Diffeomorphic Add-ons Version 5.0.0 Released

Version 5.0.0 of the DAZ Importer, MHX Runtime System and BVH and FBX Retargeter have been released. They can be downloaded from

DAZ Importer: 
https://www.dropbox.com/scl/fi/8b5z3cy6kd9pcxeo17bh1/import_daz-5.0.0.zip?rlkey=udvj5mbw4f1psmvrnt8949ed5

MHX Runtime System: 
https://www.dropbox.com/scl/fi/0nf018loxjpy0nbe26zpr/mhx_rts-5.0.0.zip?rlkey=dd33350oazwagkund3puqt7up

BVH and FBX Retargeter: 
https://www.dropbox.com/scl/fi/0ftrenp005ub78ov5fek0/retarget_bvh-5.0.0.zip?rlkey=c27jhf4pxc3scpu0ffmil197d

The add-ons have been tested on Blender 3.6, 4.5 and 5.0. They should run on Blender versions from 3.0 onwards.

The main reason for this release is compatibility with Blender 5.0. Several other bugs have also been fixed.

Conclusion

The time has come to wrap up this series of posts. I went to grad school in the early 1980s, and specialized in phase transitions and critical phenomena, often working on 2D models. A few years later conformal field theory (CFT) came along, and essentially finished my field of research. I was too young and isolated to notice at the time, but when I did notice a few years later I felt that I had missed the bandwagon. CFT explains everything worth knowing about 2D critical phenomena, but it says nothing about the much harder and physically more relevant case of critical phenomena in 3D. So I started to think that something similar might work in higher dimensions as well. It can not be conformal symmetry, because the conformal group is infinite-dimensional only in 2D. But the same group also described diffeomorphisms in 1D, and that has a natural generalization to higher dimensions.

This made me decide that a multi-dimensional Virasoro algebra had to exist and I went out to find it. That it really does exist and has natural realizations on p-jets was very satisfying, but completing this task took a long time and I ran out of funding in the process. Moreover, it is only a limited success, because important pieces of the puzzle are still missing.

• There are operators acting on a vector space, but I haven't found a nice way to identify this space with its dual. So there is no known inner product and we can not discuss unitarity.
• The vector space is too big because momenta and velocities are not identified, which they should be.
• This is a purely kinematical theory, like tensor calculus. To use it in physics we should introduce dynamics in some form, with a Lagrangian, a Hamiltonian, or equations of motion. Doing this for p-jets is surprisingly difficult, or at least it was for me.

Because of these shortcomings I haven't been able to apply this theory to physics. Nevertheless, the math exists which made the effort worthwhile in itself, and I hope that somebody will make progress on the above problems one day.

But even if much remains to be done, some physics insights can still be extracted. The Virasoro extension depends on the observer's trajectory, and therefore we need a theory with an explicit physical observer. Since the observer's position is an operator, we must measure it to know its value, and that measurements is subject to quantum fluctuations. This is really enough to rule out every approach to quantum gravity which does not take horizontal fuzziness into account, because the lack of horizontal fuzziness amounts to a hidden assumption about an infinite mass.

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Four Dimensions?

Recall how we constructed representations of the multi-dimensional Virasoro algebra last week.

• Start from a classical representation acting on a tensor field, which corresponds to a representation of the general linear algebra \(gl(d)\).
• Make a Taylor expansion around the observer's location and truncate at order \(p\), i.e. pass to \(p\)-jet space.
• Consider trajectories in \(p\)-jet space.
• Introduce a vacuum state that is annihilated by all negative frequency modes, and normal order.

The "abelian charges", i.e. the coefficients in front of the extensions, depend both on the choice of \(gl(d)\) representation and on the trunctation order \(p\). They are polynomials in \(p\) of order \(d\), which means that they diverge in the limit \(p \to \infty\). This is not surprising. We passed to \(p\)-jets because the extension becomes infinite if we start from fields. By taking the limit \(p \to \infty\) we come back to the field we started from, insofar as an infinite Taylor series can be identified with the field itself, and the extension becomes infinite again.

Even if expected, having an infinite extension is undesirable and should be avoided. How to do that? We started from a bosonic field, but if we start from a fermionic field instead, we get the same extension but negative. So one way to avoid infinities is to have a perfect symmetry between bosons and fermions. However, such a symmetry is not seen in nature, and we saw yesterday that we lose locality if the total extension vanishes.

There is another way to cancel the leading terms, only leaving abelian changes that are independent of \(p\). Add more jets of order \(p-1\), \(p-2\) etc, both bosonic and fermionic. With a clever choice of field content, we can cancel all terms that depend on \(p\). For this to work in \(d\) dimensions, we need jets down to order \(p-d\). A similar hierarchy appears in gauge theory, where the lower-order jets would come from equations of motion and gauge conditions (continuity equations). Since those add derivatives, their jet order must be smaller, so the total number of derivatives does not exceed \(p\). So we start from

• Bosons, fermions, and gauge fields at order \(p\).
• Fermionic equations of motion at order \(p-1\).
• Bosonic equations of motion at order \(p-2\).
• Gauge conditions at order \(p-3\).

Then we can arrange the field content so that the divergent terms cancel if \(d=3\). We can further identify the time parameter \(t\) with one of the coordinates, and we naturally are led to the conclusion that the dimension of spacetime must be \(d+1 = 4\).

Unfortunately, a more detailed version of this argument seems to suggest things that disagree with observation, so the results are inconclusive. Nevertheless, I find it promising that the multi-dimensional Virasoro algebra seems to predict not only the number of spacetime dimensions, but actually the observed number of dimensions.

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Locality in Quantum Gravity

Today I argue that quantum gravity must be a local theory. However, a well-known theorem asserts that there are no local observables in quantum gravity. How can these statements be reconciled?

The thing to notice is that theorems have axioms. If one of the axioms is violated, the theorem does not apply. The relevant axiom in this case is the assumption that classical and quantum gravity have the same sets of gauge symmetries. If the group of spacetime diffeomorphisms acquires an extension upon quantization, that assumption fails. The multi-dimensional Virasoro algebra is that extension on the Lie algebra level.

Hence a theory of quantum gravity with local observables can not be a QFT. It must be a theory with horizontal fuzziness, i.e. QJT.

The same argument applies to theories with infinite conformal symmetry, because the infinite conformal group is isomorphic to the diffeomorphism group in one complex dimension. Hence there can be no local observables in a theory with infinite conformal symmetry. But this is not a problem in conformal field theory, because the relevant symmetry is not infinite conformal symmetry but rather the Virasoro symmetry with a nonzero central charge. 

If there are local observables, the symmetry group acts in a non-trivial way on them. A necessary condition for locality is thus that the symmetry group has nontrivial unitary representations. Indeed, the centerless Virasoro algebra does not have any of those, and hence there can't be any local observables if the central charge vanishes. If the central charge is nonzero there are many nontrivial representations, and hence local observables are possible. The same thing happens with gravity. If the spacetime diffeomorphism group does not have an extension, it remains a gauge symmetry and there are no local observables. When it has extensions described by the multi-dimensional Virasoro algebra, it becomes an ordinary symmetry and local observables are no longer ruled out.

Another way to see the need for locality is that correlation functions diverge in the same way when points coalesce, whether gravity is present or not. Correlation functions in flat space (ignoring gravity) typically diverge when spacetime points \(x\) and \(y\) approach each other in the following manner:
\[
<\phi(x) \phi(y)> \ \cong\ |x-y|^{-2\Delta}.
\]
The distance between the two points depends on the metric \(g_{\mu\nu}\):
\[
|x-y|^2 = g_{\mu\nu} (x^\mu - y^\mu) (x^\nu - y^\nu)
\]
In flat space the metric is the constant Minkowski metric. In the presence of gravity the physical metric is the gravitational field, and it is unclear what the distance would mean. However, we don't have to use the physical metric to define this distance; any metric would do. Of course the distance and hence the correlation function depends on the choice of unphysical metric, but the leading singularity does not. This is because the scaling dimension is the eigenvalue of the dilatation operator
\[
D = x^\mu {\partial \over \partial x^\mu}
\]
This is the contraction of an upper and lower index and hence independent of the metric. The scaling dimension only depends on the smooth structure. Since the correlation functions diverge in flat space, they must diverge in the presence of gravity as well.

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Causal order

The key lesson from general relativity is background independence. There is no classical background metric, but the metric is the gravitational field which participates in the dynamics. However, this poses problems for the Hamiltonian formulation of physics. 

In its simplest form, Hamiltonian physics deals with the phase space of positions and momenta at time \(t=0\), and the choice of a time coordinate makes use of the background metric. More generally, spacetime can be foliated into spacelike surfaces, but the notion of spacelikeness again requires a background metric. There is also a formulation that probably goes back to Lagrange which identifies phase space with the space of solutions to the equations of motions, but the background metric sneaks in even here. Such an orbit is typically specified by position and velocity at time \(t=0\). So orbits are parametrized by positions and velocities, but phase space is the space of positions and momenta. In order to identify the two we need to identify velocity and momentum, i.e. a contravariant vector with a covariant one. This again requires a background metric.

In QJT the problem goes away altogether, because there is no notion of spacelikeness. All objects in the theory are separated by a timelike distance, because everything lives on the observer's trajectory, both the Taylor coefficients \(\phi_m(t)\) and the expansion point \(q(t)\). This does of course not mean that points outside the observer's trajectory do not exist, but they are not covered by QJT because they can never be directly observed by the preferred observer. 

The same situation arises in mathematics. In order to make a Taylor expansion we must specify a field and an expansion point, i.e. the observer's position. We could pick a different expansion point, and there is a relation between the two sets of Taylor coefficients. But we must pick one, i.e. we must commit to one specific observer. All points on the observer's worldline are then causally ordered.

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Diff Anomalies in Four Dimensions

An extension of the diffeomorphism algebra is a kind of diff anomaly, but according to standard wisdom there are no diff anomalies at all in four dimensions. This is correct, within the framework of QFT without horizontal fuzziness. Hence a quantum theory of gravity must go beyond QFT.

To elaborate on this point, let us consider an analogous but simpler system, namely Yang-Mills theory in four dimensions. In the Hamiltonian formulation, the constraint algebra is the algebra of maps from 3-space to some finite-dimensional Lie algebra. As has been well known to experts since the 1980s [Pressley-Segal, Loop groups, chapters 4 and 9], this algebra of maps admits two qualitatively different extensions: the central extension and the Mickelsson-Faddeev extension. Let us contrast the two.

The Mickelsson-Faddeev extension
1. is proportional to the third Casimir invariant.
2. describes gauge anomalies in QFT.
3. treats all space points on an equal footing.
4. has no non-trivial unitary representations, at least not of lowest-energy type [Pickrell 1989].
5. is therefore a bad anomaly, which must not arise in nature.
6. and indeed, cancels in the standard model.

In contrast, the central extension
1. is proportional to the second Casimir invariant.
2. does not arise within the framework of QFT without horizontal fuzziness.
3. does not treat space points equally, but depends on a privileged one-dimensional curve, "the observer's trajectory".
4. has many non-trivial unitary representations.
5. is therefore a good anomaly, which cannot be ruled out on the basis of unitarity.

The multi-dimensional Virasoro algebra is the diffeomorphism analogue of this central extension, which we may also call the multi-dimensional affine algebra.

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Gauge Anomalies and Consistency

The diffeomorphism algebra is the gauge symmetry of classical general relativity. An extension thereof is hence a kind of gauge anomaly, which according to conventional wisdom is fatal for the quantum theory. However, the claim that gauge anomalies are always inconsistent is wrong. Counterexample: the free subcritical string, which can be quantized with a ghost-free spectrum despite its conformal gauge anomaly.

Of course, this does not mean that every theory with a gauge anomaly is consistent. Some are (free subcritical string), others are not (free supercritical string, interacting subcritical string). But if the anomalous theory is consistent, some classical gauge symmetry becomes an ordinary quantum symmetry, which acts on the Hilbert space rather than reducing it (here: conformal symmetry). Conversely, some classical gauge degrees of freedom become physical after quantization (trace of the worldsheet metric).

So the crucial property is unitarity, not triviality. 

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Third Week

The first week of this series centered around the idea of horizontal fuzziness. This is really about taking quantum mechanics seriously. Quantum fluctuations do not only apply to measurements inside a detector (vertical fuzziness), but also to the measurement of the detector's location (horizontal fuzziness). Ignoring horizontal fuzziness amounts to a hidden assumption about an infinitely massive detector, which does not work in the presence of gravity.

The second week I described the multi-dimensional Virasoro algebra, i.e. the Virasoro-like extensions of the diffeomorphism algebra in \(d\) dimensions. The classical representations act on tensor fields, but that is not a good start for quantization, because normal ordering gives rise to infinite extensions due to unrestricted sums over transverse directions. Instead we must start from \(p\)-jets, i.e. Taylor series truncated at order \(p\). Since a Taylor series depends not only on the field being expanded, but also on the choice of expansion point, we naturally have a horizontal observable that displays fuzziness.

The multi-dimensional Virasoro algebra is not really physics, since it only involves kinematics but not dynamics. It is more like a quantum version of tensor calculus rather than a quantum version of gravity. Nevertheless, already at this level there are some consequences which are very much in disagreement with conventional wisdom. This is what we will discuss during this third and final week.

• Gauge Anomalies and Consistency
• Diff Anomalies in Four Dimensions
• Causal Order
• Locality in Quantum Gravity
• Four dimensions?
• Conclusion

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Multi-dimensional Virasoro Algebra and Jets

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.

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Off-shell Representations

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.

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Geometrical Formulation

Previously we introduced the multi-dimensional Virasoro algebra in a Fourier basis on the \(d\)-dimensional torus, and showed that trying to build representations in the same way as in one dimension fails. Today I will describe a more geometrical formulation which will make the connection to QJT explicit.

Infinitesimal diffeomorphisms are generated by vector fields, which are locally of the form
\[
\xi = \xi^\mu\partial_\mu 
= \sum_{\mu=0}^{d-1} \xi^\mu(x) {\partial\over\partial x^\mu},
\]
where again we use Einstein's summation convention: two indices, one up and one down, are implicitly summed over. The bracket between two vector fields is
\[
[\xi, \eta] = \xi^\mu \partial_\mu \eta^\nu \partial_\nu -
\eta^\nu \partial_\nu \xi^\mu \partial_\mu.
\]
The diffeomorphism algebra, or algebra of vector fields, is the Lie algebra generated by the Lie derivatives \({\cal L}_\xi\):
\[
[{\cal L}_\xi, {\cal L}_\eta] = {\cal L}_{[\xi,\eta]}.
\]
In this formalism, the multi-dimensional Virasoro algebra takes the form in the blog banner.
\[\begin{align}
[{\cal L}_\xi, {\cal L}_\eta] &= {\cal L}_{[\xi,\eta]}
 + {1\over{2\pi i}}\int dt\ \dot q^\rho(t)
 \Big( c_1\ \partial_\rho\partial_\nu\xi^\mu(q(t))\ \partial_\mu \eta^\nu(q(t))\ +\\
& \qquad\qquad\qquad +\ c_2\ \partial_\rho \partial_\mu \xi^\mu(q(t))\ \partial_\nu \eta^\nu(q(t)) \Big),\\
[{\cal L}_\xi, q^\mu(t)] &= \xi^\mu(q(t)), \\
[q^\mu(t), q^\nu(t')] &= 0. 
\end{align}\] 
This is strictly speaking not a Lie algebra, because \(q\) appears non-linearly in the right-hand side. But we can extend these brackets to arbitrary functionals of \(q\), and then they define a proper Lie algebra.

To verify that this is a Lie algebra (anti-symmetry and Jacobi identities), we integrate by parts and throw away the boundary terms. Hence we must also add the condition that the integral of a total derivative vanishes.
\[
\int dt {dF(t)\over dt} \equiv 0.
\]
This is the case if the integral runs over a circle, or if we only consider vector fields which decrease sufficiently fast when \(t \to \pm\infty\). 

To connect this formulation with the previous one, we note that a basis for the vector fields on the \(d\)-torus is given by the plane waves \(i\exp(im\cdot x) \partial_\mu\). Define
\[\begin{align}
L_\mu(m) &= {\cal L}(i\exp(im\cdot x) \partial_\mu) \\
S^\mu(m) &= {1\over{2\pi}}\int dt\ \dot q^\mu(t) \exp(im\cdot q(t)) 
\end{align}\] 
These operators indeed satisfy the multi-dimensional Virasoro algebra in the Fourier basis. In particular, the vanishing of total derivatives leads to the condition
\[
m_\mu S^\mu = 
{-1\over{2\pi i}} \int dt\ {d\over dt}\Big(\exp(im\cdot q(t)) \Big) 
\equiv 0.
\]
We used the same letter \(q\) here as we did for the expansion point in the Taylor series. This is not a coincidence, and we will show later that they are the same. Hence the observer's position appears already in the very definition of the multi-dimensional Virasoro algebra, which explains why we need QJT to construct representations.

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Divergencies

Representations of the algebra of diffeomorphisms on the circle is essentially QFT on the circle. It would be natural to think that the same is true in higher dimensions, but this is not the case. If we try to repeat the recipe used in one dimension, we formally arrive at an infinite central charge, due to an unrestricted sum over transverse directions. This is of course a nonsense result, and it stopped physicists, myself included, from going further. Normal ordering only works if we start from a classical representation acting on finitely many functions of a single variable, because then there are no transverse directions that cause infinities. In later posts I will show how to do that within QJT.

First we review the construction of representations of the Virasoro algebra in one dimension. Let us start with the Heisenberg algebra:
\[\begin{align}
[\phi_m, \pi_n] &= i\delta_{m+n}, \\
[\phi_m, \phi_n] &= [\pi_m, \pi_n] = 0,
\end{align}\]
where \(m\) and \(n\) are integers. The algebra of diffeomorphisms on the circle can be embedded in this algebra in the following way.
\[
L_m = i \sum_{n=-\infty}^\infty (m-n) \phi_{m-n} \pi_n
\]
The brackets with the elements of the Heisenberg algebra read
\[\begin{align}
[L_m, \phi_n] &= (m+n) \phi_{m+n}, \\
[L_m, \pi_n] &= n \pi_{m+n},
\end{align}\]
which are the transformation laws of a scalar field and a scalar density, respectively.

Embedding the diffeomorphism algebra into the Heisenberg algebra means that a representation of the latter gives rise to a representation of the former. Introduce a vacuum \(|0>\) which annihilates all negative modes.
\[
\phi_{-m} |0> = \pi_{-m} |0> = 0,
\]
for all \(m > 0\).  This does not quite give us a representation of the diffeomorphism algebra, because the vacuum energy is infinite.
\[
L_0 |0> = -i\sum_{n=1}^\infty n \phi_{-n} \pi_n |0>
= \sum_{n=1}^\infty n |0>.
\]
To avoid this problem we redefine \(L_0\) so that \(L_0 |0> = 0\). 
\[
L_0 = i \sum_{n=1}^\infty n 
  \Big( \phi_{n} \pi_{-n} - \pi_{n} \phi_{-n}\Big)
\]
This is called normal ordering, because it is equivalent to moving the infinitely many negative modes to the right. However, this changes the algebra that the generators \(L_m\) satisfy, which can be seen by calculating the bracket
\[
[L_m, L_{-m}] |0> = -\sum_{n=1}^{m-1} n(m-n)|0> = -{(m^3-m)\over6}|0>
\]
Comparing with the definition of the Virasoro algebra in the previous post we see that normal ordering leads to a central charge \(c = 2\). The central charge is positive because the construction was made with bosonic fields. Repeating the analysis with fermionic fields would lead to  \(c = -2\) instead.

Now let us try to repeat the analysis in two dimensions. The indices now become 2D integer vectors \(m = (m_0, m_1)\), and the diffeomorphism generators can be embedded in a Heisenberg algebra like this:
\[
L_\mu(m_0,m_1) = i \sum_{n_0=-\infty}^\infty 
\sum_{n_1=-\infty}^\infty (m_\mu-n_\mu) \phi(m_0-n_0,m_1-n_1) \pi(n_0,n_1)
\]
To build a representation we must introduce a vacuum which is annihilated by half the modes, e.g. those with \(m_0 < 0\). Again this leads to an infinite vacuum energy, which can be removed by moving modes with \(m_0 < 0\) to the right. But this kind of normal ordering goes wrong, because we have to reorder infinitely many terms. E.g., when we compute \([L_0(m_0,m_1), L_0(-m_0,-m_1)] |0>\) we formally get the central charge
\[
\sum_{n_0=1}^{m_0} \sum_{n_1=-\infty}^\infty n_0(m_0-n_0) = K {(m_0^3-m_0)\over6},
\]
where
\[
K = \sum_{n_1=-\infty}^\infty 1 = \infty.
\]
So the would-be central charge is infinite, due to the unrestricted sum over transverse modes. An infinite central charge is of course a sign that something has gone very wrong. Also, the diffeomorphism algebra in \(d > 1\) dimension does not have a central extension, because it would not be compatible with the Jacobi identities.

The conclusion is that representations of the Virasoro algebra in \(d > 1\) dimensions can not be built in the same way as we do when \(d=1\). This is because the natural arena is not QFT but QJT.

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