The classical representations of the diffeomorphism algebra act on tensor densities, in the following way.
\[
[{\cal L}_\xi, \phi(x)] = -\xi^\mu(x)\partial_\mu\phi(x)
- \partial_\nu \xi^\mu(x) T^\nu_\mu\phi(x),
\]
where the \(T^\mu_\nu\) are some matrices that satisfy the general linear Lie algebra \(gl(d)\), with brackets
\[
[T^\mu_\nu, T^\sigma_\tau] =
\delta^\sigma_\nu T^\mu_\tau - \delta^\mu_\tau T^\sigma_\mu.
\]
By choosing different representations of \(gl(d)\) we obtain the transformation laws for all kinds of tensor densities in \(d\) dimensions.
From this we can derive how the diffeomorphism algebra acts on Taylor series. We already know that it acts nonlinearly on the observer's trajectory:
\[
[{\cal L}_\xi, q^\mu(t)] = \xi^\mu(q(t)).
\]
The action on the Taylor coefficients is given by
\[
[{\cal L}_\xi,\phi_n(t)] = -\sum_m T^m_n(\xi(q(t))) \phi_m(t),
\]
where \(T^m_n(\xi)\) are complicated expressions that depend on the vector field \(\xi\) and its derivatives, and also on the matrices \(T^\mu_\nu\). The exact form can be found in [math-ph/9810003], [arXiv:1502.07760].
A \(p\)-jet is locally a Taylor series truncated at order \(p\). So instead of summing over all \(m\), we restrict the sum to \(m < p\). In multiple dimensions, this means that if \(m = (m_0, m_1, .., m_{d-1})\), the sum \(m_0 + m_1 + .. + m_{d-1} < p\).
\[
\phi(x,t) = \sum_m^p {1\over m!} \phi_m(t) (x-q(t))^m.
\]
The space spanned by \(\phi_m\) with \(m < p\) is preserved by diffeomorphisms, because \(T^m_n(\xi) = 0\) if \(m < n\).
Now introduce the canonically conjugate momenta \(p_\mu(t)\) and \(\pi^n(t)\),
\[\begin{align}
[q^\mu(t), p_\nu(t')] &= i\delta^\mu_\nu \delta(t-t'), \\
[\phi_m(t), \pi^n(t')] &= i\delta_m^n \delta(t-t').
\end{align}\]
We can now write down the normal-ordered Lie derivatives.
\[\begin{align}
{\cal L}_\xi &= i \int dt\ \Big( :\xi^\mu(q(t)) p_\mu(t):
- \sum_{m,n} : \pi^n(t) T^m_n(\xi(q(t))) \phi_m(t): \Big)
\end{align}\]
As in the previous post, normal ordering means that we move negative-frequency modes to the right, where they can annihilate the vacuum. Because of normal ordering, the Lie derivatives satisfy the multi-dimensional Virasoro algebra.
- The space of \(p\)-jets, spanned by the base point and the Taylor coefficients up to order \(p\), is finite-dimensional. Hence the space of trajectories in this space is spanned by finitely many functions of a single variable, so we can still do normal ordering without encountering infinities.
- The coefficients in front of the extensions ("the abelian charges") depend on the choice of \(gl(d)\) representation and the truncation order \(p\). In \(d\) dimensions, the abelian charges are polynomials in \(p\), and the leading terms are proportional to \(p^d\).
- In the limit \(p\to\infty\), i.e. when we deal with infinite jets, the abelian charges thus diverge. This is not surprising. We turned to \(p\)-jets because the original realization on fields resulted in nonsensical infinities. This is a kind of regularization. When we remove the regularization by letting \(p\to\infty\), the infinities reappear. I will discuss some ideas how to cancel these infinities in a later post.
<Previous Next>
