Scattering amplitude annihilators

TL;DR

This paper demonstrates that specific second order differential operators annihilate Yang-Mills and General Relativity tree scattering amplitudes, confirming a conjecture about hidden conformal symmetry in GR.

Contribution

It proves a conjecture that certain differential operators annihilate scattering amplitudes, revealing hidden conformal symmetry in GR.

Findings

Second order operators annihilate YM and GR amplitudes

Confirmation of a conjecture on hidden conformal symmetry in GR

New mathematical tools for analyzing scattering amplitudes

Abstract

Several second order differential operators are shown to annihilate the YM and GR tree scattering amplitudes. In particular we prove a conjecture of Loebbert, Mojaza and Plefka from their investigation of a hidden conformal symmetry in GR.

Equations

$$k_{ii}$$

$$c_{ii}$$

$$e_{ii}$$

$$X_i$$

$$Y_i$$

$$Z$$

$$A_{i}=\textstyle\sum_{j,k\in I}\big{(}\tfrac{1}{2}k_{jk}D_{c_{ji}}D_{c_{ki}}+c_{jk}D_{c_{ji}}D_{e_{ki}}+\tfrac{1}{2}e_{jk}D_{e_{ji}}D_{e_{ki}}\big{)}$$

$$B_{i}=\textstyle\sum_{j,k\in I}\big{(}c_{jk}D_{k_{ij}}D_{e_{ik}}+e_{jk}D_{c_{ij}}D_{e_{ik}}+k_{jk}D_{k_{ij}}D_{c_{ki}}+c_{kj}D_{c_{ij}}D_{c_{ki}}\\ -c_{kj}D_{k_{jk}}D_{e_{ji}}-e_{jk}D_{c_{jk}}D_{e_{ji}}-k_{jk}D_{k_{jk}}D_{c_{ji}}-c_{jk}D_{c_{ji}}D_{c_{jk}}\big{)}$$

$$\widetilde{C}_{i}=\textstyle\sum_{j,k\in I}\big{(}\tfrac{1}{2}e_{jk}D_{c_{ij}}D_{c_{ik}}+c_{kj}D_{k_{ik}}D_{c_{ij}}+\tfrac{1}{2}k_{jk}D_{k_{ij}}D_{k_{ik}}\\ -e_{jk}D_{c_{ij}}D_{c_{jk}}-c_{kj}D_{k_{jk}}D_{c_{ij}}-c_{jk}D_{k_{ji}}D_{c_{jk}}-k_{jk}D_{k_{ji}}D_{k_{jk}}\big{)}$$

$$\mathrm{Res}{M}_I=U^2{M}_{J\bigsqcup \{\bullet \}}{M}_{K\bigsqcup \{\bullet \}}$$

$$\xi \stackrel{\text{def}}{=}{\sum }_{j,j^{\prime }\in J}{k}_{j{j}^{\prime }}=0$$

$$R_X=\text{mathbbm}C[x_1,\dots ,x_m]$$

$$x_i{x}_j-x_j{x}_i{\partial }_i{\partial }_j-{\partial }_j{\partial }_i{\partial }_i{x}_j-x_j{\partial }_i-{\delta }_{ij}$$

$$I=J\bigsqcup K$$

$$J_{\bullet }=J\bigsqcup \{\bullet \}{K}_{\bullet }=K\bigsqcup \{\bullet \}$$

$$k_{ij},c_{ij},e_{ij}\text{where}i,j\in I$$

$$\underset{k}{\underbrace{\frac{1}{2}n(n-3)}}+\underset{c}{\underbrace{n(n-2)}}+\underset{e}{\underbrace{\frac{1}{2}n(n-1)}}=2n(n-2)$$

$$D_{k_{ij}},D_{c_{ij}},D_{e_{ij}}\in C(I)$$

$$[D_{k_{ij}},k_{ab}]$$

$$[D_{c_{ij}},c_{ab}]$$

$$[D_{e_{ij}},e_{ab}]$$

$$=0$$

$$[D_{k_{ij}},e_{ab}]$$

$$D_{k_{ii}}$$

$$D_{c_{ii}}$$

$$D_{e_{ii}}$$

$$\xi$$

$$\Xi$$

$$k_{j{j}^{\prime }}^{\perp }$$

$$k_{jk}^{\perp }=k_{kj}^{\perp }$$

$$k_{k{k}^{\prime }}^{\perp }$$

$$c_{i{i}^{\prime }}^{\perp }$$

$$e_{i{i}^{\prime }}^{\perp }$$

$$C(I)=\text{mathbbm}C\Xi \oplus C(I{)}^{\perp }$$

$$[\Xi ,\xi ]$$

$$[\Xi ,k^{\perp }]$$

$$[D^{\perp },\xi ]$$

$${\sum }_{j,j^{\prime }\in J}{k}_{j{j}^{\prime }}^{\perp }=0{\sum }_{j,j^{\prime }\in J}{D}_{k_{j{j}^{\prime }}}^{\perp }=0$$

Full text

1 Introduction

The scattering amplitudes of Yang-Mills (YM) and general relativity (GR) are interesting objects in physics. The dimension-neutral tree amplitudes are rational functions of many variables; see Appendix LABEL:app:formulas for examples. As with other special functions, it is natural to ask what differential identities they satisfy. We show that a number of second order differential operators annihilate the dimension-neutral YM and GR amplitudes. This paper is motivated by recent work of Loebbert, Mojaza and Plefka lmp. They investigated a potential hidden conformal symmetry of GR tree amplitudes in general spacetime dimension $d$. This led them to conjecture certain identities for the dimension-neutral amplitudes that they verified by explicit calculation for amplitudes with $n=3,4,5,6$ legs111We thank F. Loebbert, M. Mojaza and J. Plefka for making available computer code to check the $n=5$ case and for answering questions we had about their paper lmp. . They conjectured that they would continue to hold for $n\geq 7$. This paper contains:

• •

A reformulation of these identities as proper differential identities for the amplitudes. The formulation in lmp is more intricate, as discussed in Appendix LABEL:app:translation.

• •

A proof of these identities for all $n\geq 3$, by induction on $n$.

We do not take up the interesting theme of hidden conformal symmetry lmp. The identities conjectured in lmp and proved here do not readily imply identities for the amplitudes in specific dimensions such as $d=4$, an interesting question that we leave open. In this paper we exclusively work with the dimension-neutral version of the tree amplitudes; the spacetime dimension $d$ will be absent. These amplitudes are rational functions on a complex vector space of dimension $2n(n-2)$ with simple poles along a constellation of hyperplanes. The coordinates on this vector space will be denoted $k_{ij}$, $c_{ij}$, $e_{ij}$ with subscripts running over an index set222The YM amplitude requires a cyclic order on $I$, whereas the GR amplitude is permutation invariant and $I$ is an unordered set. We will not mention this further in this introduction. $I$ with $|I|=n$. The following linear relations among the coordinates cut the dimension of this vector space down to $2n(n-2)$:

[TABLE]

The amplitudes are actually polynomial in the $c_{ij}$ and $e_{ij}$ variables. To obtain the amplitudes in $d$ spacetime dimensions set $k_{ij}=k_{i}\cdot k_{j}$ and $c_{ij}=k_{i}\cdot\epsilon_{j}$ and $e_{ij}=(1-\delta_{ij})\epsilon_{i}\cdot\epsilon_{j}$ where $k_{i}$ and $\epsilon_{i}$ are $d$-dimensional vectors, the momentum and polarization of the $i$-th particle. We need a workable definition of the amplitudes for general $n$. For the purpose of this paper, the amplitudes are defined by the recursion in Definition LABEL:def:rec. This recursion is based on the factorization of residues formulated directly in $k_{ij}$, $c_{ij}$, $e_{ij}$ variables333The original recursion of this kind is BCFW recursion bcfw. The recursion we use in this paper does not involve BCFW shifts nor decay conditions under such shifts, it is only required that the amplitudes factor properly. The proof that this determines the amplitudes uniquely is along the lines of nr, which is for $d=4$, but is simpler in the present dimension-neutral case since one is on a vector space and all poles are along linear subspaces. It is useful in this paper to have the uniqueness statement in a simple form since the same argument (cf. Lemmas LABEL:lemma:uniq and LABEL:lemma:vanish) also plays a key role in proof of the main theorem, Theorem 1.. On the face of it, it is neither obvious that this recursion admits a solution, nor that the solution is unique. A sketch of existence using Feynman rules is in Remark LABEL:remark:existence, but since we are unaware of a reference that spells this out in detail for the dimension-neutral amplitudes, we logically state existence as Assumption A below. Uniqueness is shown in Lemma LABEL:lemma:uniq.

Assumption A**.**

A sequence of rational functions satisfying the recursion in Definition LABEL:def:rec exists. We refer to them as the YM respectively GR amplitudes.

In stating our main result below, we avoid standard notation for partial derivatives such as $\partial_{k_{ij}}$, $\partial_{c_{ij}}$, $\partial_{e_{ij}}$ because it is ambiguous how these act on functions on the vector space defined by (1). To illustrate, if $I=\{1,\ldots,5\}$ then $c_{12}$ and $-c_{32}-c_{42}-c_{52}$ are two representatives of one the same function, but if we act on them with $\partial_{c_{12}}$ using a naive interpretation of the partial derivative, then we obtain conflicting results, $1$ and [math] respectively. To fix this we introduce new notation $D_{k_{ij}}$, $D_{c_{ij}}$, $D_{e_{ij}}$ which are differential operators that act unambiguously on functions on the vector space defined by (1). To illustrate, if $I=\{1,\ldots,5\}$ then we use $D_{c_{12}}=\partial_{c_{12}}-\tfrac{1}{4}(\partial_{c_{12}}+\partial_{c_{32}}+\partial_{c_{42}}+\partial_{c_{52}})$ which acts unambiguously. The $D_{k_{ij}}$ are linear combinations of the $\partial_{k_{ij}}$ with constant coefficients; the $D_{c_{ij}}$ are linear combinations of the $\partial_{c_{ij}}$ with constant coefficients; the $D_{e_{ij}}$ are linear combinations of the $\partial_{e_{ij}}$ with constant coefficients. Their complete definition is in Lemma 3, and we have also included computer code in Appendix LABEL:app:code. We note that our definition of the $D$-operators is also a matter of convention; we have merely made a convenient choice of operators that span the $2n(n-2)$-dimensional space of constant first order differential operators along the vector space defined by (1). There are $2n+1$ first order differential operators that are well-known to annihilate the amplitudes: $n$ of them express gauge invariance; $n$ express homogeneity in the polarizations; and one expresses homogeneity jointly in all momenta. They are, respectively,

[TABLE]

where $h=1$ and $s=4-n$ for YM respectively $h=2$ and $s=2$ for GR. Polynomiality in the $c_{ij}$ and $e_{ij}$ variables implies further obvious annihilators. There are, in addition, several second order annihilators as we show in this paper.

Theorem 1**.**

Define the YM and GR tree amplitude by the recursion in Definition LABEL:def:rec and make Assumption A. The differential operators $A_{i}$, $B_{i}$, $C_{i}$ defined below annihilate the tree amplitudes for all $n=|I|\geq 3$ and $i\in I$:

[TABLE]

The $D$-operators are defined in Lemma 3.

That $B_{i}$, $C_{i}$ annihilate the GR amplitude was conjectured in lmp, using a more intricate formulation that we discuss in Appendix LABEL:app:translation. The formulation of these identities directly as annihilating differential operators in Theorem 1 is therefore new444It allows us, for instance, to compute a number of commutators in Section LABEL:sec:comms. They imply that every function annihilated by $X_{i}$, $Y_{i}$, $Z$, $C_{i}$ is automatically annihilated by $A_{i}$, $B_{i}$.. Our proof of Theorem 1 is by induction on $n$ and is summarized later in this introduction.

Remark 1*.*

The amplitudes in $d\geq 4$ spacetime dimensions are obtained from the dimension-neutral ones by setting $k_{ij}=k_{i}\cdot k_{j}$ and $c_{ij}=k_{i}\cdot\epsilon_{j}$ and $e_{ij}=(1-\delta_{ij})\epsilon_{i}\cdot\epsilon_{j}$ where $k_{i}$ and $\epsilon_{i}$ are elements of a $d$-dimensional complex vector space, and $\cdot$ is a nondegenerate symmetric bilinear pairing555For instance, $\mathbbm{C}^{d}$ with standard pairing $z\cdot w=z_{1}w_{1}+\ldots+z_{d}w_{d}$. The choice does not matter since all such vector spaces with nondegenerate pairing are isomorphic. . The momentum vectors $k_{i}$ are subject to $k_{i}\cdot k_{i}=0$ and momentum conservation $\sum_{i}k_{i}=0$, and the polarization vectors $\epsilon_{i}$ are subject to $k_{i}\cdot\epsilon_{i}=0$. Assume $k_{i}\neq 0$ here, so $\epsilon_{i}$ lies in a subspace of dimension $d-1$. For every $i$, the YM amplitude is linear in $\epsilon_{i}$, the GR amplitude is quadratic in $\epsilon_{i}$; this is witnessed by the annihilator $Y_{i}$. For every $k\neq 0$ with $k\cdot k=0$ abbreviate $P(k)=\{\epsilon\mid k\cdot\epsilon=0\}/\mathbbm{C}k$ which is a vector space of dimension $d-2$, and observe that $\cdot$ induces a nondegenerate symmetric bilinear pairing on $P(k)$. Then, separately for every $i$ and fixed $k_{i}\neq 0$:

• •

As a function of $\epsilon_{i}$, the YM amplitude descends to a linear form on $P(k_{i})$.

• •

As a function of $\epsilon_{i}$, the GR amplitude descends to a quadratic form on $P(k_{i})$666Equivalently, a linear form on the symmetric tensor product $S^{2}P(k_{i})$. One can decompose this into the trace and the traceless part relative to the symmetric bilinear pairing on $P(k_{i})$..

This is witnessed by $X_{i}$ and is known as gauge invariance. The amplitudes are also homogeneous jointly in all momenta, as witnessed by $Z$. In summary, the $d$-dimensional amplitudes are sections of certain vector bundles on the projective variety $k_{i}\cdot k_{i}=0$ and $\sum_{i}k_{i}=0$777Actually one only has a vector bundle away from the singular locus. ,888It would be interesting to see whether the annihilators $A_{i}$, $B_{i}$, $C_{i}$ or suitable linear combinations of them imply annihilators for the $d$-dimensional amplitudes. The latter would be differential operators on the vector bundle on which the amplitudes live, possibly taking values in another vector bundle. ,999The fact that $A_{i}$ annihilates is vacuous for YM, since $A_{i}$ involves two derivatives with respect to the polarization of the $i$-th particle. It is not vacuous for GR amplitudes, in the sense that one can write down a rational function annihilated by the obvious annihilators listed before Theorem 1 but not by $A_{i}$. . In this paper we do not work with momenta $k_{i}$ and polarizations $\epsilon_{i}$. Our primary variables are $k_{ij}$, $c_{ij}$, $e_{ij}$ and we exploit the fact that the relations (1) are linear.

Let us outline the proof of Theorem 1, which is by brute force by induction on $n$. It uses the recursive definition of the amplitudes, by which the amplitudes have simple poles along a constellation of hyperplanes, each residue being equal to a product of two lower amplitudes. Denoting by $M_{I}$ the GR amplitude for index set $I$, schematically one has

[TABLE]

for every decomposition $I=J\sqcup K$ into two subsets of two or more elements, $|J|\geq 2$ and $|K|\geq 2$. The residue is taken along the locus where $k_{J}=\sum_{j\in J}k_{j}$ satisfies $k_{J}\cdot k_{J}=0$. Translating to the coordinates we use, this locus is the hyperplane

[TABLE]

On the right hand side of (4), the bullet $\bullet$ stands for one more particle. Properly defining the right hand side of (4) requires contracting the polarizations of the respective $\bullet$ particles in $M_{J\sqcup\{\bullet\}}$ and $M_{K\sqcup\{\bullet\}}$. This contraction is effected by $U^{2}$, using the differential operator $U$ in Definition LABEL:def:u. Suppose now we are at the $n$-th induction step. Within each induction step, one first shows that the $A_{i}$ annihilate, then the $B_{i}$, then the $C_{i}$. The structure of the argument is analogous in each case, so suppose for concreteness that we want to show $C_{i}M_{I}=0$ for some $i$. This is done in two steps:

Show that $C_{i}M_{I}$ has no pole. Namely, for every decomposition $I=J\sqcup K$ as above one must check that $C_{i}M_{I}$ has no pole along the hyperplane $\xi=0$ in (5). 2. 2.

Show that the complete absence of poles in $C_{i}M_{I}$, together with other known properties of $C_{i}M_{I}$ such as homogeneity properties, imply $C_{i}M_{I}=0$.

In Step 1, note that $M_{I}$ has a simple $\xi^{-1}$ type pole, so $C_{i}$ being second order, the application $C_{i}M_{I}$ can naively have a third order pole, terms of type $\xi^{-3}$ and $\xi^{-2}$ and $\xi^{-1}$. While the $\xi^{-3}$ term is easily seen to be absent, the absence of $\xi^{-2}$ and $\xi^{-1}$ terms is by a lengthy explicit calculation that exploits only the annihilators of $M_{J\sqcup\{\bullet\}}$ and $M_{K\sqcup\{\bullet\}}$ that are known by the induction hypothesis. In a nutshell, and using schematic notation again, in Step 1 one must check that $C_{i}(\tfrac{1}{\xi}U^{2}M_{J\sqcup\{\bullet\}}M_{K\sqcup\{\bullet\}})$ has no pole. The computations establishing Step 1 are in the most technical lemma of this paper, Lemma LABEL:lemma:key. Step 2 relies on Lemma LABEL:lemma:vanish (which is the same lemma that also establishes that the recursion determines the amplitudes uniquely) and on the commutators in Lemma LABEL:lemma:comms. The YM case is entirely analogous.

The proof does not use explicit formulas for the amplitudes. Instead, the recursion for the amplitudes using (4) is effectively turned into a recursion for the annihilators. One can ask what the full annihilator of the YM respectively GR tree amplitudes is. To every rational function one can associate its annihilator, a left ideal in the Weyl algebra of polynomial differential operators. Rational functions are holonomic, meaning their annihilator is as big as allowed by the Bernstein inequality101010This says that over $\mathbbm{R}$ or $\mathbbm{C}$, and for a proper left ideal $I\subseteq D$ in the Weyl algebra $D$ in $x_{1},\ldots,x_{m},\partial_{1},\ldots,\partial_{m}$, the dimension of $D/I$ (defined to be the degree of a suitable Hilbert polynomial) is $\geq m$. Note that a rational function $g/f$ always has $m$ first order annihilators of the form $f(\partial_{i}g)-g(\partial_{i}f)-gf\partial_{i}\in D$ but they are in general of high polynomial degree. See also z.. In a Weyl algebra, every left ideal is finitely generated111111Somewhat surprisingly, by a theorem of Stafford, every left ideal in a Weyl algebra over $\mathbbm{R}$ or $\mathbbm{C}$ can in fact be generated by just two elements.. Algorithms are available to determine the annihilator of a rational function, e.g. in Macaulay2 m2; jl, and while in principle such tools can be applied to tree amplitudes, this approach does not seem to be practically viable for general $n$. It would be interesting to understand if there is a more conceptual explanation for these annihilators perhaps building on ideas in lmp; to inquire if there are corresponding identities for the amplitudes in specific dimensions such as $d=4$; and to try to apply these annihilators to derive properties of the amplitudes.

2 Preliminaries

Notation. All vector spaces and algebras are over the complex numbers. For every finite-dimensional vector space $X$ we denote:

[TABLE]

These things are defined independently of coordinates. This is important because the spaces we encounter do not have a canonical coordinate system, and we work with a linearly dependent set of coordinates. Canonically,

[TABLE]

As vector spaces, $R_{X}\simeq SX^{\ast}$ and $D_{X}\simeq SX^{\ast}\otimes SX$ where $S$ is the symmetric tensor algebra. Here $SX$ are the constant coefficient differential operators. Coordinate dependent definitions. Even though we do not commit to a particular basis, we recall the coordinate dependent definitions. By a coordinate we mean an element of $X^{\ast}$. A coordinate system is a basis $x_{1},\ldots,x_{m}\in X^{\ast}$. The polynomial ring is then

[TABLE]

Let $\partial_{1},\ldots,\partial_{m}\in X$ be the dual basis. Then $D_{X}$ is the associative algebra with identity generated by the variables $x_{1},\ldots,x_{m},\partial_{1},\ldots,\partial_{m}$ modulo the two-sided ideal generated by

[TABLE]

Linear maps. If $X,Y$ are vector spaces then every linear map $\alpha:X\to Y$ canonically induces several other linear maps:

[TABLE]

It neither induces a map $D_{X}\to D_{Y}$ nor $D_{Y}\to D_{X}$. We often find it convenient to specify a linear map by specifying the adjoint $Y^{\ast}\to X^{\ast}$. Direct sums. For a direct sum of finite-dimensional vector spaces, there are canonical isomorphisms $R_{X\oplus Y}\simeq R_{X}\otimes R_{Y}$ and $D_{X\oplus Y}\simeq D_{X}\otimes D_{Y}$ that we frequently use. This uses the tensor product of algebras, where all elements of $D_{X}$ commute with all elements of $D_{Y}$. Canonical inclusions such as $D_{X}\hookrightarrow D_{X}\otimes D_{Y},\delta\mapsto\delta\otimes 1$ are sometimes used. Index sets. Instead of a standard index set such as $\{1,\ldots,n\}$ we work with finite index sets denoted $I$, hence $n$ is replaced by $|I|$. For YM, a cyclic ordering of the elements of $I$ is required. In many calculations, $I$ is a disjoint union

[TABLE]

If ${J_{\bullet}}$, ${K_{\bullet}}$ have a cyclic order then $J\sqcup K$ acquires a cyclic order121212If the elements of ${J_{\bullet}}$ are ${\bullet},a_{1},\ldots,a_{|J|}$ and the elements of ${K_{\bullet}}$ are ${\bullet},b_{1},\ldots,b_{|K|}$ listed in cyclic order, starting with ${\bullet}$ for convenience, then on $J\sqcup K$ use the cyclic order $a_{1},\ldots,a_{|J|},b_{1},\ldots,b_{|K|}$.. Conversely, if $I$ has a cyclic order and (6a) respects this, then ${J_{\bullet}}$, ${K_{\bullet}}$ inherit a cyclic order.

3 Kinematic variables

Here we define in detail the vector space on which the amplitudes live; the differential operators $D$ used in Theorem 1; and notation that allows one to state the schematic factorization of residues in (4) in a precise way in Definition LABEL:def:rec.

Definition 2**.**

For every finite set $I$ with $|I|\geq 3$, consider first the complex ‘ambient’ vector space of dimension $3|I|^{2}$ with coordinate system

[TABLE]

Denote by $C(I)$ the linear subspace defined by the relations (1). In this paper, the elements (7) are understood to be elements of the dual space $C(I)^{\ast}$.

The dimension of $C(I)$ is

[TABLE]

where $n=|I|$. If $|I|=3$ then the $k_{ij}$ vanish identically as elements of $C(I)^{\ast}$. Note that the relations (1) do not refer to an ordering, hence there is a natural action of the group of permutations of $I$ on the vector space $C(I)$. The (7) are a linearly dependent set of coordinates, not a coordinate system on $C(I)$. Therefore we cannot define partial derivatives in the usual way. We work with the following linearly dependent set of constant coefficient differential operators.

Lemma 3**.**

There are unique

[TABLE]

that as elements of the Weyl algebra $D_{C(I)}$ satisfy

[TABLE]

and such that all ‘mixed’ commutators are zero:

[TABLE]

They span $C(I)$. They satisfy

[TABLE]

Proof*.*

The given commutators for $D_{k_{ij}}$ at first only determine an operator on the ambient vector space. But since $[D_{k_{ij}},-]$ annihilates all left hand sides of the relations (1), we obtain a unique constant coefficient differential operator $\in C(I)$ as claimed. The rest of the lemma goes the same way. ∎ We have introduced coordinates $k_{ii}$, $c_{ii}$, $e_{ii}$ and derivatives $D_{k_{ii}}$, $D_{c_{ii}}$, $D_{e_{ii}}$ that are identically zero and in some sense superfluous. These phantom objects are nevertheless useful because they allow us to write sums as in (3).

Convention (6) is in force, $I=J\sqcup K$. To this decomposition of $I$ we associate a coordinate $\xi$, and a vector $\Xi$ transversal to the hyperplane $\xi=0$. The dependence of these definitions on the decomposition $I=J\sqcup K$ is left implicit in the notation.

Definition 4**.**

Define $\xi\in C(I)^{\ast}$ and $\Xi\in C(I)$ by131313One can equivalently define $\Xi$ by $\Xi=\text{const}\cdot\sum_{j,j^{\prime}\in J}D_{k_{jj^{\prime}}}$ normalized so that $[\Xi,\xi]=2$.

[TABLE]

Also define

[TABLE]

for all $j,j^{\prime}\in J$ and $k,k^{\prime}\in K$ and $i,i^{\prime}\in I=J\sqcup K$. Set $C(I)^{\perp}=\{\xi=0\}$.

Lemma 5**.**

We have

[TABLE]

The $D^{\perp}$ span $C(I)^{\perp}$. In the Weyl algebra $D_{C(I)}$,

[TABLE]

for all $j,j^{\prime}\in J$ and $k,k^{\prime}\in K$. The $k^{\perp},c^{\perp},e^{\perp}$ and $D^{\perp}$ satisfy the linear relations (1) and (9), with $k$ replaced by $k^{\perp}$ and so forth. They additionally satisfy

[TABLE]

Same if the summation is over $k,k^{\prime}\in K$ instead.