Phase decomposition and Wightman reconstruction · Part 5

Density of product tests

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Finite sums of product Schwartz tests are dense in the joint Schwartz space. Here is the concrete
approximation used. First take a compactly supported joint test φ, and choose a product cutoff
χ(X) = Q
i χi(Xi) equal to one on its support. Approximate its Fourier transform by compactly
supported Fourier cutoffs. Inverting and multiplying by χ gives Schwartz convergence to φ. Each
approximant is an integral over a compact frequency set of the tests

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h
χi(Xi)ei⟨qi,Xi⟩i
.
χ(X)ei⟨q,X⟩=
Y
i

This is a continuous family in Schwartz topology: all derivatives vary jointly on a common compact spatial support, giving uniform control of every seminorm. Quadrature on the compact frequency set approximates its integral in each seminorm by finite sums of these product tests. Thus each approximant is in the closure of the product-test span, and so is. Finally smooth compact spatial cutoffs converge to any Schwartz test, proving density in general. At arity zero the product test is the constant one on the one-point configuration space and already spans that space. This density permits subsequent joint-test conclusions to be recovered from word-vector identities.

Spacelike exchange on joint tests and on words

Two germs anchored to the same tube at a point of its closure agree. To prove this, take a ball on which both extensions are holomorphic and agree with the tube function in that ball’s intersection with the tube. This intersection contains a point because the base point belongs to the tube closure; it is open because the tube is open. Analytic uniqueness on the connected ball makes the extensions equal there.

This also proves propagation along a continuous path of base points in the closures of two tubes. At each base point take the two germ extensions on a ball. At nearby base points their germs agree with the germs selected there, by the preceding uniqueness statement. Whether the two extensions agree is therefore locally constant along the path: agreement propagates across the ball by analytic uniqueness, in either direction. On a connected parameter interval it is constant.

Given any real configuration with the designated pair spacelike, contract all its real time coordinates to zero: x

The pair remains spacelike, sincexx. With the fixed spectator heights, the path belongs to both tube closures. Both germs exist throughout it. Their equality at = 0, proved in (GE1), therefore propagates to = 1. Thus the two local extensions agree near every spacelike configuration.

Equation (GE2) now identifies the two partial distributions locally on a sufficiently small real ball. Their difference vanishes there on every compact smooth test. Consequently its distributional support misses every point of the spacelike-pair open set, and it vanishes on all Schwartz tests whose closed support is contained in that set. Apply this to every profile. The damped distribution at converges, on each fixed test, to the undamped boundary distribution: the closed-height multiplier is continuous in Schwartz topology, and its zero-height cutoffis one near the frequency support. The same is true after the adjacent transposition. Since the two partial distributions agree for every, uniqueness of their scalar limits gives

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Wv(φ) = Wv◦σ(φ ◦σ)
if supp φ ⊂{X : Xj −Xi is spacelike}.
(LC1)

This is the joint-test exchange identity with all spectator variables present.

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For product tests φ = Q
k φk, its closed support is contained in the inverse image of supp φk
under every coordinate projection.
If the supports of φi, φj are mutually spacelike, that joint
support lies in the set in (LC1). Permuting the joint product permutes exactly those two test
factors. Place these factors between any two spectator word lists to obtain

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W
a (v, φi)(w, φj) b

 = W
a (w, φj)(v, φi) b

.
(LC2)

No support restriction is placed on the spectator tests. This is the word relation that will give locality on the common field domain.

The label quotient remains faithful at the boundary

Suppose a union label is a boundary marker: every word expectation containing its smearing, with arbitrary words before and after, is zero. Then every joint distribution with that label in any prescribed slot is zero. It vanishes first on product tests, since a word can be split just before that slot, and then on all joint tests by product-test density and continuity.

Represent in a finite quotient field space by a raw label. The full-tube emission for any distribution containing is zero, so (TI1) makes its strictly time-ordered Euclidean reading zero. This remains true after any permutation of the arguments, because the marker may occupy any slot. For a general Euclidean configuration of distinct points, sort its time coordinates. They are weakly ordered, possibly with ties. Approach the sorted heights by strictly ordered heights while keeping spatial coordinates fixed. The configurations remain distinct near the given one. Smoothness of the hierarchy there makes their readings converge to the given reading, which is therefore zero. Undoing the sorting permutation proves vanishing at every distinct configuration.

The Euclidean null-space criterion now says that belongs to its finite-level null subspace. The selected hierarchy has exactly the retained null subspaces of the original hierarchy, so its class in the original quotient is zero. Hence = 0 in the common field union. This proves faithfulness of the boundary label quotient; it is not an additional field assumption.

Boundary positivity and the common field domain

The common domain is the complex span of finite smeared field words applied to the vacuum. Field multiplication, adjoints and spacetime covariance are asserted on that domain. Its density follows from the cyclic construction, and invariance follows from adjoining another smeared field or moving every test by the spacetime action. No extension to a common bounded-operator algebra is claimed.

For each word, (HV4) with makes) Cauchy as0. Indeed expand

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∥ηb(t) −ηb(u)∥2 = ⟨ηb(t), ηb(t)⟩−⟨ηb(t), ηb(u)⟩−⟨ηb(u), ηb(t)⟩+ ⟨ηb(u), ηb(u)⟩.

Every term has the same limit by the independent two-parameter convergence, so the expression tends to zero. Completeness gives a vector. Taking independent limits in (HV4) yields

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⟨ηb, ηc⟩= W(b∗c),
η∅= Ω,
ηa·b = U(a)ηb.
(FR1)

Consequently the kernel) is Hermitian and positive: for any finite coefficients,

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2
≥0.
X
X
zαzβK(bα, bβ) =
α
zαηbα
α,β

The empty expectation is one because the chamber vacuum has norm one.

Form the vector space of finite formal sums of smeared words, with sesquilinear form, linear in its second argument. Divide by its zero-norm subspace and complete. Denote the resulting Hilbert space by , the word classes by [], and their algebraic span by. This span is dense, and Ω] is a unit vector. The common operator domain will be exactly

For a letter), left insertion on formal sums is]. The word involution gives

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⟨Llp, q⟩= ⟨p, Ll∗q⟩,
l∗= (Jx, φ).
(FR2)

This identity also proves insertion preserves the null space. If [] = 0, choose a formal sum representing the vector in; then (FR2) makes = 0. Thus

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Φ(x, φ)[b] = [(x, φ)b]
(FR3)

is a well-defined linear operator. It is bilinear in. To prove this directly, pair the proposed additive or scalar relation with every word vector. The corresponding word expectation is linear in the inserted label and test, since the joint distribution is multilinear in its labels and the product test is multilinear in its factors. All pairings vanish, and density proves the relation. There is no boundedness claim for the fields themselves.

Equation (FR2) is the adjoint relation on the common domain. Repeated application of (FR3) gives the word vectors, and their vacuum expectations are the prescribed’s. Cyclicity follows because is exactly their span. The smearing map is norm-continuous for each . For a word vector, continuity of the diagonal and mixed joint-distribution pairings implies continuity of the norm difference by its squared-norm expansion; finite linear combinations give the assertion for. The jointly continuous two-smearing Gram pairing used here follows from continuity of product tests in Schwartz topology and continuity of each fixed-label joint distribution.

The word exchange identity (LC2), paired against all word probes, gives

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[Φ(x, φ), Φ(y, ψ)]

D = 0

when the two test supports are mutually spacelike. Finally if Φ( for every, all expectations with in any slot vanish: pair its zero insertion against the reversed adjoint left spectator word. Involutivity of removes that double reversal. The boundary marker criterion already proved gives = 0. Thus the field representation retains the full prescribed label quotient.

Unitary spacetime action on the reconstructed space

For a Poincaré transformation, transform each letter by its finite-dimensional label action and inverse placement of its test. Boundary covariance preserves); the adjoint on letters commutes with this transformation. Thus

extends isometrically to . Its inverse is), and the word transformation law gives the representation law on the dense span and hence everywhere.

Its strong continuity can be proved without a topology on the full label union. Pin the finitely many labels in one finite field space and expand the multilinear distribution in a fixed basis there. Each moved expectation is a finite sum of products of continuous label coordinates and continuous fixed-label distribution pairings with moved tests. Strong continuity of test motion was proved in (BC4). Hence all mixed word-vector pairings vary continuously. The squared-norm identity gives norm continuity of], and then of finite sums. To extend to any vector, approximate by such a

sum within3; isometry controls the two approximation errors uniformly in. This proves strong continuity on the Hilbert space. The domain is invariant, the empty word fixes the vacuum, and insertion transforms by

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U(g)Φ(x, φ)U(g)−1 = Φ(ρ(g)x, g · φ)
on D.
(FR4)

The isometric attachment to the original reflection-positive chamber is also concrete: map [ in (FR1), extend by continuity, and use equality of all Gram pairings to see that it is an isometry. It sends the vacuum to Ωand intertwines translations by (FR1). The spectral and vacuum conclusions can therefore be transferred through this attachment by the arguments below.

Coordinate boosts, the spectral cone, and attachment

We supply the group-generation fact used above. For a strictly future timelike vector, its time component is positive and every spatial component satisfies 1. The coordinate boost of rapidity artanh(

sets that component to zero, since sinh cosh = 0. It preserves all other spatial components and keeps the vector in the strict future cone. Three successive coordinate boosts therefore bring any future unit timelike vector to the unit time vector. For any proper orthochronous Lorentz transformation, apply these boosts to, obtaining

A Lorentz transformation fixing preserves the time coordinate, by invariance of the Lorentz pairing. Since it also preserves the Minkowski quadratic form, it preserves the Euclidean norm. It is therefore an orthogonal spatial turn, with the original determinant. Thus is such a spatial turn and) is a finite product of coordinate boosts and spatial turns. This proves the generation statement used to obtain full boundary covariance.

For precision, the absolute-frequency cone is initially defined as the intersection, over all Lorentz transformations, of the inverse images underof the time cone in (FT3). Boundary covariance and the Fourier change-of-variables formula show that the Fourier distribution is supported in every one of these inverse images: pull a test disjoint from one inverse image through, use the time-cone support for the transformed labels, and transform back. Hence it is supported in their intersection.

This intersection has the suffix-momentum description used in (DC1). For any suffix, test the time cone against the weakly increasing step height that is zero before that suffix and one on it. Its time component is nonnegative. Doing this after every Lorentz transformation makes the suffix vector future causal: a vector with nonnegative transformed time component under every proper orthochronous Lorentz transformation is in the closed forward cone. To see the converse characterization of a single vector, if its time is negative the identity already excludes it; if its spatial norm exceeds its nonnegative time, a boost toward its spatial direction makes its time negative. Constant negative as well as positive heights force total time momentum zero. The total suffix is future causal and has zero time, so its spatial part is zero too. Conversely, zero total momentum and future-causal proper suffixes imply all the time-cone inequalities by telescoping and imply them after every Lorentz transformation. This proves the description, with no change to relative coordinates.

For the field Hilbert space let be the joint projection-valued measure of its strongly continuous translation representation, with convention

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

The translation attachment in (FR1) is an isometry into the reflection-positive chamber and intertwines translations. Equality of these scalar characteristic functions and uniqueness of Fourier transforms of finite measures give equality of the diagonal measures with the chamber’s diagonal translation measures at The latter have nonnegative time support, as established by the semigroup argument below.

Lorentz covariance of the unitary representation gives

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μU(r)v = (r∨)∗μv.
(SP1)

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This follows by conjugating translations and comparing characteristic functions, using preservation
of Hilbert inner products. If p is in the support of μv, then r∨p is in the support of the right side
of (SP1), and hence has nonnegative time component for every r. The characterization just proved
makes p future causal. Thus every diagonal measure is supported in the closed forward cone, and
polarization gives
P(V c
+) = 0.
(SP2)

This is the spectral condition for the actual field translation representation, not merely for the hierarchy distributions.

Reflection-positive heat contractions

The chamber heat flow is obtained from future-word translation, rather than postulated. On finite sums of future words let be the reflection-positive word vector and its forward translate through0. Translation and reflection of the Euclidean hierarchy give

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⟨XTtp, XTuq⟩= ⟨Xp, XTt+uq⟩.
(OS1)

Its fixed-word correlation bound implies supfor every finite sum: first bound each individual translated word norm by the square root of its reflected two-word correlation bound, then sum the finitely many coefficient-weighted bounds. Put = sup. Equation (OS1) and Cauchy–Schwarz imply

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∥XTtp∥2 = |⟨Xp, XT2tp⟩| ≤∥Xp∥c.

Taking the supremum and using0 yields Thus forward translation preserves null vectors and extends to a contraction) on the completed reflection-positive Hilbert space. Equation (OS1) makes) self-adjoint. Translation parallel to the reflection plane preserves the Gram form, extends to an isometry, and its opposite translation is its inverse. These give the commuting spatial unitary group.

The semigroup and commutation laws hold on planted words and hence everywhere. Strong continuity follows first on planted words: the squared norm of the difference of two translated vectors is a sum of continuous Euclidean correlation functions. Finite sums inherit continuity; their density and the uniform contraction bounds extend it to the Hilbert space by approximation. The same argument proves strong continuity of the spatial group. This constructs the self-adjoint contraction semigroup and commuting spatial translations required throughout the wave construction.

Holomorphic heat powers and the real-time edge

Let) be the self-adjoint contraction semigroup just constructed. Each) is positive, since , so its spectrum is contained in [0 1]. Put(1). For

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F(z) = Az

by continuous functional calculus, with 0 = 0. The function is continuous on [0 including at zero. On compact subsets of the right half-plane it varies continuously in uniform norm; point evaluations are holomorphic in. Equivalently, uniform estimates for log on smaller half-planes justify its derivatives. Thus is holomorphic in operator norm there. Functional calculus gives (HE1)

It also commutes with the spatial translations.

These powers coincide with the supplied heat flow. For positive integers, let so and. Functional calculus on the nonnegative spectrum and the scalar equality ( give). Approximate any positive by positive rational numbers; operator-norm continuity of and strong continuity of, applied to each vector, give

Every) is injective. If = 0, the semigroup law gives = 0 for The holomorphic vector function consequently vanishes on a real interval in the right half-plane, hence everywhere there by analytic uniqueness. Strong continuity at zero then gives = 0. Self-adjointness implies dense range: the orthogonal complement of the range is its kernel.

For real, (HE1) gives

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∥F(r −is)v∥= ∥E(r)v∥
(r > 0),

because both squared norms are the quadratic form of). Define an isometry on the dense range of

It extends uniquely to the Hilbert space. Checking on that dense range 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.

Therefore) is unitary, with inverse). It commutes with spatial translations. Its strong continuity holds on the range of(1) by operator-norm holomorphy of, and on all vectors by density and the isometric bound. Together these operators form the chamber’s strongly continuous real spacetime translation group. The sign is), consistent with the spectral convention

The Gaussian half-plane support argument

We prove the precise support fact used for the chamber diagonal spectral measures. Suppose a finite positive measure, a continuous real function, and a bounded function, continuous on the closed upper half-plane and holomorphic in its interior, satisfy

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h(t) =
Z
eitω(q) dμ(q),
|h(z)| ≤C.

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For b ∈R, ε > 0, multiply by e−ibz−εz2. Cauchy’s theorem on rectangles between imaginary heights
zero and s ≥0, followed by sending their vertical sides to infinity, gives

Z
R
h(t)e−ibt−εt2 dt
 ≤C
q
π/ε ebs+εs2.
(PA1)

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The vertical side integrals vanish because their norms are bounded by a constant times e−εT 2; the
horizontal integrals are absolutely integrable Gaussian majorants. Fubini and the scalar Gaussian
Fourier integral also give
Z
R
h(t)e−ibt−εt2 dt =
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 in (PA1). Equations (PA1)–(PA2) and positivity imply

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

Thus 0, impossible as0. Hence supp Apply this to the chamber’s diagonal translation measure with. Its required upperhalf-plane extension is

It is continuous to the edge, holomorphic inside because it equals, and bounded by At real it is precisely the characteristic function of the time marginal of the joint translation measure. The support argument proves nonnegative time energy. Combined with (SP1), it completes the forward-cone spectral conclusion (SP2).

The ground compression algebra and its probability space

The phase space is the spectrum of the commutative ground compression algebra. Its probability measure is obtained from the vacuum state on that algebra. Character evaluation gives the phase moments; their barycenter returns the unselected hierarchy. The countable common family of probes and the uniform phase bounds are what allow positivity, regularity and vacuum uniqueness to be asserted simultaneously. Let be the heat-fixed subspace of the unselected chamber and its orthogonal projection. Compress each slab operator to that subspace:

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m(w, d) = P0 place(d, w)

H0.

Take the norm-closed unital star algebra generated by these operators. Once its commutativity and countable norm density are established below, it is a separable commutative-algebra. Its character space Θ is compact and metrizable. To see metrizability explicitly, take a dense sequence . The map is continuous and injective: equality on the dense sequence extends to all of by continuity of each character. A continuous injection from the compact character space into a countable product of complex planes is an embedding.

The vacuum vector state has the representation

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⟨Ω0, aΩ0⟩=
Z
Θ
χ(a) dν(χ).
(PH1)

Here the Gelfand representation identifies with continuous functions on Θ, and the representation of a positive functional by a finite measure supplies. It is a probability measure sinceΩ It has full support. The vacuum is cyclic for on, as follows from the compression identity below. Commutativity then makes it separating: ifΩ = 0, thenΩΩ) = 0 on a dense set, so = 0. If a nonempty open subset of Θ had measure zero, choose a nonzero continuous bump

supported there. Its squared integral is zero, so the represented operator kills Ω, contradicting this separating property. Thus every nonempty open set has positive measure.

The compressed operator of a distinct word is independent of how it is placed in a slab. Increasing the slab width adds heat evolution on its right, which fixes the input ground vector. Increasing its initial time adds heat evolution on the left, which disappears under. Spatial translation disappears on both sides because ground vectors are spatially invariant. Positive or negative common time translations reduce to these identities by comparing the two placements. Denote the resulting algebra element by). A component hierarchy is specified by

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Sχ(w) = χ(M(w)).
(PH2)

For a future word, let be its slab insertion with ground input. This is independent of the trailing slab width, and

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

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The last equality is slab concatenation and the reflected adjoint identity. Since the future-word
vectors span densely, their ground projections do too; the middle equality gives cyclicity of Ω0 for
A. Equations (PH1)–(PH2) also give the barycenter of every distinct Euclidean word:
Z

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Θ
Sχ(w) dν(χ) = S(w).
(PH4)

Choose any common slab placement to prove this equality, apply (PH1), and use the vacuum slab expectation and translation invariance. The source constructs component hierarchies satisfying (PH2); their existence and regularity must use these same compression elements, rather than arbitrary disintegrated point values.

Almost-everywhere one-dimensional heat-fixed space

For a future word in a component chamber, set. The heat limit gives

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∥aχ∥2 = lim
s→∞∥Eχ(s)[b]χ∥2.

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Use integer s. Each expression before the limit is a continuous character reading of the reflected
translated word.
It is bounded by its value at s = 0, a continuous integrable function on the
compact character space, because heat is contractive. Dominated convergence and (PH4) therefore
give
Z

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Θ
∥aχ∥2 dν = ∥P0[b]∥2.

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On the other hand, multiplicativity of a character and (PH1) imply
Z

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Θ
|Sχ(b)|2 dν = ∥M(b)Ω0∥2 = ∥P0[b]∥2.

The vacuum pairing satisfiesΩ). Since the component vacuum has norm one,

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

Its integral is zero by the preceding identities, so it vanishes almost everywhere for this word. Choose one countable sequence of future words whose linear spans of planted vectors are dense in every component chamber; the source obtains such a sequence from the countable probe construction, rather than assuming separability of arbitrary label sets. Intersect the countably many