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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