jaj•a•person's notes

Formal calculus MOC

Formal delta

The formal delta over a field is the Laurent series1 fcalc

given by the Fourier series expansion of the Dirac delta.

Properties

Let be a vector space over . Let and . Finally let and . Then in [[Formal sums over a vector space|]]

\begin{align*} v(z)\delta(az) = v(a^{-1}) \delta (az) \end{align*}

^P2 3. $$ \begin{align*} v(z)T[\delta(az)] = v(a^{-1})T[\delta(az)]-(Tv)(a^{-1})\delta(az) \end{align*}

Erroneous nesting of equation structures^P3 Let $X(z_{1},z_{2}) \in (\End V) \D[z_{1},z_{1}^{-1},z_{2},z_{2}^{-1}\D]$ such that $\lim_{ z_{1} \to z_{2} }X(z_{1},z_{2})$ exists and $a \in \mathbb{K}^\times$. Finally let $p(z_{1},z_{2}) \in \mathbb{K}[z_{1},z_{1}^{-1},z_{2},z_{2}^{-1}]$, $T_{1} = p(z_{1},z_{2}) \frac{ \partial }{ \partial z_{1} }$, and $T_{2}=p(z_{1},z_{2})\frac{ \partial }{ \partial z_{2} }$. Then in [[Formal sums over endomorphisms|$(\End V)\{ z_{1},z_{2} \}$]] 4. $$ \begin{align*} X(z_{1},z_{2}) \delta(az_{1} / z_{2}) &= X(a^{-1}z_{2},z_{2})\delta(az_{1}/z_{2}) \\ &= X(z_{1},az_{1})\delta(az_{1} /z_{2}) \end{align*} $$ ^PA 5. $$ \begin{align*} X(z_{1},z_{2})\frac{ \partial }{ \partial z_{1} } [\delta(az_{1} / z_{2})] &= X(a^{-1}z_{2},z_{2}) \frac{ \partial }{ \partial z_{1} } [\delta(az_{1} / z_{2})] - \left( \frac{ \partial X }{ \partial z_{1} } \right)(a^{-1}z_{2},z_{2})\delta(az_{1} / z_{2}) \\ X(z_{1},z_{2})\frac{ \partial }{ \partial z_{2} } [\delta(az_{1} / z_{2})] &= X(z_{1},az_{1}) \frac{ \partial }{ \partial z_{2} } [\delta(az_{1} / z_{2})] - \left( \frac{ \partial X }{ \partial z_{2} } \right)(z_{1},az_{1})\delta(az_{1} / z_{2}) \end{align*}

^PB 6. $$ \begin{align*} X(z_{1},z_{2}) T_{1}[\delta(az_{1} / z_{2})] &= X(a^{-1}z_{2},z_{2})T_{1}[\delta(az_{1} / z_{2})] - (T_{1}X)(a^{-1}z_{2},z_{2})\delta(az_{1} / z_{2}) \ X(z_{1},z_{2})T_{2}[\delta(az_{1} / z_{2})] &= X(z_{1},az_{1})T_{2}[\delta(az_{1} / z_{2})] - (T_{2}X)(z_{1},az_{1}) \delta(az_{1} / z_{2}) \end{align*}

You can't use 'macro parameter character #' in math mode^PC Note these fail for non-integer powers. > [!check]- Proof of 1–6 > > First we prove [[#^p1|^P1]]. > Consider the special case $v(z) = v_{n}z^n$. Then > $$ > \begin{align*} > v(z)\delta(az) &= (v_{n}z^n) \left( \sum_{k \in \mathbb{Z}}a^kz^k \right) > = \sum_{k \in \mathbb{Z}}a^kv_{n}z^{k+n} \\ > &= \sum_{k \in \mathbb{Z}}a^{k-n}v_{n}z^k > = (a^{-n}v_{n})\left( \sum_{k \in \mathbb{Z}}a^kz^k \right) \\ > &= v(a^{-1})\delta(az) > \end{align*} > $$ > whence follows [[#^p1|^P1]] by linearity. > The proof of [[#^pa|^PA]] is similar. > Let $X(z_{1},z_{2}) = \sum_{m,n \in \mathbb{Z}}z_{1}^mz_{2}^n$. Then > $$ > \begin{align*} > X(z_{1},z_{2})\delta(az_{1} / z_{2}) > &= \left( \sum_{m,n \in \mathbb{Z}} x(m,n)z_{1}^mz_{2}^n \right) \left( \sum_{k \in \mathbb{Z}} a^k z_{1}^k z_{2}^{-k} \right) \\ > &= \sum_{m,n, k \in \mathbb{Z}} a^k x(m,n) z_{1}^{m+k}z_{2}^{n-k} \\ > &= \sum_{m,n,k \in \mathbb{Z}}a^{k-m} x(m,n) z_{1}^kz_{2}^{m+n-k} \\ > &= X(a^{-1}z_{2},z_{2})\delta(az_{1} / z_{2}) > \end{align*} > $$ > Then [[#^p3|^P3]] and [[#^pc|^PC]] follow by taking appropriate derivatives, and [[#^p2|^P2]] and [[#^pb|^PB]] are special cases. <span class="QED"/> # --- #state/develop | #lang/en | #SemBr

Footnotes

  1. 1988. Vertex operator algebras and the Monster, §2.1–§2.2, p. 52ff ↩

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