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.
What's missing in Quantum Gravity II: Observer Dependence
(This post was originally written in 2016, but I forgot to publish it).
In the previous post
I explained why locality in Quantum Gravity (QG) requires that the
spacetime diffeomorphism group acquires an extension upon quantization.
Now I will turn to a more physical viewpoint and argue that what also is
missing in all approaches to QG is a physical observer. At first sight
it may not be obvious that the two concepts have anything to do with
each other, but they are in fact closely related which I will discuss in
a later post. However, for the time being I will content myself with
the following trivial observation:
Every real experiment is an interaction between a system and an
observer, and the outcome depends on the physical properties of both. In
particular, the result depends on the observer's mass.
This may seem as an innocuous observation, until you realize that
neither General Relativity (GR) nor Quantum Field Theory (QFT) make
predictions that depend on the observer's mass. So clearly there must be
a more general, observer-dependent, theory that reduces to GR and QFT
in the appropriate limits. A little thought reveals that these limits
are:
- In GR, the observer's heavy mass is assumed to be zero, so the observer does not disturb the fields.
- In QFT, the observer's inert mass is assumed to be infinite, so the observer knows where he is at all times. In particular, the observer's position and velocity at equal times commute.
The assumption about the small heavy mass is very intuitive: if the observer had a large heavy mass, he would immediately collapse into a black hole, which is not what happens in a typical experiment. However, the assumption about the infinite inert mass requires some more explanation. Here my philosophy is completely operational: in order to know something, you must measure it. In particular, in order to know where he is, the observer must measure his position, e.g. with a GPS receiver. In theory, that can be done with arbitrary accuracy. However, in order to know where he will be in the future, the observer must also be able to measure his velocity at the same time, but Heisenberg's uncertainty principle tells us that there is a limit to the precision with which the position and the momentum can be simultaneously known.
More precisely, let \(\Delta x\) and \(\Delta v\) denote the uncertainties in the observer's position and velocity, and assume that momentum and velocity are related by \(p = Mv\) where \(M\) is the observer's mass. Then
\[
\Delta x \cdot \Delta v \sim \hbar/M,
\]
where \(\hbar\) is Planck's constant. Hence there are only two situations in which both the observer's position and velocity can be known arbitrarily well:
- If \(\hbar = 0\), i.e. in classical physics including GR. In this limit we can assume \(M = 0\).
- If \(M = \infty\), which only makes sense if gravity is turned off. In this limit we have QFT in flat space.
An analogous statement can be made about other interactions than gravity: field theory assumes that the observer's charge is zero and his inert mass is infinite. However, this double limit does not pose a problem for non-gravitational interactions, because charge and mass are unrelated. So even if non-gravitational physics depends on the observer's charge in principle, this dependence is merely an experimental nuisance that can be eliminated. In gravity, where charge and mass are the same, this nuisance becomes a conceptual problem.
Rovelli makes the distinction between two types of observables: partial observables, which can be measured but not predicted, and complete observables, whose time evolution can be predicted by the theory and which are subject to quantum fluctuations. In Quantum Mechanics, there are two types of partial observables: \(A\), the reading of the detector, and \(t\), time measured by a clock, which combine into the complete observable \(A(t)\). In QFT, there is a third type of partial observable: \(x\), the position measured e.g. by a GPS receiver, and the complete observables are of the form \(A(x,t)\). However, beneath this expression lies an unphysical assumption: that the pair \((x,t)\) is a partial observable that can only be measured but not predicted. A real observer obeys his own set of equations of motion, so his position at later times \(x(t)\) can be predicted. Let us now change notation and call this observable \(q(t)\) instead, because \(x\) will be reused below.
In a physically correct treatment, we combine the three partial observables of QFT into two complete observables: \(A(t)\) and \(q(t)\), the readings of the detector and GPS receiver at a certain tick of the clock. However, we immediately notice that a lot of information is gone: we no longer sample data throughout spacetime but only along the observer's trajectory. Fortunately, this is not as disastrous at it might seem at first glance, because the available local data includes not only the values of the fields \(A\) but also their derivatives \(d^m A/dx^m\) up to arbitrary order. We can assemble the local complete observables into a Taylor series,
\[
A(x,t) = \sum_m \frac{1}{m!} \frac{d^m A}{dx^m}(t) (x-q(t))^m
\]
This formula looks one-dimensional, but with multi-index notation it makes sense in higher dimensions as well. To the extent that we can identify an infinite Taylor series with the field itself, the original field has been recovered entirely, but with a twist: the Taylor series does not only depend on the field, but also on the expansion point \(q(t)\), an operator which we identify with the observer's position.
In the next installment in this series this expression will be used to make the connection to locality and the multi-dimensional Virasoro algebra from the previous post.
