Comments on Weingarten’s preferred branch

A senior colleague asked me for thoughts on this paper describing a single-preferred-branch flavor of quantum mechanics, and I thought I’d copy them here. Tl;dr: I did not find an important new idea in it, but this paper nicely illustrates the appeal of Finkelstein’s partial-trace decoherence and the ambiguity inherent in connecting a many-worlds wavefunction to our direct observations.

We start by assuming that a precise wavefunction branch structure has been specified. The idea, basically, is to randomly draw a branch at late times according to the Born probability, then to evolve it backwards in time to the beginning of the universe and take that as your initial condition. The main motivating observation is that, if we assume that all branch splittings are defined by a projective decomposition of some subsystem (‘the system’) which is recorded faithfully elsewhere (‘the environment’), then the lone preferred branch — time-evolving by itself — is an eigenstate of each of the projectors defining the splits. In a sense, Weingarten lays claim to ordered consistency [arxiv:gr-qc/9607073] by assuming partial-trace decoherenceNote on terminology: What Finkelstein called “partial-trace decoherence” is really a specialized form of consistency (i.e., a mathematical criterion for sets of consistent histories) that captures some, but not all, of the properties of the physical and dynamical process of decoherence. That’s why I’ve called it “partial-trace consistency” here and here.a   [arXiv:gr-qc/9301004]. In this way, the macrostate states stay the same as normal quantum mechanics but the microstates secretly conspire to confine the universe to a single branch.

I put proposals like this in the same category as Bohmian mechanics. They take as assumptions the initial state and unitary evolution of the universe, along with the conventional decoherence/amplification story that argues for (but never fully specifies from first principles) a fuzzy, time-dependent decomposition of the wavefunction into branches.… [continue reading]

Weinberg on the measurement problem

In his new article in the NY Review of Books, the titan Steven Weinberg expresses more sympathy for the importance of the measurement problem in quantum mechanics. The article has nothing new for folks well-versed in quantum foundations, but Weinberg demonstrates a command of the existing arguments and considerations. The lengthy excerpts below characterize what I think are the most important aspects of his view.

Many physicists came to think that the reaction of Einstein and Feynman and others to the unfamiliar aspects of quantum mechanics had been overblown. This used to be my view. After all, Newton’s theories too had been unpalatable to many of his contemporaries…Evidently it is a mistake to demand too strictly that new physical theories should fit some preconceived philosophical standard.

In quantum mechanics the state of a system is not described by giving the position and velocity of every particle and the values and rates of change of various fields, as in classical physics. Instead, the state of any system at any moment is described by a wave function, essentially a list of numbers, one number for every possible configuration of the system….What is so terrible about that? Certainly, it was a tragic mistake for Einstein and Schrödinger to step away from using quantum mechanics, isolating themselves in their later lives from the exciting progress made by others. Even so, I’m not as sure as I once was about the future of quantum mechanics. It is a bad sign that those physicists today who are most comfortable with quantum mechanics do not agree with one another about what it all means. The dispute arises chiefly regarding the nature of measurement in quantum mechanics…

The introduction of probability into the principles of physics was disturbing to past physicists, but the trouble with quantum mechanics is not that it involves probabilities.

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Three arguments on the measurement problem

When talking to folks about the quantum measurement problem, and its potential partial resolution by solving the set selection problem, I’ve recently been deploying three nonstandard arguments. To a large extent, these are dialectic strategies rather than unique arguments per se. That is, they are notable for me mostly because they avoid getting bogged down in some common conceptual dispute, not necessarily because they demonstrate something that doesn’t formally follow from traditional arguments. At least two of these seem new to me, in the sense that I don’t remember anyone else using them, but I strongly suspect that I’ve just appropriated them from elsewhere and forgotten. Citations to prior art are highly appreciated.

Passive quantum mechanics

There are good reasons to believe that, at the most abstract level, the practice of science doesn’t require a notion of active experiment. Rather, a completely passive observer could still in principle derive all fundamental physical theories simply by sitting around and watching. Science, at this level, is about explaining as many observations as possible starting from as minimal assumptions as possible. Abstractly we frame science as a compression algorithm that tries to find the programs with the smallest Kolmogorov complexity that reproduces observed data.

Active experiments are of course useful for at least two important reasons: (1) They gather strong evidence for causality by feeding a source of randomness into a system to test a causal model, and (2) they produce sources of data that are directly correlated with systems of interest rather than relying on highly indirect (and perhaps computationally intractable) correlations. But ultimately these are practical considerations, and an inert but extraordinarily intelligent observer could in principle derive general relativity, quantum mechanics, and field theoryOf course, there may be RG-reasons to think that scales decouple, and that to a good approximation the large-scale dynamics are compatible with lots of possible small-scale dynamics.[continue reading]

Comments on an essay by Wigner

[PSA: Happy 4th of July. Juno arrives at Jupiter tonight!]

This is short and worth reading:

This essay has no formal abstract; the above is the second paragraph, which I find to be profound. Here is the PDF. The essay shares the same name and much of the material with Wigner’s 1963 Nobel lecture [PDF].The Nobel lecture has a nice bit contrasting invariance principles with covariance principles, and dynamical invariance principles with geometrical invariance principles.a  

Some comments:

  • It is very satisfying to see Wigner — the titan — highlight the deep importance of the seminal work by the grandfather of my field, Dieter Zeh. Likewise for his comments on Bell:

    As to the J.S. Bell inequalities, I consider them truly important, inasmuch as they prove that in the case considered by him, one cannot define a non-negative probability function which describes the state of his system in the classical sense, i.e., gives nonnegative probabilities for all possible events….

    This is a very interesting and very important observation and it is truly surprising that it has not been made before. Perhaps some of those truly interested in the epistemology of quantum mechanics took it for granted but they did not demonstrate it.

  • I like the hierarchy of regularity that Wigner draws: data ➢ laws ➢ symmetries. Symmetries are strong restrictions on, but do not determine, laws in the same way that laws are strong restrictions on, but do not determine, data.
  • It is interesting that Wigner tried to embed relativistic restrictions into the description of initial states:

    Let me mention, finally, one effect which the theory of relativity should have introduced into the description of the initial conditions and perhaps also into the description of all states.

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Comments on Rosaler’s “Reduction as an A Posteriori Relation”

In a previous post of abstracts, I mentioned philosopher Josh Rosaler’s attempt to clarify the distinction between empirical and formal notions of “theoretical reduction”. Reduction is just the idea that one theory reduces to another in some limit, like Galilean kinematics reduces to special relativity in the limit of small velocities.Confusingly, philosophers use a reversed convention; they say that Galilean mechanics reduces to special relativity.a   Formal reduction is when this takes the form of some mathematical limiting procedure (e.g., v/c \to 0), whereas empirical reduction is an explanatory statement about observations (e.g., “special relativity can explains the empirical usefulness of Galilean kinematics”).

Rosaler’s criticism, which I mostly agree with, is that folks often conflate these two. Usually this isn’t a serious problem since the holes can be patched up on the fly by a competent physicist, but sometimes it leads to serious trouble. The most egregious case, and the one that got me interested in all this, is the quantum-classical transition, and in particular the serious insufficiency of existing \hbar \to 0 limits to explain the appearance of macroscopic classicality. In particular, even though this limiting procedure recovers the classical equations of motion, it fails spectacularly to recover the state space.There are multiple quantum states that have the same classical analog as \hbar \to 0, and there are quantum states that have no classical analog as \hbar \to 0.b  

In this post I’m going to comment Rosaler’s recent elaboration on this ideaI thank him for discussion this topic and, full disclosure, we’re drafting a paper about set selection together.c  :

I was tempted to interpret the thesis of this essay like this:

The only useful notion of theory reduction is necessarily intertwined with empirical facts about the domain of applicability.[continue reading]

Comments on Myrvold’s Taj Mahal

Last week I saw an excellent talk by philosopher Wayne Myrvold.

(Download MP4 video here.)

The topic was well-defined, and of reasonable scope. The theorem is easily and commonly misunderstood. And Wayne’s talk served to dissolve the confusion around it, by unpacking the theorem into a handful of pieces so that you could quickly see where the rub was. I would that all philosophy of physics were so well done.

Here are the key points as I saw them:

  • The vacuum state in QFTs, even non-interacting ones, is entangled over arbitrary distances (albeit by exponentially small amounts). You can think of this as every two space-like separated regions of spacetime sharing extremely diluted Bell pairs.
  • Likewise, by virtue of its non-local nature, the vacuum contains non-zero (but stupendously tiny) overlap with all localized states. If you were able to perform a “Taj-Mahal measurement” on a region R, which ask the Yes-or-No question “Is there a Taj Mahal in R?”, you always have a non-zero (but stupendously tiny) chance of getting “Yes” and finding a Taj Mahal.
  • This non-locality arises directly from requiring the exact spectral condition, i.e., that the Hamiltonian is bounded from below. This is because the spectral condition is a global statement about modes in spacetime. It asserts that allowed states have overlap only with the positive part of the mass shell.
  • This is very analogous to the way that analytic functions are determined by their behavior in an arbitrarily small open patch of the complex plane.
  • This theorem says that some local operator, when acting on the vacuum, can produce the Taj-Mahal in a distant, space-like separated region of space-time.
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How to think about Quantum Mechanics—Part 6: Energy conservation and wavefunction branches

[Other parts in this series: .]

In discussions of the many-worlds interpretation (MWI) and the process of wavefunction branching, folks sometimes ask whether the branching process conflicts with conservations laws like the conservation of energy.Here are some related questions from around the web, not addressing branching or MWI. None of them get answered particularly well.a   There are actually two completely different objections that people sometimes make, which have to be addressed separately.

First possible objection: “If the universe splits into two branches, doesn’t the total amount of energy have to double?” This is the question Frank Wilczek appears to be addressing at the end of these notes.

I think this question can only be asked by someone who believes that many worlds is an interpretation that is just like Copenhagen (including, in particular, the idea that measurement events are different than normal unitary evolution) except that it simply declares that new worlds are created following measurements. But this is a misunderstanding of many worlds. MWI dispenses with collapse or any sort of departure from unitary evolution. The wavefunction just evolves along, maintaining its energy distributions, and energy doesn’t double when you mathematically identify a decomposition of the wavefunction into two orthogonal components.

Second possible objection: “If the universe starts out with some finite spread in energy, what happens if it then ‘branches’ into multiple worlds, some of which overlap with energy eigenstates outside that energy spread?” Or, another phrasing: “What happens if the basis in which the universe decoheres doesn’t commute with energy basis? Is it then possible to create energy, at least in some branches?” The answer is “no”, but it’s not obvious.… [continue reading]

How to think about Quantum Mechanics—Part 4: Quantum indeterminism as an anomaly

[Other parts in this series: .]

I am firmly of the view…that all the sciences are compatible and that detailed links can be, and are being, forged between them. But of course the links are subtle… a mathematical aspect of theory reduction that I regard as central, but which cannot be captured by the purely verbal arguments commonly employed in philosophical discussions of reduction. My contention here will be that many difficulties associated with reduction arise because they involve singular limits….What nonclassical phenomena emerge as h → 0? This sounds like nonsense, and indeed if the limit were not singular the answer would be: no such phenomena.Michael Berry

One of the great crimes against humanity occurs each year in introductory quantum mechanics courses when students are introduced to an \hbar \to 0 limit, sometimes decorated with words involving “the correspondence principle”. The problem isn’t with the content per se, but with the suggestion that this somehow gives a satisfying answer to why quantum mechanics looks like classical mechanics on large scales.

Sometimes this limit takes the form of a path integral, where the transition probability for a particle to move from position x_1 to x_2 in a time T is

(1)   \begin{align*} P_{x_1 \to x_2} &= \langle x_1 \vert e^{-i H T} \vert x_2 \rangle \\ &\propto \int_{x_1,x_2} \mathcal{D}[x(t)] e^{-i S[x(t),x'(t)]/\hbar} = \int_{x_1,x_2} \mathcal{D}[x(t)] e^{-i \int_0^T \mathrm{d}t L(x(t),x'(t))/\hbar} \end{align*}

where \int_{x_1,x_2} \mathcal{D}[x(t)] is the integral over all paths from x_1 to x_2, and S[x(t),x'(t)]= \int_0^T \mathrm{d}t L(x(t),x'(t)) is the action for that path (L being the Lagrangian corresponding to the Hamiltonian H). As \hbar \to 0, the exponent containing the action spins wildly and averages to zero for all paths not in the immediate vicinity of the classical path that make the action stationary.

Other times this takes the form of Ehrenfest’s theorem, which shows that the expectation values of functions of position and momentum follow the classical equations of motion.… [continue reading]