What is space-time in LQG

[J. Horta]

Hi,

In quantum field theory space-time seems to play the same
role as it does in classical physics namely one expresses
quantities as functions of space and time. Is this changed=20
in a significant way in loop quantum gravity? Are fields
still viewed as functions of space-time parameters?

Thanks
Paul Colby

[Oh No]

Thus spake J. Horta <bite@me.spam>
>Hi,
>
>In quantum field theory space-time seems to play the same
>role as it does in classical physics namely one expresses
>quantities as functions of space and time. Is this changed=20
>in a significant way in loop quantum gravity? Are fields
>still viewed as functions of space-time parameters?
>
I'm no expert but my understanding is that loop quantum gravity is a
theory of gravity and nothing more. In other words, it hasn't got as far
as answering that question.

Regards

--
Charles Francis
substitute charles for NotI to email

[Igor Khavkine]

J. Horta wrote:

> In quantum field theory space-time seems to play the same
> role as it does in classical physics namely one expresses
> quantities as functions of space and time. Is this changed
> in a significant way in loop quantum gravity? Are fields
> still viewed as functions of space-time parameters?

In gravity, the fundamental dynamical variables are the components of
the connection (which are basically equivalent to the metric tensor).
These are fields on the space-time manifold, they can be seen as
functions of space-time parameters, as you say. However, since the
space-time manifold has no a priori geometry associated to it, the
space-time parameters carry no geometric information with them, only
topological information about continuity and smoothness. That is the
main difference between gravity as a field theory and other field
theories on fixed backgrounds.

Loop Quantum Gravity approaches quantization of gravity from this point
of view. The main trick is to forget that the components of the
connection can be seen as functions that can be evaluated at space-time
points. Instead, one sees the connection as a function assigning a
holonomy to any closed loop embedded in space-time. This notion of
holonomy can be generalized from loops to closed graphs embedded in
space-time. These graphs are called spin networks. The point of this
change of perspective is that the dynamical variables are no longer
viewed as fields on space-time (a way to assign values to space-time
points) but rather as a way to assign values to spin-networks embedded
in space-time.

If you also want to consider matter fields coupled to gravity in LQG,
then to fit them into the same framework you need a similar change of
perspective. There are proposals for coupling scalar fields and gauge
fields to LQG. These proposals usually start out by considering matter
fields also as functions on spin networks as opposed to space-time
points. Look for work by Ashtekar, Lewandowski, Sahlmann and
Varadarajan for specifics.

Hope this helps.

Igor