Semi-Differential Operators and the Algebra of Operator Product Expansion of Quantum Fields

Abstract

We introduce a symmetric operad whose algebras are the operator product expansion (OPE) Algebras of quantum fields. There is a natural classical limit for the algebras over this operad and they are commutative associative algebras with derivations. The latter are the algebras of classical fields. In this paper we completely develop our approach to models of quantum fields, which come from vertex algebras in higher dimensions. However, our approach to OPE algebras can be extended to general quantum fields even over curved space–time. We introduce a notion of OPE operations based on the new notion of semi-differential operators. The latter are linear operators $\varGamma :\mathcal {M}\rightarrow \mathcal {N}$ between two modules of a commutative associative algebra $\mathcal {A}$, such that for every $m \in \mathcal {M}$ the assignment $a\mapsto \varGamma (a \cdot m)$ is a differential operator $\mathcal {A}\rightarrow \mathcal {N}$ in the usual sense. The residue of a meromorphic function at its pole is an example of a semi-differential operator.

Appendix A: Proof of Theorem 10.2

We need to prove that the map $\text { OPE}_{V}$ (10.24) is a morphism of (symmetric) operads. This includes the compatibility (preservation) of the operadic composition, preservation of units and permutation equivariance. The latter is manifest according to (10.19). The preservation of the operadic units is also obvious.

We continue with the compatibility of the operadic compositions. The proof is straightforward, but there are several hidden natural maps in our constructions, which makes the proof complicated. The fact that that the map $\text { OPE}_{V}$ (10.24) preserves the operadic compositions means that

$$\begin{aligned}&\text { OPE}_{V} (\varGamma ') \Bigl ( \text { OPE}_{V} (\varGamma ''_1)\bigl ({a}_{(1,1)},\dots ,{a}_{(1,j_1)}\bigr ),\, \dots ,\, \text { OPE}_{V} (\varGamma ''_k)\bigl ({a}_{(k,1)},\dots ,{a}_{(k,j_k)}\bigr ) \Bigr ) \nonumber \\&\quad = \, \text { OPE}_{V} (\gamma _{j_1,\dots ,j_k}\left( \varGamma ';\varGamma ''_1,\dots ,\varGamma ''_k\right) ) \bigl ({a}_{(1,1)},\dots ,{a}_{(1,j_1)},\, \ldots ,\, {a}_{(k,1)},\dots ,{a}_{(k,j_k)}\bigr ) \,.\nonumber \\ \end{aligned}$$

(A.1)

In other words, if

$$\begin{aligned} b_{\ell } \,:= & {} \text { OPE}_{V} (\varGamma ''_{\ell })\bigl ({a}_{(\ell ,1)},\dots ,{a}_{(\ell ,j_{\ell })}\bigr ) \quad (\ell \, = \, 1,\dots ,k) \,, \nonumber \\ c \,:= & {} \text { OPE}_{V} (\varGamma ')\bigl (b_1,\dots ,b_k\bigr ) \,, \nonumber \\ \varGamma \,:= & {} \gamma _{j_1,\dots ,j_k}\left( \varGamma ';\varGamma ''_1,\dots ,\varGamma ''_k\right) \,, \quad \text {then:} \nonumber \\ c \,= & {} \text { OPE}_{V} (\varGamma )\bigl ({a}_{(1,1)},\dots ,{a}_{(1,j_1)},\, \ldots ,\, {a}_{(k,1)},\dots ,{a}_{(k,j_k)}\bigr ) \,. \end{aligned}$$

(A.2)

Now, combining the definition (10.25) of $\text { OPE}_{V}$ and the construction of $\varGamma _{V}$ (10.5) one has¹⁸

$$\begin{aligned}&b_{\ell } \, = \, {\bigl ((\varepsilon _{(\mathrm x_{\ell } = 0)} \circ {(\varGamma ''_{\ell })}_{\{{Q''_{\ell }}^{-1}\}}) \otimes \text {id}_{V}\bigr )}^{\widehat{\,}} \bigl ( \mathcal {F}_{{a}_{(\ell ,1)},\dots ,{a}_{(\ell ,j_{\ell })}} \bigr ) \,, \nonumber \\&\mathcal {F}_{{a}_{(\ell ,1)},\dots ,{a}_{(\ell ,j_{\ell })}} ({\mathrm x}_{(\ell ,1)},\dots ,{\mathrm x}_{(\ell ,j_{\ell })}) \, := \, Q''_{\ell } \cdot \mathcal {Y}_{{a}_{(\ell ,1)},\dots ,{a}_{(\ell ,j_{\ell })},\widehat{1}\,} ({\mathrm x}_{(\ell ,1)},\dots ,{\mathrm x}_{(\ell ,j_{\ell })}) \nonumber \\&\quad \quad \in \, V[[{\mathrm x}_{(\ell ,1)},\dots ,{\mathrm x}_{(\ell ,j_{\ell })}]] \,, \nonumber \\&Q''_{\ell } \, :=\, \bigl ( Q_{j_\ell } ({\mathrm x}_{(\ell ,1)},\dots ,{\mathrm x}_{(\ell ,j_{\ell })}) \bigr )^{N_{{a}_{(\ell ,1)},\dots ,{a}_{(\ell ,j_{\ell })}}} \,, \end{aligned}$$

(A.3)

for $\ell = 1,\dots ,k$. Here:

• (i) roughly speaking, $b_{\ell }$ is obtained by applying the differential operator $\varepsilon _{(\mathrm x_{\ell }=0)} \circ {(\varGamma ''_{\ell })}_{\{{Q''_{\ell }}^{-1}\}}$ on the formal power series $\mathcal {F}_{{a}_{(\ell ,1)},\dots ,{a}_{(\ell ,j_{\ell })}} ({\mathrm x}_{(\ell ,1)},$ $\dots ,$ ${\mathrm x}_{(\ell ,j_{\ell })})$ without acting on the coefficients $\in V$ of that series.

• (ii) $\varepsilon _{(\mathrm x_{\ell }=0)}$  :  $F(\mathrm x_{\ell })$ $\mapsto$ F(0) is the augmentation $\varepsilon$ on $\mathcal {C}$ with indicated set of generating formal variables $\mathrm x_{\ell }$ on which it acts.

• (iii) In (i) we assumed that after applying the (continued) differential operator ${(\varGamma ''_{\ell })}_{\{{Q''_{\ell }}^{-1}\}}$ to $\mathcal {F}_{{a}_{(\ell ,1)},\dots ,{a}_{(\ell ,j_{\ell })}} ({\mathrm x}_{(\ell ,1)},$ $\dots ,$ ${\mathrm x}_{(\ell ,j_{\ell })})$ then the resulting variable is $\mathrm x_{\ell }$, which afterwards is set to 0 by $\varepsilon _{(\mathrm x_{\ell }=0)}$.

• (iv) $N_{{a}_{(\ell ,1)},\dots ,{a}_{(\ell ,j_{\ell })}}$ are sufficiently large positive integer numbers, which according to the theory of vertex algebras will ensure that $\mathcal {F}_{{a}_{(\ell ,1)},\dots ,{a}_{(\ell ,j_{\ell })}} ({\mathrm x}_{(\ell ,1)},$ $\dots ,$ ${\mathrm x}_{(\ell ,j_{\ell })})$ $\in$ $V[[{\mathrm x}_{(\ell ,1)},$ $\dots ,$ ${\mathrm x}_{(\ell ,j_{\ell })}]]$ (cf. Eq. (10.18)).

It follows that

$$\begin{aligned}&\mathcal {Y}_{b_1,\dots ,b_k,\widehat{1}\,} (\mathrm y_1,\dots ,\mathrm y_k) \, \nonumber \\&\quad =\Bigl ( \bigl ( (\varepsilon _{(\mathrm x_1 = 0)} \circ {(\varGamma ''_1)}_{\{{Q''_1}^{-1}\}}) \otimes \cdots \otimes (\varepsilon _{(\mathrm x_k = 0)} \circ {(\varGamma ''_k)}_{\{{Q''_k}^{-1}\}}) \bigr ) \nonumber \\&\quad { \otimes \, \text {id}_{V[[\mathrm y_1,\dots ,\mathrm y_k]] [1/Q'] } \Bigr )}^{\widehat{\,}} \Bigl ( \mathcal {Y}_{ \mathcal {F}_{{a}_{(1,1)},\dots }({\mathrm x}_{(1,1)},\dots ),\, \ldots ,\,\mathcal {F}_{{a}_{(k,1)},\dots ,{a}_{(k,j_k)}}({\mathrm x}_{(k,1)},\dots ,{\mathrm x}_{(k,j_k)}) ,\,\widehat{1}\, } \bigl (\mathrm y_1, \nonumber \\&\quad \quad \dots ,\mathrm y_k\bigr ) \Bigr ) \,, \qquad \end{aligned}$$

(A.4)

where similarly to (A.3),

$$\begin{aligned}&Q' \,:=\, \bigl ( Q_k (\mathrm y_1,\dots ,\mathrm y_k)\bigr )^{N_{b_1,\dots ,b_k}} \quad \text {is such that} \,, \nonumber \\&\mathcal {F}_{b_1,\dots ,b_k} (\mathrm y_1,\dots ,\mathrm y_k) \, := \, Q' \cdot \mathcal {Y}_{b_1,\dots ,b_k,\widehat{1}\,} (\mathrm y_1,\dots ,\mathrm y_k) \, \in \, V[[\mathrm y_1,\dots ,\mathrm y_k]] \,. \end{aligned}$$

(A.5)

In Eq. (A.4) we encounter the first subtlety related to hidden natural maps. Before arguing for Eq. (A.4), we will explain the meaning of the substitution process

$$\begin{aligned} \mathcal {Y}_{b_1,\dots ,b_k,\widehat{1}} \, \mapsto \, \mathcal {Y}_{ \mathcal {F}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)}},\, \ldots ,\, \mathcal {F}_{{a}_{(k,1)},\dots ,{a}_{(k,j_k)}} ,\,\widehat{1}\, } \, \end{aligned}$$

and in what space of series the result belongs to. To this end, note first that the assignment $b_1 \otimes \cdots \otimes b_k$ $\mapsto$ $\mathcal {Y}_{b_1,\dots ,b_k,\widehat{1}\,} (\mathrm y_1,$ $\dots$ $\mathrm y_k)$ is a linear map $V^{\otimes k}$ $\rightarrow$ $V [[\mathrm y_1,$ $\dots ,$ $\mathrm y_k]][1/Q_k(\mathrm y_1,$ $\dots ,$ $\mathrm y_k)]$ $=$ $V [[\mathrm y_1,$ $\dots ,$ $\mathrm y_k]][1/Q']$. Applying to this map the functorial assignment W $\mapsto$ $W [[({\mathrm x}_{(i',i'')})_{i',i''}]]$ we lift the map $b_1$ $\otimes$ $\cdots$ $\otimes$ $b_k$ $\mapsto$ $\mathcal {F}_{b_1,\dots ,b_k}$ to a linear map

$$\begin{aligned} \bigl (V^{\otimes k}\bigr ) [[({\mathrm x}_{(i',i'')})_{i',i''}]] \, \rightarrow \, \bigl (V [[\mathrm y_1, \dots , \mathrm y_k]][1/Q'] \bigr ) [[({\mathrm x}_{(i',i'')})_{i',i''}]] \,. \end{aligned}$$

Composing further the latter map on the right by the maps

we obtain the linear map

In particular,

$$\begin{aligned} \mathcal {Y}_{\mathcal {F}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)}},\, \ldots ,\, \mathcal {F}_{{a}_{(k,1)},\dots ,{a}_{(k,j_k)}} ,\,\widehat{1}\,} \,\in \, \bigl (V [[\mathrm y_1, \dots , \mathrm y_k]][1/Q'] \bigr ) [[({\mathrm x}_{(i',i'')})_{i',i''}]] \,.\nonumber \\ \end{aligned}$$

(A.6)

Then the meaning of Eq. (A.4) is that the differential operators

$\varepsilon _{(\mathrm x_{\ell } = 0)}$ $\circ$ ${(\varGamma ''_{\ell })}_{\{{Q''_k}^{-1}\}}$ act now on variables $({\mathrm x}_{(i',i'')})$ in the series (A.6) without touching the coefficient series $\in$ $V [[\mathrm y_1,$ $\dots ,$ $\mathrm y_k]][1/Q']$.

Having argued (A.4) we now use another important fact from the theory of vertex algebras called “the associativity theorem”. Here we state its general form without a proof, which is a straightforward generalization of [BN06, Theorem 5.1].

Theorem A.1

Let $V$ be a vertex algebra and $Q''_{\ell }$, $\mathcal {F}_{{a}_{(\ell ,1)},\cdots ,{a}_{(\ell ,j_{\ell })}}$ be set according to Eq. (A.3) for ${a}_{(i',i'')} \in V$ $($ $i'$ $=$ 1,  $\dots ,$ k, $i''$ $=$ 1,  $\dots ,$ $j_{i'}$ $)$. Then, if we set for short ${\widetilde{\mathrm x}}_{(i',i'')}$ $:=$ ${\mathrm x}_{(i',i'')}+\mathrm y_{i'}$ we have

$$\begin{aligned}&Q''_1 \cdots Q''_k \nonumber \\&\quad \times \, \mathcal {Y}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)},\ldots ,{a}_{(k,1)},\dots ,{a}_{(k,j_k)},\widehat{1}\,} \bigl ({\widetilde{\mathrm x}}_{(1,1)},\dots ,{\widetilde{\mathrm x}}_{(1,j_1)},\ldots , {\widetilde{\mathrm x}}_{(k,1)},\dots ,{\widetilde{\mathrm x}}_{(k,j_k)}\bigr ) \nonumber \\&\quad \quad \in \, V [[({\widetilde{\mathrm x}}_{(i',i'')})_{i',i''}]]\bigl [1/Q_{j_1|\cdots |j_k}\bigl (({\widetilde{\mathrm x}}_{(i',i'')})_{i',i''}\bigr )\bigr ] \,, \end{aligned}$$

(A.7)

where $Q_{j_1|\cdots |j_k}$ is introduced in Eq. (7.20). Furthermore, we have

$$\begin{aligned}&\mathcal {Y}_{ \mathcal {F}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)}}({\mathrm x}_{(1,1)},\dots ,{\mathrm x}_{(1,j_1)}),\, \ldots ,\, \mathcal {F}_{{a}_{(k,1)},\dots ,{a}_{(k,j_k)}}({\mathrm x}_{(k,1)},\dots ,{\mathrm x}_{(k,j_k)}) \,,\,\widehat{1}\,} \bigl (\mathrm y_1,\dots ,\mathrm y_k\bigr ) \nonumber \\&\quad =\, \iota _{{\mathrm x}_{(1,1)},{\mathrm x}_{(1,2)}\dots ,{\mathrm x}_{(k,j_k)}} \, Q''_1 \cdots Q''_k \, \nonumber \\&\quad \quad \times \, \mathcal {Y}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)}, \ldots ,{a}_{(k,1)},\dots ,{a}_{(k,j_k)},\widehat{1}\,} \bigl ({\widetilde{\mathrm x}}_{(1,1)},\dots ,{\widetilde{\mathrm x}}_{(1,j_1)},\ldots , {\widetilde{\mathrm x}}_{(k,1)},\dots ,{\widetilde{\mathrm x}}_{(k,j_k)}\bigr ) \,,\nonumber \\ \end{aligned}$$

(A.8)

where $\iota _{{\mathrm x}_{(1,1)},{\mathrm x}_{(1,2)}\dots ,{\mathrm x}_{(k,j_k)}}$ stands for the (formal) Taylor expansion in ${\mathrm x}_{(i',i'')}:$

$$\begin{aligned}&\iota _{{\mathrm x}_{(1,1)},{\mathrm x}_{(1,2)}\dots ,{\mathrm x}_{(k,j_k)}} \,: V [[({\widetilde{\mathrm x}}_{(i',i'')})_{i',i''}]]\bigl [1/Q_{j_1|\cdots |j_k}\bigl (({\widetilde{\mathrm x}}_{(i',i'')})_{i',i''}\bigr )\bigr ] \nonumber \\&\quad \rightarrow \, \bigl (V [[\mathrm y_1, \dots , \mathrm y_k]][1/Q'] \bigr ) [[({\mathrm x}_{(i',i'')})_{i',i''}]] \,. \end{aligned}$$

(A.9)

Remark .1

Roughly, Eq. (A.8) can be written as

$$\begin{aligned}&\mathcal {Y}_{\mathcal {Y}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)},\,\widehat{1}\,}({\mathrm x}_{(1,1)},\dots ,{\mathrm x}_{(1,j_1)}),\, \ldots ,\, \mathcal {Y}_{{a}_{(k,1)},\dots ,{a}_{(k,j_k)},\,\widehat{1}\,}({\mathrm x}_{(k,1)},\dots ,{\mathrm x}_{(k,j_k)}) \,,\,\widehat{1}\,} \bigl (\mathrm y_1,\dots ,\mathrm y_k\bigr ) \nonumber \\&\quad \approx \, \mathcal {Y}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)}, \ldots ,{a}_{(k,1)},\dots ,{a}_{(k,j_k)},\widehat{1}\,} \bigl ({\widetilde{\mathrm x}}_{(1,1)},\dots ,{\widetilde{\mathrm x}}_{(1,j_1)},\ldots , {\widetilde{\mathrm x}}_{(k,1)},\dots ,{\widetilde{\mathrm x}}_{(k,j_k)}\bigr ) \,,\quad \end{aligned}$$

(${\widetilde{\mathrm x}}_{(i',i'')}$ $:=$ ${\mathrm x}_{(i',i'')}+\mathrm y_{i'}$), where “$\approx$” stands for the identification after the expansion (A.9).

Continuing we the proof of Theorem 10.2, we apply Eq. (A.8) to the right hand side of Eq. (A.4). We note that any differential operator commutes with the Taylor expansion since any operator of multiplication by a formal variable or any partial derivative commute with the Taylor expansion. As a result, we can move the differential operators $\varepsilon _{(\mathrm x_{\ell }=0)} \circ {(\varGamma ''_{\ell })}_{\{{Q''_{\ell }}^{-1}\}}$ in (A.4) inside the Taylor expansion on the right hand side of Eq. (A.8). We claim that then Eq. (A.4) takes the following form:

$$\begin{aligned}&\mathcal {Y}_{b_1,\dots ,b_k,\widehat{1}\,} (\mathrm y_1,\dots ,\mathrm y_k) \,=\, (\varepsilon _{(\mathrm x_1=0)} \otimes \cdots \otimes \varepsilon _{(\mathrm x_{\ell }=0)}) \nonumber \\&\quad \circ \, { \bigl ( {({(\varGamma ''_1)}_{\{{Q''_1}^{-1}\}} \otimes \cdots \otimes {(\varGamma ''_k)}_{\{{Q''_k}^{-1}\}})}^{ext} \otimes \text {id}_{V}\bigr )}^{\widehat{\,}} \Bigl ( Q''_1 \cdots Q''_k \, \nonumber \\&\quad \times \, \mathcal {Y}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)},\ldots ,{a}_{(k,1)},\dots ,{a}_{(k,j_k)}\,,\,\widehat{1}\,} \bigl ({\mathrm x}_{(1,1)}+\mathrm y_1, {\mathrm x}_{(1,2)}+\mathrm y_1,\ldots ,{\mathrm x}_{(k,j_k)}+\mathrm y_k\bigr ) \Bigr ) \,.\nonumber \\ \end{aligned}$$

(A.10)

Indeed:

• when we move the differential operator ${(\varGamma ''_1)}_{\{{Q''_1}^{-1}\}}$ $\otimes$ $\cdots$ $\otimes$ ${(\varGamma ''_k)}_{\{{Q''_k}^{-1}\}}$ $=$ $(\varGamma ''_1$ $\otimes$ $\cdots$ $\otimes$ ${\varGamma ''_1)}_{\{{(Q''_1 \cdots Q''_k)}^{-1}\}}$ before the Taylor expansion in all ${\mathrm x}_{(i',i'')}$ it now acts according to its extension ${(\cdots )}^{ext}$ due to Theorem 3.1.¹⁹

• The differential operators ${(\varGamma ''_{\ell })}_{\{{Q''_{\ell }}^{-1}\}}$ set all the variables ${\mathrm x}_{(\ell ,s)}$ equal to $\mathrm x_{\ell }$, which then is set to zero by $\varepsilon _{(\mathrm x_{\ell }=0)}$. Thus, the Taylor expansion disappears in the right hand side of Eq. (A.10).

We can further remove all the $Q''_{\ell }$ from ${(\varGamma ''_{\ell })}_{\{{Q''_{\ell }}^{-1}\}}$ (following the construction of Theorem 5.6). Thus, we obtain

$$\begin{aligned}&\mathcal {Y}_{b_1,\dots ,b_k,\widehat{1}\,} (\mathrm y_1,\dots ,\mathrm y_k) \,=\, (\varepsilon _{(\mathrm x_1=0)} \otimes \cdots \otimes \varepsilon _{(\mathrm x_{\ell }=0)}) \nonumber \\&\quad \circ \, \bigl ( {(\varGamma ''_1 \otimes \cdots \otimes \varGamma ''_k)}^{ext} \bigr )_{V} \Bigl ( \mathcal {Y}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)},\ldots ,{a}_{(k,1)},\dots ,{a}_{(k,j_k)}\,,\,\widehat{1}\,} \bigl ({\mathrm x}_{(1,1)}+\mathrm y_1, \nonumber \\&\quad \quad \quad {\mathrm x}_{(1,2)}+\mathrm y_1,\ldots ,{\mathrm x}_{(k,j_k)}+\mathrm y_k\bigr ) \Bigr ) \,, \end{aligned}$$

(A.11)

where

$$\begin{aligned} \bigl ( {(\varGamma ''_1 \otimes \cdots \otimes \varGamma ''_k)}^{ext} \bigr )_{V} \,:=\, {\bigl ( {(\varGamma ''_1 \otimes \cdots \otimes \varGamma ''_k)}^{ext} \otimes \text {id}_{V} \bigr )}^{\widehat{\,}} \end{aligned}$$

is the same type of extension as those constructed in Theorem 10.1.

Now, in the right hand side of Eq. (A.11) we can omit all the $\varepsilon _{(\mathrm x_{\ell }=0)}$ just by replacing $\mathrm y_1,$ $\dots ,$ $\mathrm y_k$ by $\mathrm x_1,$ $\dots ,$ $\mathrm x_k$, respectively. We obtain:

$$\begin{aligned}&\mathcal {Y}_{b_1,\dots ,b_k,\widehat{1}\,} (\mathrm x_1,\dots ,\mathrm x_k) \nonumber \\&\quad =\, \bigl ( {(\varGamma ''_1 \otimes \cdots \otimes \varGamma ''_k)}^{ext} \bigr )_{V} \Bigl ( \mathcal {Y}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)},\ldots ,{a}_{(k,1)},\dots ,{a}_{(k,j_k)}\,,\,\widehat{1}\,} \bigl (({\mathrm x}_{(i',i'')})_{i',i''}\bigr ) \Bigr ) \,.\nonumber \\ \end{aligned}$$

(A.12)

Finally,

$$\begin{aligned} c \,= & {} \bigl (\varepsilon _{(\mathrm x= 0)} \circ (\varGamma ')_{V}\bigr ) \Bigl (\mathcal {Y}_{b_1,\dots ,b_k,\widehat{1}\,} (\mathrm x_1,\dots ,\mathrm x_k)\Bigr ) \nonumber \\= & {} \Bigl (\varepsilon _{(\mathrm x= 0)} \circ \bigl ( \varGamma ' \circ {(\varGamma ''_1 \otimes \cdots \otimes \varGamma ''_k)}^{ext} \bigr )_{V} \Bigr ) \Bigl ( \mathcal {Y}_{{a}_{(1,1)},\dots ,{a}_{(1,j_1)},\ldots ,{a}_{(k,1)},\dots ,{a}_{(k,j_k)}\,,\,\widehat{1}\,} \Bigr ) \end{aligned}$$

and this is exactly (A.2).