ.

This isometry is enough to recover the diagonal translation measures; its range need not be the entire chamber at this step. For the actual field spectral measure define

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P Z(B) = VW P(B)V −1
W ,
μZ
VW v(B) = ⟨VW v, P Z(B)VW v⟩= μv(B).

The characteristic-function reader and finite-measure Fourier uniqueness therefore identify the same attachment measures on both carriers. The Lorentz action and dual frequency map recover (SP1), while the retained coordinate-boost generation and suffix-cone argument recover the same cone support. In particular the spectral projection readout is

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V −1
W P Z(V c
+)VW = P(V c
+) = 0.

Thus (SP2) is preserved for the translation representation of the actual completed field space, with its stated chamber-semigroup nonnegative-time dependency.

19.34 Heat contractions from reflection-positive words

For finite sums of future words let be their chamber vectors and their forward translation by0. Reflection and translation of the Euclidean hierarchy give

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⟨XTtp, XTuq⟩= ⟨Xp, XTt+uq⟩.
(OS1)

The fixed-word correlation bound implies = sup: bound each translated word norm by the square root of its reflected two-word correlation bound and sum the finitely many coefficient-weighted bounds. Then

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∥XTtp∥2 = |⟨Xp, XT2tp⟩| ≤∥Xp∥c.

Taking the supremum gives, hence(including = 0). Forward translation preserves null vectors and extends to a contraction) on the completion. OS1 makes it self-adjoint. Translation parallel to the reflection plane preserves the Gram form and has its opposite as inverse, giving the commuting spatial unitaries. Semigroup and commutation laws hold on planted

words and extend by density. Strong continuity on planted words follows by expanding squared norm differences into continuous Euclidean correlations; finite sums and uniform contraction bounds extend it to the completion. The spatial group is strongly continuous by the same argument. Thus the wave construction uses a recovered self-adjoint contraction semigroup, rather than a separately postulated heat evolution. The lifted source is the planted-word Gram form with its translation operations; the quotient reader preserves OS1 and its null ideal, recovering exactly these completed operators.

19.35 Holomorphic heat powers and the energy-support reader

Each is positive with spectrum in [0 1]. Set(1) and define, for Re

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F(z) = Az,
0z = 0.

The function is continuous on [0 1], uniformly continuous in on compact subsets of this half-plane. Bounds on log on smaller half-planes justify differentiation in the uniform norm, so functional calculus gives operator-norm holomorphy and

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∥F(z)∥≤1,
F(z + w) = F(z)F(w),
F(z)∗= F( ̄z).
(HE1)

These operators commute with spatial translations. For positive integers, put then and. The scalar equality ( on the nonnegative spectrum gives). Rational approximation, norm continuity of and strong continuity of give) for every = 0, the semigroup makes = 0 for. Analytic uniqueness for makes it zero throughout the half-plane; strong continuity at zero then gives = 0. Self-adjointness implies dense range because its orthogonal complement is the kernel. HE1 givesfor 0: both squared norms are the quadratic form of Hence

defines an isometry on a dense range and extends uniquely. Checking there with HE1 gives

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V (s)F(z) = F(z −is) = F(z)V (s),
V (s)V (t) = V (s + t),
V (0) = 1.

Thus) is unitary with inverse) and commutes with spatial translations. Norm holomorphy gives strong continuity on ran(1); density and isometry extend it to all vectors. This recovers the chamber real spacetime translations. The sign) agrees with the convention

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Word translation, heat powers and edge recovery.
On the source future-word carrier retain
the actual translation Tt and pull back the chamber Gram form through its word readout. It gives
exactly (OS1), the same finite-word supremum c, and the same estimate c ≤∥Xp∥.
Its null
preimage is invariant under Tt, so descent and completion produce EZ(t) = VRE(t)V −1
R , where VR
is the chamber section unitary. In particular,

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∥EZ(t)VRv −EZ(u)VRv∥= ∥E(t)v −E(u)v∥,
(EZ(t))∗= EZ(t),
∥EZ(t)∥≤1.

The dense planted-word continuity and contraction approximation therefore recover strong continuity without a new heat-flow choice. Spatial translations recover by the same Gram-preserving word operations.

For (1), continuous functional calculus commutes with unitary recovery, giving throughout

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F Z(z) = (AZ)z = VRAzV −1
R ,
F Z(t) = EZ(t),
ker EZ(t) = VR ker E(t) = {0}
(t > 0).

Thus positivity, (HE1), holomorphy and dense range retain their source proofs and domains. Define the edge reader initially on that dense range:

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V Z(s)EZ(1)VRv = F Z(1 −is)VRv = VRF(1 −is)v.

The equal-norm identity makes this the same isometric extension as . Consequently its group law, spatial commutation, strong continuity and the sign 1 recover exactly the chamber real-time edge used in the wave and spectral constructions.

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Retain also the source’s explicit support proof. Suppose a finite positive measure μ, continuous
real ω and bounded function h, continuous on the closed upper half-plane and holomorphic inside,
R eitω(q)dμ(q) and |h(z)| ≤C. For real b and ε > 0, integrate h(z)e−ibz−εz2 around
satisfy h(t) =
rectangles between heights 0, s. The vertical sides vanish as their real coordinates tend to infinity,
with Gaussian bound proportional to e−εT 2. The horizontal Gaussian majorants are integrable, and
Cauchy’s theorem gives

Z
 ≤C
q
h(t)e−ibt−εt2dt
π/ε ebs+εs2.
(PA1)
R

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Fubini and the scalar Gaussian Fourier integral give
Z
R
h(t)e−ibt−εt2dt =
q
π/ε
Z
e−(ω(q)−b)2/(4ε)dμ(q).
(PA2)

If a support point had 0, the open set would have finite positive measure. Choose 0. Positivity and PA1–PA2 imply

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ae−b2/(16ε) ≤
Z
e−(ω−b)2/(4ε)dμ ≤Ce−b2/(4ε).

Thus 0, impossible as0. Consequently supp. For the chamber diagonal translation measure take and

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h(z) = ⟨v, V (Re z)E(Im z)v⟩,
Im z ≥0.

It is continuous to the edge and bounded by; inside it equalsand is holomorphic. Its edge is the characteristic function of the time marginal. This proves nonnegative time support and, with SP1, completes SP2.

The lifted heat object retains, the continuous functional calculus, the dense range and the diagonal finite measures. Its complex-power readout recovers) on the positive axis and the uniquely extended unitary edge on the dense range. The Gaussian test reader PA2 retains positivity and the numerical exponents needed for support recovery; attachment then transports these very measures to the field space.

Gaussian support-test recovery. For the chamber section unitary, the retained diagonal finite spectral measure and half-plane reader satisfy

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μZ
VRv = μv,
hZ(z) = ⟨VRv, V Z(Rz)EZ(Iz)VRv⟩= h(z).

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The same scalar test e−ibz−εz2 therefore recovers both contour and measure integrals in (PA1)–
(PA2), including their common factor
p
π/ε. In particular the positive Gaussian measure reader
is
GZ
b,ε =
Z
e−(ω(q)−b)2/(4ε) dμZ(q) = Gb,ε.

For a hypothetical negative support value, the retained optimizer) reads the exact completed-square value

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bs + εs2 = −b2
4ε,
μZ(U)e−b2/(16ε) ≤GZ
b,ε ≤Ce−b2/(4ε).

The set and its positive measure are unchanged under the diagonal measure reader. The source thus recovers the same contradiction as0, and the same nonnegative-time support for the actual chamber translation measure. With the retained attachment and Lorentz pushforward it recovers (SP2). The Gaussian test operation changes neither the spectral convention nor the zero-energy atom.

19.36 Ground compression and the component probability space

In the unselected chamber let be the heat-fixed space, with projection. Retain the compressed slab operators place(

Their norm-closed unital star algebra is. With the commutativity and countable norm density established below, this is a separable commutative-algebra. Its compact character space Θ is metrizable: for a dense sequence is a continuous injection into a countable product of complex planes. Equality there extends by continuity from the dense sequence to every algebra element; compactness makes the injection an embedding. Gelfand representation and the measure representation of the vacuum positive functional give

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⟨Ω0, aΩ0⟩=
Z
Θ
χ(a) dν(χ),
ν(Θ) = 1.
(PH1)

The measure has full support. Indeed the compression identity below makes Ω cyclic. In a commutative algebra cyclicity implies separation:Ω = 0 impliesΩΩ = 0 on a dense set and hence = 0. A nonempty open null set would support a nonzero continuous bump whose squared integral is zero. Its represented operator would annihilate Ω, contradicting separation.

Compression of a distinct word is independent of slab placement. Increasing the width adds heat on the right, fixing the ground input; increasing the initial time adds heat on the left, removed by. Spatial translations disappear on both sides by spatial invariance of ground vectors, proved below. Compare two placements to cover positive or negative common time shifts. Write this element) and set (PH2)

For a future word, its ground-input slab insertion is independent of trailing width, and slab concatenation with the reflected adjoint gives

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XbΩ0 = [b],
M(b)Ω0 = P0[b],
M(b∗c) = X∗
b Xc.
(PH3)

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Ground projections of the dense planted-word span are dense in H0, proving the cyclicity just
used. Choose a common slab placement and apply PH1, its vacuum expectation and translation
invariance to obtain the barycenter
Z

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Θ
Sχ(w) dν(χ) = S(w).
(PH4)

The component hierarchies and their regularity use these same compression elements, not arbitrary disintegrated point readings. For a future word in a component chamber put. The heat limit gives

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∥aχ∥2 = lim
s→∞∥Eχ(s)[b]χ∥2.

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Take integer s.
Each prelimit is a continuous character reading of a reflected translated word
and is bounded by its s = 0 value, a continuous integrable function on compact Θ. Dominated
convergence and PH4 yield
Z

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Θ
∥aχ∥2dν = ∥P0[b]∥2.

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Character multiplicativity and PH1 also give
Z

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Θ
|Sχ(b)|2dν = ∥M(b)Ω0∥2 = ∥P0[b]∥2.

SinceΩ) and the vacuum has unit norm,

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∥aχ −Sχ(b)Ω0,χ∥2 = ∥aχ∥2 −|Sχ(b)|2 ≥0.

Its integral is zero, so it vanishes almost everywhere for this word. The countable probe construction supplies one sequence of future words whose planted spans are dense in every component chamber. Intersect their countably many full-measure sets. For each remaining character, all these ground projections lie in the vacuum line; their projected span is dense in the ground space. The line is closed, so the heat-fixed space equals it. This supplies an admissible selected character. The earlier full-support continuity argument upgrades the corresponding final reconstructed uniqueness statement to every character.

For the source lift retain the complete compression algebra and its vacuum state before taking any character. The realization is its concrete action on; observable readers are the characters and their barycenter PH1. Gelfand representation recovers every algebra element from all character readings. PH3 recovers the planted Gram kernel, and PH4 recovers the unselected hierarchy from the probability space. A single character generally loses the other components; that loss is explicit and is reversed only by retaining the whole character family and its measure. Rank one here concerns the heat-fixed subspace and makes no positive spectral-gap assertion.

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Compression algebra, character and probability recovery.
Restrict the chamber section
unitary to its heat-fixed space, writing V0 : H0 →HZ
0 . The concrete algebra readout is the isometric
unital star isomorphism

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α(a) = V0aV −1
0
,
α(m(w, d)) = mZ(w, d),
α(M(w)) = MZ(w).

It preserves the norm closure and all supplied commutativity and countable-density dependencies. The induced character homeomorphism and recovered probability are

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h : Θ −→ΘZ,
h(χ) = χ ◦α−1,
νZ = h∗ν.

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Indeed h(χ)(α(a)) = χ(a) for every algebra element; the inverse is composition with α. These
equalities on all evaluations give continuity in both directions. For ΩZ
0 = V0Ω0, the vacuum state
reads
⟨ΩZ
0 , α(a)ΩZ
0 ⟩=
Z
ΘZ χZ(α(a)) dνZ(χZ) =
Z
Θ
χ(a) dν(χ).

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Thus probability normalization, full support, and (PH1)–(PH2) recover under one character-family
map. The entire compressed word is retained before applying a character.
For ground-input word maps set XZ
b = VRXbV −1
0
. Then

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(XZ
b )∗XZ
c = α(X∗
b Xc) = MZ(b∗c),
SZ
h(χ)(w) = Sχ(w).

These are the precise (PH3)–(PH4) Gram and barycenter readers. On each component the section unitary also preserves the nonnegative defect. Its zero integral and the countable full-measure intersection therefore transfer through unchanged. Retaining the same countable probe words preserves density in every component and recovers the almost-everywhere heat-fixed vacuum line, with the later every-character upgrade kept as its separate step.

19.37 Spatial ground invariance and compressed commutativity

Positive contraction spectral calculus gives = lim) strongly: integer powers of converge to 1(1)) by dominated convergence, and the semigroup and contraction bounds cover general times. Its range is exactly the common heat-fixed space. For two distinct Euclidean words, reflection and placement in the positive half-space give, for large enough, fixed vectors and offsets with x (GI1)

Along any sequence, a fixed temporal displacement changes this by a quantity tending to zero. Both heat vectors converge to, and the possibly varying spatial unitaries preserve the norm of their difference. For fixed with 0, choose a Euclidean orthogonal transformation sending to and sending to a vector with positive time component. Reflection in the bisecting hyperplane between and does this, with the positive component following from preservation of their inner product. Rotate the words and displacements; the rotated large separation still has time tending to infinity. The temporal comparison and covariance give

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S(p τte0+bq) −S(p τte0q) −→0.
(GI2)

For spatial, apply GI2 to and subtract the temporal comparison adding to Taking heat limits in GI1 yields. Density gives. Spatial invariance was not assumed in comparing the varying large separations. For distinct-point cluster words, with pairwise ordered time separations of displacements) tending to infinity,

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r
Y
!
S
k=1
τdk(N)wk
−→⟨Ω0, M(w1) · · · M(wr)Ω0⟩.
(GI3)

Place each cluster in the positive half-space by a fixed shift, compensated in its displacement. Induct backwards from the last cluster using bounded insertion operators and

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U(aN)E(tN)vN −→P0v
(vN →v, tN →∞).

The error is at most, by contractivity and spatial ground invariance. Between successive cluster insertions there is a heat duration tending to infinity and an arbitrary spatial translation. Vacuum pairing finishes the induction. Large separations also ensure eventual distinctness of the combined configuration.

For four clusters, choose time/spatial slopes (0 3) and multiply by Their time order is; interchanging the two Euclidean axes changes it to. Finite-

permutation symmetry and covariance equate the expectations. Apply GI3. Remove the simultaneous rotation by a second GI3 application with all four clusters along, whose order is unchanged by the interchange. Thus

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⟨Ω0, M(a)M(b)M(c)M(d)Ω0⟩= ⟨Ω0, M(a)M(c)M(b)M(d)Ω0⟩.
(GI4)

Take the exterior words as a reflected reversed future probe and another future probe. PH3 and the adjoint identity turn GI4 into equal matrix elements of) and) on the dense projected future-word span. These bounded operators are equal. Slab reflection makes the generators closed under adjoints; hence star polynomials commute, and norm continuity extends commutativity to. This is the actual algebra used by PH1, not a separately selected commutative replacement. The Ground/Lean, Moments and Permute procedures thus supply spatial invariance, arbitrary transverse cluster-train convergence and the four-slope exchange. The four-cluster construction proves the needed exchange of two compressed factors. The resulting spatial ground invariance, cluster limit and commutative compression algebra will now supply one compact phase space and one countable family of future probes for all its characters. Norm-separable generators and the character hierarchy are supplied by the Work/B and Cells constructions below.

Ground-limit and ordered cluster-product recovery. For the chamber unitary and its ground restriction, the strong heat limit and spatial action recover as

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P Z
0 = VRP0V −1
R ,
UZ(a)P Z
0 = VRU(a)P0V −1
R
= P Z
0 .

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The last equality is read only after the retained GI1–GI2 comparison establishes spatial invariance.
Before that comparison, the source reader preserves exactly the varying-unitary norm bound on
the difference of the two heat vectors; it imposes no invariance premise. For the backwards cluster
induction, set vZ
N = VRvN and vZ = VRv. The quantitative reader is

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∥UZ(aN)EZ(tN)vZ
N −P Z
0 vZ∥≤∥vN −v∥+ ∥E(tN)v −P0v∥−→0.

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Consequently every fixed bounded cluster insertion and vacuum pairing recover the same ordered
GI3 product, even for arbitrary transverse displacements.
Using α(a) = V0aV −1
0
on the compression algebra, the four-slope reader preserves the exact
order of all four factors and gives

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⟨ΩZ
0 , α(M(a))[α(M(b)), α(M(c))]α(M(d))ΩZ
0 ⟩= 0.

The exterior projected future probes remain dense under, so their matrix elements recover the zero commutator of the two actual bounded compressed generators. Adjoint recovery and norm closure then give

This retains the original compression algebra, its four-slope limit and its spatial-ground dependency, and supplies the commutativity used in the character-probability construction.

19.38 A common countable probe family and character positivity

For every arity and degree tuple), the parameter space of distinct positive-half-space positions and raw coefficient vectors in those finite-dimensional degree spaces is separable metrizable. Choose a countable dense subset in each. Countably many, together with the