Unitarity in Spin Foams: Canonical LQG vs. Path Integrals

[atyy]

In canonical LQG, unitarity is presumably guaranteed by the canonical formalism. How does one check for unitarity in the spin foam (path integral) formalism? Do the new spin foams pass the necessary tests?

 

[Physics Monkey]

One very common test you can apply is to check that the trace of one is an integer i.e. the number of states in the hilbert space. Often this can be computed using a path integral over some closed manifold. Not sure what the analogue is in spin foams for gravity, but this should be a sensible test for spin foam descriptions of gauge theories. Certain euclidean topological quantum field theories in 3+1d seem to be ruled out as low energy theories for unitary quantum systems since they don't have an integer number of states (= number of ground states).

 

[genneth]

If I understand the literature (Rovelli and co.) correctly, this is a subtle problem, even classically. Normally (i.e. non-strong gravitational field), one has (or can choose) a global time and define unitarity relative to that. In the presence of strong GR it becomes more difficult to define what this could mean. The ADM formulation requires a global Cauchy surface and well-defined lapse and shift functions --- something which can turn out to be tricky, as the numericists discovered for simulating black holes. Even canonically one has to be very careful to maintain symplectic volume.

On the other hand, unitarity must be restored for any observing observing a closed system, for reasons of pure logic. The problem is that it seems difficult to construct a closed system gravitationally without requiring simple asymptotics. I seem to think that calculations exist for graviton (i.e. weak GR limit) propagators and even 2 particle scattering --- I assume these are indeed unitary; though clearly vacuous because they true only in the weak curvature limit anyway.

I don't know what the cosmology people do; but I suspect that since they integrate out all but the largest scales anyway they probably don't worry about unitarity since states can disappear into high frequencies.

 

[tom.stoer]

The problem is to define unitarity. In QFT this is done using the time evolution operator U(t',t) in the interaction picture. It is clear that every calculation, regularization etc. must preserve a unitary U(t',t).

b/c there is no t in ADM and LQG, there is no U(t',t).

 

[atyy]

negru had comments in marcus's bibliography thread, which I thought might be related to the topic of this thread.

https://www.physicsforums.com/showpost.php?p=3879834&postcount=1705

Unfortunately not available online, a talk given today at Princeton Institute for Advanced Studies:
Monday, April 23, 2012
High Energy Theory Seminar
“Loop Quantum Gravity: Recent Results and Open Problems”
Location: Bloomberg Lecture Hall
Time: 2:30 PM
Speaker(s): Carlo Rovelli, Centre de Physique Théorique de Luminy, Aix-Marseille University, France
Description: The loop approach to quantum gravity has developed considerably during the last few years, especially in its covariant ('spinfoam') version. I present the current definition of the theory and the results that have been proven. I discuss what I think is still missing towards of the goal of defining a consistent tentative quantum field theory genuinely background independent and having general relativity as classical limit.
http://www.princeton.edu/physics/events/viewevent.xml?id=347

https://www.physicsforums.com/showpost.php?p=3881844&postcount=1706

I was at the talk at IAS, everyone was pretty confused by what he was doing. One point brought up was that there exist other models, like YM in 5d i think, whose discrete version has the correct classical limit and is uv and ir finite, but does not make sense quantum mechanically. And there was no concrete argument for why lqg would be a better example. Some numerical checks are needed, and he said they are very hard to do but people are working on them.

 

[julian]

The Hamiltonian constraint operator in its usual form is non-hermitian, implying evolution is not unitary...but this is all OK because evolution with respect to the time coordinate has no physical meaning.

The reason it is non-Hermitian is that it only adds links at vertices but doesn't remove them.

 

[julian]

The Hamiltonian constraint operator in its usual form is non-hermitian, implying evolution is not unitary...but this is all OK because evolution with respect to the time coordinate has no physical meaning.

The reason it is non-Hermitian is that it only adds links at vertices but doesn't remove them.

When I made this comment the model I had in mind was the spinfoam model proposed by Reisenberger and Rovelli: "Spin foams as Feynman diagrams" - arXiv:gr-qc/0002083. There are mathematical difficulties in formulating the exponetial of a non-Hermitian operator. This was one of the motivations for formulating spinfoams from a different perspective.

 

[julian]

In canonical LQG, unitarity is presumably guaranteed by the canonical formalism. How does one check for unitarity in the spin foam (path integral) formalism? Do the new spin foams pass the necessary tests?

Reisenberger and Rovelli spinfoam was a direct attempt at relating the canonical approach to the 'path-integral' approach. However, because of issues to do with the constraint algebra and non-hermiticity of the Hamiltonian constraint this approach was formal. The Master constraint circumvents these problems and Thiemann et al have been making inroads into directly connecting the canonical formulation with the spinfoam one.

 

[julian]

genneth is right in the argument of logical consistency...Rovelli's formulation of background scatering amplitudes, for example - experiment on a closed system should result in conservation of relational probabiltity...but then there is Pullin and Gambini interpretation of QM in which there is a fundamental decoherence due to the requirement of the use of real material reference systems, which themselves are subject to quantum fluctuations, that would lead pure states to evolve to mixed states, but corrections would be negligile.

 

[julian]

arXiv:0911.3431:

"On the Relation between Rigging Inner Product and Master Constraint Direct Integral Decomposition"

"Canonical quantisation of constrained systems with first class constraints via Dirac’s operator constraint method proceeds
by the theory of Rigged Hilbert spaces, sometimes also called Refined Algebraic Quantisation (RAQ). This method can work
when the constraints form a Lie algebra. When the constraints only close with nontrivial structure functions, the Rigging map
can no longer be defined.
To overcome this obstacle, the Master Constraint Method has been proposed which replaces the individual constraints by a
weighted sum of absolute squares of the constraints. Now the direct integral decomposition methods (DID), which are closely
related to Rigged Hilbert spaces, become available and have been successfully tested in various situations.
It is relatively straightforward to relate the Rigging Inner Product to the path integral that one obtains via reduced phase
space methods. However, for the Master Constraint this is not at all obvious. In this paper we find sufficient conditions under
which such a relation can be established. Key to our analysis is the possibility to pass to equivalent, Abelian constraints, at
least locally in phase space. Then the Master Constraint DID for those Abelian constraints can be directly related to the Rigging
Map and therefore has a path integral formulation."arXiv:0911.3428: "On the Relation between Operator Constraint –, Master Constraint –, Reduced Phase Space – and Path Integral Quantisation""Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space formulation. In this article we review this rather involved procedure in full generality. Moreover, we demonstrate that the reduced phase space path integral formulation formally agrees with the Dirac’s operator constraint quantisation and, more specifically, with the Master constraint quantisation for first class constraints. For first class constraints with non trivial structure functions the equivalence can only be established by passing to Abelian(ised) constraints which is always possible locally in phase space. Generically, the correct configuration space path integral measure deviates from the exponential of the Lagrangian action. The corrections are especially severe if the theory suffers from second class secondary constraints. In a companion paper we compute these corrections for the Holst and Plebanski formulations of GR on which current spin foam models are based."

 

[julian]

"...sometimes also called Refined Algebraic Quantisation (RAQ). This method can work when the constraints form a Lie algebra. When the constraints only close with nontrivial structure functions, the Rigging map can no longer be defined."

Like I said problems to do with the constraint algebra - the Master constraint doesn't suffer from these problems.
"...absolute squares of the constraints." - in other words the Master constraint is Hermitian.

 

[marcus]

Nice to see Thiemann's work reviewed like this! Thanks Julian!