Free will and retrocausality at Cambridge II: Conspiracy vs. Retrocausality; Signaling and Fine-Tuning

Expect (with moderate probability) substantial revisions to this post, hopefully including links to relevant talks from the Cambridge conference on retrocausality and free will in quantum theory, but for now I think it's best just to put this out there.

Conspiracy versus Retrocausality

One of the main things I hoped to straighten out for myself at the conference on retrocausality in Cambridge was whether the correlation between measurement settings and "hidden variables" involved in a retrocausal explanation of Bell-inequality-violating quantum correlations are necessarily "conspiratorial", as Bell himself seems to have thought.  The idea seems to be that correlations between measurement settings and hidden variables must be due to some "common cause" in the intersection of the backward light cones of the two.  That is, a kind of "conspiracy" coordinating the relevant hidden variables that can affect the meausrement outcome with all sorts of intricate processes that can affect which measurement is made, such as those affecting your "free" decision as to how to set a polarizer, or, in case you set up a mechanism to control the polarizer setting according to some apparatus reasonably viewed as random ("the Swiss national lottery machine" was the one envisioned by Bell), the functioning of this mechanism.  I left the conference convinced once again (after doubts on this score had been raised in my mind by some discussions at New Directions in the Philosophy of Physics 2013) that the retrocausal type of explanation Price has in mind is different from a conspiratorial one.

Deflationary accounts of causality: their impact on retrocausal explanation

Distinguishing "retrocausality" from "conspiratorial causality" is subtle, because it is not clear that causality makes sense as part of a fundamental physical theory.   (This is a point which, in this form, apparently goes back to Bertrand Russell early in this century.  It also reminds me of David Hume, although he was perhaps not limiting his "deflationary" account of causality to causality in physical theories.)  Causality might be a concept that makes sense at the fundamental level for some types of theory, e.g. a version ("interpretation") of quantum theory that takes measurement settings and outcomes as fundamental, taking an "instrumentalist" view of the quantum state as a means of calculating outcome probabilities giving settings, and not as itself real, without giving a further formal theoretical account of what is real.  But in general, a theory may give an account of logical implications between events, or more generally, correlations between them, without specifying which events cause, or exert some (perhaps probabilistic) causal influence on others.  The notion of causality may be something that is emergent, that appears from the perspective of beings like us, that are part of the world, and intervene in it, or model parts of it theoretically.  In our use of a theory to model parts of the world, we end up taking certain events as "exogenous".  Loosely speaking, they might be determined by us agents (using our "free will"), or by factors outside the model.  (And perhaps "determined" is the wrong word.)   If these "exogenous" events are correlated with other things in the model, we may speak of this correlation as causal influence.  This is a useful way of speaking, for example, if we control some of the exogenous variables:  roughly speaking, if we believe a model that describes correlations between these and other variables not taken as exogenous, then we say these variables are causally influenced by the variables we control that are correlated with them.  We find this sort of notion of causality valuable because it helps us decide how to influence those variables we can influence, in order to make it more likely that other variables, that we don't control directly, take values we want them to.  This view of causality, put forward for example in Judea Pearl's book "Causality", has been gaining acceptance over the last 10-15 years, but it has deeper roots.  Phil Dowe's talk at Cambridge was an especially clear exposition of this point of view on causality (emphasizing exogeneity of certain variables over the need for any strong notion of free will), and its relevance to retrocausality.

This makes the discussion of retrocausality more subtle because it raises the possibility that a retrocausal and a conspiratorial account of what's going on with a Bell experiment might describe the same correlations, between the Swiss National lottery machine, or whatever controls my whims in setting a polarizer, all the variables these things are influenced by, and the polarizer settings and outcomes in a Bell experiment, differing only in the causal relations they describe between these variables.  That might be true, if a retrocausalist decided to try to model the process by which the polarizer was set.  But the point of the retrocausal account seems to be that it is not necessary to model this to explain the correlations between measurement results.  The retrocausalist posits a lawlike relation of correlation between measurement settings and some of the hidden variables that are in the past light cone of both measurement outcomes.  As long as this retrocausal influence does not influence observable past events, but only the values of "hidden", although real, variables, there is nothing obviously more paradoxical about imagining this than about imagining----as we do all the time---that macroscopic variables that we exert some control over, such as measurement settings, are correlated with things in the future.   Indeed, as Huw Price has long (I have only recently realized for just how long) been pointing out, if we believe that the fundamental laws of physics are symmetric with respect to time-reversal, then it would be the absence of retrocausality, if we dismiss its possibility, and even if we accept its possibility to the limited extent needed to potentially explain Bell correlations, its relative scarcity, that needs explaining.  Part of the explanation, of course, is likely that causality, as mentioned above, is perhaps a notion that is useful for agents situated within the world, rather than one that applies to the "view from nowhere and nowhen" that some (e.g. Price, who I think coined the term "nowhen") think is, or should be,  taken by fundamental physical theories.  Therefore whatever asymmetries---- these could be somewhat local-in-spacetime even if extremely large-scale, or due to "spontaneous" (i.e. explicit, even if due to a small perturbation) symmetry-breaking --- are associated with our symmetry-breaking experience of directionality of time --- may be the explanation for why we introduce the causal arrows we do into our description, and therefore why we so rarely introduce retrocausal ones.  At the same time, such an explanation might well leave room for the limited retrocausality Price would like to introduce into our description, for the purpose of explaining Bell correlations, especially because such retrocausality does not allow backwards-in-time signaling.

Signaling (spacelike and backwards-timelike) and fine-tuning. Emergent no-signaling?

A theme that came up repeatedly at the conference was "fine-tuning"---that no-spacelike-signaling, and possibly also no-retrocausal-signaling, seem to require a kind of "fine-tuning" from a hidden variable model that uses them to explain quantum correlations.  Why, in Bohmian theory, if we have spacelike influence of variables we control on physically real (but not necessarily observable) variables, should things be arranged just so that we cannot use this influence to remotely control observable variables, i.e. signal?  Similarly one might ask why, if we have backwards-in-time influence of controllable variables on physically real variables, things are arranged just so that we cannot use this influence to remotely control observable variables at an earlier time?  I think --- and I think this possibility was raised at the conference --- that a possible explanation, suggested by the above discussion of causality, is that for macroscopic agents such as us, with usually-reliable memories, some degree of control over our environment and persistence over time, to arise, it may be necessary that the scope of such macroscopic "observable" influences be limited, in order that there be a coherent macroscopic story at all for us to tell---in order for us even be around to wonder about whether there could be such signalling or not.  (So the term "emergent no-signalling" in the section heading might be slightly misleading: signalling, causality, control, and limitations on signalling might all necessarily emerge together.) Such a story might end up involving thermodynamic arguments, about the sorts of structures that might emerge in a metastable equilibrium, or that might emerge in a dynamically stable state dependent on a temperature gradient, or something of the sort.  Indeed, the distribution of hidden variables (usually, positions and/or momenta) according to the squared modulus of the wavefunction, which is necessary to get agreement of Bohmian theory with quantum theory and also to prevent signaling (and which does seem like "fine-tuning" inasmuch as it requires a precise choice of probability distribution over initial conditions), has on various occasions been justified by arguments that it represents a kind of equilibrium that would be rapidly approached even if it did not initially obtain.  (I have no informed view at present on how good these arguments are, though I have at various times in the past read some of the relevant papers---Bohm himself, and Sheldon Goldstein, are the authors who come to mind.)

I should mention that against such arguments for "emergent" no-signalling, the point was made at the conference that one might expect "fluctuations" away from the equilibria, metastable structures, or steady states, but we don't observe small fluctuations away from no-signalling---the law seems to hold with certainty. This is an important point, and although I suspect there may be adequate rejoinders, I don't see at the moment what these might be like.

Free will and retrocausality in the quantum world, at Cambridge. I: Bell inequalities and retrocausality

I'm in Cambridge, where the conference on Free Will and Retrocausality in the Quantum World, organized (or rather, organised) by Huw Price and Matt Farr will begin in a few hours.  (My room at St. Catherine's is across from the chapel, and I'm being serenaded by a choir singing beautifully at a professional level of perfection and musicality---I saw them leaving the chapel yesterday and they looked, amazingly, to be mostly junior high school age.)  I'm hoping to understand more about how "retrocausality", in which effects occur before their causes, might help resolve some apparent problems with quantum theory, perhaps in ways that point to potentially deeper underlying theories such as a "quantum gravity". So, as much for my own use as anyone else's, I thought perhaps I should post about my current understanding of this possibility.

One of the main problems or puzzles with quantum theory that Huw and others (such as Matthew Leifer, who will be speaking) think retrocausality may be able to help with, is the existence of Bell-type inequality violations. At their simplest, these involve two spacelike-separated regions of spacetime, usually referred to as "Alice's laboratory" and "Bob's laboratory", at each of which different possible experiments can be done. The results of these experiments can be correlated, for example if they are done on a pair of particles, one of which has reached Alice's lab and the other Bob's, that have previously interacted, or were perhaps created simultaneously in the same event. Typically in actual experiments, these are a pair of photons created in a "downconversion" event in a nonlinear crystal.  In a "nonlinear"  optical process photon number is not conserved (so one can get a "nonlinearity" at the level of a Maxwell's equation where the intensity of the field is proportional to photon number; "nonlinearity" refers to the fact that the sum of two solutions is not required to be a solution).  In parametric downconversion, a photon is absorbed by the crystal which emits a pair of photons in its place, whose energy-momentum four-vectors add up to that of the absorbed photon (the process does conserve energy-momentum).   Conservation of angular momentum imposes correlations between the results of measurements made by "Alice" and "Bob" on the emitted photons. These are correlated even if the measurements are made sometime after the photons have separated far enough that the changes in the measurement apparatus that determine which component of polarization it measures (which we'll henceforth call the "polarization setting"), on one of the photons, are space-like separated from the measurement process on the other photon, so that effects of the polarization setting in Alice's laboratory, which one typically assumes can propagate only forward in time, i.e. in their forward light-cone, can't affect the setting or results in Bob's laboratory which is outside of this forward light-cone.  (And vice versa, interchanging Alice and Bob.)

Knowledge of how their pair of photons were prepared (via parametric downconversion and propagation to Alice and Bob's measurement sites) is encoded in a "quantum state" of the polarizations of the photon pair.  It gives us, for any pair of polarization settings that could be chosen by Alice and Bob, an ordinary classical joint probability distribution over the pair of random variables that are the outcomes of the given measurements.  We have different classical joint distributions, referring to different pairs of random variables, when different pairs of polarization settings are chosen.   The Bell "paradox" is that there is no way of introducing further random variables that are independent of these polarization settings, such that for each pair of polarization settings, and each assignment of values to the further random variables, Alice and Bob's measurement outcomes are independent of each other, but when the further random variables are averaged over, the experimentally observed correlations, for each pair of settings, are reproduced. In other words, the outcomes of the polarization measurements, and in particular the fact that they are correlated, can't be "explained" by variables uncorrelated with the settings. The nonexistence of such an explanation is implied by the violation of a type of inequality called a "Bell inequality". (It's equivalent to to such a violation, if "Bell inequality" is defined generally enough.)

How I stopped worrying and learned to love quantum correlations

One might have hoped to explain the correlations by having some physical quantities (sometimes referred to as "hidden variables") in the intersection of Alice and Bob's backward light-cone, whose effects, propagating forward in their light-cone to Alice and Bob's laboratories, interact their with the physical quantities describing the polarization settings to produce---whether deterministically or stochastically---the measurement outcomes at each sites, with their observed probabilities and correlations. The above "paradox" implies that this kind of "explanation" is not possible.

Some people, such as Tim Maudlin, seem to think that this implies that quantum theory is "nonlocal" in the sense of exhibiting some faster-than-light influence. I think this is wrong. If one wants to "explain" correlations by finding---or hypothesizing, as "hidden variables"---quantities conditional on which the probabilities of outcomes, for all possible measurement settings, factorize, then these cannot be independent of measurement settings. If one further requires that all such quantities must be localized in spacetime, and that their influence propagates (in some sense that I'm not too clear about at the moment, but that can probably be described in terms of differential equations---something like a conserved probability current might be involved) locally and forward in time, perhaps one gets into inconsistencies. But one can also just say that these correlations are a fact. We can have explanations of these sorts of fact---for example, for correlations in photon polarization measurements, the one alluded to above in terms of energy-momentum conservation and previous interaction or simultaneous creation---just not the sort of ultra-classical one some people wish for.

Retrocausality

It seems to me that what the retrocausality advocates bring to this issue is the possibility of something that is close to this type of classical explanation. It may allow for the removal of these types of correlation by conditioning on physical quantities. [Added July 31: this does not conflict with Bell's theorem, for the physical quantities are not required to be uncorrelated with measurement settings---indeed, being correlated with the measurement settings is to be expected if there is retrocausal influence from a measurement setting to physical quantities in the backwards light-cone of the measurement setting.] And unlike the Bohmian hidden variable theories, it hopes to avoid superluminal propagation of the influence of measurement settings to physical quantities, even unobservable ones.  It does this, however, by having the influence of measurement settings pursue a "zig-zag" path from Alice to Bob: in Alice's backward light-cone back to the region where Alice and Bob's backward light-cones intersect, then forward to Bob's laboratory. What advantages might this have over superluminal propagation? It probably satisfies some kind of spacetime continuity postulate, and seems more likely to be able to be Lorentz-invariant. (However, the relation between formal Lorentz invariance and lack of superluminal propagation is subtle, as Rafael Sorkin reminded me at breakfast today.)

Probable signature of gravitational waves from early-universe inflation found in cosmic microwave background by BICEP2 collaboration.

Some quick links about the measurement, announced today, by the BICEP2 collaboration using a telescope at the South Pole equipped with transition edge sensors (TESs) read out with superconducting quantum interference devices (SQUIDs), of B-modes (or "curl") in the polarization of the cosmic microwave background (CMB) radiation, considered to be an imprint on the CMB of primordial graviational waves stirred up by the period of rapid expansion of the universe (probably from around 10-35--10-33 sec).  BICEP2 estimates the tensor-to-scalar ratio "r", an important parameter constraining models of inflation, to be 0.2 (+0.7 / -0.5).

Note that I'm not at all expert on any aspect of this!

Caltech press release.

Harvard-Smithsonian Center for Astrophysics press release.

Main paper:  BICEP2 I: Detection of B-mode polarization at degree angular scales.

Instrument paper: BICEP2 II: Experiment and three-year data set

BICEP/Keck homepage with the papers and other materials.

Good background blog post (semi-popular level) from Sean Carroll

Carroll's initial reaction.

Richard Easther on inflation, also anticipating the discover (also fairlybroadly accessible)

Very interesting reaction from a particle physicist at Résonaances.

Reaction from Liam McAllister guesting on Lubos Motl's blog.

Reaction from theoretical cosmology postdoc Sesh Nadathur.

NIST Quantum Sensors project homepage.

Besides a microwave telescope to collect and focus the relevant radiation, the experiment used transition-edge sensors (in which photons can trigger a quantum phase transition) read out by superconducting quantum interference devices (SQUIDs).  I don't know the details of how that works, but TE sensors have lots of applications (including in quantum cryptography), as do SQUIDs;  I'm looking forward to learning more about this one.

 

Some ideas on food and entertainment for those attending SQUINT 2014 in Santa Fe

I'm missing SQUINT 2014 (bummer...) to give a talk at a workshop on Quantum Contextuality, Nonlocality, and the Foundations of Quantum Mechanics in Bad Honnef, Germany, followed by collaboration with Markus Mueller at Heidelberg, and a visit to Caslav Brukner's group and the IQOQI at Vienna.  Herewith some ideas for food and entertainment for SQUINTers in Santa Fe.

Cris Moore will of course provide good advice too.  For a high-endish foodie place, I like Ristra.  You can also eat in the bar there, more casual (woodtop tables instead of white tablecloths), a moderate amount of space (but won't fit an enormous group), some smaller plates.  Pretty reasonable prices (for the excellent quality).  Poblano relleno is one of the best vegetarian entrees I've had in a high-end restaurant---I think it is vegan.  Flash-fried calamari were also excellent... I've eaten here a lot with very few misses.  One of the maitres d' sings in a group I'm in, and we're working on tenor-baritone duets, so if Ed is there you can tell him Howard sent you but then you have to behave ;-).  The food should be good regardless.  If Jonathan is tending bar you can ask him for a flaming chartreuse after dinner... fun stuff and tasty too.  (I assume you're not driving.)  Wines by the glass are good, you should get good advice on pairing with food.

Next door to Ristra is Raaga... some of the best Indian food I've had in a restaurant, and reasonably priced for the quality.

I enjoyed a couple of lunches (fish tacos, grilled portobello sandwich, weird dessert creations...) at Restaurant Martin, was less thrilled by my one foray into dinner there.  Expensive for dinner, less so for lunch, a bit of a foodie vibe.

Fish and chips are excellent at Zia Café (best in town I think), so is the green chile pie--massive slice of a deep-dish quiche-like entity, sweet and hot at the same time.

I like the tapas at El Mesón, especially the fried eggplant, any fried seafood like oysters with salmorejo, roasted red peppers with goat cheese (more interesting than it sounds).  I've had better luck with their sherries (especially finos) better than their wines by the glass.  (I'd skip the Manchego with guava or whatever, as it's not that many slices and you can get cheese at a market.)  Tonight they will have a pretty solid jazz rhythm section, the Three Faces of Jazz, and there are often guests on various horn.  Straight-ahead standards and classic jazz, mostly bop to hard bop to cool jazz or whatever you want to call it.  "Funky Caribbean-infused jazz" with Ryan Finn on trombone on Sat. might be worth checking out too... I haven't heard him with this group but I've heard a few pretty solid solos from him with a big band.  Sounds fun.  The jazz is popular so you might want to make reservations (to eat in the bar/music space, there is also a restaurant area I've never eaten in) especially if you're more than a few people.

La Boca and Taverna La Boca are also fun for tapas, maybe less classically Spanish.  La Boca used to have half-price on a limited selection of tapas and $1 off on sherry from 3-5 PM.  Not sure if they still do.

Il Piatto is relatively inexpensive Italian, pretty hearty, and they usually have some pretty good deals in fixed-price 3 course meals where you choose from the menu, or early bird specials and such.

Despite a kind of pretentious name Tanti Luci 221, at 221 Shelby, was really excellent the one time I tried it.  There's a bar menu served only in the bar area, where you can also order off the main menu.  They have a happy hour daily, where drinks are half price.  That makes them kinda reasonable.  The Manhattan I had was excellent, though maybe not all that traditional.

If you've got a car and want some down-home Salvadoran food, the Pupuseria y Restaurante Salvadoreño, in front of a motel on Cerillos, is excellent and cheap.

As far as entertainment, get a copy of the free Reporter (or look up their online calendar).  John Rangel and Chris Ishee are two of the best jazz pianists in town;  if either is playing, go.  Chris is also in Pollo Frito, a New Orleans funk outfit that's a lot of fun.  If they're playing at the original 2nd street brewery, it should be a fun time... decent pubby food and brews to eat while you listen.  Saxophonist Arlen Asher is one of the deans of the NM jazz scene, trumpeter and flugelhorn player Bobby Shew is also excellent, both quite straight-ahead.  Dave Anderson also recommended.  The one time I heard JQ Whitcomb on trumpet he was solid, but it's only been once.  I especially liked his compositions.  Faith Amour is a nice singer, last time I heard her was at Pranzo where the acoustics were pretty bad.  (Tiny's was better in that respect.)

For trad New Mexican (food that is) I especially like Tia Sophia's on Washington (I think), and The Shed for red chile enchiladas (and margaritas).

Gotta go.  It's Friday night, when all good grad students, faculty, and postdocs anywhere in the worlkd head for the nearest "Irish pub".

 

 

Answer to question about Bekenstein BH entropy derivation

I had a look at Jacob Bekenstein's 1973 Physical Review D paper "Black holes and entropy" for the answer to my question about Susskind's presentation of the Bekenstein derivation of the formula stating that black hole entropy is proportional to horizon area.  An argument similar to the one in Susskind's talk appears in Section IV, except that massive particles are considered, rather than photons, and they can be assumed to be scalar so that the issue I raised, of entropy associated with polarization, is moot.  Bekenstein says:

we can be sure that the absolute minimum of information lost [as a particle falls into a black hole] is that contained in the answer to the question "does the particle exist or not?"  To start with, the answer [to this question] is known to be yes.  But after the particle falls in, one has no information whatever about the answer.  This is because from the point of view of this paper, one knows nothing about the physical conditions inside the black hole, and thus one cannot assess the likelihood of the particle continuing to exist or being destroyed.  One must, therefore, admit to the loss of one bit of information [...] at the very least."

Presumably for the particle to be destroyed, at least in a field-theoretic description, it must annihilate with some stuff that is already inside the black hole (or from the outside point of view, plastered against the horizon). This annihilation could, I guess, create some other particle. In fact it probably must, in order to conserve mass-energy.  My worry in the previous post about the entropy being due to the presence/absence of the particle inside the hole was that this would seem to need to be due to uncertainty about whether the particle fell into the hole in the first place, which did not seem to be part of the story Susskind was telling, and the associated worry that this would make the black hole mass uncertain, which also didn't seem to be a feature of the intended story although I wasn't sure. But the correct story seems to be that the particle definitely goes into the hole, and the uncertainty is about whether it subsequently annihilates with something else inside, in a process obeying all relevant conservation laws, rendering both of my worries inapplicable. I'd still like to see if Bekenstein wrote a version using photons, as Susskind's presentation does. And when I feel quite comfortable, I'll probably post a fairly full description of one (or more) versions of the argument. Prior to the Phys Rev D paper there was a 1972 Letter to Nuovo Cimento, which I plan to have a look at; perhaps it deals with photons. If you want to read Bekenstein's papers too, I suggest you have a look at his webpage.

Question about Susskind's presentation of Bekenstein's black hole entropy derivation

I'm partway through viewing Leonard Susskind's excellent not-too-technical talk "Inside Black Holes" given at the Kavli Institute for Theoretical Physics at UC Santa Barbara on August 25.  Thanks to John Preskill,  @preskill, for recommending it.

I've decided to try using my blog as a discussion space about this talk, and ultimately perhaps about the "Harlow-Hayden conjecture" about how to avoid accepting the recent claim that black holes must have an information-destroying "firewall" near the horizon.  (I hope I've got that right.)  I'm using  Susskind's paper "Black hole complementarity and the Harlow-Hayden conjecture"  as my first source on the latter question.  It also seems to be a relatively nontechnical presentation (though much more technical than the talk so far)... that should be particularly accessible to quantum information theorists, although it seems to me he also does a good job of explaining the quantum information-theoretic concepts he uses to those not familiar with them.

But first things first.  I'm going to unembarassedly ask elementary questions about the talk and the paper until I understand.  First off, I've got a question about Susskind's "high-school level" presentation, in minutes 18-28 of the video, of Jacob Bekenstein's 1973 argument that in our quantum-mechanical world the entropy of a black hole is proportional to its area (i.e. the area of the horizon, the closed surface inside which nothing, not even light, can escape).   The formula, as given by Susskind, is

S = (\frac{c^3}{4 \hbar G}) A,

where S is the entropy (in bits) of the black hole, and A the area of its horizon.  (The constant here may have been tweaked by a small amounts, like 4 \pi or its inverse, to reflect considerations that Susskind alluded to but didn't describe, more subtle than those involved in Bekenstein's argument.)

The argument, as presented by Susskind, involves creating the black hole out of photons whose wavelength is roughly the Schwarzschild radius of the black hole.  More precisely, it is built up in steps; each step in creating a black hole of a given mass and radius involves sending in another photon of wavelength roughly the current Schwarzschild radius.  The wavelength needs to be that big so that there is no information going into the hole (equivalently, from the point of view outside the hole, getting "plastered" (Susskind's nice choice of word) against the horizon) about where the photon went in.  Presumably there is some argument about why the wavelength shouldn't be much bigger, either...perhaps so that it is sure to go into the hole, rather than missing.  That raises the question of just what state of the photon field should be impinging on the hole...presumably we want some wavepacket whose spatial width is about the size of the hole, so we'll have a spread of wavelengths centered around some multiple (roughly unity) of the Schwarzschild radius.  Before there is any hole, I guess I also have some issues about momentum conservation... maybe one starts by sending in a spherical shell of radiation impinging on where we want the hole to be, so as to have zero net momentum.  But these aren't my main questions, though of course it could turn out to be necessary to answer them in order to answer my main question.  My main question is:  Susskind says that each such photon carries one bit of information: the information is "whether it's there or not".  This doesn't make sense to me, as if one is uncertain about how many photons went into creating the hole, it seems to me one should have a corresponding uncertainty about its mass, radius, etc...  Moreover, the photons that go in still seem to have a degree of freedom capable of storing a bit of information:  their polarization.  So maybe this is the source of the one bit per photon?  Of course, this would carry angular momentum into the hole/onto the horizon, so I guess uncertainty about this could generate uncertainty about whether or not we have a Schwarzschild or a Kerr (rotating) black hole, i.e. just what the angular momentum of the hole is.

Now, maybe the solution is just that given their wavelength of the same order of the hole, there is uncertainty about whether or not the photons actually get into the hole, and so the entropy of the black hole really is due to uncertainty about its total mass, and the mass M in the Bekenstein formula is just the expected value of mass?

I realize I could probably figure all this out by grabbing some papers, e.g. Bekenstein's original, or perhaps even by checking wikipedia, but I think there's some value in thinking out loud, and in having an actual interchange with people to clear up my confusion... one ends up understanding the concepts better, and remembering the solution.  So, if any physicists knowledgeable about black holes (or able and willing to intelligently speculate about them...) are reading this, straighten me out if you can, or at least let's discuss it and figure it out...

Bohm on measurement in Bohmian quantum theory

Prompted, as described in the previous post, by Craig Callender's post on the uncertainty principle, I've gone back to David Bohm's original series of two papers "A suggested interpretation of the quantum theory in terms of "hidden" variables I" and "...II", published in Physical Review in 1952 (and reprinted in Wheeler and Zurek's classic collection "Quantum Theory and Measurement", Princeton University Press, 1983).  The Bohm papers and others appear to be downloadable here.

Question 1 of my previous post asked whether it is true that

"a "measurement of position" does not measure the pre-existing value of the variable called, in the theory, "position".  That is, if one considers a single trajectory in phase space (position and momentum, over time), entering an apparatus described as a "position measurement apparatus", that apparatus does not necessarily end up pointing to, approximately, the position of the particle when it entered the apparatus."

It is fairly clear from Bohm's papers that the answer is "Yes". In section 5 of the second paper, he writes

"in the measurement of an "observable," Q, we cannot obtain enough information to provide a complete specification of the state of an electron, because we cannot infer the precisely defined values of the particle momentum and position, which are, for example, needed if we wish to make precise predictions about the future behavior of the electron. [...] the measurement of an "observable" is not really a measurement of any physical property belonging to the observed system alone. Instead, the value of an "observable" measures only an incompletely predictable and controllable potentiality belonging just as much to the measuring apparatus as to the observed system itself."

Since the first sentence quoted says we cannot infer precise values of "momentum and position", it is possible to interpret it as referring to an uncertainty-principle-like tradeoff of precision in measurement of one versus the other, rather than a statement that it is not possible to measure either precisely, but I think that would be a misreading, as the rest of the quote, which clearly concerns any single observable, indicates. Later in the section, he unambiguously gives the answer "Yes" to a mutation of my Question 1 which substitutes momentum for position. Indeed, most of the section is concerned with using momentum measurement as an example of the general principle that the measurements described by standard quantum theory, when interpreted in his formalism, do not measure pre-existing properties of the measured system.

Here's a bit of one of two explicit examples he gives of momentum measurement:

"...consider a stationary state of an atom, of zero angular momentum. [...] the \psi-field for such a state is real, so that we obtain

\mathbf{p} = \nabla S = 0.

Thus, the particle is at rest. Nevertheless, we see from (14) and (15) that if the momentum "observable" is measured, a large value of this "observable" may be obtained if the \psi-field happens to have a large fourier coefficient, a_\mathbf{p}, for a high value of \mathbf{p}. The reason is that in the process of interaction with the measuring apparatus, the \psi-field is altered in such a way that it can give the electron particle a correspondingly large momentum, thus transferring some of the potential energy of interaction of the particle with its \psi-field into kinetic energy."

Note that the Bohmian theory involves writing the complex-valued wavefunction \psi(\mathbf{x}) as R(\mathbf{x})e^{i S(\mathbf{x})}, i.e. in terms of its (real) modulus R and (real) phase S. Expressing the Schrödinger equation in terms of these variables is in fact probably what suggested the interpretation, since one gets something resembling classical equations of motion, but with a term that looks like a potential, but depends on \psi. Then one takes these classical-like equations of motion seriously, as governing the motions of actual particles that have definite positions and momenta. In order to stay in agreement with quantum theory concerning observed events such as the outcomes of measurements, m theory, one in addition keeps, from quantum theory, the assumption that the wavefunction \psi evolves according to the Schrödinger equation. And one assumes that we don't know the particles' exact position but only that this is distributed with probability measure given (as quantum theory would predict for the outcome of a position measurement) by R^2(\mathbf{x}), and that the momentum is \mathbf{p} = \nabla S. That's why the real-valuedness of the wavefunction implies that momentum is zero: because the momentum, in Bohmian theory, is the gradient of the phase of the wavefunction.

For completeness we should reproduce Bohm's (15).

(15) \psi = \sum_\mathbf{p} a_{\mathbf{p}} exp(i \mathbf{p}\cdot \mathbf{x} / \hbar).

At least in the Wheeler and Zurek book, the equation has p instead of \mathbf{p} as the subscript on \Sigma, and a_1 instead of a_\mathbf{p}; I consider these typos, and have corrected them. (Bohm's reference to (14), which is essentially the same as (15) seems to me to be redundant.)

The upshot is that

"the actual particle momentum existing before the measurement took place is quite different from the numerical value obtained for the momentum "observable,"which, in the usual interpretation, is called the "momentum." "

It would be nice to have this worked out for a position measurement example, as well. The nicest thing, from my point of view, would be an example trajectory, for a definite initial position, under a position-measurement interaction, leading to a final position different from the initial one. I doubt this would be too hard, although it is generally considered to be the case that solving the Bohmian equations of motion is difficult in the technical sense of complexity theory. I don't recall just how difficult, but more difficult than solving the Schrödinger equation, which is sometimes taken as an argument against the Bohmian interpretation: why should nature do all that work, only to reproduce, because of the constraints mentioned above---distribution of \mathbf{x} according to R^2, \mathbf{p} = \nabla S---observable consequences that can be more easily calculated using the Schrödinger equation?
I think I first heard of this complexity objection (which is of course something of a matter of taste in scientific theories, rather than a knockdown argument) from Daniel Gottesman, in a conversation at one of the Feynman Fests at the University of Maryland, although Antony Valentini (himself a Bohmian) has definitely stressed the ability of Bohmian mechanics to solve problems of high complexity, if one is allowed to violate the constraints that make it observationally indistinguishable from quantum theory. It is clear from rereading Bohm's 1952 papers that Bohm was excited about the physical possibility of going beyond these constraints, and thus beyond the limitations of standard quantum theory, if his theory was correct.

In fairness to Bohmianism, I should mention that in these papers Bohm suggests that the constraints that give standard quantum behavior may be an equilibrium, and in another paper he gives arguments in favor of this claim. Others have since taken up this line of argument and done more with it. I'm not familiar with the details. But the analogy with thermodynamics and statistical mechanics breaks down in at least one respect, that one can observe nonequilibrium phenomena, and processes of equilibration, with respect to standard thermodynamics, but nothing like this has so far been observed with respect to Bohmian quantum theory. (Of course that does not mean we shouldn't think harder, guided by Bohmian theory, about where such violations might be observed... I believe Valentini has suggested some possibilities in early-universe physics.)

A question about measurement in Bohmian quantum mechanics

I was disturbed by aspects of Craig Callender's post "Nothing to see here," on the uncertainty principle, in the New York Times' online philosophy blog "The Stone," and I'm pondering a response, which I hope to post here soon.  But in the process of pondering, some questions have arisen which I'd like to know the answers to.  Here are a couple:

Callender thinks it is important that quantum theory be formulated in a way that does not posit measurement as fundamental.  In particular he discusses the Bohmian variant of quantum theory (which I might prefer to describe as an alternative theory) as one of several possibilities for doing so.  In this theory, he claims,

Uncertainty still exists. The laws of motion of this theory imply that one can’t know everything, for example, that no perfectly accurate measurement of the particle’s velocity exists. This is still surprising and nonclassical, yes, but the limitation to our knowledge is only temporary. It’s perfectly compatible with the uncertainty principle as it functions in this theory that I measure position exactly and then later calculate the system’s velocity exactly.

While I've read Bohm's and Bell's papers on the subject, and some others, it's been a long time in most cases, and this theory is not something I consider very promising as physics even though it is important as an illustration of what can be done to recover quantum phenomena in a somewhat classical theory (and of the weird properties one can end up with when one tries to do so).  So I don't work with it routinely.  And so I'd like to ask anyone, preferably more expert than I am in technical aspects of the theory, though not necessarily a de Broglie-Bohm adherent, who can help me understand the above claims, in technical or non-technical terms, to chime in in the comments section.

I have a few specific questions.  It's my impression that in this theory, a "measurement of position" does not measure the pre-existing value of the variable called, in the theory, "position".  That is, if one considers a single trajectory in phase space (position and momentum, over time), entering an apparatus described as a "position measurement apparatus", that apparatus does not necessarily end up pointing to, approximately, the position of the particle when it entered the apparatus.

Question 1:  Is that correct?

A little more discussion of Question 1.  On my understanding, what is claimed is, rather, something like: that if one has a probability distribution over particle positions and momenta and a "pilot wave" (quantum wave function) whose squared amplitude agrees with these distributions (is this required in both position and momentum space? I'm guessing so), then the probability (calculated using the distribution over initial positions and momenta, and the deterministic "laws of motion" by which these interact with the "pilot wave" and the apparatus) for the apparatus to end up showing position in a given range, is the same as the integral of the squared modulus of the wavefunction, in the position representation, over that range.  Prima facie, this could be achieved in ways other than having the measurement reading being perfectly correlated with the initial position on a given trajectory, and my guess is that in fact it is not achieved in that way in the theory.    If that were so it seems like the correlation should hold whatever the pilot wave is.  Now, perhaps that's not a problem, but it makes the pilot wave feel a bit superfluous to me, and I know that it's not, in this theory.  My sense is that what happens is more like:  whatever the initial position is, the pilot wave guides it to some---definite, of course---different final position, but when the initial distribution is given by the squared modulus of the pilot wave itself, then the distribution of final positions is given by the squared modulus of the (initial, I guess) pilot wave.

But if the answer to question 1 is "Yes", I have trouble understanding what Callender means by "I measure position exactly".  Also, regardless of the answer to Question 1, either there is a subtle distinction being made between measuring "perfectly accurately" and measuring "exactly" (in which case I'd like to know what the distinction is), or these sentences need to be reformulated more carefully.  Not trying to do a gotcha on Callender here, just trying to understand the claim, and de Broglie Bohm.

My second question relates to Callender's statement that:

It’s perfectly compatible with the uncertainty principle as it functions in this theory that I measure position exactly and then later calculate the system’s velocity exactly

Question 2: How does this way of ascertaining the system's velocity differ from the sort of "direct measurement" that is, presumably, subject to the uncertainty principle? I'm guessing that by the time one has enough information (possibly about further positions?) to calculate what the velocity was, one can't do with it the sorts of things that one could have done if one had known the position and velocity simultaneously.  But this depends greatly on what it would mean to "have known" the position and/or velocity, which --- especially if the answer to Question 1 was "Yes"--- seems a rather subtle matter.

So, physicists and other readers knowledgeable on these matters (if any such exist), your replies with explanations, or links to explanations, of these points would be greatly appreciated.  And even if you don't know the answers, but know de Broglie-Bohm well on a technical level... let's figure this out!  (My guess is that it's well known, and indeed that the answer to Question 1 in particular is among the most basic things one learns about this interpretation...)

Nagel and DeLong I: Common Sense

Brad DeLong has been hammering --- perhaps even bashing --- away at Thomas Nagel's new book Mind and Cosmos (Oxford, 2012).  Here's a link to his latest blow. I think Nagel's wrong on several key points in that book, but I think Brad is giving people a misleading picture of Nagel's arguments.  This matters because Nagel has made very important points --- some of which are repeated in this book, though more thoroughly covered in his earlier The Last Word (Oxford, 1997) --- about the nature of reason, defending the possibility of achieving, in part through the use of reason, objectively correct knowledge (if that is the right word) in areas other than science, and giving us some valuable ideas about how this can work in particular cases, for example, in the case of ethics, in The Possibility of Altruism [Princeton, 1979].

In his latest salvo Brad suggests that "If you are going to reject any branch of science on the grounds that it flies in the face of common sense, require[s] us to subordinate the incredulity of common sense, is not based ultimately on common sense, or is a heroic triumph of ideological theory over common sense--quantum mechanics is definitely the place to start…".  This is preceded by some quotes from Nagel:

  • But it seems to me that, as it is usually presented, the current orthodoxy about the cosmic order is the product of governing assumptions that are unsupported, and that it flies in the face of common sense…
  • My skepticism is… just a belief that the available scientific evidence, in spite of the consensus of scientific opinion, does not… rationally require us to subordinate the incredulity of common sense…
  • Everything we believe, even the most far-reaching cosmological theories, has to be based ultimately on common sense, and on what is plainly undeniable…
  • I have argued patiently against the prevailing form of naturalism, a reductive materialism that purports to capture life and mind through its neo-Darwinian extension…. I find this view antecedently unbelievable— a heroic triumph of ideological theory over common sense…

Now there are things I disagree with here, but Nagel is clearly not claiming that no theory that is not itself a piece of common sense is acceptable. Indeed, the second bullet point makes it clear that he allows for the possibility that scientific evidence could "rationally require" him to subordinate the incredulity of common sense. It is his judgment that it does not in this case. Now---at least with regard to the possibility of an explanation by evolutionary biology of the emergence of life, consciousness, and reason on our planet and in our species, which is what I think is at issue--- I don't share his incredulity, and I also suspect that I would weigh the scientific evidence much more heavily against such incredulity, if I did share some of it.  But Nagel is not commited to a blanket policy of "reject[ing] scientific theories because they fail to match up to your common sense." Regarding the third bullet point, it's perhaps stated in too-strong terms, but it's far from a claim that every scientific theory can directly be compared to common sense and judged on that basis. The claim that scientific theories are "ultimately based in common sense and on what is plainly undeniable" does not imply that this basis must be plain and direct. Logic and mathematics develop out of common-sense roots, counting and speaking and such... science develops to explain "plainly undeniable" results of experiments, accounts of which are given in terms of macroscopic objects... Some of this smacks a bit too much of notions that may have proved problematic for positivism ("plainly undeniable" observation reports?)... but the point is that common sense carries some weight and indeed is a crucial element of our scientific activities, not that whatever aspect of "common sense" finds quantum theory hard to deal with must outweigh the enormous weight of scientific experience and engineering practice, also rooted "ultimately" according to Nagel in common sense, in favor of that theory.

Just for the record I don't find that the bare instrumentalist version of quantum theory as an account of the probabilities of experimental results "flies in the face of common sense" --- but it does seem that it might create serious difficulty for the conception of physical reality existing independent of our interactions with it. At any rate it does not seem to provide us with a picture of that sort of physical reality (unless you accept the Bohm or Everett interpretations), despite what one might have hoped for from a formalism that is used to describe the behavior of what we tend to think of as the basic constituents of physical reality, the various elementary particles or better, quantum fields.  But if someone, say Nagel, did believe that this all flies in the face of common sense, it would be open to him to say that in this case, we are permitted, encourage, or perhaps even required to fly in said face by the weight of scientific evidence.

As I've said, I disagree on two counts with Nagel's skepticism about an evolutionary explanation of mind and reason: it doesn't fly in the face of my common sense, and I weigh the evidence as favoring it more strongly than does Nagel. Part of my disagreement may be that what Nagel has in mind is an evolutionary explanation that is commited to a "reductive materialism that purports to capture life and mind through its neo-Darwinian extension." Whereas I have in mind a less reductive approach, in which consciousness and reason are evolutionarily favored because they have survival value, but we do not necessarily reduce these concepts themselves to physical terms. In my view, biology is rife with concepts that are not physical, nor likely to be usefully reduced to physical terms--- like, say, "eye". As with "eye", there may be no useful reduction of "consciousness" or "perception" or "thought" or "word" or "proposition", etc.., to physics, but I don't think that implies that the appearance of such things cannot have an evolutionary explanation. (Nor, just to be clear, does it imply that these things are not realized in physical processes.) So I might share Nagel's incredulity that such things could have a "materialist" explanation, if by this he means one in terms of physics, but not his incredulity about evolutionary explanations of the appearance of mind and reason. To me, it seems quite credible that these phenomena form part of the mental aspect of structures made of physical stuff, though we will never have full explanations for all the phenomena of consciousness and the doings of reason, in terms of this physical structure.

(David Deutsch's recent book The Beginning of Infinity is one excellent source for understanding such non-reductionism---see in particular its Chapter 5, "The Reality of Abstractions".)

I'll likely make several more posts on this business, both on other ways in which I think Brad and others have mischaracterized Nagel's arguments or misplaced the emphasis in their criticisms, and on why this matters because some important points that Nagel has made on matters closely related to these, that I think have value, are in danger of being obscured, caricatured, or dismissed under the influence of the present discussion by Brad and others.

Thomas Nagel's "Mind and Cosmos"

I've just finished reading Thomas Nagel's newish book, "Mind and Cosmos" (Oxford, 2012).  It's deeply flawed, but in spite of its flaws some of the points it makes deserve more attention, especially in the broader culture, than they're likely to receive in the context of a book that's gotten plenty of people exercised about its flaws.  I'm currently undecided about whether to recommend reading his book for these points, as they are probably made, without the distracting context and possibly better formulated, equally well elsewhere, notably in Nagel's  "The Last Word" (Oxford, 1997).  The positive points are the emphasis on the reality of mental phenomena and (more controversially) their ireducibility to physical or even biological terms, the unacceptability of viewing the activities of reason in similarly reductive terms, and a sense that mind and reason are central to the nature of reality.  Its greatest flaws are an excessively reductionist view of the nature of science, and, to some degree in consequence of this, an excessive skepticism about the potential for evolutionary explanations of the origins of life, consciousness, and reason.

One of the main flaws of Nagel's book is that he seems --- very surprisingly --- to view explananations in terms of, say, evolutionary biology, as "reductively materialist".  He seems not to appreciate the degree to which the "higher" sciences involve "emergent" phenomena, not reducible---or not, in any case, reduced---to the terms of sciences "below" them in the putative reductionist hierarchy.  Of course there is no guarantee that explanations in terms of these disciplines' concepts will not be replaced by explanations in terms of the concepts of physics, but it has not happened, and may well never happen.  The rough picture is that the higher disciplines involve patterns or structures formed, if you like, out of the material of the lower ones, but the concepts in terms of which we deal with these patterns or structures are not those of physics, they are higher-order ones.  And these structures and their properties---described in the language of the higher sciences, not of physics---are just as real as the entities and properties of physics.  My view --- and while it is non-reductionist, I do not think it is hugely at variance with that of many, perhaps most, scientists who have considered the matter carefully --- is that at a certain very high level, some of these patterns have genuine mental aspects.  I don't feel certain that we will explain, in some sense, all mental phenomena in terms of these patterns, but neither does it seem unreasonable that we might.  ("Explanation" in this sense needn't imply the ability to predict perfectly (or even very well), nor, as is well known, need the ability to predict perfectly be viewed as providing us with a full and adequate explanation---simulation, for example, is not necessarily understanding.)   Among scientists and philosophers who like Nagel hold a broadly "rationalist" worldview David Deutsch, in his books The Fabric of Reality and especially The Beginning of Infinity, is much more in touch with the non-reductionist nature of much of science.

Note that none of this means there isn't in some sense a "physical basis" for mind and reason.  It is consistent with the idea that there can be "no mental difference without a physical difference", for example (a view that I think even Nagel, however, agrees with).

This excessively reductionist view of modern science can also be found among scientists and popular observers of science, though it is far from universal.   It is probably in part, though only in part, responsible for two other serious flaws in Nagel's book.  The first of these is his skepticism about the likelihood that we will arrive at an explanation of the origin of life in terms of physics, chemistry, and perhaps other sciences that emerge from them---planetary science, geology, or perhaps some area on the borderline between complex chemistry and biology that will require new concepts, but not in a way radically different from the way these disciplines themselves involve new concepts not found in basic physics.  The second is his skepticism that the origins of consciousness and reason can be explained primarily in terms of biological evolution.  I suspect he is wrong about this.  The kind of evolutionary explanation I expect is of course likely to use the terms "consciousness" and "reason" in ways that are not entirely reductive.   I don't think that will prevent us from understanding them as likely to evolve through natural selection.   I expect we will see that to possess the faculty of reason, understood (with Nagel) as having the---fallible, to be sure!---power to help get us in touch with a reality that transcends, while including, our subjective point of view, confers selective advantage.  Nagel is aware of the possibility of this type of explanation but --- surprisingly, in my view --- views it as implausible that it should be adaptive to possess reason in this strong sense, rather than just some locally useful heuristics.

The shortcomings in his views on evolution and the potential for an evolutionary explanation of life, consciousness, and reason deserve more discussion, but I'll leave that for a possible later post.

The part of Nagel's worldview that I like, and that may go underappreciated by those who focus on his shortcomings, is, as I mentioned above, the reality of the mental aspect of things, and the need to take seriously the view that we have the power, fallible as it may be, to make progress toward the truth about how reality is, about what is good, and about what is right and wrong.  I also like his insistence that much is still unclear about how and why this is so.  But to repeat, I think he's somewhat underplaying the potential involvement of evolution in an eventual understanding of these matters.  He may also be underplaying something I think he laid more stress on in previous books, notably The View from Nowhere and the collection of papers and essays Mortal Questions: the degree to which there may be an irreconcilable tension between the "inside" and "outside" views of ourselves.  However, his attitude here is to try to reconcile them. Indeed, one of the more appealing aspects of his worldview as expressed in both Mind and Cosmos and The Last Word is the observation that my experience "from inside" of what it is to be a reasoning subject, involves thinking of myself as part of a larger objective order and trying to situate my own perspective as one of many perspectives, including those of my fellow humans and any other conscious and reasoning beings that exist, upon it.  It is to understand much of my reasoning as attempting, even while operating as it must from my particular perspective, to gain an understanding of this objective reality that transcends that perspective.

So far I haven't said much about the positive possibilities Nagel moots, in place of a purely biological evolutionary account, for explaining the origin of life, consciousness, and reason.  These are roughly teleological, involving a tendency "toward the marvelous".  This is avowedly a very preliminary suggestion.  My own views on the likely role of mind and reason in the nature of reality, even more tentative than Nagel's, are that it is less likely that it arises from a teleological tendency toward the marvelous than that a potential for consciousness, reason, and value is deeply entwined with the very possibility of existence itself.  Obviously we are very far from understanding this.  I would like to think this is fairly compatible with a broadly evolutionary account of the origin of life and human consciousness and reasoning on our planet, and with the view that we're made out of physical stuff.