State-independent consistent sets

In May, Losada and Laura wrote a paperM. Losada and R. Laura, Annals of Physics 344, 263 (2014). a   pointing out the equivalence between two conditions on a set of “elementary histories” (i.e. fine-grained historiesGell-Mann and Hartle usually use the term “fine-grained set of histories” to refer to a set generated by the finest possible partitioning of histories in path integral (i.e. a point in space for every point in time), but this is overly specific. As far as the consistent histories framework is concerned, the key mathematical property that defines a fine-grained set is that it’s an exhaustive and exclusive set where each history is constructed by choosing exactly one projector from a fixed orthogonal resolution of the identity at each time. b  ). Let the elementary histories \alpha = (a_1, \dots, a_N) be defined by projective decompositions of the identity P^{(i)}_{a_i}(t_i) at time steps t_i (i=1,\ldots,N), so that

(1)   \begin{align*} P^{(i)}_a &= (P^{(i)}_a)^\dagger \quad \forall i,a \\ P^{(i)}_a P^{(i)}_b &= \delta_{a,b} P^{(i)}_a \quad \forall i,a,b\\ \sum_{a} P^{(i)}_a (t_i) &= I \quad  \forall i,k \\ C_\alpha &= P^{(N)}_{a_N} (t_N) \cdots P^{(1)}_{a_1} (t_1) \\ I &= \sum_\alpha C_\alpha = \sum_{a_1}\cdots \sum_{a_N} C_\alpha \\ \end{align*}

where C_\alpha are the class operators. Then Losada and Laura showed that the following two conditions are equivalent

  1. The set is consistent“Medium decoherent” in Gell-Mann and Hartle’s terminology. Also note that Losada and Laura actually work with the obsolete condition of “weak decoherence”, but this turns out to be an unimportance difference. For a summary of these sorts of consistency conditions, see my round-up. c   for any state: D(\alpha,\beta) = \mathrm{Tr}[C_\alpha \rho C_\beta^\dagger] = 0 \quad \forall \alpha \neq \beta, \forall \rho.
  2. The Heisenberg-picture projectors at all times commute: [P^{(i)}_{a} (t_i),P^{(j)}_{b} (t_j)]=0 \quad \forall i,j,a,b.

However, this is not as general as one would like because assuming the set of histories is elementary is very restrictive. (It excludes branch-dependent sets, sets with inhomogeneous histories, and many more types of sets that we would like to work with.) Luckily, their proof can be extended a bit.… [continue reading]

Consistency conditions in consistent histories

[This is akin to a living review, which may improve from time to time. Last edited 2015-4-27.]

This post will summarize the various consistency conditions that can be found discussed in the consistent histories literature. Most of the conditions have gone by different names under different authors (and sometimes even under the same author), so I’ll try to give all the aliases I know; just hover over the footnote markers.

There is an overarching schism in the choice of terminology in the literature between the terms “consistent” and “decoherent”. Most authors, including Gell-Mann and Hartle, now use the term “decoherent” very loosely and no longer employ “consistent” as an official label for any particular condition (or for the formalism as a whole). Zurek and I believe this is a significant loss in terminology, and we are stubbornly resisting it. In our recent arXiv offering, our rant was thus:

…we emphasize that decoherence is a dynamical physical process predicated on a distinction between system and environment, whereas consistency is a static property of a set of histories, a Hamiltonian, and an initial state. For a given decohering quantum system, there is generally a preferred basis of pointer states [1, 8]. In contrast, the mere requirement of consistency does not distinguish a preferred set of histories which describe classical behavior from any of the many sets with no physical interpretation.

(See also the first footnote on page 3347 of “Classical Equations for Quantum Systems”Gell-Mann and Hartle a   which agrees with the importance of this conceptual distinction.) Since Gell-Mann and Hartle did many of the investigations of consistency conditions, some conditions have only appeared in the literature using their terminology (like “medium-strong decoherence”).… [continue reading]

Follow-up questions on the set-selection problem

Physics StackExchange user QuestionAnswers asked the question “Is the preferred basis problem solved?“, and I reproduced my “answer” (read: discussion) in a post last week.  He had some thoughtful follow-up questions, and (with his permission) I am going to answer them here. His questions are in bold, with minor punctuation changes.

How serious would you consider what you call the “Kent set-selection” problem?

If a set of CHs could be shown to be impossible to find, then this would break QM without necessarily telling us how to correct it. (Similar problems exist with the breakdown of gravity at the Planck scale.) Although I worry about this, I think it’s unlikely and most people think it’s very unlikely. If a set can be found, but no principle can be found to prefer it, I would consider QM to be correct but incomplete. It would kinda be like if big bang neucleosynthesis had not been discovered to explain the primordial frequency of elements.

And what did Zurek think of it, did he agree that it’s a substantial problem?

I think Wojciech believes a set of consistent histories (CHs) corresponding to the branch structure could be found, but that no one will find a satisfying beautiful principle within the CH framework which singles out the preferred set from the many, many other sets. He believes the concept of redundant records (see “quantum Darwinism”) is key, and that a set of CHs could be found after the fact, but that this is probably not important. I am actually leaving for NM on Friday to work with him on a joint paper exploring the connection between redundancy and histories.… [continue reading]

Kent’s set-selection problem

Unfortunately, physicists and philosophers disagree on what exactly the preferred basis problem is, what would constitute a solution, and how this relates (or subsumes) “the measurement problem” more generally. In my opinion, the most general version of the preferred basis problem was best articulated by Adrian Kent and Fey Dowker near the end their 1996 article “On the Consistent Histories Approach to Quantum Mechanics” in the Journal of Statistical Physics. Unfortunately, this article is long so I will try to quickly summarize the idea.

Kent and Dowker analyzed the question of whether the consistent histories formalism provided a satisfactory and complete account of quantum mechanics (QM). Contrary to what is often said, consistent histories and many-worlds need not be opposing interpretations of quantum mechanics Of course, some consistent historians make ontological claims about how the histories are “real”, where as the many-world’ers might say that the wavefunction is more “real”. In this sense they are contradictory. Personally, I think this is purely a matter of taste. a  . Instead, consistent histories is a good mathematical framework for rigorously identifying the branch structure of the wavefunction of the universe Note that although many-worlders may not consider the consistent histories formalism the only way possible to mathematically identify branch structure, I believe most would agree that if, in the future, some branch structure was identified using a completely different formalism, it could be described at least approximately by the consistent histories formalism.  Consistent histories may not be perfect, but it’s unlikely that the ideas are totally wrong. b  . Most many-world’ers would agree that unambiguously describing this branch structure would be very nice (although they might disagree on whether this is “necessary” for QM to be a complete theory).… [continue reading]