We saw in a previous post that we can not construct representations in multiple dimensions by quantizing fields. So how could it be done? This was a problem that I struggled with for many years. In retrospect the answer is obvious, but I could only arrive at it after studying the seminal work of Rao and Moody.
A vector field is of the form \(\xi^\mu(x) \partial_\mu\), so we can embed it in a Heisenberg algebra generated by the oscillators \(q^\mu\) and \(p_\nu\). The nonzero brackets are
\[
[q^\mu, p_\nu] = i\delta^\mu_\nu
\]
and the embedding is
\[
{\cal L}_\xi = i \xi^\mu(q) p_\mu.
\]
More precisely, this is an embedding into the corresponding enveloping algebra.
This realization is not very interesting, because this Heisenberg algebra is finite-dimensional and therefore no extension arises upon normal ordering. But with a slight modification we can obtain non-trivial representations of the multi-dimensional Virasoro algebra. Namely, let the oscillators depend on a parameter \(t\) which lives on the circle. The nonzero brackets are now given by
\[
[q^\mu(t), p_\nu(t')] = i\delta^\mu_\nu \delta(t-t'),
\]
where \(\delta(t)\) is the delta function. The Lie derivatives
\[
{\cal L}_\xi = i \int dt\ \xi^\mu(q(t)) p_\mu(t)
\]
have the following brackets with the oscillators
\[
[{\cal L}_\xi, q^\mu(t)] = \xi^\mu(q(t)).
\]
Since the oscillators depend on a circle variable, we can expand them in a Fourier series.
\[\begin{align}
q^\mu(t) &= \sum_{k=-\infty}^\infty {\hat q}^\mu(k) \\
p_\nu(t) &= \sum_{k=-\infty}^\infty {\hat p}_\nu(k).
\end{align}\]
The vacuum is defined to annihilate all negative frequency modes, i.e.
\[
{\hat q}^\mu(-k) |0> = {\hat p}_\nu(-k) |0> = 0,
\]
for all \(k > 0\). Define \(p^>_\mu(t)\) as the sum over positive frequency modes and \(p^<_\mu(t)\) as the negative frequency part, i.e.
\[
p^>_\mu(t) = \sum_{k=0}^\infty {\hat p}_\mu(k).
\]
Normal ordering amounts to moving negative frequency modes to the right.
\[
:\xi^\mu(q(t)) p_\mu(t): = p^>_\mu(t) \xi^\mu(q(t)) + \xi^\mu(q(t)) p^<_\mu(t).
\]
It we normal order the expression for the Lie derivatives we obtain a representation of the multi-dimensional Virasoro algebra with \(c_1 = 1, c_2 = 0\).
There are two crucial observations:
- The Heisenberg algebra is infinite-dimensional, but its basis consists of finitely many functions of a single variable \(t\). This is the case where we can do normal ordering without introducing infinities, because there are no transverse modes that can give rise to infinite sums.
- The embedding is nonlinear, and therefore the extension is non-central. The Virasoro extensions are functionals of the observer's trajectory, which do not commute with diffeomorphisms that move the observer's trajectory around. It is central in one dimension, because on the circle there is only a single trajectory, namely the circle itself.
