Phase decomposition and Wightman reconstruction · Part 2

Bounded slab operators in every character chamber

The operator estimates needed in reconstruction hold at every character of the compression algebra. Their proof does not require discarding exceptional characters. We give the power argument, including the step that removes a character-dependent constant. Let map the algebraic span of future words into its reflection-positive Hilbert completion. Let satisfy

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⟨XAv, Xw⟩= ⟨Xv, XBw⟩.

Set. Suppose 0 and, for every , there is a finite constant such that

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∥XT nv∥≤CvD2n
(n ≥0).
(ST1)

The constant need not be uniform in. The adjoint relation implies symmetry of, and then of each power, relative to this Gram form. Put

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dn = ∥XT nv∥D−2n,
s = sup
n≥0
dn.

The supremum is finite by (ST1). Symmetry and Cauchy–Schwarz give

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d2
n = D−4n|⟨Xv, XT 2nv⟩| ≤∥Xv∥d2n ≤∥Xv∥s.

Taking the supremum shows, hence. Finally,

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∥XAv∥2 = |⟨Xv, XTv⟩| ≤∥Xv∥∥XTv∥≤D2∥Xv∥2.
(ST2)

Thus descends through the null space and extends to a bounded operator of norm at most on the Hilbert completion. Apply this to insertion of a slab word and insertion of its reflected word. Translation invariance of the hierarchy gives the displayed adjoint relation. Starting with a future word, denote by the word obtained by adjoining successive reflected/unreflected slab pairs. It suffices to have

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| Gram(vn, vn)| ≤QvD4n.
(ST3)

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For a scalar multiple av, the Hilbert norm bound is |a|√QvD2n. A finite linear combination has
(ST1) by the triangle inequality, with constant equal to the corresponding finite sum. Hence a
diagonal estimate on individual words really does produce a bounded operator on their entire
dense span. This proves the slab operator construction without assuming boundedness beforehand.
To obtain (ST3) in a character chamber, let U be the slab operator in the unselected chamber
and let Xv be its ground-space word map. Concatenation says

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Xenter(w,v) = UXv,

while reflection givesfor the reflected slab. Therefore

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Xvn = (U∗U)nXv,
∥Xvn∥≤∥Xv∥∥U∥2n.

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Let χ be the chosen continuous character of the closed compression algebra. The character chamber
receives the scalar Gram values of that algebra, and the compressed two-sided word is X∗
vXv.
Consequently
| Gramχ(v, v)| = |χ(X∗
vXv)| ≤∥χ∥∥Xv∥2.

For any, substitution of proves (ST3) with. The power argument removes this prefactor, producing a character-chamber slab operator of norm at most

All the operators obtained for differentagree on the dense future-word span, where their action is prescribed by insertion. They are therefore the same bounded operator. Letting proves the exact estimate (ST4)

This also covers= 0. Every unselected arbitrary-copy estimate thus transfers with its original constants to every character, uniformly over the character space.

For each character construct its reflection-positive chamber, these bounded slab maps, and the derivative calculus inherited by the character hierarchy. The transferred arbitrary-copy estimates give the slab laws required to form its contour and strict waves. Apply the field reconstruction to those data to obtain its Hilbert space and fields, with the common conjugation and rotation labels. At the previously selected character, choose the already constructed chamber, slab maps, contour and fields themselves. This is legitimate because they satisfy the same requirements; consequently no comparison between independently chosen copies is needed there. The resulting family supplies the common phasewise operator bounds used in the strict-wave polynomial estimates and in passage to the tested Wightman boundaries. The remaining contour and field construction is developed below; vacuum uniqueness is a separate conclusion and is not assumed in this step.

Centered insertions and the strict-time waves

Write) for the chamber’s heat contraction,) for its holomorphic contraction semigroup on 0, and) for spatial translation. Thus) for 0; translations commute with the heat flow and fix the unit vacuum Ω. For a field label and 0, define its centered, buffered insertion by the bounded slab operator

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Bp(v) = place
2p, [(0, p; v)]

.
(SW1)

Initially belongs to one finite field space. Its matrix element between any two future-word vectors is the Euclidean hierarchy with that insertion between the reflected left spectators and the right spectators translated forward by 2. The spectators and insertion have distinct positions. Multilinearity of the hierarchy in the insertion therefore gives linearity of. If two finite-level labels represent the same element of the common field union, the derivative-plan inclusion identity makes these matrix elements equal. Equality on the dense future-word span then makes the bounded operators equal. Hence (SW1) defines a linear map on the common field union, rather than a map depending on a representative.

The arbitrary-copy slab bound applied to this one-insertion word gives, for each fixed union label, constants 0 and with

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∥Bp(v)∥≤Qvp−dv
(0 < p < 1).
(SW2)

The word has both boundary buffers equal to, which is why that estimate applies at this width. Width zero is unused and may be assigned the zero map. For and

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R(w, s, x) = F(−iw −s)U(x),
Iw > s.

Its norm is at most one, because 0. Let have strictly increasing imaginary parts, let, and take

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0 < p < 1,
4p < Izj −Izi
(i < j).
(SW3)

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The buffered matrix element is

Ω (SW4) ForΩ Ω. All factors are defined under (SW3), and this expression is multilinear in the union labels. There are two exact identities behind the construction. Slab stretching and concatenation give

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Bp+u(v) = E(u)Bp(v)E(u).
(SW5)

When the larger width is admissible, the semigroup identity and commutation with spatial translation give (SW6)

Insert (SW5) into the larger-width version of (SW4). Each intervening pair of heat contractions combines with its larger-width propagator according to (SW6); the two exterior contractions disappear against Ω. Thus. Comparing the smaller and larger of any two admissible widths proves complete width independence.

At purely imaginary times, the same product is exactly the original Euclidean hierarchy at the points (). To see this directly, build its ket recursively from the final insertion backwards. The centered insertion adds the next plant at time; the positive heat interval and spatial translation move the remaining word into its prescribed relative position. Slab insertion and chamber translation act on planted word vectors by those very operations. Taking the vacuum inner product gives the expectation of the complete word, translated by a common vector. Translation invariance removes this vector, and the quotient-label identity identifies the result with the hierarchy on the common union. This proves the Euclidean identity for every admissible Define) on the strict-time region. An admissible exists because there are finitely many positive imaginary gaps; the choice is irrelevant by width independence. A single such remains admissible on a neighborhood of any fixed strict-time point. On that neighborhood all propagators in (SW4) vary holomorphically in the complex time variables and continuously in the spatial variables. Bounded composition therefore proves the same properties for particular this argument requires no continuity of the arbitrarily chosen width as a function of

Repeated contraction bounds and Cauchy–Schwarz show

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|Fn(z, x; v)| ≤∥Ω∥2
n
Y
i=1
∥Bp(vi)∥.

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If 0 < k ≤1 and every imaginary gap is at least k, choose p = k/8, which satisfies (SW3). From
(SW2),
|Fn(z, x; v)| ≤Cvk−mv,
mv =
X
i
dvi,
Cv = ∥Ω∥2 Y
i
(Qvi8dvi).
(SW7)

The bound is uniform in all real times and spatial positions. At arities zero and one the pairwise gap condition is empty; the same choice of is still allowed. No uniformity in arity or in the labels is asserted.

Tested Cauchy–Riemann equations

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For a real spacetime configuration X, let
fv,y(X) = Fn
(ti(X) + iyi)i, (xi(X))i; v

,
Ψv(y; φ) =
Z
fv,y(X)φ(X) dX,

where and is a joint Schwartz test. A neighborhood of any such has one common positive lower gap. This follows by decreasing the finitely many strictly positive gaps slightly; equivalently subtract a sufficiently small multiple of the increasing height vector. Equation (SW7) therefore bounds, uniformly in and nearby, by a fixed constant. The integral defining Ψ is absolutely convergent, and dominated convergence proves continuity in

For a height direction, write for the spacetime direction whose-th time coordinate is and whose spatial coordinates vanish. The one-complex-variable slice

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hX,u(ζ) = fv,u+(Iζ)a(X + (Rζ)ea)

is holomorphic wherever its heights remain strictly ordered. A single disc is admissible for every and all in a sufficiently small height neighborhood. The same common bound holds on these discs. Cauchy’s estimate gives

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|∂eafv,u(X)| ≤C/r.
(CR1)

The radius is independent of the noncompact real configuration; this is essential for integration. Holomorphy of the slice also gives

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d
dsfv,y+sa(X) = i ∂eafv,y+sa(X).

Bound (CR1) permits differentiation under the joint test integral. Integration by parts along the real direction then yields

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d
dsΨv(y + sa; φ) = −i Ψv(y + sa; ∂eaφ).
(CR2)

Both the original integrand and the derivative integrand are absolutely integrable by the Schwartz decay and the uniform bounds above. These hypotheses justify integration by parts directly on the whole real space. Thus (CR2) holds for arbitrary joint tests and arbitrary height directions staying in the strict cone, not just for tensor tests or a single approach ray. Together with (SW7), it supplies the tested Taylor and tempered boundary arguments used earlier.

The frequency identity and the time spectral cone

Use the Fourier normalization of the construction, denoting its Schwartz transform by H ; it includes the fixed coordinate rescaling that removes the 2 from the following derivative identity. If is the pure-time spacetime vector belonging to a height vector, that identity is

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−i∂d(H g) = −H (⟨d, ·⟩g).
(FT1)

For compactly supported smooth, multiplication bydefines a Schwartz test, with the same compact support, for every real. Moreover

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d
dsgs = ⟨d, ·⟩gs

in the Schwartz topology. Indeed the divided difference of the scalar exponential is smooth at zero. Its products with, and every derivative of those products, are continuous jointly in the parameter and the point on a common compact support. Uniform convergence of every weighted derivative follows from compactness; outside that support all those derivatives vanish. This proves the asserted Schwartz derivative, including at an arbitrary parameter value.

Let be the tempered boundary functional of the strict waves. For strictly increasing consider H

The tested Cauchy–Riemann identity and (FT1) show) = 0: the derivative due to the heights is ΨH), and the derivative of the test is its negative. This product rule is valid also near the boundary. To justify moving tests there, the tested Taylor bound supplies one continuous Schwartz seminorm with Ψ

for every strict height in the boundary approach region. Thus a test difference tending to zero in Schwartz topology contributes a quantity tending to zero uniformly in. On the interior, the ordinary locally dominated derivative argument applies. Constancy of) and the boundary limit at0 give

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c
Wv(g) := Wv(H g) = Ψv

y; H
e⟨ey,q⟩g(q)


.
(FT2)

This equality uses absolute frequency coordinates; it is not an assertion only in relative coordinates. Now let be weakly increasing and let have compact support in. Fix a strictly increasing height vector with unit gaps, and apply (FT2) at0. These heights retain gaps at least one. The tests

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gt(q) = e⟨eY (t),q⟩g(q)

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tend to zero in Schwartz topology. For completeness, each derivative of order j is bounded in a
weighted Schwartz seminorm by a finite sum of terms
 
j
l
!
e−t(t∥ey∥)lpi,j−l
(0 ≤l≤j),

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where the constants include the fixed factor e⟨er,q⟩g(q). Every term tends to zero. The unit-gap
version of (SW7) bounds the right side of (FT2) by a fixed continuous Schwartz seminorm of H gt,
so it tends to zero. Its left side is independent of t, and hence c
Wv(g) = 0.
If a frequency point q has ⟨ey, q⟩< 0 for some weakly increasing y, rescale y so that this
pairing equals −2. An open neighborhood of q then has pairing less than −1, and the preceding
compact-test argument makes the distribution vanish there. It follows that

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supp c
Wv ⊂
\
y1≤···≤yn
{q : ⟨ey, q⟩≥0}.
(FT3)

The weak ordering, including constant height vectors of either sign, is essential: it enforces total time-frequency conservation as well as the ordered partial-energy inequalities. The passage from this time cone to the relativistic spectrum also uses the rotation and covariance arguments developed next.

Removing the outer angular flux and obtaining all Euclidean rotations

Continuous Euclidean rotations are obtained from shell Ward identities and control of the outer angular flux. They are not inserted as an extra invariance assumption. The corresponding label representation must descend to the physical quotient and agree with the finite-degree free comparison. Only after this step can the analytic tube identities be converted into the real boost action used in reconstruction.

The lattice symmetries give only signed coordinate permutations. Continuous rotations of the physical quotient require the conserved stress and its shell maps constructed below. Write for the translation shell map and) for the angular shell map, where is a real skew 4-by-4 matrix. Before their normalization, the angular Ward expression is

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AK(v; x) =
X
l
{S(v1, . . . , Y (K)vl, . . . ; x) + (Kxl)αS(v1, . . . , pαvl, . . . ; x)} .
(RT1)

Its vanishing cannot be inferred by simply enlarging a spherical cutoff: the correlations have subpower infrared bounds, rather than the decay that such an argument would require. The source removes the outer flux through tested spectral estimates. We give the details of that passage here. The tests in this argument have the following precise meaning. Fix a nonnegative even smooth compactly supported function, equal to one on [ 1], and let

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cw(p) = w−1c(p/w),
w > 0.

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The source uses this central frequency profile and real Schwartz profiles with the same normalization.
Its j-th moment is
R pjf(p) dp; substitution gives the moment of fw as wj times that of f. In the
Fourier convention used here,

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∂j
x bf(0) = (−i)j
Z
pjf(p) dp.

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Differentiation under the integral is justified by the integrable Schwartz majorant |p|j|f(p)|. If
moments below m vanish, repeated integration of the derivative along a segment bounds | bf(x)| ≤
C|x|m on a fixed ball. This is the origin-jet cancellation used in the compensated profiles.
For four one-dimensional tests fa, the tested stress pair is the product-test extension of

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Rab,cd(f) =
Z
⟨Jab(x)Jcd(0)⟩
Y
i
fi(xi) d4x.
(RT0a)

When a factor vanishes sufficiently far in Taylor order at zero this is the ordinary integral. For arbitrary Schwartz factors, subtract the finite product Taylor polynomial near zero; the polynomial short-distance bound makes the remainder integrable. This defines the extension without presuming rotation invariance. Pair exchange, tensor symmetry and signed covariance follow by the corresponding changes of variables in the remainder integral.

Its tested conservation defect is

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∆b,cd(f) =
X
i
Rib,cd(f0, . . . , f′
i, . . . , f3).

Let!, and let have factor at and elsewhere. Set

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wab,cd = ∆b,cd(f(a)),
R◦
ab,cd(f) = Rab,cd(f) −wab,cd
Y
i
fi(0).
(RT0b)

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The subtracted term is supported at the single contact x = 0. Its differentiated value on f(a) is
exactly wab,cd, because g(l)
j (0) = 1{j=l}. The resulting defect therefore vanishes on all first-order
product jets. It vanishes on the order-zero and order-two jets by total inversion: the three remaining
tensor indices give an odd sign whereas an even total test order gives an even sign. These finite
contact subtractions preserve the pair and signed symmetries; reflection in a single axis kills a
coefficient unless its four tensor indices occur in pairs. Thus the compensated conservation defect

has no jets of total degree below three. This precise contact remainder, rather than an assumed exact conservation law at coincident points, enters the spectral band estimates. For a fixed exterior word on coordinate line, put

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Fdb(y) = ⟨Jdb(y)
Y
l
Bl(xl)⟩.

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Realize that exterior word as a finite word vector and pair it with the one-stress vector.
The
hierarchy’s permutation and conjugation laws identify this pairing with Fdb whenever the points
are distinct. This is why the spectral bounds apply to an arbitrary exterior word, not only to the
stress two-point function. The actual outer term is
Z
(∂dφab,l
λ )(y)Fdb(y) d4y,
d ̸= a,
(RT0c)

for the compensated angular primitive . Its frequency widths are in axes on the remaining line axis, and otherwise; it carries the primitive amplitude. The band subtractions described next give the actual estimate for (RT0c), including its integrability. The finite excluded insertion set has Lebesgue measure zero and does not change this integral. Compact shell conservation identifies the difference between the summed local charges and this outer integral. Its vanishing therefore proves (RT1).

For each coordinate, reflection positivity supplies its own positive joint spectral measures , supported on0, for finite linear combinations of reflected words. They include atoms at = 0. Mixed readings are obtained by polarization and obey the weighted Cauchy–Schwarz inequality

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t
Z
X dνB
i + t−1
Z
Z dνA
i

if |f|2 ≤XZ,
t > 0.
(RT2)
|⟨B, f(P)A⟩| ≤1
2

This follows by applying 2 to the two spectral vectors; approximation by simple spectral functions proves it for the displayed measurable weights. No identification of spectral measures for different axes is needed.

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All transverse moments needed below are finite after positive time damping. To prove this,
take the characteristic function of the finite damped measure and apply k pairs of forward and
backward differences in a transverse direction j. Its integral multiplier, up to the cancelling sign of
the difference pair, is

2k
eihpj −1
.

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h

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The fundamental theorem of calculus bounds the difference expression by the corresponding 2k-th
derivative of the smooth separated correlation. Fatou’s lemma as h →0 gives the moment of |pj|2k.
The same argument for moved word clusters uses their actual positive time gaps. For a cluster
collapsed in the i direction this yields, for some finite D, C depending on the word and k,
Z
e−tE|pj|2k dνA
i ≤Ct−D,
0 < t ≤1.
(RT3)

Thus moment orders can be chosen before shrinking the spectral bands. Here is the interpolation that controls low energy. For distinct nodes, let be their Lagrange basis polynomials, 0. Polynomial interpolation gives

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1
x + y −
X
Lj(x)
zj + y =
1
x + y
Y
zj −x
zj + y ,
y ≥0,
x + y ̸= 0.
(RT4)
j<m
j<m

For completeness, multiply the left side by. It is a polynomial of degree at most, zero at every, and equal to one at. These + 1 values determine it and give the product on the right. Rescale. Multiplying by the even Laplace numerator 2 gives the exact response after subtracting the finite high-band interpolant. For moment-controlled profiles the subtracted moments are exactly those furnished by (RT4). The residual response is nonnegative for0, and the choice of the central profile gives

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Rm,a(E) ≥
cm
E + a,
0 ≤E ≤1,
0 < a ≪1,
(RT5)

with At zero this is the continuous value of the full response, including the profile’s zero-energy contribution; one does not divide a bare factor 2 by and discard an atom. The numerator factors in (RT4) stay of one sign because the high nodes lie outside the primary profile band. Their fixed ratios give; their denominators give the stated reciprocal bound.

The reason this positive response is small is a sign identity for the same tested position correlation. Suppose has reflection parities of opposite sign on axes. Write

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Pi(h; f) =
Z
Li(h)(pi)
Y
l∈Q\{i}
fl(pl) dνA
i .

Here are nonnegative even Schwartz frequency profiles and is the real even Laplace response of the inverse Fourier profile in direction. The two spectral formulas for that one tested correlation give. If some transverse axes are absent, insert profiles equal to one on larger and larger intervals; bounded dominated convergence gives the same identity for that subset. Split each active profile into its Lagrange interpolant and residual. Both residuals have nonnegative response. Consequently the primary residual on axis is bounded by the sum of the two cross terms containing an interpolant. The interpolants are finite sums of profiles in a high band.

For clarity the choice of orders in this repeated operation is important. Put, and scale each frequency profile as 0. A bound will be called a bound of exponent in this paragraph. If a high band has exponent 0, a transverse 2-th moment from (RT3) bounds its far remainder with exponent at most

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−2ka′ + K|Q \ {i, j}| + max(K, dD),
(RT6)

where and is the earlier chosen band exponent. Given any 0, choose so that the negative term exceeds the other fixed losses, and then choose 0 so small that At each switch the coordinate just treated is removed from the list still requiring a bound. If its parity agrees with the next required coordinate, insert an opposite-parity coordinate first. A finite induction on that list terminates; the terminal case has only nonshrinking transverse widths and is bounded by (RT3). Choose moment orders and admissible band exponents backwards through this finite list. This proves an exponent, for every 0, for the residual response. Equation (RT5) therefore gives

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Q
l∈Q\{i} f(N)
l
(pl)
pi + λq
dνA
i ≤Cηλη,
q < 0,
(RT7)
Z
0≤pi≤1

when the collapsed cluster has both parities on A word on a coordinate line is collapsed on the other axes but can have a genuine time gap on . Subtract there a shifted positive-time profile. Its response is positive, dominates a fixed multiple 1, and its damped transverse moments cost at most