Radial jets, calibrated labels and the free Maxwell comparison hierarchy · Part 3

imaginary parts proves that every invariant complex-valued real bilinear form is a complex multiple of the fixed color inner product.

Searchable transcription

A monomial of weight at most five has at most two fields, since each variable has weight at least
two. With two fields its jet orders are either (0, 0) or (0, 1); the latter has weight five. Decompose
a polynomial into its constant, linear and these quadratic pieces by ordinary field number and
engineering weight. Invariance of each piece follows from the linearity of the color action. The
linear piece vanishes by (FC1). Polarizing the (0, 0) piece, and separating the two jet orders in
the (0, 1) piece, gives invariant bilinear forms. Resolve their spatial tensor coordinates; (FC1) then
makes every color contraction a multiple of the color inner product. Consequently all weight-four
labels are spanned by the unit and ⟨Fab, Fcd⟩; through weight five one adds ⟨Fab, ∂eFcd⟩. This is a
spanning assertion about the actual invariant labels before further free equation relations.
Put Qab,cd = P
c F c
abF c
cd, with products interpreted as in (FM3), and define

Searchable transcription

Na =
X
i
Qai,ai,
Ra =
X
i,k: i̸=a, k̸=a
Qik,ik,
Pab =
X
i
Qai,bi.
(FC2)

The sums in are over ordered pairs. Consider centered real symmetric tensor fields transforming under the signed coordinate group as rank-two tensors. Reflection in one axis multiplies by the product of its two axis signs. Thus a diagonal component permits only products in which every axis occurs evenly; for two antisymmetric pairs this forces the same unoriented plane twice. The permutations fixing axis have just two orbits: planes containing and planes disjoint from They give. A constant could also occur, but centeredness removes it: the Wick quadratic one-point functions are zero and the unit expectation is one.

For the off-diagonal component, the parity character is the product of the signs of The two planes must therefore share exactly one other axis, and be, up to antisymmetry and exchange of the two factors. Permutations fixing identify the possible remaining axes, giving precisely. The characters of the other products differ in at least one reflection, so averaging against the required character kills them. Permuting axes then determines every tensor component from 00 and 01. Reality makes the coefficients real. The whole centered signed tensor space is therefore spanned by

Searchable transcription

Aab = δabNa,
Bab = δabRa,
Cab = 1{a̸=b}Pab.
(FC3)

Searchable transcription

For a real polar vector of weight at most five, fix component zero and average under its stabilizing
permutations and reflection character. Constants and FF terms have the wrong parity. In an
Fab∂eFcd term, parity requires that the symmetric difference of the two plane-index sets be {0, e}.
If e = 0, the planes coincide, giving the first two sums below. Otherwise they share a remaining
index; the location of the derivative leg gives the last two sums. Thus its zeroth component is
spanned by
n =
X
i̸=0
⟨F0i, ∂0F0i⟩,
c =
X
⟨Fij, ∂0Fij⟩,

Searchable transcription

i,j̸=0
i̸=j
a =
X
⟨F0i, ∂jFji⟩,
h =
X
⟨Fji, ∂jF0i⟩.
(FC4)
i,j̸=0
i̸=j
i,j̸=0
i̸=j

The free divergence equation gives = 0. Bianchi gives Summing over ordered distinct nonzero gives 2. Hence only are needed. A signed vector is determined by its zeroth component, since exchanging axis zero with any axis transports that component. The polar vector space has real dimension at most two. These relations hold on all solution towers, so they are zero in the actual free quotient by (FM4).

The lower rank bounds are obtained by explicit separated correlations, not by declaring the formal expressions independent. Let) be (FM2) and set

Searchable transcription

Qab,ik;ce,fm(z) = Hab,ce(z)Hik,fm(z) + Hab,fm(z)Hik,ce(z).
(FC5)

The correlation of with at 0 is (dim g), because the two cross matchings force the same color. At, writing, the Hessian and third derivative of the Green function are

Searchable transcription

ν−1∂r∂j∂iG0(et) = 8δijδtr −48δtiδtjδtr + 8(δriδtj + δrjδti).
(FC6)

These follow by differentiating. Substituting them into the four signed Kronecker terms in (FM2), and applying the two-factor product rule to (FC5), gives every sample below. The common nonzero factor removed is = (dim g Take the divergence component zero of (FC3), correlate it at with and at zero, divide by, and take real parts. The two resulting linear functionals have columns

Searchable transcription

A
B
C
Q01,01
−256
0
128
Q12,12
0
−512
−128
.
(FC7)

Searchable transcription

For example the first A entry is P3
k=0 ∂0Q0k,0k;01,01(e0)/ν2 = −256; the corresponding B entry is
the same sum over ordered i, k ̸= 0, which is zero. For C sum ∂rQrk,0k;01,01 over r ̸= 0, k. The
other row uses 12, 12. These finite sums and (FC6) give respectively the three displayed columns. A
third, undifferentiated sample correlates tensor component 01 at e0 with Q02,12 at zero; its columns
are (0, 0, −16), directly from (FC5). It separates the last tensor direction.
Thus A, B, C are independent, so the tensor space has dimension exactly three.
The two
divergence samples in (FC7) are independent; combined with the upper bound two from (FC4),
they identify the entire polar vector space and prove surjectivity of divergence. Rank-nullity gives
a one-dimensional conserved kernel. Solving (FC7) yields its generator

Searchable transcription

1
2A −1
4B + C = T free.
(FC8)

Searchable transcription

For its diagonal entry use P
i,k Qik,ik = 2Na + Ra; its off-diagonal entry is Pab. This is exactly
(FM6), already shown conserved. The last sample is nonzero on it, so the generator is nonzero.
Every conserved centered real signed tensor in this weight-four space is its real scalar multiple.
Any later nonzero scalar normalization, including either sign, preserves the rank and the conserved-
line statements; the normalization of shell fluxes is a separate argument.
There is a raw degree-five vector direction which must be kept until the equation quotient is
taken. In polynomial notation its components are

Searchable transcription

Eb =
X
i,a
pairDeriv(b, i, a, a, i),
(FC9)

that is, the contraction of with the equation, in the normalized radial-jet coordinates used above. The two other raw vector directions are the polynomials called momentum1 and momentum2 in the preceding classification. Every real signed-equivariant raw vector has a unique expression. To see the independence after imposing the Maxwell equations, take a unit color vector and the two solution jets with respectively

Searchable transcription

(F (0), F (1)) = (leg12, leg0;12),
(leg01, leg0;01 −leg2;21).
(FC10)

Here the legs are radial potential jets, with the normalization made explicit below. Both satisfy the linear divergence equation and Bianchi identity. The first reads3 in component zero; the second reads = 1. An equation polynomial reads zero on both. It follows first that then that = 0. Conversely vanishes on every solution jet, directly by the divergence equation. A non-solution jet with nonzero divergence shows that itself is a nonzero raw polynomial. Thus the raw vector space has dimension three, and the kernel of its map to the free vector quotient is exactly the one-dimensional space

This statement is independent of coefficient coordinates. For a coherent degree-five frame, apply its inverse coordinate map to each component of a real signed-equivariant coefficient vector. This is a real linear isomorphism to the raw polynomial vector space, because the coordinate map intertwines signed permutations and conjugation. Composing with the free quotient projection gives a surjection whose kernel is the inverse-coordinate image of. A section is obtained by applying the corrected polynomial lift to each quotient component. That lift commutes with both symmetries, and its quotient projection is the identity. These facts justify the vector kernel subtraction used later: it removes precisely the equation direction, and does not discard either of the two nonzero free vector classes.

To make the raw coordinates completely explicit, write

Searchable transcription

Pab;kcd = ⟨Fab, ∂kFcd⟩,
(V1)b =
X
i
Pbi;bbi,
(V2)b =
X
i,j
Pij;bij,
Eb =
X
i,a
Pbi;aai.
(FC11)

The pairings are the actual invariant polynomial labels, evaluated on radial potential jets; imposing the Maxwell equation is a later quotient operation. The basic legs in (FC10) are the homogeneous potentials

Searchable transcription

(legab e)i(z) = 1
2(δibza −δiazb)e,
(legk;ab e)i(z) = 1
3(δibza −δiazb)zke.
(FC12)

Searchable transcription

Taking their exterior derivatives defines the curvature legs, including the factor 1/3 in the first
curvature derivative. This factor explains the 4/3 sample in (FC10). A constant curvature is auto-
matically divergence free. For a linear curvature the Maxwell condition is exactly P
i curl1(i, i, b) = 0
for every b; direct differentiation of (FC12) checks it for the two combinations in (FC10). Higher
homogeneous jets can be set to zero.

Here is also the raw upper-dimension argument, before using those solution samples. A weight-five label decomposes as a constant, an invariant linear term, an term and an term. Simplicity kills the linear term as above. Total axis inversion fixes the two’s but negates a polar vector, so both its constant and terms vanish. Resolve its remaining invariant bilinear color form into the color inner product. To recover its spatial coefficients it suffices to use the mixed evaluations m (FC13)

where occupies jet order zero and order one. Take, in component zero, the three pairs

Searchable transcription

(leg01 e, leg0;01 e),
(leg01 e, leg2;21 e),
(leg21 e, leg2;01 e).

If all three readings vanish, every mixed coefficient vanishes. Indeed reflections require the symmetric difference of its two plane-index sets to equal, where is the derivative index. For = 0, equal planes containing zero and equal planes avoiding zero are the two possible types. For = 0, the planes share one other index, and the two placements of the derivative factor are the other types. Permutations fixing zero identify all coefficients of each type; antisymmetry fixes their

signs. The three displayed tests kill three types. The cyclic identity leg + leg + leg paired with leg, kills the fourth. The legs span each radial jet space by the expansion

Searchable transcription

2
X
2
X
A = 1
i,j
legij(curl0(i, j)A),
B = 1
i,j,k
legk;ij(curl1(k, i, j)B).

Bilinearity therefore kills the entire mixed part. Reality recovers the complex coefficient from its real part. The three evaluations form an injective real map, proving the raw upper bound three. The two solution evaluations in (FC10) and the nonsolution pair (leg leg), on which = 1, prove independence of. Thus (FC11) is a basis and every raw vector has unique coordinates.

For later use, the divergence of the raw centered tensor with coefficients respectively diagonal mountain, diagonal total and off-diagonal shear, has these coordinates:

Searchable transcription

Div(w) = 2(w0 −w2)V1 + (2w1 + 1
2w2)V2 + w2E.
(FC14)

Searchable transcription

The product rule gives 2V1 for the first tensor and 2V2 for the second. For the shear use Bianchi and
interchange the ordered indices to obtain P
a,i Pai;abi = 1
2V2,b; deleting the a = b terms subtracts
2V1,b, while the other product derivative is Eb.
This proves (FC14).
Its coordinate matrix is
invertible, with inverse

Searchable transcription

w2 = y2,
w0 = y0/2 + y2,
w1 = y1/2 −y2/4.

It defines the actual scalar contraction on raw vectors:

Searchable transcription

ρ(Div(w)) = −4(w0 + 4w1) = −2y0 −8y1.
(FC15)

Searchable transcription

If Q = 1
4
P
i,j⟨Fij, Fij⟩, the tensor trace is 4(w0 + 4w1)Q; hence ρ reads minus that trace in Q-units.
It kills E and the traceless conserved tensor w = (1, −1/4, 1). Since (FC10) proves that the raw
equation kernel is exactly RE, ρ annihilates that entire kernel. This is the scalar sensor used later,
with its sign and normalization fixed. The identification of the free quotient here uses the Maxwell
polynomial evaluation and the reflected Gram norm proved in (FM3)–(FM4) and in the detector’s
reflected polynomial calculation; it is not an additional assumption of nondegeneracy.

Source–readout formulation. This subsection is a classification of invariant label spaces in low weight, and it contains the simplest instance of the descent criterion in the whole construction. The source object is a raw vector, a signed equivariant element of the label algebra before the equation quotient; the readout is its class in the free vector quotient.

Proposition 16.5 (Raw vectors and the equation kernel). Every real signed-equivariant raw vector of weight five is uniquely, and the kernel of the map to the free vector quotient is exactly

Proof. Upper bound: a weight-five label is a constant, an invariant linear term, an term and an term; simplicity kills the linear term and total inversion the constant and terms of a polar vector. The mixed coefficients are recovered by

Searchable transcription

m(A, B; v) = Re{v(A + B) −v(A) −v(B) + v(0)}

at order zero, at order one) on the three pairs (leg leg), (leg leg (leg leg reflections require the symmetric difference of the two plane-index sets

Searchable transcription

to be {0, k}, giving four types, of which the three tests kill three and the cyclic identity
leg0;12 + leg1;20 + leg2;01 = 0 paired with leg12 e the fourth; the legs span the jet spaces by
A = 1
2
P
i,j legij(curl0(i, j)A), B = 1
2
P
i,j,k legk;ij(curl1(k, i, j)B). Independence: the two solution
jets
(F (0), F (1)) = (leg12, leg0;12),
(leg01, leg0;01 −leg2;21)

satisfy the divergence and Bianchi equations; the first reads3 in component zero and the second = 1, while an equation polynomial reads zero on both, so = 0 then vanishes on every solution jet and equals one on the non-solution pair (leg leg

Searchable transcription

The scalar contraction ρ of (FC15) is a reader on raw vectors that is constant on the fibres
of the equation quotient: it kills E and the traceless conserved tensor w = (1, −1
4, 1), for which
Div(w) = E, and hence the whole raw equation kernel. It therefore descends to the physical vector
space. This is the scalar sensor used later, with sign and normalization fixed by (FC14)–(FC15): it
reads minus the trace in units of Q = 1
4
P
i,j⟨Fij, Fij⟩. The classification itself uses simplicity of the
color Lie algebra through (FC1) and does not use connectedness of the group; this is a hypothesis
of the supplied construction and is retained. The coordinates (y0, y1, y2) ↔(w0, w1, w2) of (FC14)
are an explicit inverse pair, so the divergence map from invariant symmetric tensors of weight four
to raw vectors of weight five is a bijection, and the fibre of the equation quotient is the single line
RE.

16.7 Free shell charges and separation of the Ward kernel

We give the polynomial calculations underlying the free charge separation used in the rotation and trace arguments. This also specifies the exterior tests needed when a charge acts on high-degree fields.

16.8 The compact stress shell identity

Let be a smooth compactly supported function equal to one near a distinguished point, whose support avoids all spectator points. For a current, define its outward shell functional by

Searchable transcription

Z
(∂aχ)(y)Ja(y) d4y.
(FS1)
Eχ(J) = −
X
a

The integrand is supported in a compact annulus away from all insertions. All exchanges of finite Wick sums and this integral are consequently justified by smoothness there. A one-leg source at produces the Maxwell pulse obtained by applying its jet functional to the covariance. For a smooth source-free Maxwell field on a neighborhood of the support of, Green’s formula for the bilinear Maxwell stress current gives

Searchable transcription

EχJv(pulsep,x, f) = −samplex(p, δvf).
(FS2)

Indeed the divergence vanishes offthe source by the Maxwell and Bianchi equations. Integrating the differentiated fundamental covariance against the cutofftransfers the derivatives to the smooth field. The fundamental-solution identity leaves precisely its jet at. The boundary term at infinity is absent because the cutoffis compact. The variation of a two-form is . For the three vector fields used here this variation preserves the Maxwell equations. The stress is symmetric and traceless, so its contraction with the symmetric derivative of a constant, skew-linear, or radial vector field contributes zero to the divergence.

Here is a regularized derivation of the fundamental-solution step in (FS2). With the covariance normalization), put

Searchable transcription

gt(y) =
c4
t2 + |y|2 ,
ρt(y) =
8c4t2
(t2 + |y|2)3 ,
t > 0.
(FS2a)

Searchable transcription

R ρ1 = 1: the sphere area is
Direct differentiation gives −∆gt = ρt.
The radial integral gives
R ∞
2π2 and
0 r3(1 + r2)−3dr = 1/4. Also t4ρt(ty) = ρ1(y). Hence for every continuous compactly
R ρt(y −x)h(y)dy →h(x), by the change of variable y = x + tz and the integrable
supported h,
majorant ρ1(z)∥h∥∞.

Searchable transcription

For a plane source il, form the potential Ab = δib∂lgt−δlb∂igt, and take its exterior derivative. It
satisfies the Bianchi identity by commutation of derivatives, while its divergence is δlb∂iρt −δib∂lρt.
The divergence of its bilinear stress with the source-free field f is therefore the contraction of
this expression with f; the terms from the symmetric derivative of v cancel by tracelessness and
the other terms cancel by skew symmetry.
Integration by parts against χ transfers the source
derivatives to the smooth compact test made from χ, v, f. The preceding approximate-identity
limit evaluates their difference at x, giving −δvfil(x). On the shell the regulated exterior derivative
converges smoothly to the pulse, since that compact shell avoids x. Its flux thus converges to the
left side of (FS2). This proves the plane-source case with the stated sign.

Differentiate the source center in any direction while keeping the cutoffand vector field fixed. All differentiated shell integrands have a common bound on a fixed compact annulus, so differentiation passes through the integral. A center derivative raises the source leg by that direction; on the right it differentiates the sample of. Induction gives (FS2) for all derivative legs; linearity gives every solution-jet functional. The wave property of for the three motions ensures these samples respect the Maxwell jet relations.

For a constant vector, (FS2) gives minus the derivative-leg sample. For), with skew, it gives minus the infinitesimal tensor rotation. For, a homogeneous solution jet of derivative order has Euler contribution, and the two form indices contribute two more. Thus the one-leg radial charge is + 2) times that jet. This specifies the engineering weight, including the two contributed by curvature rather than its derivatives.

Searchable transcription

When both pulses are centered at x, use cutoffindependence and homogeneity instead of mul-
tiplying distributions at their common pole. Two cutoffs equal to one near x have a difference
supported away from x; source-free integration by parts makes their fluxes equal. Replace the
cutoffby its dilation by a factor two about x. A jet-order-j pulse scales by 2−4−j; the other scales
by 2−4−l; the volume and cutoffderivative together contribute 23, and a motion homogeneous of
degree e = 0 or 1 contributes 2e. Thus the common flux Z equals 2e−5−j−lZ. That coefficient is
strictly smaller than one, so Z = 0. When neither pulse is centered at x, their fields are smooth
throughout the support of the cutoff, and the source-free divergence identity gives zero. When
precisely one pulse is centered there and the other is at a spectator point, (FS2) gives minus the
covariance with the transformed distinguished leg.

Now insert the quadratic Wick stress in an arbitrary Wick monomial word. Double Wick extraction expresses its correlation as one half the sum over ordered choices of two occurrences, times the bilinear pulse current for those occurrences and the Wick pairing of all remaining occurrences. There are exactly the three cases just considered. The terms with both distinguished occurrences or with neither distinguished occurrence vanish. In each remaining term exactly one occurrence of the distinguished monomial is transformed. The two orders of the selected pair cancel the one half. Summing over its occurrences is the Leibniz rule for the derivation on that monomial. By

multilinearity this proves, for arbitrary polynomial words,

Searchable transcription

X
!
*
+
b
vb(y)⟨Tab(y)P(x)
Y
l
Pl(xl)⟩
(DvP)(x)
Y
l
Pl(xl)
Eχ
= −
.
(FS3)

Here is respectively translation, rotation, or the engineering-weight derivation. This proves the three shell laws, including their common minus sign, and includes every contraction term. In particular conservation of the free stress alone was not being used as a substitute for its charge normalization.

Source–readout formulation. The shell functional is a reader of the stress insertion that depends on the auxiliary cutoff only through the set of enclosed insertions: two cutoffs equal to one near and avoiding the spectators give equal values. The cutoffis thus a presentation choice with trivial effect on the readout, and (FS3) says that the reader returns the derivation on the enclosed label: the charge is recovered from the hierarchy and is not an extra datum.

16.9 Low-degree vector charges

Searchable transcription

Let Sj be the real space of homogeneous order-j Maxwell solution jets and Vj = S ∗
j .
Their
elements are the real one-leg test functionals. A variable at order j has engineering weight j + 2.
For p ∈Vj, write Bc(p) for its linear polynomial in color c, and set

Searchable transcription

[p, q] =
X
c
Bc(p)Bc(q).

Searchable transcription

This bracket denotes a color-contracted quadratic polynomial, not a Lie bracket. For a linear map
D : Sj →Vl, its polynomial is
C(D) =
X
ν
[eν, Deν],
(FS4)

Searchable transcription

where (eν) is any basis and (eν) its dual. This is basis-independent, and C(p ⊗q) = [p, q]. If j ̸= l,
assigning independent jets in these two orders in one nonzero color proves that C is injective.
Consider a vector row Ti on the invariant polynomial filtration, killing the unit, increasing
filtration by at most one, real under conjugation, and signed-covariant. Suppose it is silent, meaning
X

Searchable transcription

l
⟨P1(x1) · · · (TiPl)(xl) · · · Pn(xn)⟩= 0
(FS5)

for every allowed exterior word and separated configuration. At input weight four the invariant space is spanned by the unit and [] withV. At output weight at most five the only further possibility is a color-contracted pair of orders zero and one. Total inversion is even on weight four and odd on such a pair. The vector law anticommutes with total inversion, and therefore removes the weight-four part of the output. Reality permits real coefficients in (FS4). Consequently

Searchable transcription

Ti[p, p] = C(Di(p)),
Di(p) : S0 →V1.
(FS6)

Let) denote the real single-color covariance between two test legs at displacement To extract the constraint on (FS6), put two quadratic insertions at and zero, and a third at a fixed nonzero point, and multiply (FS5) by. The covariance of jet orders zero and one is homogeneous of degree5. The triangle terms where the transformed order-one leg is on either near insertion have a finite limit; the term with that leg at the distant insertion tends to zero. The

double contractions contribute 4 dim g. Choose the distant one-leg functional so that its covariance with a zero-order test is evaluation on a prescribedS. Cancelling the common nonzero factor gives (FS7)

The availability of these evaluations follows from covariance separation of solution jets. First suppose a test functional of order has zero covariance with every order-zero plane test at every nonzero displacement. Its pulse is then zero offits source, in every color component. Pair its radial stress shell with the homogeneous polynomial Maxwell solution having any prescribed order- solution jet. The shell integral is zero because the pulse is zero there; (FS2) makes it times the test evaluated on that arbitrary jet. The functional is zero. For partner order vanishing against all such partners makes the radial derivative of its covariance with any order- partner zero. That covariance is homogeneous of degree); Euler’s rule makes the radial derivative that nonzero scalar times the covariance. Thus it vanishes, and induction reduces to order zero. Finite-dimensional duality now makes these covariance evaluations span the required full dual. Thus no inverse of a covariance matrix at a single momentum is being assumed. This also proves that the solution-jet space is nonzero at every order: otherwise separation against that order would force the nonzero plane evaluation at order zero to vanish. If) = 0, (FS7) first makes) = 0 for= 0. Given) = 0, add a test with nonzero value at, apply that result to the sum, and subtract. Covariance separation gives = 0. Hence kerker). Choose with= 0; subtracting) from proves (FS8)

Set(0) = 0. For two nonzero linear functionals there is an on which both are nonzero: use a witness for one, a witness for the other, and their sum if needed. Substitution in (FS7) yields

Searchable transcription

Kz(fi(p), q) = Kz(fi(q), p).
(FS9)

Pair the difference) with an arbitrary one-leg test, use (FS9), and cancel by linearity in its other slot. Separation makes the difference zero. The same argument with a scalar proves homogeneity. Thus VV is real linear. Signed covariance of, injectivity of (FS4), and cancellation of a nonzero test functional give simultaneous signed covariance of the family

We classify this family explicitly. Write for evaluation of a zero jet in the-plane and for the corresponding first-jet functional. Under reflection of axis, the latter has character equal to the product of the three signs belonging to Project onto a reflection character by the four commuting projections (1 The character of) leaves only . Average the permutation interchanging axes two and three. The Maxwell relation gives, so this fixed subspace is the line spanned by. Similarly the character of) leaves only. Antisymmetrize the interchange of axes one and two; the Bianchi relation leaves the line spanned by. Hence these two values have scalar coefficients

Use (FS9) with, and The explicit differentiated covariance (FM2), evaluated below after (WE4), gives the same nonzero value for and Therefore. Permutations fixing zero give all remaining plane evaluations, which span V permutations moving zero give the other directions. We conclude

Searchable transcription

fi = a∂i,
Ti[p, p] = a[p, ∂ip].
(FS10)

Since], the scalar of the translation derivation is2. Squares span the symmetric quadratic polynomials by polarization, and both rows kill the unit. This proves the entire degree-four assertion, with its factor two, rather than just a classification of invariant labels.