Phase decomposition and Wightman reconstruction · Part 4

It is continuous in the Schwartz topology. Here is the bound that accounts for this assertion. For an invertible linear map, translation, and seminorm) = sup, the chain rule and) give

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pi,k
φ(A( · −b))

 ≤(∥A−1∥+ ∥b∥)i∥A∥k 2i max
r≤i
s≤k
pr,s(φ).
(BC4)

The derivatives vary continuously on compact sets as vary, and (BC4), with higher weighted seminorms of the fixed test, controls the complement. This proves convergence in every Schwartz seminorm. For a boost in direction, pointwise differentiation of inverse placement gives

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d
ds(gs · φ)

s=0
= −Ba,0φ.
(BC5)

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This is also a derivative in Schwartz topology. In fact its pointwise derivative at arbitrary s is
gs·(−Ba,0φ), a strongly continuous Schwartz family by (BC4). The fundamental theorem of calculus
after evaluation at X writes the difference between the difference quotient and this derivative as
Z 1

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D(us) −D(0)
(X) du.
0

The same equality holds after each spatial derivative. Taking each weighted supremum bounds its seminorm by sup(0)), which tends to zero by continuity on the compact parameter interval. This proves (BC5) strongly without integrating in the Schwartz space itself. Pass (BC3) to the boundary. Its extra height terms are the finite sum

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i
X
i
yi Ψv(y; ∂xi,aφ);

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each tends to zero, because the tested boundary limit exists for that fixed derivative test and yi →0.
Therefore
X

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i
Wv[i←Kavi](φ) = Wv(Ba,0φ).
(BC6)

Choose one finite field space containing all labels, using the directed inclusions. Consider

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H(s) = W(ρ(gs)vi)i(gs · φ).

Differentiate at zero. The label derivative is the sum on the left of (BC6), and the test derivative is its negative by (BC5). Their cancellation gives(0) = 0. This product rule needs no topology on the entire field union: expand the multilinear distribution in a basis of the one finite field space, differentiate the finitely many scalar coordinate products and the pairings with fixed basis labels, and sum. At an arbitrary parameter value repeat the same argument with the already moved labels and test. The group law gives) = 0 everywhere, hence

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W(ρ(gs)vi)i(gs · φ) = Wv(φ).
(BC7)

Real translations and spatial rotations follow first at strict heights from their Euclidean identities by one-complex-parameter uniqueness, and then at the boundary by change of variables, whose Jacobian has modulus one. The transformations satisfying the boundary covariance identity form a subgroup: composition follows from the representation law and test pushforward law, and inverses follow by applying the identity to inversely transformed labels and tests. Spatial rotations and coordinate boosts generate the specified Lorentz group, so (BC7) gives full covariance once that generation statement is established.

Dual cone and a smooth frequency cutoff

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The absolute-frequency cone has a useful elementary dual description. Write p = (p1, . . . , pn), with
total momentum zero, and Pj = Pn
i=j pi. Its defining remaining condition is that each proper
suffix momentum belongs to the closed forward cone.
If y has all successive differences in the
corresponding closed dual forward cone, summation by parts gives

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n
X
⟨p, y⟩=
j=2
⟨Pj, yj −yj−1⟩≥0.
(DC1)

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The endpoint terms vanish because the total momentum and the empty suffix are zero. Self-duality
of the forward cone proves the inequality term by term. The identity also covers the empty tuple.
There is a quantitative interior form. If y is in the open tube-height cone, choose r > 0 such
that B(y, r) is contained in that cone. For ∥u −y∥< r/2 and nonzero p in the frequency cone, the
point
u −
r
2∥p∥p

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lies in B(y, r). Apply (DC1) there to obtain

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⟨p, u⟩≥(r/2)∥p∥.
(DC2)

For = 0 the estimate is immediate. This gives one positive exponential damping margin on a whole neighborhood of any interior height.

A smooth cutoffnear the frequency cone is constructed explicitly. Convolve the indicator of its open distance-two neighborhood with a nonnegative, unit-integral smooth bump supported in the ball of radius 12. Denote the result by. It equals one on the distance-one neighborhood of the cone: every point sampled by the convolution then has distance less than 32 from the cone. Its support is contained in the distance-52 neighborhood, and its closed support lies in the distance-three neighborhood. All derivatives are globally bounded, since derivatives can be put on the compactly supported bump and the indicator has absolute value at most one:

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∥Dkη∥∞≤∥Dkb∥L1.

The smoothing and support assertions use the actual Euclidean metric on the full absolute-frequency space, including its dimension-zero case. For a closed tube height and a frequency in this closed support, choose a cone point within distance three of. Equation (DC1) and Cauchy–Schwarz imply

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−⟨y, x⟩≤⟨y, z −x⟩≤3∥y∥.

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Hence on ∥y∥≤R,
∥Dl
xe−⟨y,x⟩∥≤e3RRl.

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Leibniz’s formula and the cutoffderivative bounds show that η(x)e−⟨y,x⟩has uniformly controlled
derivatives for bounded closed heights. More generally the polynomial-growth cutoffestimates used
in the source give
∥Dk
x(ηe−⟨y,·⟩)(x)∥≤Ck,R(1 + ∥x∥)Nk.
(DC3)

Outside the closed support all its derivatives vanish. Thus these multipliers act continuously on Schwartz tests, with bounds uniform over bounded closed heights; interior heights additionally have the exponential margin (DC2). These are the cutoffand damping estimates needed for the subsequent tube construction.

Holomorphic tube functions from the frequency distribution

The tube is an open complex-time domain with strictly ordered imaginary heights. Positive heights provide the damping needed for the frequency distributions and Hilbert vectors. Removing them is a tested tempered boundary limit; it is not substitution of a pointwise value at zero height. The Gram argument below shows that independently heated word vectors give one positive boundary form on the common word space.

Here the radian Fourier transforms on, are explicitly

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H g(q) =
Z
e−i⟨x,q⟩g(x) dx,
Pf(q) = (2π)−N
Z
ei⟨q,x⟩f(x) dx.

They are inverse continuous maps of the Schwartz space. For a closed tube height, set

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Dyg(q) = η(q)e−⟨y,q⟩g(q).

The estimates (DC3) and Leibniz’s rule bound each seminorm of, uniformly for in bounded subsets of the closed height cone, by finitely many seminorms of Joint smoothness of the multiplier gives convergence of every jet on compact frequency sets as varies. The same bounds with one extra polynomial weight control the complements. It follows that is continuous in the Schwartz topology even at closed-cone boundary heights.

For a full complex configuration in the open tube, define the lens

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λZ(q) = η(q)ei⟨Z,q⟩= η(q)ei⟨X,q⟩−⟨Y,q⟩.

It is a genuine Schwartz test. More precisely, (DC2), together with the distance-three support bound, gives constants such that

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−⟨Y, q⟩≤A −c∥q∥
(q ∈supp η);
A = 3(∥Y ∥+ c) is sufficient.

Each frequency derivative of the exponential contributes at most a power of its linear functional norm. Derivatives of have polynomial growth. Leibniz’s formula and

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(1 + t)keA−at ≤k!a−keA+a
(t ≥0, a > 0)

therefore prove, for every, a finite bound

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sup
q (1 + ∥q∥)ke(c/2)∥q∥∥Dj
qλZ(q)∥< ∞.
(TL1)

The exponential margin retained in (TL1) justifies complex differentiation of the lens as a Schwartzvalued function. For completeness, if is a small real-linear, complex-valued frequency functional, with min( 1), the scalar exponential remainder and its derivatives satisfy

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∥Dj(eL(q) −1 −L(q))∥≤∥L∥2(1 + ∥q∥)2ea∥q∥.
(TL2)

For = 0 this is the exponential-series remainder. For = 1 differentiate once and bound For2, derivatives of the affine terms vanish and the bound follows from, using 1. Multiply (TL2) by the lens and use (TL1) and Leibniz’s rule. Every Schwartz seminorm of is at most Dividing byproves the complex Fréchet derivative (TL3)

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in the actual Schwartz topology, jointly in all complex coordinates.
For the tempered frequency functional c
Wv, define

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Fv(Z) = (2π)−N c
Wv(λZ).
(TL4)

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Continuity and linearity of c
Wv transfer (TL3) to complex differentiability of Fv. Finite-dimensional
complex analysis makes it holomorphic on the full open tube. At this point no spectral support
assertion is needed for holomorphy itself; the support condition will make the result independent
of the cutoff.
The identification with a damped distribution requires an integral argument in Schwartz topol-
ogy. At a fixed interior Y , put g(q) = η(q)e−⟨Y,q⟩, a Schwartz test by (TL1), and gX(q) = ei⟨q,X⟩g(q).
Its Fourier transform is the translate (H g)( · −X). For any Schwartz function h,

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pi,j
h( · −X)

 ≤2i h
pi,j(h) + ∥X∥ip0,j(h)
i
.
(TL5)

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This follows from ∥q∥≤∥q−X∥+∥X∥, splitting according to which summand is larger. Continuity
of a tempered distribution bounds it by finitely many Schwartz seminorms. Thus X 7→c
Wv(gX) is
continuous and has at most polynomial growth. Its product with any joint Schwartz test φ(X) is
absolutely integrable.
For compactly supported φ, integrate gXφ(X) over its compact support. Uniform control of
every jet permits the continuous functional c
Wv to pass through this integral. Fourier inversion
gives
c
Wv
DY (Pφ)

 =
Z
Fv(X + iY )φ(X) dX.
(TL6)

For arbitrary Schwartz, use its smooth compact cutoffs. The left side converges by continuity on Schwartz space; the right side converges by (TL5), polynomial growth, and dominated convergence. This proves (TL6) for full joint tests, with an absolutely convergent right side. It identifies the holomorphic emission with the actual damped boundary distribution at every fixed interior height, without a fictitious norm on Schwartz space or an exchange of unbounded integrals.

Identification of the tube continuation and its covariance

Spectral support now removes the auxiliary cutoff. A distribution supported in the closed frequency cone annihilates a Schwartz test that vanishes on a neighborhood of that cone. For compactly supported, the two tests

agree on the distance-one neighborhood of the cone. Their difference has closed support disjoint from it; hence the supported distribution gives them the same value. This argument uses the compact support of when defining the second test. It does not assert that an unrestricted exponential is a Schwartz multiplier.

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At a fixed strictly ordered temporal height y, the strict restriction fv,y is bounded and continuous
by (SW7). Its integral against a joint test therefore defines a tempered distribution uy. In (FT2)
use the compact test e−⟨ey,q⟩g(q). Its exponential lift cancels exactly. The preceding cutoffremoval
then gives
c
Wv(Deyg) = buy(g)

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for all compactly supported smooth g. Both sides are continuous on Schwartz space, so smooth
compact cutoffs extend equality to every Schwartz test. Fourier inversion and (TL6) show
Z
Fv(X + iey)φ(X) dX =
Z
fv,y(X)φ(X) dX.

Both functions are continuous. Equality against all compact smooth tests gives almost-everywhere equality, and full support of Lebesgue measure upgrades it to pointwise equality:

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Fv(X + iey) = fv,y(X).
(TI1)

Thus the constructed full-tube function continues the very strict waves previously constructed. Let be a real Lorentz transformation in the specified group, andits dual for the Euclidean frequency pairing, so that. The tube-height cone is invariant under, and the frequency cone is invariant under. Compare the two Schwartz lenses

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λZ(q),
λrZ(r∨q).

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Their exponential factors agree everywhere. Their cutofffactors need not agree globally, but both
equal one near every cone point p: use the intersection of the unit ball about p and the inverse
image of the unit ball about r∨p. Consequently their difference has closed support disjoint from
the cone and is annihilated by c
Wv. Fourier transformation of boundary covariance, followed by this
lens comparison, gives
F(ρ(r)vi)i(rZ) = Fv(Z).
(TI2)

This proves real Lorentz covariance inside the tube without requiring an invariant cutoff. The same local support argument compares any two permitted cutoffs and proves that their emissions agree.

Analytic germs at an adjacent spacelike pair

Fix adjacent argument slots + 1, and write for their transposition. Choose height profiles 0 with exactly this adjacent gap flattened: , and all other ordered gaps strictly

positive. One explicit choice is + 1), where except. At two arguments this is the zero profile. Both the original and transposed imaginary configurations lie in the closed tube-height cone, and their complex configurations are in the closures of the respective tubes.

At a real configuration with spacelike difference, setv, and choose the spatial unit vector v. The infinitesimal boost

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Cev = (v0e, e · v)

is strictly future timelike, since its spatial norm is and its time component is. Let project onto the spatial- and time plane. The complex boost is

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Ae(z)Z = Z + sinh z CeZ + (cosh z −1)SeZ.

For, its imaginary difference at any adjacent pair is

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Y + sin b V + (cos b −1)W,
(LG1)

where is the original imaginary difference, applied to the real difference, and For every unflattened adjacent gap, is already strictly timelike, so (LG1) remains so in a neighborhood of = 0 and of the given configuration. At the flattened pair, is strictly timelike. Use the exact identity

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sin b V + (cos b −1)W = sin b

V −
sin b
1 + cos bW

.

The vector in parentheses is strictly timelike throughout a neighborhood of (), and sin 0 for sufficiently small 0. Adding it with that positive coefficient to a closed forward-cone vector enters the open cone. Since there are finitely many adjacent gaps, one product neighborhood and one small positive angle work for all of them, uniformly down to angle zero. Apply the same calculation to the transposed configuration with the opposite imaginary boost; at the flattened pair the two reversals of sign 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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The labels also have an analytic boost action. The following elementary Banach-algebra ar-
R t
gument supplies it.
For a continuous one-parameter group u(t), put V (t) =
0 u(s) ds.
Then
V ′(t) = u(t) and V (t)/t →1. Choose nonzero a so that g = V (a)/a is invertible. The group law
gives
u(t) = a−1[V (t + a) −V (t)]g−1.
u(t)V (a) = V (t + a) −V (t),

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The right side proves differentiability. If B = u′(0), differentiating the group law gives u′(t) = u(t)B,
whence
d
dt[u(t)e−tB] = 0. Evaluation at zero yields u(t) = etB. For each finite-dimensional field
space this gives the entire complex extension u(z) = ezB of its boost representation.
Fix the finite field space containing the finitely many labels. For a boundary configuration Z0,
take the small imaginary-boost endpoint it just constructed and define near Z0

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

The endpoint lies in the tube, so (LG2) is holomorphic on a full neighborhood of. It agrees with the original emission on nearby points already in the original tube. To prove this agreement, for such a point consider

This is holomorphic in a neighborhood of the entire imaginary-boost path. For moving labels, expand in the finite field-space basis; each coefficient product is holomorphic and each fixed-label emission is holomorphic. At real, (TI2) makes) equal to the original emission at. The one-variable identity theorem propagates this constant value along the connected path from zero to. This proves the claimed agreement and hence the analytic germ.

For the transposed tube repeat the argument with the opposite path and with labels permuted by. The permutation map is complex linear and commutes with real boosts, so the same argument applies. We have thus constructed both holomorphic germs at every spacelike adjacent pair, with the common spectator-height sequence tending to zero. The next argument identifies the two germs and passes their equality to the boundary exchange relation.

Identifying the germs and realizing their partial boundaries

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First consider a real configuration whose time coordinates are zero, with the adjacent pair spacelike,
and the spectator profile y chosen above. The associated Euclidean configuration q = ((xk, yk))k
has no collisions: every pair other than the flattened pair has different heights, and the flattened
pair has different spatial positions because it is spacelike.
Use the totally real Wick slice
M(q) = (qk, iq0
k)k.

Its image spans the full complex configuration space over. Indeed each complex configuration is ), by taking spatial real and imaginary parts and, in the time coordinates, respectively the imaginary part and the negative real part.

Let andbe the two holomorphic germs previously constructed at). On the open Euclidean side with strictly ordered heights, equals the original smooth Euclidean hierarchy . On the different open side where the transposed heights are strictly ordered, equals the same function: the strict-wave identification gives the permuted Euclidean hierarchy there, and permutation symmetry identifies it with. Both open sides accumulate at, since the tied heights can be separated in either direction.

For each order, differentiate on the relevant open side and approach. Continuity of the -th derivatives of the analytic pullbacks and of the smooth hierarchy proves

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Dm(G+ ◦M)(q) = DmSv(q) = Dm(G−◦M)(q).

The chain rule identifies these real derivatives with restrictions of the complex-linear derivatives of to arguments in the image of. Complex multilinearity and the spanning property extend equality to all complex arguments. Thus all complex derivatives agree. The convergent analytic Taylor series give on a neighborhood of (GE1)

Notice that equality was not assumed on a common open part of the two Euclidean sides; equality of their limiting jets is what proves (GE1).

We also need to know what the germs represent as distributions. The following local argument supplies that fact. Suppose is an interior tube function, is a holomorphic germ through and the interior slices have a weak distributional limit as their heights approach . Choose a small ball of real configurations about and a small height neighborhood such that is defined and bounded in their product, and agrees with at the interior points there. For a joint compact smooth test supported in that ball, dominated convergence with the majorant gives

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Q(φ) =
Z
G(X′ + iY )φ(X′) dX′.
(GE2)

The height approach is nonempty because the base point belongs to the closure of the tube. Holomorphy gives continuity of, and the already established damped-distribution boundary theorem supplies the weak limit. Thus (GE2) identifies that particular partial boundary; it does not merely define a new distribution.

Apply this argument separately to the original and transposed tubes. In the second application reindex all real integration variables by the adjacent transposition. This map preserves Lebesgue measure, is its own inverse, and sends the test to its corresponding pullback; it also permutes the imaginary heights. Therefore the transposed partial boundary is represented locally by the second germ in precisely the same original variables. This proves the local representation of both partial distributions by their respective germs. Combined with (GE1), it proves their local equality at the zero-real-time spacelike configurations. The extension from these configurations to general spacelike pairs and the final zero-height limit are the next steps.

Independently heated word vectors and their Gram limit

The positivity argument uses actual vectors in the reflection-positive chamber. For a positive ordered height profile, choose a width smaller than its initial height and its gaps. The buffered shot vector is

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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,

where. At arity zero it is Ω. Recursive contraction estimates give

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∥ξp,y,v(X)∥≤∥Ω∥
Y
i
∥Bp(vi)∥.
(HV1)

It is strongly continuous in: the heat factors vary in operator norm while their parameters remain in the open half-plane; the spatial translations are strongly continuous; induction handles their application to the recursively varying ket.

If the real times vanish, the shot is exactly the planted future word at the Euclidean positions ). This follows by the insertion and translation recursion used for (SW4), now with the initial reference width zero. Its inner product with a second shot, of independent width and positive profile, is the Euclidean hierarchy of the first word reflected and reversed, followed by the second. The first block of heights is (), the second is; the first labels are, with the induced adjoint on union labels.

The same identity holds for arbitrary real times. Multiply the first array of real times by ̄ and the second by, in a complex parameter. Antilinearity of the first Hilbert slot makes the scalar inner product holomorphic in. There is an open horizontal strip containing both zero and one where all initial-height and gap inequalities for these fixed widths hold. The hierarchy side is holomorphic on that strip as well. At purely imaginary both shots are Euclidean planted words, with the corresponding shifted heights, so the preceding Gram identity holds there. The identity theorem gives it at = 1. Thus

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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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In particular no equality of the two widths was needed.
The real coordinates in the reversed
first block are reversed without a further geometric reflection; the imaginary heights have already
changed sign.
For a smeared word b = ((vi, φi))n
i=1, let φb(X) = Q
i φi(Xi), take yi(t) = it with slots numbered
from one, and take p(t) = min(t, 1)/8. Define

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ηb(t) =
Z
φb(X)ξp(t),y(t),v(X) dX,
t > 0.

This is an absolutely integrable Hilbert-valued integral by (HV1) and Schwartz integrability. Translating every real argument by a common translates the shot by the chamber’s real-time translation unitary. For the empty word this uses invariance of Ωunder both the spatial group and the real-time edge group: invariance under heat first extends to the holomorphic half-plane by analytic uniqueness, and then to its real-time edge. For nonempty words the initial propagator changes by that edge translation, whereas all relative propagators are unchanged. Change of variables in the integral therefore gives (HV3)

Apply (HV2) under the two integrals. Their absolute integrability follows from the same fixed-height bounds. The map () is a real linear isometry and preserves Lebesgue measure. The product test becomes ), exactly the test of the reversed adjoint word followed by. Hence the all-height boundary theorem gives the independent two-parameter limit lim (HV4)

The joined profile is strictly ordered for all 0 and tends to zero as the pair tends to zero, which justifies using that boundary theorem. The independence of the parameters is stronger than a diagonal limit and is what will make the shot vectors Cauchy.