starblade13

Well, what would happen if gravity didn't act on any abstract state
space, but on real space? How does this change things?


They should be independant of coordinates. However, some topological
theory assert that spacetime is quantized at the fundamental level, so
the field COULD depend on the 'distance' (IE how many 'jumps' are
made) from one 'point' to the other 'point'.


Well, my quantum mechanics book says that the Schrodinger Equation has
to be taken from first principles. Then again, it is an old book, a
kind of hand-me-down of sorts.

(...Starblade Riven Darksquall...)

davidoff404

The problem is of course that gravity *does* act on physical space. The
difficulty is in describing the union of its action on physical space
together with its action on the infinite dimensional Hilbert spaces
required in quantum theory.


Let's straighten something out. Coordinate independence means that the
physics looks the same to all observers, regardless of what coordinate
system they choose to employ. The question of a theory's topology and
any notions of "distance" seems to be a little vague to you.

A manifold is basically a topological space equipped with some nice
extra properties involving differentiability and smoothness. As a
result, one can think of the topology underlying a particular manifold
as being simply a set of of points. Distance, however, is a whole other
matter. To introduce a measure of distance, you need to have a metric
and thus impose extra structure on the manifold (the manifold being a
level of abstraction removed from the underlying topological space anyway).

The ``abstract state space'' in question is the space of quantum
states, that is, the space of wave functions. If gravity doesn't
affect wave functions, then you haven't quantized gravity.

Steve Carlip

starblade13

Why are people trying to use abstract space states? How successful
have they been with this? Not very, I assume, otherwise we'd already
have a quantum theory of gravity. Then why do they continue insisting
that it must be done this way?

(...Starblade Riven Darksquall...)

starblade13

Well, do they assume that timespace is continuous or not? When you
quantize the fundamental particles, you don't have to worry about it
either way. But when you quantize gravity, shouldn't timespace be, well, quantized?


I agree, distance is vague, because it depends not only on the points
but how many points lie between those points. Most likely it will end
up that there isn't any specific distance between points. So we'll
have to label points qua points rather than labelling them with
respect to their coordinates.


Well if timespace structure was statistical in nature, then that would
just muddle things up, because you'd have to figure out which metric
is being used, and at which point, and so on.

Why don't we just use something like 'the set of points' with each
point in timespace having a unique label, then have each have
properties such as which other point it was connected to?

Then rather than working with manifolds, we're working with something
much different, and then from there determining what kind of manifold
it approximates towards being.

That seems like the perfect way to do it.

(...Starblade Riven Darksquall...)

dubious

Starblade Darksquall:

Because that happens to be how quantum mechanics works. If you
think using quantum mechanics to create quantum theories hasn't
been very successful, consider the chances of creating a quantum
theory assuming something which is incompatible with a quantum
theory.

Also, since we already have a quanntum theory of everything else
called the standard model, which works better than it has any reason
to work, I'd say quantum theory has been pretty successful.


When drawing a square, why do people insist you draw all four
sides the same length _and_ have right angles at all of the vertices?
Because if you didn't, you wouldn't have a square and calculating the
distance between opposite corners assuming you did draw a square wouldn't
make sense, much less give you the right answer. Apply that to quantum
theory and abstract spaces.

dubious

Starblade Darksquall:


Yes, plus one option you left out. Some physicists will say probably,
some will say probably not and a lot will say they don't know but are
pretty certain it's one or the other.

If the problem is to find a quantum theory of gravity, then the
answer has to be (1) a quantum theory, and (2) of gravity, right?

A quantum theory is a theory of wave functions, also known as
``states.'' These wave functions, or states, form an ``abstract
vector space'' -- this is basically just the statement that you can
add them. This is known as the principle of superposition, and
is part of the basic definition of a quantum theory. If you don't
have wave functions, or you can't add them, then you don't have
a quantum theory. Finally, if your quantum theory is also a theory
of gravity, then gravity has to affect the wave functions; I assume
this is self-evident, isn't it?

If you don't think ``it must be done this way,'' then you are no
longer talking about a quantum theory of gravity. There's
nothing particularly deep here -- you just have to know a little
bit (a *very* little bit) about what a ``quantum theory'' means.

Steve Carlip

davidoff404

Please, please, please, stop referring to it as 'timespace'. The proper
term is 'spacetime' since it's less ambiguous (if one was pedantically
inclined, the phrase 'timespace' could probably refer to R^1).

One doesn't quantize fundamental particles per se, but rather certain
properties they may have such as spin or angular momentum. It's true
that in a limited sense one doesn't have to care too much about whether
spacetime is quantized since the realm of applicability of quantum
mechanics is confined situations where the underlying structure of
spacetime is of no consequence. However, it's a BIG question in modern
theories as to the exact nature of quantization on a large scale (in a
sense, it's actually one of the goals of a TOE).

Distance doesn't really depend on "how many points lie between" two
distinct points; after all, one can always include infinitely

You already have to do that in standard general relativity and 3+1 decompositions thereof.

We already do. Spacetime coordinates are called events and they're
related to "nearby points" by whatever metric you're using.

Why are you so prejudiced against the use of a manifold?

davidoff

starblade13

What? "Quantization on a large scale"? I thought quantization was a
small scale phenomenom. :P


Well in a continuous, differentiable manifold that's true, and
distance is something that's almost metaphysical.


In GR, the metric is deterministic. But what I'm saying is that the
metric would no longer be deterministic.


But the set of points nearby becomes an infinite set if spacetime were
continuous. What if spacetime were quantized? Then the set would be finite.

I'm prejudiced against the use of a CONTINUOUS manifold, not against
any kind of manifold. Sorry, I should've clarified that.

(...Starblade Riven Darksquall...)