In the two first posts (1, 2) in this series I argued that the missing ingredients in Quantum Gravity (QG) are locality and observer dependence, respectively. Now it is time to make the connection between these concepts. But before making this connection at the end of this post, we need some rather massive calculations. The main reference is my CMP paper (essentially the same as arXiv:math-ph/9810003). That paper is somewhat difficult to follow, even for myself, so I recently updated it using more standard notation, arXiv:1502.07760.
Local operators in QG are only possible if the spacetime diffeomorphism algebra in $d$ dimensions, also known as the Lie algebra of vector fields $vect(d)$, acquires an extension upon quantization. This turns $vect(d)$ into the Virasoro algebra in $d$ dimensions, $Vir(d)$. In particular, $Vir(1)$ is the ordinary Virasoro algebra, the central extension of the algebra of vector fields on the circle, $vect(1)$. $Vir(1)$ is also a crucial ingredient in Conformal Field Theory (CFT), which successfully describes critical phenomena in two dimensions.
There is a simple recipe for constructing off-shell representations of $Vir(1)$:
1. Start with classical densities, which in the context of CFT are called primary fields.
2. Introduce canonical momenta for all fields.
3. Introduce a vacuum state which annihilates all negative-frequency operators, both the fields and their momenta.
4. Normal order to remove an infinite vacuum energy.
The last step introduces a central charge, turning $vect(1)$ into $Vir(1)$.
Intuitively one could try to duplicate these steps in higher dimensions, but that does not work. The reason is that normal ordering introduces an unrestricted sum over transverse directions, which makes the putative central extension infinite and thus meaningless. Instead we introduce a new step after the first one. The recipe now reads:
1. Start with classical tensor densities.
2. Expand all fields in a Taylor series around a privileged curve in $d$-dimensional spacetime, and truncate the Taylor series at order $p$.
3. Introduce canonical momenta for the Taylor data, which include both the Taylor coefficients and the points on the selected curve.
4. Introduce a vacuum state which annihiles negative frequency states.
5. Normal order to remove an infinite vacuum energy.
A $p$-jet is locally the same thing as a Taylor series truncated at order $p$, and we will use the two terms synonymously. Since the space of $p$-jets is finite-dimensional, the space of trajectories therein is spanned by finitely many functions of a single variable, which is precisely the situation where normal ordering can be done without introducing infinities coming from transverse directions.
Locally, a vector field is of the form $\xi^\mu(x) \partial_\mu$, where $x = (x^\mu)$ is a point in $d$-dimensional space and $\partial_\mu = \partial/\partial x^\mu$ the corresponding partial derivative. The bracket of two vectors fields reads
\[
[\xi, \eta] = \xi^\mu \partial_\mu \eta^\nu \partial_\nu -
\eta^\nu \partial_\nu \xi^\mu \partial_\mu.
\]
$vect(d)$ is defined by the brackets
\[
[{\cal L}_\xi, {\cal L}_\eta] = {\cal L}_{[\xi,\eta]}.
\]
The extension $Vir(d)$ depends on two parameters $c_1$ and $c_2$, both of which reduce to the central charge in one dimension:
\[
[{\cal L}_\xi, {\cal L}_\eta] = {\cal L}_{[\xi,\eta]} +
\frac{1}{2\pi i} \oint dt\ \dot q^\rho(t) \Big(
c_1 \partial_\rho \partial_\nu \xi^\mu(q(t)) \partial_\mu \eta^\nu(q(t)) \\
\qquad\qquad + c_2 \partial_\rho \partial_\mu \xi^\mu(q(t)) \partial_\nu \eta^\nu(q(t)) \Big), \\
{[}{\cal L}_\xi, q^\mu(t)] = \xi^\mu(q(t)), \\
{[}q^\mu(t), q^\nu(t')] = 0.
\]
In particular when $d=1$, vectors only have a single component and we can ignore the spacetime indices. The extension then reduces to
\[
\frac{1}{2\pi i} \oint dt\ \dot q(t) \xi^{\prime\prime}(q(t)) \eta^\prime(q(t))
= \frac{1}{2\pi i} \oint dq\ \xi^{\prime\prime}(q) \eta^\prime(q),
\]
which has the ordinary Virasoro form. The terms proportional to $c_1$ and $c_2$ where first written down by Rao and Moody and myself, respectively.
To carry out the construction explicitly for tensor-valued $p$-jets is quite cumbersome, but in the special case that we deal with $-1$-jets (which depend only on the expansion point, and not on any Taylor coefficients at all), formulas become manageable. Consider the Heisenberg algebra defined by the brackets
\[
[q^\mu, p_\nu] = i\delta^\mu_\nu, \qquad
[q^\mu, q^\nu] = [p_\mu, p_\nu] = 0.
\]
Clearly we can embed $vect(d)$ into the universal enveloping algebra of this Heisenberg algebra as follows:
\[
{\cal L}_\xi = i\xi^\mu(q) p_\mu.
\]
This embedding immediately yields a representation of $vect(d)$ on $C^\infty(q)$, the space of smooth functions of $q$, and also on the dual space $C^\infty(p)$. However, neither of the representations are particularly interesting because they do not exhibit any extension. However, with a slight modification of the construction, we immedately obtain something interesting. Consider the infinite-dimensional Heisenberg algebra where everything also depends on an extra variable $t$, which lives on the circle. It is defined by the brackets
\[
[q^\mu(t), p_\nu(t')] = i\delta^\mu_\nu \delta(t-t'), \qquad
[q^\mu(t), q^\nu(t')] = [p_\mu(t), p_\nu(t')] = 0.
\]
The new embedding of $vect(d)$ reads
\[
{\cal L}_\xi = i \oint dt\ \xi^\mu(q(t)) p_\mu(t).
\]
This operator acts on the space $C^\infty[q(t)]$ of smooth functionals of $q(t)$, and, after moving the momentum operator to the left, on the dual space $C^\infty[p(t)]$. Neither of these representations yields any extension.
However, there is a more physical way to split the Heisenberg algebra into creation and annihilation operators. Since the oscillators depend on a circle parameter $t$, they can be expanded in a Fourier series, and we postulate that the components of negative frequency annihilate the vacuum state. If $p^>_\mu(t)$ and $p^<_\mu(t)$ denote the positive and negative frequency parts of $p_\mu(t)$, respectively, and analogously for $q^\mu(t)$, the state space can be identified with $C^\infty[q_>(t), p^>(t)]$. We still have a problem with a infinite vacuum energy, but this can be taken care of by normal ordering. Hence we replace the expression for ${\cal L}_\xi$ by
\[
{\cal L}_\xi = i \oint dt\ :\xi^\mu(q(t)) p_\mu(t): \\
\equiv i \oint dt\ \Big(\xi^\mu(q(t)) p^<_\mu(t) + p^>_\mu(t) \xi^\mu(q(t)) \Big).
\]
This expression satisfies $Vir(d)$ with parameters $c_1 = 1$, $c_2 = 0$.
Finally, we can make the connection between locality and observer dependence. The off-shell representations of $Vir(1)$ act on quantized fields on the circle, and may therefore be viewed as one-dimensional Quantum Field Theory (QFT). In higher dimensions $Vir(d)$ act on quantized $p$-jet trajectories instead, so the theory deserves the name Quantum Jet Theory (QJT). $p$-jets (Taylor series) do not only depend on the function being expanded but also on the choice of expansion point, i.e. the observer's position.
The following diagram summarizes the argument:
Locality => Virasoro extension => $p$-jets => observer dependence.
