What's Missing in Quantum Gravity IV: Objections

Today I sum up some of the more obvious objections to the arguments presented in the previous posts, and explain why these objections are not relevant.

1. The diffeomorphism algebra in $d$ dimensions has no central extension at all when $d>1$

This is correct, and would appear to be fatal blow to the existence of a multidimensional Virasoro algebra since the the ordinary Virasoro algebra is precisely the central extension of the diffeomorphism algebra on the circle. However, we saw in the previous post that the Virasoro extension in higher dimensions is not central, i.e. it does not commute with all diffeomorphisms. The Virasoro algebra in $d$ dimensions is essentially an extension of the diffeomorphism algebra by its module of closed $(d-1)$-forms. When $d=1$, a closed zero-form is a constant function, and the extension is a constant. In higher dimensions, a closed $(d-1)$-form does not commute with diffeomorphisms, but we still have a well-defined Lie algebra extension, not just a central one.

2. In QFT, there are no diff anomalies at all in four dimensions

Again this statement is correct and apparently fatal, because an extension of the diffeomorphism algebra is a diff anomaly by definition. However, the caveat is the phrase "in QFT". Virasoro-like extensions of the diffeomorphism algebra in four dimensions certainly exist, but they cannot arise within the framework of QFT. The reason is simple: as we saw in the previous post, the extension is a functional of the observer's trajectory $q^\mu(t)$, and since the observer does not explicitly appear in QFT, such a functional can not be written down within that framework. To formulate the relevant anomalies, a more general framework which explicitly involves the observer is needed.

3. Diff anomalies are gauge anomalies which are always inconsistent

In contrast to the first two objections, this statement is blatantly wrong. Counterexample: the free subcritical string, which according to the no-ghost theorem can be consistently quantized despite its conformal gauge anomaly. Of course, this does not mean that every kind of gauge anomaly can be rendered consistent; the free supercritical string and the interacting subcritical string are examples where the conformal anomaly leads to negative-norm states in the physical spectrum and hence to inconsistency. But the crucial condition is unitarity rather than triviality.

A necessary condition for a gauge anomaly to be consistent is clearly that the algebra of anomalous gauge transformation possesses non-trivial unitary representations. The Mickelsson-Faddeev (MF) algebra, which describes gauge anomalies in Yang-Mills theory, fails this criterion. It was shown by Pickrell a long time ago that the MF algebra has no "nice" non-trivial unitary representations at all. Therefore, Nature must abhor this kind of gauge anomaly, which of course is in agreement with experiments; gauge anomalies in the Standard Model do cancel. But this argument has no bearing on the situation where the algebra of gauge transformations does possess "nice" unitary representations.

4. Gauge symmetries are redundancies of the description rather than proper symmetries

This is only true classically, and quantum-mechanically in the absense of a gauge anomaly. A gauge anomaly converts a classical gauge symmetry into a ordinary quantum symmetry, which acts on the Hilbert space rather than reducing it. As an example we consider again the free string in $d$ dimensions. Classically, the physical dofs are the $d-2$ transverse modes, and this remains true in the critical case. In the subcritical case, however, there are $d-1$ physical dofs, because in addition to the transverse modes the longitudinal mode has become physical; time-like vibrations remain unphysical.

5. There are no local observables in Quantum Gravity

This is essentially the same objection as the previous one, and it assumes that there are no diff anomalies. There can be no local observables in a theory with proper diffeomorphism symmetry because the centerless Virasoro algebra has no nontrivial unitary representations. If the diffeomorphism algebra acquires a nontrivial extension upon quantization, there is no reason why local observables should not exist.

Note that the same argument applies to Conformal Field Theory (CFT). There are no local observables in a theory with infinite conformal symmetry, but that is not a problem in CFT because the relevant symmetry is not infinite conformal but Virasoro; the central charge makes a difference.

Barefoot dancer

Since the latest release of the DAZ-Blender importer I have mainly been tweaking materials. It is pretty important that the importer produces attractive materials automatically, because DAZ meshes contain lots of materials, and modifying all of them manually involves a lot of work. E.g., I think that a human uses some seventeen different materials for different parts of the body. This has some advantages in DAZ Studio, because you can easily add makeup by modifying the lip, eyelash, fingernail or toenail materials, but having so many materials does complicate editing afterwards in Blender

As a benchmark I will use the barefoot dancer is a standard product that ships (or used to ship, at least) with DAZ Studio. You can find pictures of her with Google, but since I don't know if it is legal to include other people's art on your blog (it is at least impolite), you have to look them up yourself.

So here is what I did. First I created to DAZ Studio (.duf) files, one containing the character in rest pose and the other the stage (Shaded haven). Then I created two Blender files. In the first I imported the stage, which required some tweaking because the importer does not always get object transformations right (working on this). Into the other I imported the character and merged the rigs (character, hair and each piece of clothing have separate rigs). Actually, every import was made twice, once with Blender Render being the render engine, in order to get the Blender Internal (BI) materials, and once with Cycles to get Cycles materials.

Finally, the set file was linked into the character file and the animation was loaded. I also loaded expression and chose something suitable, and created a simple three-point lighting setup. Here is a render using Blender Internal.

If you change the render engine to Cycles, the importer creates a Cycles material instead. Now, I have never really used Cycles and I don't understand how to control lights in it - it seems like most of the lighting comes from a world surface light whereas the actual lamps play a very marginal role. But even if the lighting is extremely dull, I think the materials look reasonably good.

The renders were made with the unstable version of the DAZ Importer, which can be downloaded from
https://bitbucket.org/Diffeomorphic/import-daz/downloads.

 

What's Missing in Quantum Gravity III: The Connection Between Locality and Observer Dependence

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.

What's missing in Quantum Gravity I: Locality

I'm planning to put up some pretty nice renders of characters imported from DAZ studio into Blender, but before doing that I will write about something that I have thought about for a long time and think that I have some rather unique insights about (and something that motivates the name of this blog): how to construct a quantum theory of gravity. For a more exhaustive discussion of the topic in this and the next post, see arXiv:1407.6378 [gr-qc].

In the beginning of the previous century, two great theories were found that describe all of physics: Einstein's General Relativity (GR), which describes gravity, and Quantum Theory (including Quantum Mechanics and Quantum Field Theory (QFT)), which describes everything else. The problem is that these theories seem to be mutually incompatible. But we know that Nature exists and hence cannot be inconsistent, so a consistent theory of Quantum Gravity (QG) has to exist, and it must reduce to GR and QFT in appropriate limits.

For almost a century many great minds have tried to solve this conundrum. The most popular approaches in recent decades have been string theory and, to a lesser extent, loop quantum gravity. However, these and other approaches all have problems of their own, especially since the LHC recently has ruled out supersymmetry beyond reasonable doubt (which, incidentally, is not the same thing as beyond every doubt).

My own proposal is to go back to basics, and simple postulate that QG combines the main properties of GR and QFT: gravity, quantum theory, and locality.

Now, this may seem as a very natural and innocent assumption, but it is in fact very controversial. The reason is that there is a well-known theorem stating that there are no local observables at all in QG, unless classical and quantum gravity have different sets of gauge symmetries. Actually, if you have seen this statement before, it was certainly without the caveat, but it must be mentioned because it is the weak spot of the no-go theorem. In order to prove this, one assumes that the group of all spacetime diffeomorphisms, which is the gauge symmetry of classical GR, remains a gauge symmetry after quantization.

So, we need something that converts spacetime diffeomorphisms from a classical gauge symmetry to an ordinary quantum symmetry, which acts on the Hilbert space rather than reducing it. This something is called an extension. It is well known that the diffeomorphism algebra (the infinitesimal version of the diffeomorphism group) on the circle admits a central extension called the Virasoro algebra. This is a celebrated part of modern theoretical physics. It first appeared in string theory, but later found an experimentally successful application in condensed matter, in the theory of two-dimensional phase transition.

So the diffeomorphism algebra in one dimension has an extension, but we know that spacetime has four dimensions (at least). So we need a multi-dimensional Virasoro algebra, which is a nontrivial extension of the diffeomorphism algebra in higher (and in particular four) dimensions. There are several arguments why such an extension cannot exist, and I will address those in a later post, but exists it does. In fact, there are even two of them, that were discovered 25 years ago by Rao and Moody and myself, respectively.

DAZ Importer

My 3D app is Blender, but I have played with DAZ Studio for quite some time, and found that it is a quite capable character creator. Since I have previously done something similar for the MakeHuman project, I wrote a Python script that imports native DAZ files (with file extensions .duf or .dsf) into Blender. In fact, the script is quite old and I have been using it privately for a long time, but recently I decided that it was time to complete it. The primary reason for writing this script is that I want to use it myself, but if somebody is interested it is now available for download.

The script repository is located at https://bitbucket.org/Diffeomorphic/import-daz

To download a zip file with the development version, go to  https://bitbucket.org/Diffeomorphic/import-daz/downloads.

There is also some documentation about the intended use at http://diffeomorphic.blogspot.se/p/daz-importer.html.

All assets needed to make the picture above were imported from DAZ Studio: the characters, the clothes, the poses, and the environment. The only manual intervention was to set up the lighting and make some minor tweaks to the woman's pose.

Hello World

Diffeomorphic is my new blog where I will collect my thoughts on physics, 3D graphics, history, languages, and other stuff that might interest me. Mostly but not exclusively in English.