Phase decomposition and Wightman reconstruction · Part 5

where projects onto the spatial- and time plane. The time component of is and its spatial norm is, so it is strictly future timelike. Under an adjacent imaginary difference becomes + sin + (cos (LG1)

Strict gaps remain strict in a neighborhood of = 0. At the flat gap,

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sin b V + (cos b −1)W = sin b

V −
sin b
1 + cos bW

.

The parenthesis remains strictly timelike near (0); for small 0 its addition to a closed-cone enters the open cone. Finitely many gaps permit one product neighborhood and one angle, uniformly down to zero. For the transposed order use the opposite imaginary boost: both sign reversals at the flat gap cancel. Thus the original tube and the transposed tube each have a small imaginary-boost path into their interiors, with this path property holding for nearby points of the corresponding tube, not just at the boundary point itself.

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Here is the label continuation, retaining the source’s finite-dimensional hypothesis. For a contin-
R t
0 u(s) ds. Then V ′(t) = u(t)
uous one-parameter group u(t) in a unital Banach algebra, set V (t) =
and V (t)/t →1. Choose nonzero a with g = V (a)/a invertible. The group law yields

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u(t)V (a) = V (t + a) −V (t),
u(t) = a−1[V (t + a) −V (t)]g−1.

Thus is differentiable. With(0) it satisfies = 0 and(0) = 1 give . On the finite field space containing the chosen labels this supplies the entire extension At the chosen boundary point take the small endpoint above and put

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G(Z) = F(u(it)vk)k(Ae(it)Z).
(LG2)

Its endpoint is interior, hence is holomorphic on a full neighborhood. For a nearby original-tube point consider

Expansion of the moving labels in the fixed finite basis makes this holomorphic near the whole boost path. For real, TI2 makes it constant with value). The one-variable identity theorem propagates that constant along the connected path to. Thus is an actual extension of the original emission. The opposite path, with permuted labels, gives the second germ: permutation is complex linear and commutes with real boosts. Both germs use the same spectator height sequence tending to zero. The next argument identifies the two germs and passes their equality to the boundary exchange relation.

In the lifted object retain the ordered labels, transposition, height profile, finite label space and the entire boost action together with the lens data. The readout LG2 is compatible with the original tube readout by the path argument, and restriction recovers the original emission. The germ is therefore independent of its path presentation on overlaps by analytic uniqueness; the retained order still records which boundary exchange is to be established next.

Recovery of the boost-path germ. The source germ record retains the adjacent transposition , physical boundary point, spectator-height profile, direction, endpoint angle, and one finite label carrier containing the word. The configuration reader is the actual complex boost Directly taking its imaginary part at gives (LG1), so the same closed-to-open cone path and

its product neighborhood are retained. The angle and neighborhood are chosen with the quantifiers supplied for that boundary configuration and profile; the record does not replace them by a global uniform angle over all profiles. On the horizontal image of the finite coefficient section, put and Since, its entire power series and recovery give

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pduZ(z)jd = ezB,
uZ(z)jd = jdu(z),
uZ(z1)uZ(z2) = uZ(z1 + z2).

Here the identity at = 0 is the identity on that horizontal image. This preserves the source’s finite-dimensional analytic label action. Define the source germ reader using those lifted labels and the same boosted point:

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GZ(Z) = FZ
(uZ(it)jdvk)k(Ae(it)Z) = F(u(it)vk)k(Ae(it)Z) = G(Z).

The same equality holds for) throughout its retained connected path neighborhood. Real boost covariance and the one-variable identity theorem therefore recover its constant value at the endpoint before restricting to the original tube. Permutation commutes with the componentwise lifted boost action, so the opposite path recovers the transposed germ with the same spectator profile. These explicit readers preserve the local holomorphic extension and its overlap uniqueness; identification of the two orders remains the subsequent germ-comparison step.

19.29 Germ identification and actual partial boundaries

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First set all real times to zero and retain the common spectator profile with only the selected
adjacent heights tied. The Euclidean configuration q = ((xk, yk))k has no collisions: untied slots
have different heights and the tied spacelike pair has different spatial positions. Define the totally
real Wick slice
M(q) = (qk, iq0
k)k.

Its image spans the entire complex configuration space: every point is), using the real and imaginary spatial parts and, for time, the imaginary part and negative real part respectively. The germ equals the smooth Euclidean hierarchy on the strictly ordered side. On the opposite open side equals the permuted hierarchy by strict-wave identification, and hence by permutation symmetry. Both open sides accumulate at, since the tied heights can be separated in either direction. For every, continuity of derivatives up to the tied-height point gives

The chain rule identifies these with restrictions of the complex-linear derivatives. Complex multilinearity and the spanning property give equality on all complex arguments. Equality of the convergent Taylor series therefore proves

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G+ = G−
near M(q).
(GE1)

This uses equality of limiting jets, not an assumed common open Euclidean side. To identify the distributions represented by these germs, let be an interior tube function and a holomorphic extension through . Suppose the interior height slices converge weakly to . Choose a small real ball about and a small height neighborhood whose product lies in the germ domain and on which is bounded by. For every compact smooth joint test supported

in that ball, agreement with at interior points and dominated convergence with majorant give

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Q(φ) =
Z
G(X′ + iY )φ(X′) dX′.
(GE2)

The approach is nonempty since the base point is in the tube closure; the weak limit is the one already supplied by the damped-distribution boundary theorem. Thus (GE2) identifies that particular partial boundary; it does not merely define a new distribution. Apply GE2 to each order. In the transposed case reindex all real variables by: it preserves Lebesgue measure, is its own inverse, pulls back the test and permutes the heights. Both partial boundaries are consequently represented by the two germs in the same original variables. GE1 gives their local equality at these zero-real-time spacelike configurations. The extension to general spacelike configurations and passage to zero height remain the subsequent source steps. Apply this argument separately to the original and transposed tubes.

The lifted germ record retains the Wick slice, both ordered domains, all limiting jets, the common smooth hierarchy and the tested boundary functional. Its realization is the pair of analytic restrictions; its observable readout is GE2. The spanning argument proves recovery of the entire germ from the retained jets, while restriction and the specified weak limit recover the original partial distributions. Reindexing is an invertible source operation, so no orientation or test-variable information is lost by comparing the two orders.

Explicit germ recovery from Wick-slice jets. For a complex tangent argument, retain its unique real decomposition) is the retained real jet, its full complex derivative is recovered by the finite formula

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DmG(M(q))[H1, . . . , Hm] =
X
ε∈{0,1}m
i|ε|Jm[w1,ε1, . . . , wm,εm],
wa,0 = ua,
wa,1 = va.

The zeroth jet is the scalar value. Thus the equal limiting jets from the two different Euclidean sides recover all equal complex derivatives. The analytic germ readout is their convergent local Taylor series,

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∞
X
1
m!DmG(M(q))[H, . . . , H].
G(M(q) + H) =
m=0

This is the explicit recovery map giving (GE1) from the supplied smooth hierarchy jets and the analytic germs already constructed. For the partial boundary retain the original weak-limit functional , the local germ ball and the admissible interior height approach. Scalar source recovery gives on its compact tests

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QZ
Y (φ) = QY (φ) =
Z
G(X + iY )φ(X) dX.

The same local bound controls both readers, so this is the actual partial limit (GE2). If is the transposed reader in its own coordinates, its recovered original-variable functional is

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(σ∗Q−
Y )(φ) = Q−
Y (φ ◦σ),
| det σ| = 1,
σ−1 = σ.

The heights and ordered labels are permuted with these same variables. It follows that the two recovered partial boundaries are represented in one coordinate system by andrespectively, and (GE1) gives their local equality with the source’s zero-real-time scope. The later passage to general spacelike points and zero height remains attached to its subsequent construction.

19.30 Independent heating, Gram recovery and product-test density

For a positive ordered height profile, choose smaller than its initial height and all its gaps. With , retain the actual chamber vector

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ξp,y,v(X) = R(z1, p, x1)Bp(v1)R(z2 −z1, 2p, x2 −x1)Bp(v2) · · ·
R(zn −zn−1, 2p, xn −xn−1)Bp(vn)Ω.

At arity zero it is Ω. Recursive contractions give

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∥ξp,y,v(X)∥≤∥Ω∥
Y
i
∥Bp(vi)∥.
(HV1)

These vectors vary strongly continuously in: heat factors vary in operator norm in the open half-plane, spatial translations vary strongly, and induction applies to the recursively varying ket. At zero real times the vector is the planted future word at (), by the insertion/translation recursion of SW4 with reference width zero. Its Gram product with an independently chosen is the Euclidean hierarchy of the reflected reversed first word followed by the second. For arbitrary real times multiply the first time array by ̄ and the second by. Antilinearity of the first slot makes the Gram product holomorphic in on an open horizontal strip containing 0 1 where both widths satisfy their height inequalities. The hierarchy is holomorphic there too. On purely imaginary both words are Euclidean planted words with shifted heights. Analytic uniqueness yields

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⟨ξp,y,v(X), ξq,z,w(Y )⟩= f(Jvn,...,Jv1,w1,...,wm),(−yn,...,−y1,z1,...,zm)
(Xn, . . . , X1, Y1, . . . , Ym).
(HV2)

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The widths need not agree. The first real block is reversed without another geometric reflection;
its imaginary heights already changed sign.
For a smeared word b = ((vi, φi))n
i=1, put φb(X) = Q
i φi(Xi), yi(t) = it (the integer slot index
times the positive real t), p(t) = min(t, 1)/8, and

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ηb(t) =
Z
φb(X)ξp(t),y(t),v(X) dX.

HV1 and Schwartz integrability make this an absolutely integrable Hilbert-valued integral. Common real translation changes only the initial propagator; all relative ones are unchanged. At arity zero, heat invariance of Ωextends by holomorphic uniqueness to the half-plane and then its real-time edge. Changing integration variables gives

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ηa·b(t) = U(a)ηb(t).
(HV3)

Integrating HV2 is justified by the same fixed-height bounds. Reversal of the first block is a measure-preserving real linear isometry; its test is ), exactly that of. The joined height profile is strictly ordered for all independent 0 and tends to zero with the pair. The independence of the parameters is stronger than a diagonal limit and is what will make the shot vectors Cauchy. The all-height boundary theorem gives

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lim
t↓0
u↓0
⟨ηb(t), ηc(u)⟩= W(b∗c).
(HV4)

The two-parameter conclusion, rather than only a diagonal limit, is retained for the subsequent Cauchy argument.

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Finite sums of product Schwartz tests are dense in joint Schwartz space.
For a compactly
supported joint test φ, take a product cutoffχ = Q
i χi equal to one on its support. Compact
frequency cutoffs of its Fourier transform, inverted and multiplied by χ, converge to φ in Schwartz
topology. Each is an integral over a compact frequency set of product tests

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χ(X)ei⟨q,X⟩=
Y
i
χi(Xi)ei⟨qi,Xi⟩.

This family is continuous in every Schwartz seminorm: its derivatives vary jointly on common compact spatial support. Quadrature approximates the integral in each seminorm by finite sums of products. A diagonal choice over the countable seminorms gives Schwartz convergence. Finally smooth compact spatial cutoffs approximate any Schwartz test. For arity zero, the constant one on the one-point space already spans it.

The lifted heated-word source retains independently, the chamber vectors and their joint Gram kernel. Readout HV2 recovers the hierarchy, and HV4 recovers its boundary word functional. The bound HV1 supplies the integration domain, while product-test density recovers joint tested distributions from these word readouts. No diagonal identification of heating parameters enters this recovery.

Independent heated-vector and joint-test recovery maps. Use the chamber section unitary and the already lifted buffered factors. Form the source shot in exactly the displayed order, with the initial factor). Cancellation of adjacent inverse unitaries gives

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ξZ
p,y,v(X) = VHξp,y,v(X),
∥ξZ
p,y,v(X)∥= ∥ξp,y,v(X)∥.

Thus (HV1) and strong continuity recover on the same positive-height and fixed-width domains. For independent widths and profiles the Gram reader is

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⟨ξZ
p,y,v(X), ξZ
q,z,w(Y )⟩= ⟨ξp,y,v(X), ξq,z,w(Y )⟩.

It recovers the complete reversed-adjoint/second-word ordering in (HV2), including the negative reversed first heights and unchanged reversed real coordinates. Antilinearity in the first slot retains the ̄ continuation used to establish that identity. Boundedness of permits it to pass through the absolutely integrable Hilbert-valued smearing integral, so

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ηZ
b (t) = VHηb(t),
ηZ
a·b(t) = UZ(a)ηZ
b (t),
lim
t,u↓0⟨ηZ
b (t), ηZ
c (u)⟩= W(b∗c).

The limit uses the full two-parameter approach and the same joined height profile. No equality of widths or heating parameters is imposed. The measure-preserving reversed-variable map and conjugated first test are retained before this limit.

Finally let be the finite product-test sums obtained by the source’s compact frequency quadrature and spatial cutoffconstruction, with in all joint Schwartz seminorms. Every retained continuous joint distribution reader recovers by

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QZ(φ) = lim
r QZ(φr) = lim
r Q(φr) = Q(φ).

This recovery is independent of the approximating product sums by continuity. It includes the one-point test space at arity zero and preserves the passage from word-vector identities to full joint tests.

19.31 Spacelike exchange with all spectator variables

Two germs anchored to the same tube at a closure point agree: on a small connected ball both equal the tube function on a nonempty open intersection, so analytic uniqueness applies. Along a continuous path in two tube closures, nearby selected germs therefore agree with the restrictions of the germs at the original base point. Equality of the two extensions is locally constant along the path, in either direction, and hence constant on a connected interval. For any designated spacelike pair, contract all real times by

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Xk(s) = (xk, stk),
0 ≤s ≤1.

Spacelikeness persists sincexx. With the fixed spectator heights the path is in both tube closures, and the germs exist throughout. GE1 at = 0 propagates to = 1. GE2 now identifies the partial distributions locally on each sufficiently small real ball in the spacelike-pair open set. Their difference has support disjoint from this open set, so it vanishes on Schwartz tests whose closed support is contained there. One can see the last assertion by compact smooth cutoffs and their Schwartz convergence, using the local vanishing on each compact support. For each profile, the partial distributions agree. Their closed-height multipliers converge in Schwartz topology at zero height, where the cutoffis one near the frequency support; the damped boundary theorem therefore gives the undamped distribution on every fixed test, in both orders. Uniqueness of scalar limits yields

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

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For a product test, its closed support lies in the inverse image of each factor support under the
coordinate projections. Mutually spacelike supports of the selected factors therefore meet the LC1
condition. Permuting them exchanges precisely those factors. Inserting arbitrary spectator words
on both sides gives
W
a (v, φi)(w, φj) b

 = W
a (w, φj)(v, φi) b

.
(LC2)

There is no restriction on spectator supports. The lifted source retains the full joint distribution and the path of both germs; readout first takes the specified partial boundary and then the zero-height limit. LC1 proves compatibility with adjacent permutation on its spacelike domain, and LC2 recovers the word relation required on the common field domain. It does not discard spectator variables or replace the tested relation by pointwise boundary values.

Recovery of spacelike exchange with spectators. Let be the open real configuration set where the selected adjacent difference is spacelike. Retain its full joint test space and define the exchange-difference reader ∆

At each retained spectator profile, use the two actual partial boundary readers in the original variables to define ∆. The source germ/path recovery gives the same ∆ and makes it zero on each compact test in. Smooth compact cutoffs extend that zero to Schwartz tests with closed support contained there. The fixed-test zero-height recovery is explicitly

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∆Z
v (φ) = lim
m→∞∆Z
v,m(φ) = lim
m→∞∆v,m(φ) = 0.

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For each order this limit uses Dey(m)g →ηg in Schwartz space and c
Wv(ηg) = c
Wv(g) by spectral
support.
Thus the reader preserves the order of partial-boundary realization, profile limit and
adjacent exchange. On product tests the unchanged coordinate projections recover exactly the
mutual-spacelike condition on the two selected factors. All other factors remain arbitrary, recovering
the whole LC2 word identity with both spectator lists.

19.32 Faithful boundary labels and reconstruction on a common domain

Suppose a union label is a boundary marker: every expectation containing its smearing, with arbitrary words before and after, vanishes. Splitting a word at that slot gives vanishing on product tests; their density and distributional continuity give vanishing on all joint tests. Represent by a raw label in a finite quotient field space. The associated full-tube emission vanishes, so TI1 makes every strictly time-ordered Euclidean reading zero. This remains true after any permutation of the arguments, because the marker may occupy any slot. At any distinct Euclidean configuration sort its heights and approach them by strictly ordered heights with spatial coordinates fixed. Distinctness persists locally; smoothness gives zero at tied heights too. Undo the sorting permutation. The boundary-marker reader can be expressed as the kernel

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M = {x : W(a(x, φ)b) = 0 for every a, b, φ}.

The label section and scalar word recovery preserve every defining pairing of this set. Product-test density and tube/Euclidean recovery then give the same raw Euclidean null preimage for each finite label representative; the retained quotient map reads its class as zero. Consequently the source boundary-marker kernel on physical union labels is also zero. This uses the supplied null-space criterion and its retention on the selected hierarchy.

The Euclidean null-space criterion puts in its finite-level null space. The selected hierarchy retains exactly the original null spaces, so its original quotient class, and hence in the union, is zero. Thus boundary faithfulness is recovered from the retained Euclidean quotient rather than imposed as a new field assumption.

HV4 makes each) Cauchy: the four terms in

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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)⟩

have the same independent two-parameter limit. Completeness supplies, and HV3–HV4 give

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

Thus the Hermitian kernel) is positive, since

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2
≥0.
X
X
zαzβK(bα, bβ) =
α
zαηbα
α,β

The empty expectation is one. Take finite formal sums of smeared words, with this form linear in the second slot, quotient by its zero-norm subspace, and complete to . Write [] for a word class, = span, and Ω]. The dense common field domain is exactly; the vacuum is a unit vector. For), formal left insertion satisfies

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

If is null, choose the formal sum; Cauchy–Schwarz for the positive form and FR2 imply = 0. Hence insertion descends, and

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Φ(x, φ)[b] = [(x, φ)b]: D →D
(FR3)

is well defined. Pairing every proposed additive or scalar relation with all word vectors proves bilinearity in, by the multilinearity of the joint distributions and product tests and density.

FR2 gives the adjoint identity on this common domain. Iteration of FR3 recovers all word vectors and their prescribed vacuum expectations; their span gives cyclicity. For fixed, the map is norm-continuous: expand the squared norm difference into diagonal and mixed joint-distribution pairings. The two-smearing pairings are jointly continuous by continuity of product tests in Schwartz topology. Finite sums of word vectors give the general assertion. No field boundedness is asserted.

LC2 paired against all word probes gives

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[Φ(x, φ), Φ(y, ψ)]

D = 0

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for mutually spacelike test supports.
If all Φ(x, φ) vanish on D, pair the zero insertion with
the reversed adjoint left spectator word to obtain vanishing expectations with x in every slot.
Involutivity of J removes the double reversal. The marker criterion above then implies x = 0,
proving faithfulness of the prescribed label quotient.
For the source lift define the word realization on finite sums by V0(P
b cb[b]) = P
b cbηb. FR1
proves that its kernel is exactly the word null space and that it is isometric there.
It extends
to a unitary from HW onto the closed span of the retained chamber limits ηb; no assertion that
this span equals a larger chamber is needed here. The word source, its null quotient and the left-
concatenation reader therefore recover the Hilbert space and unbounded operators with precisely
their actual domain. The operator identity and adjoint pairing are transported on that domain,
not inferred from compression of bounded approximants.

Positive word-quotient and common-domain recovery. Let W be the formal finite word span and. The lifted heated vectors recover their actual limits under the bounded section unitary:

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X
!
2
= K
X
b
cbb,
X
b
cbb
ηZ
b = lim
t↓0 VHηb(t) = VHηb,
b
cbηZ
b
.

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Thus the source word realization has exactly kernel N and recovers the same positive word quotient.
On its completion, write VW : HW →HZ
W for the resulting unitary and DZ = VW D. The exact
operator reader is

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ΦZ(x, φ)VW [b] = VW [(x, φ)b],
V −1
W ΦZ(x, φ)VW = Φ(x, φ)
on D.

It uses null preservation from (FR2) before completion; its domain is precisely the transported algebraic word span. The adjoint statement on that span is the pairing identity

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⟨ΦZ(x, φ)VW p, VW q⟩= ⟨VW p, ΦZ(Jx, φ)VW q⟩,
p, q ∈D.

For a fixed word-domain vector and a convergent test sequence, unitarity also recovers the exact smearing norm difference

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∥ΦZ(x, φr)VW p −ΦZ(x, φ)VW p∥= ∥Φ(x, φr −φ)p∥−→0.

Its continuity follows from the retained diagonal and mixed Gram pairings, rather than an operator-norm bound. The same intertwining recovers the LC2 commutator on this domain, cyclic word generation, and the zero boundary-marker criterion for faithfulness. Thus the positive kernel, completion, unbounded field actions and their precise common domain all use one word quotient.