Unifying general relativity with topological field theory

[marcus]

From the abstract:

"An action principle is described which unifies general relativity and topological field theory.

An additional degree of freedom is introduced, and depending on the value it takes the theory has solutions that reduce it to (1) general relativity in the Palatini form, (2) general relativity in the Ashtekar form, (3) F Λ F theory..., (4) BF theory..."

This is a new paper by Smolin and Starodubtsev, "General relativity with a topological phase: an action principle" posted yesterday, 18 November.

http://arxiv.org/hep-th/0311163

If anyone would care to elucidate any part of this, it would be much appreciated. Several of us (IIRC selfAdjoint, Ambitwistor, nonunitary...others?) have mentioned TQFT, BF theory. It would be really helpful if we had some entry-level description here clarifying basic things like "what is topological field theory" what makes it different, specifically topological, what are the connections to spin foams and other other current research areas (I know Baez has a good paper making the connection---but there has never been an summary of these things here at PF as far as I know.) Anyone have a few general words giving perspective on why Smolin/Staro's unification (by an action principle) is or is not interesting?

 

[marcus]

changing this parameter allows a continuous variation from one theory to another, one can even imagine bubbles of space in which gen rel governs and a boundary outside of which another form of the law rules.

fascinating. and some of these are closely allied to spin-network or spin-foam models.

some Greek in a selfAdjoint post on another thread
"en arche en to arachnion" (in the beginning was the web)
reminded me of lines from a wonderful play set in Fifteenth Century England
which concerns among other things a woman wrongly condemned to burn the next morning as a witch (a fate which by good fortune she escapes). At one point she describes the one night she has left to live as a small silver coin to spend

JENNET: I've only one small silver night to spend.
Show me no luxuries. It will be enough
If you spare me a spider, and when it spins I'll see
The six days of Creation in a web
And a fly caught on the seventh. And if the dew
Should rise in the web, I may well die a Christian.

 

[selfAdjoint]

Was that "The Lady's Not for Burning" by Christopher Fry? You have a wonderful education - and a very retentive memory.

 

[marcus]

Originally posted by selfAdjoint
Was that "The Lady's Not for Burning" by Christopher Fry? You have a wonderful education - and a very retentive memory.

sure was, compliments to you likewise---great play, a video made for UK television is available with K. Brannagh

you were urging renaissance of the group reps thread. I cordially agree. Lethe would be a good emcee for that, I think, if he were willing

[edit: Ranyart thanks for the TQFT links! I just saw the post of yours immediately following this one. Especially the historical survey paper
http://arxiv.org/hep-th/0011260
by Albert Schwarz "Topological Quantum Field Theories"]

 

[ranyart]

This may be a good place to start:http://arxiv.org/PS_cache/hep-th/pdf/0011/0011260.pdf

The second hyperlink at the end of the paper details some interesting technical details:http://arxiv.org/abs/hep-th/9709192

The concept of 'how-where-when' one applies transitional quantum/relativistic algabra's is of great importance, you have to state that contained within a boundery looking outwards, you are describing the boundery from this observation point.

Likewise if one is observing from a boundery looking inwards, then there has to be a transitional point that has 'both' or an equivilence of interpretation?

Q)If one Place's a Quantum Theorist at a far away location say, at the Equilibrium point between two Galaxies, and also places a Relativistic Theorist at the finite half-way point between atoms, can both these theorists agree, (upon their return to present day locations) on a unified description that combines both observations?

 

[nonunitary]

I haven't seen in detail the paper by Smolin and Starodubtsev, but
what I will try to do is to give some introductory remarks about
what is a topological theory and what not.

To begin with, the term topological comes mainly from analyzing
the action that defines the theory. The difference between a
diffeomorphism invariant and a diffeomorphism covariant theory is
whether it can be written independently of a background metric or
not. Let us consider an Abelian gauge field, described by a
connection A (a 1-form) with curvature F (2-form). There two
actions that one can write. The simplest one is to construct a
4-form (we need 4-forms to be integrated to define the action in
4D), by wedging F with itself. Then we have $\int F \wedge F$.
This theory is diffeo-invariant (and thus topological), but is not
very interesting. The other action one can write is the one that
uses a background metric $g$. One can either construct the dual
2-form *F and then wedge it with $F$, or "contract the indices" of
FF with g getting $ggFF$. They are equivalent and the resulting
action is precisely the Maxwell theory. This is not a
diffeo-invariant but it is diffeo-covariant, since the metric is
not dynamical. It is a background structure. Note that in the real
world, Maxwell theory IS diffeo-invariant because when we couple
it to gravity (and thus getting the Einstein-Maxwell theory), then
the resulting theory is diffeo-invariant since both fields are
dynamical.

Ok, back to the pure Abelian field. The first case is sometimes
regarded as a topological theory (note I haven't used the word
quantum yet), and the second is not.

Sometimes people make the distinction between these two classes of
theories based on a different criteria, namely the number of local
degrees of freedom. These are basically the free data that one can
prescribe (initially) to recover all solutions. For a scalar field
one has to give an initial condition and initial velocity at each
point of the "t=0 slice" and then divide by 2. Thus the scalar
field has one local degree of freedom. For theories that have some
type of gauge invariance as in Maxwell the counting is not so easy
and one has to do a detailed Hamiltonian treatment to count.
Sometimes when the counting gives zero then the theory is said to
be topological: the degrees of freedom are global and finite.
Examples: BF theories, Chern-Simons theory in 3D, gravity in 3D.

There are theories that are diffeo-invariant and still have local
degrees of freedom, like Chern-Simons theories in higher
dimensions. I guess they would be referred to as topological or
not depending on who you ask.

I haven't said anything about TQFT but I will leave that for a
latter posting.

 

[marcus]

Originally posted by nonunitary
I haven't seen in detail the paper by Smolin and Starodubtsev, but
what I will try to do is to give some introductory remarks about
what is a topological theory and what not.
...
I haven't said anything about TQFT but I will leave that for a
latter posting.

Hi Non-unitary,

I have been hoping that at some convenient time you would continue the discussion. I did not reply because I was waiting for more.

 

[nonunitary]

Hi Non-unitary,

I have been hoping that at some convenient time you would continue the discussion. I did not reply because I was waiting for more.

Marcus,

I apologize for not posting anything. I have been very busy
and hope to have some time to say something about this topic.

nonunitary

 

[marcus]

Originally posted by nonunitary
Marcus,

I apologize for not posting anything. I have been very busy
and hope to have some time to say something about this topic.

nonunitary

No problem, the threads that grow slowly enough that I can learn from them work better for me. Whenever you feel like talking about TQFT and have time that is fine. I have been seeing signs of more (but still tentative) connections developing between quantum gravity and TQFT, so I am going to try to understand this better and would appreciate any help.