Quantum gravity confusion redux

I.Vecchi

Googling around I recently ended up on a beautiful old spr thread
([1]) about QG, which I found extremely interesting both for its
scientific content and for the hopeful mood it conveys (the overall
mood in QG is not really hopeful these days, is it? My perception is
that it changed quite suddenly around one year ago).

The thread also contains a vivid exemplification of the semantic
problems arising in GR ([2],cf.[3]), which I like to juxtappose to the
following recent Zeilinger ([4]) quote:

"Physics will in the future put less emphasis on equations and
mathematics but more on verbal understanding".

Scary, huh?

IV

[1]
http://groups.google.com/group/sci.p...520204717c753c
[2]
http://groups.google.com/group/sci.p...a0bb6897aa8ccc
"... there might be two U. C. Riversides with clock towers, or U. C.
Riversides with dubious entities that we might or might not consider a
clock tower."
[3]
http://groups.google.com/group/sci.p...ece53891b06e6d
[4] http://xxx.lanl.gov/abs/quant-ph/0505187 , quoted by Peter Zoller.

Eugene Stefanovich

I.Vecchi wrote:
> Googling around I recently ended up on a beautiful old spr thread
> ([1]) about QG, which I found extremely interesting both for its
> scientific content and for the hopeful mood it conveys (the overall
> mood in QG is not really hopeful these days, is it? My perception is
> that it changed quite suddenly around one year ago).
>
> The thread also contains a vivid exemplification of the semantic
> problems arising in GR ([2],cf.[3]), which I like to juxtappose to the
> following recent Zeilinger ([4]) quote:
>
> "Physics will in the future put less emphasis on equations and
> mathematics but more on verbal understanding".
>
> Scary, huh?
>
> IV
>
> [1]
> http://groups.google.com/group/sci.p...520204717c753c
> [2]
> http://groups.google.com/group/sci.p...a0bb6897aa8ccc
> "... there might be two U. C. Riversides with clock towers, or U. C.
> Riversides with dubious entities that we might or might not consider a
> clock tower."
> [3]
> http://groups.google.com/group/sci.p...ece53891b06e6d
> [4] http://xxx.lanl.gov/abs/quant-ph/0505187 , quoted by Peter Zoller.
>

Thank you for the fascinating read. It strikes me how the
idea of GR spacetime is similar to long forgotten ideas of the
electromagnetic ether. In both cases we have some all-penetrating
"substance" that affects physical objects, but cannot be detected
itself. GRist will tell you that the "spacetime as rubber sheet" analogy
popularized in documentaries is an oversimplification, but I have
a feeling that this analogy plays a major role in their thinking
process. They talk about spacetime being bent, curved, twisted,
"foamed", even torn apart; they speak about including spacetime
topologies in path integrals as if these were different states of some
real object. John Baez talks (though jokingly) about applying
Newton's third law to the spacetime (matter must act on spacetime just
as spacetime is "acting" on matter).

On the other hand, the spacetime idea is conspicuously absent from the
vocabulary of quantum mechanics (or QFT). It seems that x and t
arguments of quantum fields psi(x,t) have no direct relationship to
the measurable time and positions of particles/events. After all,
when we calculate something related to experiment, like the Hamiltonian
or the S-matrix, these arguments get integrated out. In the Hilbert
space of quantum
mechanics we do have the observable of particle position R,
but it is not
distinguished among other observables, like momentum, energy, spin, etc.
(Should I mention that there is no time operator in quantum mechanics?)
The eigenvalues of R are triples of real numbers, but so are other
eigenvalues of mutually commuting operators: momentum, or
energy/spin-squared/spin-z. There is no more (and no less) sense
in combining triples (x,y,z) into some kind of manifold or continuum
than in considering "momentum space" or "energy-angular momentum space"
or any number of other "spaces". Can we introduce the "curvature"
of the "momentum space"?

It seems to me that quantum mechanics rejects the idea of the
preferred status of the position observable, and/or spacetime continuum.
I find rather odd the attempts to quantize gravity by including "states"
of the GR "rubber sheets" into QM Hilbert space. IMHO,
space (or spacetime)
is not an observable object, it can't be measured, there are no
observables associated with it, so it has no place in the Hilbert space.

If there is one lesson we can learn from quantum mechanics this lesson
should be: "never ask questions about something you are not measuring"
(e.g., whether the electron passed through the left slit or through the
right slit?). Since spacetime curvature is something we cannot directly
observe, I think our life would be much easier if we just abandon the
attempts to describe how the spacetime "looks like", and focus
instead on directly measurable observables of real particles or
systems of particles.

I anticipate your objections about how wonderful are predictions of
GR regarding light bending or binary pulsars. However, successful
predictions of QM and QFT are much broader in scope and few orders
of magnitude more accurate. So, I would place my bet on QM rather than
on GR.

Eugene.

P.S. I have zero credentials in quantum gravity research, so my
thoughts can be safely ignored.

John Baez

In article <1131438756.304268.219710@g43g2000cwa.googlegroups.com>,
I.Vecchi <vecchi@weirdtech.com> wrote:

>Googling around I recently ended up on a beautiful old spr thread
>([1]) about QG, which I found extremely interesting both for its
>scientific content and for the hopeful mood it conveys (the overall
>mood in QG is not really hopeful these days, is it? My perception is
>that it changed quite suddenly around one year ago).

I remember the thread you mention:

http://groups.google.com/group/sci.p...520204717c753c

and it was indeed a lot of fun. It concerned a key conceptual
problem of quantum gravity: the problem of describing quantum
systems without using a fixed background metric on spacetime.

This conceptual problem is largely ignored in string theory,
but people working on other approaches to quantum gravity have
thought about it for many years:

A. Ashtekar and J. Stachel, Editors; conceptual Problems of
Quantum Gravity. Proceedings of the 1988 Osgood Hill Conference
(Birkhauser, N. Y., 1991), 602 pages.

and eventually a good rough understanding was worked out.

The problem was then to take this understanding and use it to
come up with a specific theory of quantum gravity! All attempts
to do this raise difficult "technical" challenges - "technical"
in the sense that they involve a bit less verbal reasoning, and
a lot more detailed math.

Loop quantum gravity was the first theory to meet a bunch of
these technical challenges. It did so starting in the late
1980s, and into the mid 1990s. This is perhaps responsible
for the optimistic mood you detect.

Loop quantum gravity moved on to the next stage: the problem of
"getting the right semiclassical limit". This means showing the
theory gives results that match those of general relativity at
length scales far larger than the Planck scale. Doing this is
a really crucial basic test for any theory of quantum gravity.
It's preliminary to the really fun part: getting *new* predictions.

This is where things bogged down and people became less optimistic:

http://math.ucr.edu/home/baez/dynamics/

The story in string theory is very different: they made progress
on different questions and became bogged down in different ways.
The mood in string theory is also not very optimistic these days.
But that's another story.

People in modern urban societies tend to be very impatient. They
think of 10 or 20 years as a long time. So, they find it upsetting
to imagine that it might take a century or more to solve the problem
of quantum gravity. I think it's good to relax a bit. The problem
of quantum gravity is not going away; the only danger is that we'll
destroy ourselves or lose interest in science before we solve this
problem. If we manage to keep our society going and stay interested
in science, I feel sure we'll make serious progress on quantum gravity -
eventually. And even if *we* don't, *someone* probably will: there's
lots of room out there.

So, when I get myself in a sufficiently high-level frame of mind,
surveying the whole history of the universe and imagining its
immensity, I'm very optimistic about the problem of quantum gravity.
We (the people reading this now) probably won't be around to see it
solved, but at least we can have fun trying. That should be enough.

Hontas Farmer

Eugene Stefanovich wrote:

> If there is one lesson we can learn from quantum mechanics this lesson
> should be: "never ask questions about something you are not measuring"
> (e.g., whether the electron passed through the left slit or through the
> right slit?). Since spacetime curvature is something we cannot directly
> observe, I think our life would be much easier if we just abandon the
> attempts to describe how the spacetime "looks like", and focus
> instead on directly measurable observables of real particles or
> systems of particles.
>
The mass of a particle can be taken as a direct measure of the "curvature of
spacetime" due to that particle. Using the tired old rubber sheet analogy
the deeper a dent a particle is in the harder it will be to move, it will
have more inertia. Mass is a measure of inertia and a curvature tensor is
a mathematical expression of inertia.


> I anticipate your objections about how wonderful are predictions of
> GR regarding light bending or binary pulsars. However, successful
> predictions of QM and QFT are much broader in scope and few orders
> of magnitude more accurate. So, I would place my bet on QM rather than
> on GR.
>
GR is a classical theory of course it is going to be less precise. There is
no argument that QM is more fundamental. That QM is more fundamental is
the reason a theory of quantum gravity is so sought after.

> Eugene.
>
> P.S. I have zero credentials in quantum gravity research, so my
> thoughts can be safely ignored.

--
"...we advanced from the telegraph to telephones to e-mail?" -Mr ef'n
conductor- George Carlin. www.geocities.com/hontasfx

thomas_larsson_01@hotmail.com

I.Vecchi skrev:

> Googling around I recently ended up on a beautiful old spr thread
> ([1]) about QG, which I found extremely interesting both for its
> scientific content and for the hopeful mood it conveys (the overall
> mood in QG is not really hopeful these days, is it? My perception is
> that it changed quite suddenly around one year ago).
>

My own heretical idea is that you can learn something about
QG in 4D by trying to generalize the key aspects of QG in
2D, which are well understood. Actually, 2D QG proper is
quite boring, because the Einstein action in 2D is a
topological invariant - the Euler characteristic. However,
we are really interested in QG coupled to other fields,
e.g. the standard model. The D-dimensional free bosonic
string in the Polyakov formulation is nothing but 2D
gravity coupled to scalar fields, and thus it may be
regarded as the simplest possible toy model of gravity.

To me, the key lesson from 2D gravity is something that
both string theory and LQG has missed:
______________________________________________________
| |
| Some gauge components of the metric become physical |
| after quantization. |
|______________________________________________________|

In classical 2D gravity, the metric is purely gauge,
because it has 3 components and there are 3 gauge
symmetries, 2 diffeomorphisms and 1 Weyl rescaling of the
metric. But after quantization, the trace of the metric
becomes physical. Depending on your formalism, this
phenomenon manifests itself in different ways.

In lightcone quantization, the physical Hilbert space
becomes unexpectedly large, because you cannot factor out
the conformal symmetry unless the number of scalar fields
D = 26, due to the conformal anomaly. Provided that D < 26,
the theory is consistent in spite of the anomaly, because
the Hilbert space has a positive-definite inner product
(the spectrum is ghost-free).

Alternatively, one can introduce the trace of the metric as
a dynamical field already at the classical level; it is
then known as the Liouville field. The parameters can be
fine-tuned so that the Liouville field has central charge
c = 26 - D, cancelling the matter and ghost contributions,
and proper conformal invariance is restored. However, the
physical Hilbert space is still larger than expected
classically, because it now includes the Liouville field.

The moral is that a field which is classically a pure gauge
becomes physical after quantization, due to a gauge
(conformal) anomaly. This shows that, contrary to popular
belief, some (not all) gauge anomalies are indeed
consistent, and that the quantum theory as a result has
more degrees of freedom than its classical limit. In
particular, this does happen in 2D gravity, which strongly
suggests that it will happen in 4D gravity as well.

Gravity is not Weyl invariant except in 2D, so the
conformal anomaly does not directly generalize to higher
dimensions. However, if you start by gauge-fixing Weyl
symmetry rather than diffeomorphisms, the conformal anomaly
reappears as a diffeomorphism anomaly instead, see e.g.
http://www.arxiv.org/abs/hep-th/9501016 . This immediately
suggests what is missing from all approaches to QG:
_________________________________________________________
| |
| Just as Weyl/diffeomorphism anomalies make the trace |
| of the metric physical in 2D gravity, diffeomorphism |
| anomalies make gauge components of the metric (beyond |
| the two graviton polarizations) physical in 4D gravity. |
|_________________________________________________________|
|

In order to implement this program, I discovered how to
generalize the Virasoro algebra beyond 1D and how to build
quantum representations. Not alone, but as the only one who
contributed to this discovery with a motivation from
physics. Note that the existence of a multi-dimensional
Virasoro algebra is not a priori obvious, in view of two
no-go theorems:

1. The diffeomorphism algebra has no central extension
except in 1D.

2. In field theory, there are no gravitational anomalies
in 4D.

However, it turns out that these theorems, although
correct, depend on unnecessarily strong assumptions. The
Virasoro extension is not central (commutes with
everything) except in 1D, and one has to go slightly beyond
field theory proper, by introducing and quantizing the
observer's trajectory, in order to formulate the extension.

Of course, the discovery of the multi-dimensional Virasoro
algebra is by itself not enough to quantize gravity, just
as the discovery of the usual Virasoro algebra was not
enough to quantize the free string. But it is a quite
non-trivial, and IMO crucial, step. Besides, it is
undoubtedly the discovery of my life-time.

2D gravity permits an unrelated but interesting
observation. The Liouville field can be viewed as a
Goldstone boson for the conformal symmetry. So what in
lightcone quantization looks like anomalous symmetry
breaking, looks like spontanous symmetry breaking in
Liouville theory. Without having thought too much about it,
I think that this suggests an interesting idea:
_______________________________________________________
| |
| Anomalous and spontanous breaking of gauge symmetries |
| are closely related, perhaps different sides of the |
| same coin. |
|_______________________________________________________|

In particular, the algebra of gauge transformations in 4D
admits an extension which makes it into a higher-
dimensional analog of affine Kac-Moody algebra. This is a
kind of gauge anomaly, but not a conventional one because
it is proportional to the second rather than to the third
Casimir operator. It is tempting to speculate that the
symmetry breaking induced by these gauge anomalies is
related to the Higgs effect, much like conformal anomalies
are related to the Liouville field.

Finally, let me point out that my key observation, that
infinite-dimensional constraint algebras generically
acquire anomalies at the quantum level, has passed the most
stringent test conceivable: it was repeated approvingly by
Lubos Motl at
http://motls.blogspot.com/2005/09/wh...nstein-ii.html
Alas, I suspect that Motl didn't realize from whom the
formulation originated, see my comment at
http://www.haloscan.com/comments/lum...0241393188648/

I.Vecchi

Eugene Stefanovich ha scritto:

> Thank you for the fascinating read.

Well, its authors are a remarkable lot. Some of them still post here,
some unfortunately don't do it anymore.

> ... It strikes me how the
> idea of GR spacetime is similar to long forgotten ideas of the
> electromagnetic ether. In both cases we have some all-penetrating
> "substance" that affects physical objects, but cannot be detected
> itself.

Other people seem to share your insight and beef it up with historical
references ([1], #4).

> GRist will tell you that the "spacetime as rubber sheet" analogy
> popularized in documentaries is an oversimplification, but I have
> a feeling that this analogy plays a major role in their thinking
> process. They talk about spacetime being bent, curved, twisted,
> "foamed", even torn apart; they speak about including spacetime
> topologies in path integrals as if these were different states of some
> real object. John Baez talks (though jokingly) about applying
> Newton's third law to the spacetime (matter must act on spacetime just
> as spacetime is "acting" on matter).
>
> On the other hand, the spacetime idea is conspicuously absent from the
> vocabulary of quantum mechanics (or QFT). It seems that x and t
> arguments of quantum fields psi(x,t) have no direct relationship to
> the measurable time and positions of particles/events. After all,
> when we calculate something related to experiment, like the Hamiltonian
> or the S-matrix, these arguments get integrated out. In the Hilbert
> space of quantum
> mechanics we do have the observable of particle position R,
> but it is not
> distinguished among other observables, like momentum, energy, spin, etc.
> (Should I mention that there is no time operator in quantum mechanics?)

I would see that rather as a limitation of current QM, confronting us
with the key question "what is a clock?" (cf. the references in [2]).

...

>
> It seems to me that quantum mechanics rejects the idea of the
> preferred status of the position observable, and/or spacetime continuum.
> I find rather odd the attempts to quantize gravity by including "states=
"
> of the GR "rubber sheets" into QM Hilbert space. IMHO,
> space (or spacetime)
> is not an observable object, it can't be measured, there are no
> observables associated with it, so it has no place in the Hilbert space.

Formulating a viable measurement theory for space-time is indeed
crucial.
As Wigner writes under the heading "Quantum Limitations of the Concepts
of General Relativity([3]), "In relativity theory, the state is
described by a metric which consists of a network of points in
space-time, that is, a network of events, and the distances between
these events. If we wish to translate these general statements into
something concrete, we must decide what events are, and how we measure
the distance between events"

I think the last sentence is basic to any QG theory.

In [3] Wigner then builds a relevant model, first noting that "the
establishment of of a close network of points in space-time requires a
reasonable energy density, a dense forest of world lines wherever the
network is to be established" and noting further that "it is desirable
... to reduce all measurements in space-time by to measurements by
clocks". Then "the simplest framework in space-time ... is a set of
clocks which ... tick off periods and these ticks form the network of
events which we wanted to establish". Wigner then goes on to
investigate the properties of such a network and the constraints that
physical clocks impose on the theory, analysing the relationship
between the mass of the clock and its accuracy and concludes that "the
essentially nonmicroscopic nature of the general relativistic concepts
seems ... inescapable" .

Wigner then draws a parallel between the situation in GR and in QFT and
writes that in the latter "if we analyse the way in which we 'get away'
with the use of an absolute space concept , we simply find that we do
not. In our experiments we surround the microscopic objects with a very
macroscopic framework and observe *coincidences* between the particles
emanating from the microscopic system and parts of the framework. ...
There is therefore a boundary in our system ..." , indeed the infamous
boundary.

A similar line of thought glimmers in a very interesting post by Peter
Peldan ([4]) about diffeormorphism invariance in QG, in the thread we
are discussing. There was no reply, but it's never too late.

...

> I have zero credentials in quantum gravity research, so my
> thoughts can be safely ignored.

... or daringly heeded. Audaces fortuna iuvat.

Cheers,

IV

[1] Dosch, Müller and Sieroka "Quantum Field Theory, its Concepts
Viewed from a Semiotic Perspective" at
http://philsci-archive.pitt.edu/archive/00001624 .
[2]
http://groups.google.com/group/sci.p...e33639e73f3acb
[3] E. Wigner "Relativistic Invariance and Quantum Phenomena" Rev. Mod.
Phys. 29, 255 (1957)
[4]
http://groups.google.com/group/sci.p...ddaf9b47103c1c

----------------------------

"It is the free in symbols acting spirit which constructs itself in
physics a frame to which he refers the manifold of phenomena. He does
not need for that imported means like space and time and particles of
substance; he takes everything from himself."
Hermann Weyl "Wissenschaft als symbolische Konstruktion der Menschen"
1949, quoted in [1], those were the days when people like Weyl
confronted their ideas with those of people like Cassirer and Panovsky.
The Hoelderlinian German original is also available at [1].

Ilja Schmelzer

"Eugene Stefanovich" <eugenev@synopsys.com> schrieb
> Thank you for the fascinating read. It strikes me how the
> idea of GR spacetime is similar to long forgotten ideas of the
> electromagnetic ether. In both cases we have some all-penetrating
> "substance" that affects physical objects, but cannot be detected
> itself.

Moreover, it is quite easy to find an ether interpretation of the
metric.

All you need are harmonic coordinates and a choice of a
timelike time coordinate between them.

g^00 sqrt(-g) = rho > 0
g^0i sqrt(-g) = rho v^i
g^ij sqrt(-g) = rho v^i v^j - p^ij

where rho is the ether density, v^i its velocity and p^ij its
pressure/stress tensor. The harmonic condition becomes the continuity and
Euler equation.

To obtain the harmonic equation as a physical (Euler-Lagrange) equation
we have to add two cosmological terms to GR.

L = L_GR + (Y g^00 + X g^ii) sqrt(-g)

Even more, this Lagrangian may be derived from simple axioms, where
the key axiom is that continuity and Euler equations appear as the Noether
conservation laws in a special form of Noether's theorem.
See gr-qc/0205035 for details. Thus, the Lagrangian and therefore the
relativistic symmetry (EEP) may be explained. The unexplained "conspiracy"
of the Lorentz ether no longer exists.

> After all,
> when we calculate something related to experiment, like the Hamiltonian
> or the S-matrix, these arguments get integrated out. In the Hilbert
> space of quantum
> mechanics we do have the observable of particle position R,
> but it is not
> distinguished among other observables, like momentum, energy, spin, etc.
> (Should I mention that there is no time operator in quantum mechanics?)

A strong hint that the non-measurability of absolute time is not a strong
argument against absolute time.

> Since spacetime curvature is something we cannot directly
> observe,

At least in principle spacetime curvature is easily observable.
We measure the radius and circumsphere of a circle, and
if u = 2 pi r does not hold we have measured curvature.

But we have to remember that "curvature" is just a mathematical
word. "Curvature" may appear on very flat things, if we measure
them with distorted rulers.

Ilja

Eugene Stefanovich

"Ilja Schmelzer" <Ilja.Schmelzer@FernUni-Hagen.de> wrote in message
news:dl9k03$onq$1@tamarack.fernuni-hagen.de...
> > Since spacetime curvature is something we cannot directly
> > observe,
>
> At least in principle spacetime curvature is easily observable.
> We measure the radius and circumsphere of a circle, and
> if u = 2 pi r does not hold we have measured curvature.

My point was that the "circle" we are measuring is made of some "real
stuff", i.e., particles interacting (or not interacting) with each
other. Quantum mechanically, our measurements refer to expectation
values of the position operators of these particles in the Hilbert
space. So, in order to describe these measurements you just need the
Hilbert space of the system, the operators of observables (e.g.,
position) of particles belonging to the system, and the unitary
representation of the Poincare group in the Hilbert space which
expresses the properties of invariance for measurements made from
different reference frames.

This is sufficient for a complete description of the system (circle) and
all possible measurements that we can make there, including the
circumference-to-radius ratio. In this description, the space-time
continuum does not play any role at all. If we found, for example, that
the ratio u/r is different from 2 pi, I would rather attribute it to
some pecularities of interparticle interactions in the circle, than to
"curvature" of some mysterious spacetime that evades direct observation.

Eugene.

Igor Khavkine

Eugene Stefanovich wrote:
> "Ilja Schmelzer" <Ilja.Schmelzer@FernUni-Hagen.de> wrote in message
> news:dl9k03$onq$1@tamarack.fernuni-hagen.de...

> > At least in principle spacetime curvature is easily observable.
> > We measure the radius and circumsphere of a circle, and
> > if u = 2 pi r does not hold we have measured curvature.
>
> My point was that the "circle" we are measuring is made of some "real
> stuff", i.e., particles interacting (or not interacting) with each
> other. Quantum mechanically, our measurements refer to expectation
> values of the position operators of these particles in the Hilbert
> space. So, in order to describe these measurements you just need the
> Hilbert space of the system, the operators of observables (e.g.,
> position) of particles belonging to the system, and the unitary
> representation of the Poincare group in the Hilbert space which
> expresses the properties of invariance for measurements made from
> different reference frames.

Space-time is not Poincare symmetric. That is what our telescopes tell
us. That is what our satelites tell us. That is what the cosmic
microwave background tells us.

> This is sufficient for a complete description of the system (circle) and
> all possible measurements that we can make there, including the
> circumference-to-radius ratio. In this description, the space-time
> continuum does not play any role at all. If we found, for example, that
> the ratio u/r is different from 2 pi, I would rather attribute it to
> some pecularities of interparticle interactions in the circle, than to
> "curvature" of some mysterious spacetime that evades direct observation.

You are free to choose an interpretation of experimental data, and so
is everyone else. The tough part is predicting what the outcome of the
next experiment will be, where the circle will be moved, tilted,
rotated, or the particle will go faster, slower, be of different
species, etc. Success in theoretical physics is not measured by
interpretation.

Igor

Ilja Schmelzer

"Eugene Stefanovich" <eugene_stefanovich@usa.net> schrieb
> "Ilja Schmelzer" <Ilja.Schmelzer@FernUni-Hagen.de> wrote
> > > Since spacetime curvature is something we cannot directly
> > > observe,
> >
> > At least in principle spacetime curvature is easily observable.
> > We measure the radius and circumsphere of a circle, and
> > if u = 2 pi r does not hold we have measured curvature.
>
> My point was that the "circle" we are measuring is made of some "real
> stuff", i.e., particles interacting (or not interacting) with each
> other. Quantum mechanically, our measurements refer to expectation
> values of the position operators of these particles in the Hilbert
> space.

My point was about classical gravity.

What is observable in quantum gravity is, of course, a quite different
question. On the other hand, the correspondence principle requires
that the classical measurement I have described appears to be the
limit of some quantum measurement.

> If we found, for example, that
> the ratio u/r is different from 2 pi, I would rather attribute it to
> some pecularities of interparticle interactions in the circle, than to
> "curvature" of some mysterious spacetime that evades direct observation.

I don't like the mystery related with words like "curved spacetime" too.
It suggests not only some embedding in some higher-dimensional space
but also that the current measurement of distance and duration has
fundamental, almost philosophical character.

Therefore I fully agree that we can attribute with u/r =!= 2pi to something
much less mysterious. In the ether interpretation I have described in
my last posting spatial curvature is the same effect we know as
"inner stress" in condensed matter theory. No mystery at all.

(My only objection was that the thing which was given the mysterious
name "spacetime curvature" is, nonetheless, something we can measure,
as far as we have rulers and clocks.)

Ilja

Eugene Stefanovich

Ilja Schmelzer wrote:

>>My point was that the "circle" we are measuring is made of some "real
>>stuff", i.e., particles interacting (or not interacting) with each
>>other. Quantum mechanically, our measurements refer to expectation
>>values of the position operators of these particles in the Hilbert
>>space.
>
>
> My point was about classical gravity.
>
> What is observable in quantum gravity is, of course, a quite different
> question. On the other hand, the correspondence principle requires
> that the classical measurement I have described appears to be the
> limit of some quantum measurement.

What I said is applicable to the classical case too.
In order to translate, you should change
"Hilbert space" -> "phase space",
"operator" -> "real function on the phase space"
"state vector" -> "point in the phase space"
"expectation value" -> "value of the function at a point"


Eugene.

Eugene Stefanovich

Igor Khavkine wrote:

> Space-time is not Poincare symmetric. That is what our telescopes tell
> us. That is what our satelites tell us. That is what the cosmic
> microwave background tells us.

These observations tell us that 1) there is gravity and 2) the force
of gravity is different from the Newtonian 1/r^2. These two things
you can take to the bank. Regarding "curved spacetime"... This
hypothesis worked pretty well so far, but it also has a few defects.
One of them is the fundamental incompatibility with the laws of
quantum mechanics.

Even if it will be definitively proven that gravity is a spacetime
curvature (which I doubt), the Poincare group will be here to stay
with us. If you consider an isolated system in empty space,
i.e. two interacting
masses (instead of a mass in an external field), this system is
clearly invariant with respect to Poincare transformations:
space and time translations, rotations, and boosts of the system as
a whole.

Eugene.