Solving the Quantum Zeno problem with path integral QG

[marcus]

Schroeren's new paper "Decoherent histories of spin networks" made me aware of some work by two people at Cambridge DAMTP and London Imperial's Blackett Lab on the Quantum Zeno (QZ) effect in the path integral e.g. decoherent histories (DH) context.

Schroeren's paper: http://arxiv.org/abs/1206.4553
Gell-Mann and Hartle's recent DH paper http://arxiv.org/abs/1106.0767
Halliwell and Yearsley's recent QZ paper: http://arxiv.org/abs/1205.3773
Pitfalls of Path Integrals: Amplitudes for Spacetime Regions and the Quantum Zeno Effect
J.J.Halliwell, J.M.Yearsley
(Submitted on 16 May 2012)
Path integrals appear to offer natural and intuitively appealing methods for defining quantum-mechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect -- the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and in particular, no sensible classical limit. In this paper we describe this frequently-occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a "softer" way. The resulting amplitudes then involve a new coarse graining parameter, which may be taken to be a timescale ε, describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as ε >> 1/ E, where E is the energy scale of the incoming state.
24 pages, 6 figures.

 

[marcus]

I may have to say this several times before I get it right, or somebody else helps describe the situation. The thing is that conventional quantum mechanics does not work for cosmology because no classical observer (or measuring device) outside the System. The world is not divisible into a quantum piece and a classical piece.

But the whole way conventional QM was thought out is based on that split. So several people have worked on what James Hartle has called "generalized quantum mechanics" that has all the functions/predictions etc but in addition is compatible with cosmology.

And they use the path integral approach. "Sum over histories."
And this includes histories of geometry--how geometry might evolve.

The Quantum Zeno (QZ) problem is a difficulty that arises when you want to partition the set of all histories into separate subsets or EVENTS---such as trajectories that pass thru a certain window, or geometries that pass thru a certain stage of development or shape.

Being able to define the probabilities of specific EVENTS is what takes the place of having a classical observer. The condition defining an event can involve a particle trajectory or spacetime geometry.

The QZ problem arises when you try to impose a condition on histories---if you define an event naively, or too rigidly, then it would seem that you have to keep checking and checking and checking---projecting projecting projecting---and the state gets trapped.
======================
But you have to be able to partition the space of histories, so being able to impose conditions is really essential. You have to be able to define the set of all histories which do (or do not) meet a specified criterion.

David Schroeren explains this both generally and in the context of spinfoam QG.

Also Halliwell and Yearsley. The secret seems to be to invent a way to impose a condition "softly" so that it does not trap the state and cause a kind of Zeno paralysis.

The references are really useful
Hartle explains why the old QM paradigm has to be abandoned with its quantum/classical split. And he describes the basic setup: http://arxiv.org/abs/gr-qc/9304006 however this is over 100 pages
Hartle and Gell-Mann elaborate and condense this down to a short 2011 paper: http://arxiv.org/abs/1106.0767
Halliwell and Yearsley point out how Zeno paralysis can threaten:http://arxiv.org/abs/1205.3773
and suggest a solution.
Schroeren then begins to adapt the approach to Spinfoam QG: http://arxiv.org/abs/1206.4553

 

[Demystifier]

The thing is that conventional quantum mechanics does not work for cosmology because no classical observer (or measuring device) outside the System.

It depends on what one means by "cosmology". If it means "everything", then it's true. But no MEASUREMENT in cosmology is a measurement of "everything". In observational cosmology, there are always many degrees of freedom which are not observed and thus can play the role of an observer. So, as long as the goal of theoretical cosmology is to explain or predict OBSERVATIONS, I see no problem with conventional quantum mechanics.

 

[marcus]

... I see no problem with conventional quantum mechanics.

But Jim Hartle does see a problem.
And I gather quite a few other people agree with him including his co-author Murray Gell-Mann.
So it's controversial--and has been a topic of discussion for 20 years or so.

This is really out of my league but I will try to explain as best I can, although maybe YOU could do a better job at presenting the other side's nontrivial argument.
It would be interesting to hear YOU explain Hartle's reasoning, say from the 1993 writeup of his Les Houches lectures.

I guess that a key point is that in a conventional setup whatever plays the role of the observer or the measuring instrument is presumed to be CLASSICAL.

So you mention that whatever observation is to be made only involves some of the angels and there are always going to be plenty of other angels ("degrees of freedom" which we do not know what they are or of which we have at best a foggy notion) which can play the role of an observer.

And I imagine that Hartle might shake his head and say that you cannot make a clean division and that anyway the other angels are not a classical deterministic system as Bohr and the others would like them to be.

Hartle does not say the problem can not be solved. In fact he proposes a solution. He just thinks that a somewhat different formalism needs to be developed---not quite the conventional, but still quantum mechanical in spirit.

For all I know what he is proposing as a "generalized quantum mechanics" could be similar in some respects to what YOU have in mind.

 

[marcus]

Demy, to be more specific as to what we're talking about, if I google
"Hartle spacetime quantum"
here are the top two hits:Generalizing Quantum Mechanics for Quantum Spacetime
arxiv.org › gr-qc
by JB Hartle - 2006 - Cited by 11 - Related articles
Feb 2, 2006 – Title: Generalizing Quantum Mechanics for Quantum Spacetime. Authors: James B. Hartle (University of California, Santa Barbara). (Submitted ...

Spacetime Quantum Mechanics and the Quantum Mechanics of ...
arxiv.org › gr-qc
by JB Hartle - 1993 - Cited by 194 - Related articles
Title: Spacetime Quantum Mechanics and the Quantum Mechanics of Spacetime. Authors: James B. Hartle. (Submitted on 5 Apr 1993 (v1), last revised 6 Apr ...

The second is the one I was looking at earlier. I see an analogy with Riemann in 1850. One thing Riemann did was devise tools for us to experience geometry from the inside.
You don't have to be outside the space to see and understand the shape. If you are a 2D creature living in a sphere (the hollow 2D surface) you can measure the curvature entirely from within that world without ever conceiving of a third spatial dimension.

I think Hartle is trying to teach us how to experience quantum mechanics as quantum mechanical creatures ourselves in other words "from the inside."

His method is to look for nearly decoherent partitions of the histories of the (all-encompassing) system.
Partition the histories into sets which are so nearly uncorrelated/non-interfering that we can assign them probabilities. And the probabilities are nice---they add up almost right.

Such partitions (something happened or it didn't happen) allow us, quantum creatures that we are, to make sense of our world---at least to a certain extent.
From within that world---we don't have to get outside it and pretend to be classical.

And you might say, "Oh, I knew how to do that all along!" Perhaps someone said that to Riemann about experiencing geometry from the inside. Perhaps they were right. But Riemann was saying to focus on the devices he had devised to do it with. Focus attention on the tools to measure curvature internally.

And maybe that is the main thing Hartle is saying: "focus on how we partition the histories of the geometry we (quantum creatures) live in."

I'd like to know in what ways you see it differently. I've just sampled this discussion (because of David Schroeren's new paper) and am not sure about it.

 

[Demystifier]

So you mention that whatever observation is to be made only involves some of the angels and there are always going to be plenty of other angels ("degrees of freedom" which we do not know what they are or of which we have at best a foggy notion) which can play the role of an observer.

Yes, that's what I am saying.

And I imagine that Hartle might shake his head and say that you cannot make a clean division and that anyway the other angels are not a classical deterministic system as Bohr and the others would like them to be.

Even if it's true, I don't see how it would be specific to quantum cosmology.

For all I know what he is proposing as a "generalized quantum mechanics" could be similar in some respects to what YOU have in mind.

Yes, I am also proposing a "generalized quantum mechanics" (similar to that of Hartle) in some of my papers, and I cite Hartle in those papers. See
http://xxx.lanl.gov/abs/0905.0538
http://xxx.lanl.gov/abs/0912.1938

It is far from that I don't see a problem with conventional QM. I certainly do. But in my opinion
1) The problem exists even without quantum cosmology.
2) Quantum cosmology as a theoretical description of EVERYTHING makes the problem even harder, but
3) OBSERVATIONAL quantum cosmology does not include everything, so it does not make the problem harder than in other quantum phenomena.

And I believe Hartle would agree with all these 3 points.

 

[marcus]

...
It is far from that I don't see a problem with conventional QM. I certainly do. But in my opinion
1) The problem exists even without quantum cosmology.
2) Quantum cosmology as a theoretical description of EVERYTHING makes the problem even harder, but
3) OBSERVATIONAL quantum cosmology does not include everything, so it does not make the problem harder than in other quantum phenomena.

And I believe Hartle would agree with all these 3 points.

Those are good points. I agree as well. To repeat what you said in my own words, there are problems with conventional QM beyond the QC one (which however is simple to explain and understand.)

QC makes it "even harder". Indeed, in my view, this is the easiest most direct approach to explaining how necessary it is to develop a "generalized QM". The SPLIT between the quantum system and the classical observer is artificial, does not fit nature, and fails as a theory of the universe.

One can give other justifications for proceeding on this path, surely. (I would be glad to hear yours.) But for me this is the simplest argument to make for it.

Hopefully if Hartle's or some other "generalized QM" develops which removes this split and is satisfactory for QC, then it be an improvement in other ways as well---an all-round better framework.

About "observational" I am mainly interested in what QC might provide by way of theoretical models of the early universe. (There's also the related problem of modeling the black hole interior.)

I'm not sure how to define "observational" in this context. Hartle often makes the point that in the early universe a classical observer or instrument did not and could not exist.

I think one can make a simple tautology that observational astronomy of any sort is done in a context where there are observers and instruments which we can consider to be classical.
So of course you have a valid point and I agree.

 

[Demystifier]

One can give other justifications for proceeding on this path, surely. (I would be glad to hear yours.) But for me this is the simplest argument to make for it.

I think it is not difficult to find various arguments that standard QM is not perfect. What is much more difficult and challenging is to propose something better. Personally, as a better formulation of QM I prefer approaches of the Bohmian type. For an overview of my recent results in that direction see
http://xxx.lanl.gov/abs/1205.1992

 

[marcus]

I think it is not difficult to find various arguments that standard QM is not perfect. What is much more difficult and challenging is to propose something better. Personally, as a better formulation of QM I prefer approaches of the Bohmian type. For an overview of my recent results in that direction see
http://xxx.lanl.gov/abs/1205.1992

Thanks for the reference. I see that this is being published as a chapter in the book
"Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology", edited by X. Oriols and J. Mompart (Pan Stanford Publishing, 2012).
It is an interesting title. I will keep an eye out for it.

At the moment I want to focus on the Hartle scheme for generalized QM, partly because it is being considered as a framework for formulating spinfoam QG and I can really only deal with one new thing at a time.

It's really new. You could help me understand Hartle's approach if you are at all familiar with it (as I think you must be.)

My understanding is that the basic language of his generalized QM involves PARTITIONS of the space of histories. A partition being a set of disjoint subsets whose union is the whole.
The operation of refinement gives a partial ordering on the set of all partitions.
A coarse-graining (as Hartle uses the term) is simply the inverse of a refinement.

I assume that's right so far, please let me know if you see that I'm mistaken about something. And then some partitions have the feature of being approximately decoherent so that among other things the usual rule of additive probability holds in an approximate sense.

 

[marcus]

I guess I should give the links for the two Hartle articles I mentioned in post #5

...google
"Hartle spacetime quantum mechanics"
here are the top two hits:

Generalizing Quantum Mechanics for Quantum Spacetime
arxiv.org › gr-qc
by JB Hartle - 2006 - Cited by 11 - Related articles
Feb 2, 2006 – Title: Generalizing Quantum Mechanics for Quantum Spacetime. Authors: James B. Hartle (University of California, Santa Barbara). (Submitted ...

Spacetime Quantum Mechanics and the Quantum Mechanics of ...
arxiv.org › gr-qc
by JB Hartle - 1993 - Cited by 194 - Related articles
Title: Spacetime Quantum Mechanics and the Quantum Mechanics of Spacetime. Authors: James B. Hartle. (Submitted on 5 Apr 1993 (v1), last revised 6 Apr ...

The links are
http://arxiv.org/abs/gr-qc/0602013
http://arxiv.org/abs/gr-qc/9304006
======================
There's also a nice quote from this paper: arxiv 0801.0688
It brings to mind the thought that quantum mechanics notices even the motion of a mote of dust in intersteller space or as one once said "even the fall of a sparrow." (Hamlet Act 5, scene 10)
This is from page 13 of that paper:
==quote Hartle==
Sets of histories obeying [usual prob. sum rule] are said to (medium) decohere. As L. Diosi has shown [29], medium decoherence is the weakest of known conditions that are consistent with elementary notions of the independence of isolated systems. Medium-decoherent sets are thus the ones for which quantum mechanics makes predictions of consistent probabilities through (A.6)...
An important mechanism of decoherence is the dissipation of phase coherence between branches into variables not followed by the coarse graining. Consider by way of example, a dust grain in a superposition of two positions deep in interstellar space [34]. In our universe, about 10¹¹ cosmic background photons scatter from the dust grain each second. The two positions of the grain become correlated with different, nearly orthogonal states of the photons. Coarse grainings that follow only the position of the dust grain at a few times therefore correspond to branch state vectors that are nearly orthogonal and satisfy (A.8).
==endquote==
The universe is constantly disambiguating itself because of the CMB cosmic microwave background, if by no other mechanism.
Here's a further passage from page 14:
==quote Hartle==
Measurements and observers play no fundamental role in this general formulation of usual quantum theory. The probabilities of measured outcomes can be computed and are given to an excellent approximation by the usual story. But, in a set of histories where they decohere, probabilities can be assigned to the position of the Moon when it is not receiving the attention of observers and to the values of density fluctuations in the early universe when there were neither measurements taking place nor observers to carry them out.
==endquote==

In other words this way of formulating quantum theory is just as good as the others for handling measurements and predictions and where an observer is involved. It gives the same answers.
But the ontology does not depend on the existence of an observer or on an act of measurement.
Sets of all possible histories, whether this be of a mote of dust or the geometry of the early universe, can be partitioned. Component subsets of histories can under favorable circumstances be sufficiently disconnected that their probabilities add up right...
Look, Mom, no observer!
Here's the article's abstract in case it might prove useful for this discussion.
http://arxiv.org/abs/0801.0688
Quantum Mechanics with Extended Probabilities
James B. Hartle
(Submitted on 4 Jan 2008)
The quantum mechanics of closed systems such as the universe is formulated using an extension of familiar probability theory that incorporates negative probabilities. Probabilities must be positive for sets of alternative histories that are the basis of fair settleable bets. However, in quantum mechanics there are sets of alternative histories that can be described but which cannot be the basis for fair settleable bets. Members of such sets can be assigned extended probabilities that are sometimes negative. A prescription for extended probabilities is introduced that assigns extended probabilities to all histories that can be described, fine grained or coarse grained, members of decoherent sets or not. All probability sum rules are satisfied exactly. Sets of histories that are recorded to sufficient precision are the basis of settleable bets. This formulation is compared with the decoherent (consistent) histories formulation of quantum theory. Prospects are discussed for using this formulation to provide testable alternatives to quantum theory or further generalizations of it.
15 pages, 2 figures.

 

[Demystifier]

You could help me understand Hartle's approach if you are at all familiar with it (as I think you must be.)

I was interested in his approach mainly because I saw some partial similarity with my approach. Thus, you should not be surprised that I understand well only those aspects of his approach which are similar to mine. As a consequence, I am not the best person to answer you specific questions on his approach not related to that of mine.