Simplicity Constraints in Spin Foams: Physical Meaning & Motivation

[zwicky]

Hello everybody.

I have a question about the physical meaning of the simplicity constraints that is often used in spin foams. For example, in http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.1780v4.pdf, eq. (34), it is written as
K=-\gamma L
where K are the boost and L the rotations.

Is there a physical motivation besides the mathematical one given in the article?

What kind of constraints is this?
Is it related to a gauge fixing?
What is the physical meaning of making the boosts proportional to the rotations? I mean, boost are related with a non compact parameter in contrast with the spatial rotations.
To me the boost ant the rotations are in principle different things, so I just can't see why this relation is so important.

Thank you very much in advance.

Zwicky.

 

[marcus]

Hi Zwicky,
there are others better able to respond to your question. Maybe some will put a word in! In the meanwhile, to get started: For me the best short explanation is on page 5 of the October 2010 "simple model" paper

http://arxiv.org/abs/1010.1939

It is the section "Relation with GR". Starting with point #3 at the top of the righthand column.

It explains how the Einstein-Hilbert action, written in the equivalent Holst form, is related to the BF action
But this by itself would not give local degrees of freedom. So the action is to be minimized only over a restricted class of B two-forms. Namely those B which arise from a tetrad e.

See the small unnumbered equation right after equation (24) is the simplicity constraint. I have to go out...back now.

This restriction on the B field (called the "simplicity constraint") is what turns on the local degrees of freedom.

BF then connects with spinfoam. This is how a bridge between SF and the Einstein-Hilbert action is established, have to go... Maybe I can edit the following later and make it legible:
======quote Section 3 on page 5======
3. GR’s action can be written in the form [62]

S= ∫(e∧e)*∧F + (1/γ)∫e∧e∧F. (23)

The first term is the standard Einstein-Hilbert action S[g_μν]=∫√gR, written in first order form and in terms of a tetrad e and an SL2C connection with curvature F . The second term is a parity violating term that does not affect the equations of motion and leads to the real Ashtekar variables. This action is the BF action

S_BF = ∫B∧F (24)

where the two-form field B is restricted to the form B=(e∧e)* + (1/γ)(e∧e). A constraint on B forcing it to have this form is called “simplicity constraint”. Now, (5) is as a modification of Ooguri’s BF partition function [6]...(25)...obtained by restricting the sum precisely to the states where such simplicity constraints hold [63, 64].

These constraints turn the (topologically invariant) BF partition function into the (non topologically invariant) partition function for GR. Because of the restriction in the representations summed over and the SU2 integrations, (5) relaxes the BF flatness condition implemented in (25) by the delta function on the holonomy around each face, turning local degrees of freedom on.

==endquote==

αβγδεζηθικλμνξοπρσςτυφχψω...ΓΔΘΛΞΠΣΦΨΩ∏∑∫∂√ ∧± ÷←↓→↑↔~≈≠≡≤≥½∞(⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅)

 

[zwicky]

Thank you very much for your comments Marcus!

Well, yes, according tho the reference the simplicity constraint enforces B to be related with the tedrad in such a way we end with GR, otherwise we just end with a BF theory. But still I have the same question, what is the physics, if there is one, of that constraint? from the mathematical point of view it seems like things are "crystalline" because we need to add an extra ingredient to a topological field theory let the degrees of freedom be. But how the simplicity constraint works? The other version of the simplicity constraint is the one I cited before, i.e.,
K=-\gamma L
I found this formula really puzzling, first of all, because fo the presence of the Immirzi parameter (can we recover GR without dealing this parameter?), is it assumed a sort of gauge fixing? (we are choosing a class of preferred boost satisfying ans specific condition).

Cheers,

Zwicky

Zwicky.

 

[marcus]

It's certainly reasonable to wonder about the physical significance of that equation. I can't offer any good intuition regarding it--maybe someone else can. I'm glad to see you understand much of the context here. Since no one else has jumped in, I will continue to nibble away at the topic. Let's look again at equation (23), the Holst action for GR.

S= ∫(e∧e)*∧F + (1/γ)∫e∧e∧F.
==quote==
The first term is the standard Einstein-Hilbert action S[g_μν]=∫√gR, written in first order form and in terms of a tetrad e and an SL2C connection with curvature F . The second term is a parity violating term that does not affect the equations of motion and leads to the real Ashtekar variables. This action is the BF action
==endquote==

The second term in the Holst action is the one with the Immirzi. The Immirzi is there just so the action will reproduce the real Ashtekar variables.

In your post you asked "can we recover GR without dealing with this parameter?" I think it must be there because we are recovering GR by way of the real Ashtekar variables, an approach using the equivalence of SL2C to two copies of SU2: one of which might pictured as associated with rotations, if you wish, and one with boosts. The Immirzi gives a way of associating SU2 with a subspace of the direct sum---and representations of SU2 with certain ones of SL2C. Unless I'm mistaken at this stage we don't have to worry about the physical significance because what we are looking at is to some extent merely an algebraic device---a formality. It just has to do with the Holst form of the GR action, and the requirements for recovering GR in real Ashtekar variables. That's simply my take on it, hopefully someone will correct me if I'm wrong.

The Immirzi parameter may have some physical significance---this has been considered in several papers---but at this stage it plays the role of an algebraic convenience. I'll go out on a limb and say that any physical significance is not entering into what we are looking at, and may be safely ignored.

 

[zwicky]

Thanks Marcus, I will think about that.