Measurements in GR

Thread view:

Enable EMail Alerts

Start New Thread

Thread rating:

Gen Zhang - 09 Aug 2007 23:58 GMT

As is often the case, I've realised something obvious, but don't know
where to head next -- so naturally I'm asking on
sci.physics.research... Essentially, I'm wondering about what we're
really measuring in GR, and whether they make sense in a quantum
situation. I'm going to ramble for a bit to explain what I know, and
I'll ask a question at the end -- so I beg for a little patience.

My understanding of quantum measurements goes something like this:

We have our system of interest, with state vector x in the space X. x
evolves unitarily. We bring it into interaction with our ancilla or
measuring apparatus, which exists in Y, initially with state y. So the
combined state is xy (tensor product) in space X*Y (again, tensor
product). The combine state evolves unitarily, until we decouple the
two systems. Now we peek at our ancilla and get the result. From the
point of view of just the system X, a non-unitary projection has
occurred. I've shortened the chain of interactions somewhat, since in
practice we need to have quantum interactions all the way to
ourselves.

The important thing is that although we use the mathematics of linear
operators, and conceptually call it a collapse, what we've really done
is to introduce a measure apparatus which had to physically interact.
In the classical limit, our ancilla (hopefully) doesn't affect things
too much, and our little abstraction works so well that we can neglect
the fact that something had to interact for there to be a measurement.

Now, classically in GR we care about the metric, and measurements of
said metric is thus our primary concern. I believe it's a known
problem that actually we can't always measure the metric in vacuum --
we need to introduce some matter, at which point things usually become
frighteningly simple (at least conceptually, numerically it's usually
a complete pain).

But can we actually ever measure the metric when we don't have matter?
Perhaps the situations where we seem to be able to are actually just
where we've accidentally hit upon some limit where a line can be
drawn, and we can magic away our matter, and pretend that we've
directly measured the metric?

So obviously, we wonder about the quantum case. Most quantum efforts
have been focused on matter-free situations, as there are technical
advantages to do so. However, some of the quantum measurements leave
me a little uncertain. For instance, LQG has a framework which results
in the quantisation of area and volume. Conceptually promising, since
it leads to a nice theory about the entropy of black holes; but I
wonder what the area operator is really measuring? To state a simple
and naive objection, to measure geometrical scales that small, we'd
need energies on the Planck order. Is it really possible to neglect
the gargantuan amounts of energy in this case, and pretend that we're
in flat space? Put another way: I've heard people say that perhaps at
Planck scale, there isn't a good clock, so the idea of time is ill-
defined -- perhaps there's not really a good ruler too?

So I guess I'd like to read more about work where matter is considered
with gravity, and not just in a region where perturbative expansions
still work. Although string theory purports to have everything, it
doesn't have a background-free structure yet. LQG has the background
independence, but so far it's just got the background.

So the question: does anyone know of the any work, where matter is
explicitly used to measure gravity, on the Planck scale?

Thanks in advance, and for your indulgence,
Gen Zhang

Reply to this Message

thomas_larsson_01@hotmail.com - 11 Aug 2007 03:25 GMT

> Now, classically in GR we care about the metric, and measurements of
> said metric is thus our primary concern. I believe it's a known
> problem that actually we can't always measure the metric in vacuum --
> we need to introduce some matter, at which point things usually become
> frighteningly simple (at least conceptually, numerically it's usually
> a complete pain).

I think that Rovelli's GPS coordinates
http://www.arxiv.org/abs/gr-qc/0110003 use a minimal amout matter:
four GPS satellites, modelled as particles. An observer can
measure the four components of the spacetime location of an event
by reading his GPS receiver, so these are gauge-invariant, physical
coordinates.

> So obviously, we wonder about the quantum case. Most quantum efforts
> have been focused on matter-free situations, as there are technical
> advantages to do so. However, some of the quantum measurements leave
> me a little uncertain.

Any physical experiment is an interaction between a detector and a
system, and the outcome of the experiment depends on the physical
properties of both. We typically want to eliminate the detector
dependence from the results as far as possible, but not further than
that. I claim that in the presence of gravity, one detector
property can not be eliminated: its mass M.

Starting from the more fundamental detector-system physics, we could
eliminate M by going to one of two limits: M -> 0 or M -> infinity.
If M -> 0, we have no clue about the future location of the
experiment: the commutator between the detector's position and
velocity is proportional to hbar/M. If M -> infinity, the detector
will interact strongly with gravity and collapse into a black hole.
Neither alternative is realistic, and a theory of quantum gravity
must therefore depend on the observer's mass.

However, if we ignore gravity, we can set M = infinity. The detector
will then just sit still and detect, and we can forget about it.
This limit should be described by QCD. OTOH, if we put hbar = 0 and
then let M -> 0, the detector will be a test particle which moves
along classical geodesics without disturbing the gravitation field;
this is GR.

>From this perspective, it is obvious why QFT and GR are

incompatible; they correspond to different limits for the
detector's mass. Moreover, any theory which does not explicitly
introduce the detector's mass, be it a theory of fields, particles,
strings, branes, loops or whatnot, can not be the correct theory of
quantum gravity.

> So the question: does anyone know of the any work, where matter is
> explicitly used to measure gravity, on the Planck scale?

I'm working on a formulation of QFT where the degrees of freedom
are the physical ones measured by a real, local detector: the
detector's worldline (measured e.g. by GPS coordinates) and the
values of the fields and its mixed partial derivatives inside the
detector (i.e. on its worldline). The detector is material in the
sense that its mass enters its equations of motion. Earlier
attempts in this direction can be found in the arxiv, e.g.
http://www.arxiv.org/abs/hep-th/0701164, but unfortunately the
formalism was then very cumbersome.

Reply to this Message

Gen Zhang - 15 Aug 2007 16:28 GMT

On Aug 11, 3:25 am, thomas_larsson...@hotmail.com wrote:

> > Now, classically in GR we care about the metric, and measurements of
> > said metric is thus our primary concern. I believe it's a known
[quoted text clipped - 8 lines]
> by reading his GPS receiver, so these are gauge-invariant, physical
> coordinates.

That is actually the exact model I had in mind...

> > So obviously, we wonder about the quantum case. Most quantum efforts
> > have been focused on matter-free situations, as there are technical
[quoted text clipped - 30 lines]
> strings, branes, loops or whatnot, can not be the correct theory of
> quantum gravity.

That is an exceptionally nice way to think about it -- I wonder why
this view isn't more prevalently known? (I'm only an undergrad, so
I've got a fairly limited view of what the current frontiers are --
mostly from reading papers that I can only half understand and JB's
weekly finds, which are usually a only a little better.)

> > So the question: does anyone know of the any work, where matter is
> > explicitly used to measure gravity, on the Planck scale?
[quoted text clipped - 7 lines]
> attempts in this direction can be found in the arxiv, e.g.http://www.arxiv.org/abs/hep-th/0701164, but unfortunately the
> formalism was then very cumbersome.

I've tried reading that paper, and to be honest it is very much beyond
me. My current mathematical level is fairly low -- simple differential
geometry and elementary linear algebra, I'm afraid I'm only a
physicist, and not a maths student. Still, the basic idea I do grasp,
even if I don't even understand the technical issues that you're
trying to solve.

There is one thing that has been bothering me -- Rovelli, in his book
and elsewhere, go to great lengths to create a manifestly covariant
Hamiltonian approach to classical mechanics, and even to quantum
mechanics. The classical particle and field theories are nice and
simple, and the quantum particle theory is pretty straightforward.
However, having never formally been taught QFT, I can't follow all the
technical details of his covariant quantum field theory, so I can't
really make any sensible comments. The puzzling point is that he then
completely ignores all this work, and go on to describe LGQ, in its
3+1 foliated form. Still -- I found the formalism that he used to be
much more understandable, in fact much more so than the Dirac version
of constrained Hamiltonian mechanics with the different classes of
constraints and the sophisticated algebra that then results. I'm going
to try and see if your "jets" can be cast in that language.

Thanks for the pointers, and hopefully you won't mind me stalking your
work from now on ;-)
Gen Zhang

Reply to this Message

thomas_larsson_01@hotmail.com - 19 Aug 2007 20:38 GMT

> Thanks for the pointers, and hopefully you won't mind me stalking your
> work from now on ;-)

Thank you. I could certainly do with a few stalkers. Let me say a
few words, mainly because it is to my own benefit to put my
thoughts on print.

I imagine that there are three kinds of detectors in a physical
experiment: a clock which measures time t, a GPS receiver which
measures position q, and a detector which measures the field and
finitely many derivatives f_m. From the partial observables t, q,
f_m we can form two types of complete observables, q(t) (position
at time t) and f_m(t) (field and derivatives at time t). Being
complete observables, these are subject to quantum fluctuations and
thus given by self-adjoint operators.

From these operators one can construct the composite operator

f(x,t) = sum_m 1/m! f_m(t) (x-q(t))^m

which behaves as a quantum field. However, this field is a secondary
construct - it is the Taylor data f_m(t) and q(t) which can be
measured in a physical, local detector. In contrast, the device
needed to measure the field itself is much more complex. Since a
single detector can only measure the field at a single point on a
simultaneity surface - namely where the detector's worldline
intersects the surface - we need an array of detectors with
synchronized clocks. Hence measurements in QFT is a mess.

What makes my setup different from QFT is that I assume that also
the detector has dynamics; it is natural to assume that

M d^2q/dt^2 = -eF

where M and e are the detector's mass and charge and F is a force
coming from the fields. Moreover, q(t) is an operator-valued curve;
the reading of the GPS receiver is subject to quantum fluctuations.
IOW, the detector's position and velocity do not commute at equal
times:

[q, dq/dt] = i hbar/M.

One expects to recover QFT in the limit e -> 0 (so we can ignore
the backreaction of the observer on the fields) and M -> infinity
(so q(t) becomes a classical curve without fluctuations). There
are no obstructions to this limit except for gravity, where the
detector satisfies a geodesic equation

M d^2q/dt^2 = - M Gamma dq/dt dq/dt.

Thus for gravity e = M, and it is impossible to take the joint
limit e -> 0, M -> infinity.

So this is what I think is the problem with quantum gravity: QFT
has to be amended with physical, local detector whose position is
subject to quantum fluctuations, because that is what we have in
every real experiment.

Reply to this Message

Igor Khavkine - 13 Aug 2007 00:48 GMT

> My understanding of quantum measurements goes something like this:
>
[quoted text clipped - 28 lines]
> drawn, and we can magic away our matter, and pretend that we've
> directly measured the metric?

The situation in GR is very similar to what you described for quantum
theory. You can imaging a measurement apparatus that is itself part of
the theory and weakly interacts with the space-time geometry. The
measurement is taken in the limit where the influence of the apparatus
itself is negligible. The results of such measurements can be
mathematically computed without ever formally introducing the measuring
device itself. This is nothing but the age old "test particle"
idealization. Conceptually, in quantum gravity, you'd have to combine
the measurement idealizations discussed for both quantum mechanics and
for GR.

> So obviously, we wonder about the quantum case. Most quantum efforts
> have been focused on matter-free situations, as there are technical
[quoted text clipped - 9 lines]
> Planck scale, there isn't a good clock, so the idea of time is ill-
> defined -- perhaps there's not really a good ruler too?

Here's a sketch of what the area and volume operators measure in LQG.
LQG states are constructed in several steps. At one point one constructs
states describing all possible metrics on a manifold. Recall, however,
that two metrics on the same manifold that can be related by a
diffeomorphism are considered equivalent. This means that the sate space
constructed so far is too large, it distinguishes between different
metrics that define the same geometry. So, the next step is to quotient
this state space by diffeomorphisms of the manifold, the result is the
so-called diff-invariant state space.

Before quotienting by diffeomorphisms, states correspond to graphs
embedded in the manifold, embedded spin networks. At the same time, one
can also embed surfaces into the manifold. These embedded graphs and
surfaces may intersect. The number of punctures (and some other
information adorning the graph) of the surface by the spin network
determines its area. It is this area operator that has discrete
spectrum. Upon quotienting by diffeomorphism, embedded spin networks
loose almost all information about their embedding and become abstract
graphs, (abstract) spin networks. Similarly, an embedded surface also
loses almost all information about its embedding. However, the relative
positions of a surface and a spin network should be preserved, including
which links puncture the surface. So, given a spin network describing
some geometry, one can define surfaces in it by simply specifying the
links that a surface intersects.

The argument for the volume operator is very similar. For more precision
and detail, you can look at Rovelli's book on Quantum Gravity, namely
Chapter 6 and Section 6.7.

> So I guess I'd like to read more about work where matter is considered
> with gravity, and not just in a region where perturbative expansions
> still work. Although string theory purports to have everything, it
> doesn't have a background-free structure yet. LQG has the background
> independence, but so far it's just got the background.

There are many proposals for coupling matter to LQG or spin foams. You
can find some of them with Google. However, to my knowledge, none of
them are very mature (with the exceptional case of 3D gravity).

I don't know if any of them are good candidates for constructing
hypothetical Planck scale rulers or clocks. However, I wouldn't be too
surprized that one can construct (in principle) Planck scale measuring
devices using some external few-level quantum system coupled directly to
the relevant geometric operator, even though such a construction would
look extremely artificial.

Hope this helps.

Igor

Reply to this Message

Natural Science Forum / Physics / Research / August 2007

Measurements in GR