Even more philosophy of physics conferences, Summer 2010

If you know about the usual summer conferences and are still looking for more:

Hannover: Philosophy of Physics in Germany - Current State and Perspectives (11-12 Jun 2010). If you happen to be in or around Hannover next week (which unfortunately I am not), stop by to check this out! Meinard Kuhlmann (of recent Pittsburgh-fellow fame) organized this formidable program.

Scheduled Speakers: Claus Beisbart, Arianna Borrelli, Matthias Egg, Michael Esfeld, Brigitte Falkenbur), Cord Frieb), Mathias Frisch, Ulrich Gähde, Reiner Hedrich, Carsten Held, Rafaela Hillerbrand, Paul Hoyningen-Huen), Andreas Hüttemann, Meinard Kuhlmann, Dennis Lehmkuhl, Christoph Lehner, Holger Lyre, Paul Näger, Thorben Petersen, Wolfgang Pietsch, Helmut Desks, Wolfgang Rhode, Gregor Schiemann, Francisco Soler -Gil, Manfred Stöckler, Michael Stöltzner.

London: Emergence in Physics (13-14 Jul, 2010). This looks like a great meeting to check out after the BJPS/FoP weekend. It's organized by the distinguished Prof. Eleanor Knox, the former Oxford student, recently-hired IP faculty, and deft climber of Swiss mountain-tops.

Currently scheduled speakers: Bob Batterman, Jeremy Butterfield, Roman Frigg, Stephan Hartmann, Eleanor Knox, David Wallace.

Vienna: What Exists in the Quantum World? (19-24 Jul, 2010). This is a great idea for a workshop, and includes a great mix of philosophers and physicists. Applications to join are due by Monday though, so do hurry if you'd like to go!

Senior participants: Markus Aspelmeyer, Gennaro Auletta, Tina Bilban, Časlav Brukner, Jeffrey Bub, Vladimir Chaloupka, Raymond Chiao, Daniel Greenberger, Alexei Grinbaum, Richard Healey, Michael Horne, Tarja Kallio-Tamminen, Henry Krips, Franck Laloë, Xiaosong Ma, Stefano Osnaghi, Sorin Paraoanu, Sven Ramelow, Stig Stenholm, Anton Zeilinger.

Overheard at New Directions in Foundations of Physics '10

If you missed New Directions in Foundations of Physics conference earlier this month, here are a few memorable one-liners. You'll notice that many are from Bill Unruh, who seemed to be in rare form that weekend.

In this case it was more important to be right than to be correct. -- Bill Unruh

Unruh was discussing Hawking's original calculation of the thermal radiation emitted by black holes. He ended with a very interesting discussion of "dumb-holes," along the lines discussed recently on this blog.

Your transparencies should never be more clear than your thoughts. -- Bill Unruh

Here, Unruh was struggling to focus one of those new overhead-projectors, the kind that project an actual live video of your transparency. He had just explained his intention to present us with a rough-and-ready example, so the timing was impeccable.

Yes, but... but still, I... I mean, I really shouldn't tell *you* that you can't move faster than light. I'm teaching my grandmother to suck eggs, here. -- Bill Unruh, to Charlie Misner

As I recall, Unruh had mentioned that you couldn't escape a black hole unless you traveled faster than light. Misner was gently remindung Unruh that, after all, we have little empirical evidence about the interior of black holes.

Unruh: But those are just words!
Tumulka: Yeah... I don't know how to convey it better than with words

Here, Roderich Tumulka was explaining some feature of his relativistic GRW theory. I wish I could recall exactly what provoked Unruh to say this, but alas, my memory falters. At any rate, Tumulka admirably deflected his heckling.

I like to say that Feynman won the Nobel prize in 1965 for showing that BQP is contained in PSPACE. -- Scott Aaronson

According to a standard picture, BQP is the class of computations quantum computer can do, and PSPACE is just the class of computations solvable in polynomial space. Aaronson was pointing out that, if we use the Feynmann path-integral approach to calculating the probability that a quantum computer "accepts," then we're required to sum up only exponentially many terms -- and this sum is computable in PSPACE.

 If we were physicists, we would have announced decades ago our discovery that all these classes [like BQP, PSPACE, NP, etc.] are distinct. -- Scott Aaronson

Aaronson had just finished explaining how questions like the distinction between P and NP remain a major open problem in mathematics and computer science. Apparently, he considers the standard of "discovery" in physics to be a notch lower.

Hopefully I'll see some of you at one of the remaining conferences this summer!

An argument for hidden variables

Detlef Dürr, Shelly Goldstein, and Nino Zanchí once gave a very interesting argument for hidden variables. I'll give their argument a careful reconstruction. But first, here's what they say.

According to the quantum formalism, measurements performed on a quantum system with definite wave function ψ typically yield random results. Moreover, even the specification of the wave function of the composite system including the apparatus for performing the measurement will not generally diminish this randomness. However, the quantum dynamics governing the evolution of the wave function over time, at least when no measurement is being performed, and given, say, by Schrödinger's equation, is completely deterministic. Thus, insofar as the particular physical processes which we call measurements are governed by the same fundamental physical laws that govern all other processes, one is naturally led to the hypothesis that the origin of the randomness in the results of quantum measurements lies in random initial conditions, in our ignorance of the complete description of the system of interest -- including the apparatus -- of which we know only the wave function.

I'm going to interpret their "randomness" to mean the lack of a determinate value at a given time. Now, here's my reconstruction.

1. If the laws governing a physical process are deterministic, and its initial conditions are completely specified, then our description of the physical process is guaranteed to have determinate values at any given time.
2. The wave-function description of the measurement processes is *not* guaranteed to have determinate values at any given time.
3. All physical processes are governed by the same fundamental physical laws (and hence by equally deterministic equations of motion).
4. So, since Schrödinger evolution is governed by deterministic laws, the measurement process must be governed by deterministic laws as well.
5. But since the measurement process does not have determinate values, this implies by (1) that the initial conditions of the measurement process are not completely specified, when given by the wave function alone.

This argument is very different than the kind of locality complaint espoused by Einstein. And it's part of what leads these authors to adopt Bohmian mechanics, which supplements the "unspecified initial conditions" allowed by quantum mechanics with exact particle positions.

As much as I'd like to be convinced, I just don't understand the motivation for premise (3). The Schrödinger equation is deterministic. But the authors want to conclude that whatever basic fundamental law governs both Schrödinger evolution and measurement must therefore also be deterministic. Why?

The authors don't say in this article. And the following seems to be a counterexample: Measurement in quantum mechanics is indeterministic. But because of Ehrenfest's theorem, it still (on average) satisfies a deterministic law. So, a single indeterministic law appears able to give rise to deterministic law-like behavior. Therefore, deterministic law-like behavior (e.g., Schrödinger evolution) doesn't imply that all more fundamental laws are also deterministic. So premise (3) fails. Right?

Accuracy, Applicability, and Tarskian Semantics

by Erik Curiel (Guest Post)

First is a concise statement of my problem with contemporary accounts of those semantics, as based on the idea of truth as, in some sense, prior to that of meaning. Second is my problem with Tarskian semantics in particular, which seems to be far and away the most popular formal theory of semantics used to construct particular accounts of the semantics of scientific theories, no matter what else the philosopher using it thinks about semantics

I. Scientific Semantics as Based on Notions Like Truth

Carnap, in the Introduction to Semantics (ch.B, §7, p.22) concisely expresses the seductive intuition that grounds essentially all contemporary thought on the semantics of scientific theories:

... to understand a sentence, to know what is asserted by it, is the same as to know under what conditions it would be true.

As appealing as this idea is, its naive application leads to severe problems. This is so no matter the details of the architectonic form of one's account of a theory and its semantics, whether it falls, e.g., under the purview of either the syntactical or the semantical account of scientific theories and their semantics, or some other view entirely, so long as the foundation of that view takes as ineliminable a concept such as truth that must be grounded on accuracy of prediction. My gripe is not with any particular conception of truth, nor with the idea of truth itself. Truth is just the notion that specific instances of the generic form of semantics I oppose most commonly employ in their respective foundations---that genus of semantics that attributes semantic content to a theoretical representation based on the accuracy of the fit of its predictions to the results of the empirical, quantitative measurements made on the system it purports to model. In other words, my argument is with accounts of semantics that make semantic content devolve in the end upon the accuracy of a theory's models, irrespective of how exactly it is that the accuracy comes into play in fleshing out the theory's semantic relations and content (as justifying referential relations, as characterizing adequacy, as being required for truth, or what have you).

The heart of the problem is that such accounts cannot differentiate inaccuracy from inapplicability as a defect in a theoretical representation of a physical system: a semantics grounded on a notion like truth can rule a model of a system inadmissible only on the grounds that it does not model the behavior of the system accurately enough. That, however, is too coarse-grained a measure of the way models can fail to provide semantically sound representations of physical systems.

Consider the example of a model of a body of liquid as provided by the classical theory of fluid mechanics. When the liquid is not too viscous, is in a state near hydrodynamical and thermodynamical equilibrium and the level of precision and accuracy one demands of the model is not at too fine a spatiotemporal scale, then the classical theory yields excellent models of the liquid's behavior over a wide range of states and environments. When the state of the liquid, say, begins to approach turbulence, the representation the theory provides begins to break down. It does so, however, in a subtle way, one that cannot be wholly accounted for by adverting merely to the fact that the theory's model becomes inaccurate. In particular, there is a regime in which the dynamical equations of motion of the theory no longer provide accurate predictions by any reasonable measure, and yet all the quantities the theory attributes to the liquid, and all the kinematical constraints the theory jointly imposes on those quantities (e.g., the continuity of mass-density, the conservation of energy, etc.), will still be satisfied. In a strong sense, then, the theory can still provide a meaningful -- and appropriate -- model of the liquid even though the model is not adequately accurate.

A semantics whose fundamental terms require, by way of relation to empirical phenomena, no more than accuracy in prediction (as do all those grounded on truth, referential relations, and so on), however, cannot admit such models as part of the theory, period, for the models are not accurate. This view is inadequate for (at least) two reasons. First, it does not allow us, within the scope of the theory itself, to understand why such models are not sound even though all the quantities the theory attributes to the system are well defined and the values of those quantities jointly satisfy all kinematical
constraints the theory requires. Second, we miss something fundamental about the meaning of various theoretical terms by rejecting such models out of hand merely on the grounds of their inaccuracy. It is surely part of the semantics of the term `hydrostatic pressure', e.g., that its definition as a physical quantity treated by classical fluid mechanics breaks down when the fluid approaches turbulence; because, however, the theory's equations of motion stop being accurate long before, in a precise sense, the quantity loses definition in the theory, any semantics that rejects the inaccurate models in which the term still is well defined will not be able to account for that part of the term's meaning. Thus, an adequate semantics for physical theory must be grounded on notions of meaning derived from relations in some sense prior to the accuracy of the theory's representations of the dynamical behavior of the physical systems it treats, relations that govern the applicability of the theory's representational resources to the system at issue.

II. Tarskian Semantics

Let's take, at a minimum, Tarskian semantics as applied to scientific theories to require the following:

1. a theory is (characterized by) the collection of its (Tarskian) models
2. the semantic content of the theory is completely exhausted by the association of each model to the (possible) systems it adequately represents

In particular, no semantic content of intrinsic physical significance can accrue to the theory in virtue of relations among its models.

It is usual to take a model to be fully characterized by a solution to the theory's equations of motion, and, indeed, I see no other reasonable way to go. Tarskian semantics then has the consequence that no structure intrinsic to the family of all solutions to the equations of motion can have semantic content of intrinsic physical significance. This seems prima facie wrong. Families of models (classes of solutions to the equations of motion) may have on their own semantic content that forms part of the semantic content of the theory but that is not formulable in a traditional Tarskian semantics. For example, the claim that the equations of motion have a well set initial-value formulation in the sense of Hadamard indubitably informs part of a theory's semantic content, but it is one that, in its essence, consists of relations among models and cannot be reduced to the interpretation of a single model. Thus, the simple aggregation of the meaning of all individual models does not exhaust the semantic content of a theory.

Erik Curiel is a philosopher at London School of Economics, specializing in philosophy of physics, philosophy of science, and ancient philosophy. For more on Curiel's work, visit his homepage.

Improving the Peer Review Process

Peer-reviewed journals have the great potential to improve the quality of published papers. Most scholars value them for this reason. But how can we make the process better? Bee has been doing a very nice series on improving peer review over at Backreaction. My favorite two of her many suggestions are:

• Online interface for anonymous author-reviewer communication. Why keep the slow (and frankly archaic) editor-mediated communication between author and reviewer, when everyone has access to the interwebs? An anonymized online interface would be quicker, easier, and more useful. In particular, it would allow for quick clarificatory questions, and even back-and-forth discussions of important results between author and reviewer, before a finalized report is submitted to the editor.

• Incentives for high-quality reviews. Most journals don't offer you an incentive to do a good job in a timely manner. A monetary compensation for well-done reviews is the obvious thing to do. Authors might even be willing to pay a submission fee for this cause, if it improves the quality and timeliness of the report (I know I would!).

Journals in many disciplines, including philosophy and HPS, could immediately adopt the first idea to great benefit. The second idea requires some structural changes to raise the funds. To this end, I'd only suggest:

• Stop printing paper-copies of journals. It's a waste of money. Everybody prefers to access articles electronically anyway. The primary role of a journal should be as a peer-review agency. Spend the money on incentives for high-quality reviews instead.

Any other ideas out there?

Why we need superselection rules

One motivation for superselection rules comes from a passage in von Neumann's famous Mathematical Foundations of Quantum Mechanics:

There corresponds to each physical quantity of a quantum mechanical system, a unique hypermaximal Hermitian operator, as we know..., and it is convenient to assume that this correspondence is one-to-one -- that is, that actually each hypermaximal operator corresponds to a physical quantity. (von Neumann 1932 [1955], p.313)

The problem is that von Neumann's "convenient assumption," that self-adjoint operators correspond to observable quantities, can't possibly be true. Here's a (probably too)* simple example to illustrate. Suppose we have a 2-dimensional Hilbert space (it might be describe the z-spin of a single particle), generated by the states ψ₀ and ψ₁. And let A be a self-adjoint operator for which these are eigenstates:

Construct two new states, φ₀ and φ₁, given by:

Now let B be a self-adjoint operator for which these new states are eigenstates, say,

The problem is that, although both A and B are self-adjoint, they can't both at the same time represent observable quantities. For suppose the eigenstates of A correspond to observable measurements; then the eigenstates of B are not observable states, since they are linear superpositions of the eigenstates of A. But it's not possible to observe superpositions of physics measurements. And the same holds conversely if the eigenstates of B correspond to observable measurements. So, which one is the real observable? It seems we're in a pickle.

One simple way to resolve the pickle is to impose a rule, which says constructed operators like B are not allowed to be observables. That's a superselection rule.

A typical way to implement such a rule is to notice that we can think of our Hilbert space as a direct sum of one-dimensional Hilbert spaces H₀ and H₁, one containing ψ₀ and the other containing ψ₁:

We can motivate this decomposition as long as we agree that both ψ₀ and ψ₁ correspond to readings of a measurement device. We can then impose a superselection rule:

Linear superpositions of states from distinct sectors in such a direct sum are not physically realizable except as a mixture, and hence cannot be eigenstates.

Wightman attributes this particular rule to Wigner, in his excellent history of superselection rules. John Earman has recently written on some of the philosophical details. Of course, it may not be true that "we agree" on what states correspond to readings of measurement devices. And in such cases, the question of which superselection rules are justified remains a difficult problem in the foundations of physics.

* Added.

Wigner's elegant characterization of time reversal

There's a nice post at The Eternal Universe illustrating discrete symmetries like time reversal. The father of this idea, Eugene Wigner, actually gave a very elegant characterization in his 1931 book:

The following four operations, carried out in succession on an arbitrary state, will result in the system returning to its original state. The first operation is time inversion, the second time displacement by t, the third again time inversion, and the last on again time displacement by t. (Wigner [1931] 1959, p.326.)

I've illustrated this assumption below, using the analogy:

• Reversal of time = Flipping of toy car;
• Evolution through time = Forward motion of car.

Wigner’s claim is then that whatever time reversal means, the initial state of the car below is the same as its final state.

The beauty is, if you write down this assumption in terms of quantum mechanical operators (and assume energy is positive), then you can quickly see an important property of time reversal -- its antiunitarity (roughly, it involves conjugation). In terms of operators, Wigner assumed that:

This immediately implies that

But Taylor expanding the exponentials, we then have that:

Now, T can't be unitary, since then it would follow that Ti = iT, and we could cancel the i to get that THT^-1 = -H. That's false if energy is always positive. But Wigner's theorem says that if a symmetry operator is not unitary, then it must be antiunitary -- so T must be antiunitary.

Wigner's argument seems to have been dropped from most modern textbooks. Perhaps the reason is that not all physical interactions are T-reversible, and Wigner's assumption (illustrated here with cars) takes for granted that they are. On the other hand, all known physical interactions are CPT-reversible. So, a version of this argument still works if we reinterpret Wigner's "time reversal" as "CPT reversal."

Conferences for Philosophers of Physics, Summer 2010

April 30 - May 2. Washington, D.C. New Directions in Foundations of Physics. Speakers: Michel Janssen, Tony Duncan, Elise Crull, Fernando Brandao, Bill Unruh, Dan Browne, Scott Aaronson, Valerio Scarani, Miguel Navascues, Caslav Brukner, Roderich Tumulka.

May 7-9. Toronto, Ontario. Models and Simulations 4. Keynotes: Leonard Smith and Jos Uffink.

May 7-10. London, Ontario. Logic, Mathematics and Physics / UWO Philosophy of Science Conference. Speakers: Kevin Kelly, Rhonda Martens, Curtis Wilson, George Smith, John Earman, Allan Gibbard, Jim Joyce, Alan Hájek, Brian Skyrms.

July 5-7. Aberdeen, Scotland. 16th European Meeting on Foundations of Physics.

July 8-9. Dublin, Ireland. British Society for Philosophy of Science Annual Meeting.






Unfortunately, the MS4 and UWO conferences in Ontario are scheduled to overlap. (Communication error, Ontarians?) But on the bright side, FoP rescheduled so as not to clash with BSPS, so it's now possible to attend both.

Anything else we should be attending this summer?

David Albert on symmetries of motion in quantum mechanics

In Time and Chance, David Albert writes that since the Schrödinger equation involves a first (instead of a second) derivative, "the dynamical laws that govern the evolutions of quantum states in time cannot possibly be invariant under time-reversal" (p.132). I've always struggled with his argument for this claim, which he gives in a footnote on the same page. Here's what he writes.

The idea is this: suppose that the instantaneous microscopic state of a certain physical system at time t is also that system's complete dynamical condition at t, and suppose that the dynamical equations of motion of that system are invariant under time-reversal. Then whatever it is that those equations entail about times other than t is patently going to have no alternative whatsoever but to be symmetrical about t. Suppose (moreover) that the equations of the motion of this system are invariant under time-translations ... . Then (if you think it over) the state of this system is going to have no alternative but to be entirely unchanging in time. And so any theory for which instantaneous states are also invariably complete dynamical conditions, and for which the equations of motion are invariant under time-reversal, and for which the equations of motion are invariant under time-translation, is necessarily a theory according to which nothing ever happens. (Albert 2000, 132.)

I've thought it over, and I think I've finally got it.

The key to the passage is this: Albert takes time-reversal to transform the state ψ(t) to the state ψ(-t). This is a highly non-standard view; in particular, time reversal in quantum mechanics is normally taken to involve conjugation as well. But consider the consequences of Albert's view for the Schrödinger equation -- it means that if ψ(t) is a solution, then so is ψ(-t):

But substituting t → -t into the original Schrödinger equation, we find that:

Adding these two equations, we now have that:

and hence that ψ(-t) = 0. In other words, the state of the world is constant, and "nothing ever happens." And indeed, this result is more general than the Schrödinger equation -- as Albert suggests, it seems to hold for any deterministic first order equation of motion, with only a single first derivative, which is both time-translation and time-reversal invariant (according to Albert's definition).

Now, the conclusion that "nothing ever happens" is obviously false. So, one of these premises must be false as well. Albert rejects the premise that quantum mechanics is time reversal invariant. But of course, there is a plausible alternative. We can reject Albert's picture of time reversal instead.

Bob Batterman moves to University of Pittsburgh

Bob Batterman, currently at the University of Western Ontario, has been hired by the University of Pittsburgh's Department of Philosophy. One more reason to love history and philosophy of science at the Cathedral of Learning!

This is quite a coup for Pittsburgh. But it's also a timely move as far as philosophy of physics is concerned, given that John Earman is expected to retire this year.

Welcome to Pittsburgh, Bob!

1907 Crisis in Mathematical Physics According to Poincaré

With 20-20 hindsight, we all agree that Einstein's discoveries of 1905 revolutionized nearly every area of fundamental physics. But what did scientists think at the time? One telling source is Poincaré's 1907 account of the "new crisis" in physics (available here, on the newly released Popular Science archive). Poincaré identifies five fundamental principles he thought were in danger of being overturned:

1. Carnot's principle of heat transfer. Brownian motion was thought to violate Carnot's principle of heat transfer, since it apparently involved an unlimited source of motion. Poincaré wrote, "to see the world return backward, we no longer have need of the infinitely keen eye of Maxwell's demon; our microscope suffices."
2. The principle of relativity. Although Einstein had recently defended this principle, Poincaré wasn't convinced, and in particular worried about the prohibition on superluminal signaling. Anticipating a coming revolution in gravity, he wrote: "are such signals inconceivable, if we admit with Laplace that universal gravitation is transmitted a million times more rapidly than light?"
3. Newton's third law (of action-reaction). Electrodynamics seemed to be suggesting that not every action corresponds to an equal and opposite reaction. In particular, the action of one electric charge on another doesn't necessarily give rise to a simultaneous reaction.
4. Lavoisier's principle of fixed mass. Alluding to Einstein, Poincaré wrote that electrodynamics suggests a body's mass might increase with velocity, refuting principle of fixed mass: "And now certain persons think that it seems true to us only because in mechanics merely moderate velocities are considered."
5. Mayer's principle of energy conservation. Finally, the recent discovery of radiation by the Curies suggested to Laplace Poincaré that radium might be a limitless source of energy, and hence that energy is not locally conserved.

What I find striking about this list is Poincaré's recognition of the deep and difficult consequences of taking classical electrodynamics seriously -- and in particular, of retaining the principle of relativity. Of course, only 3 and 4 were actually overturned, and a version of 4 may still be salvageable (by replacing "mass" with "rest mass"). And it's somewhat surprising that as late as 1907, Poincaré isn't mentioning Einstein by name.

But then, I suppose it's never clear what the revolution will bring until well after it's over.

Translation of the long-lost Descartes letter

Amazingly, a long-lost letter by Descartes was recently recovered from the Haverford College outside Philadelphia. The letter's discoverer, historian Erik-Jan Bos at the University of Utrecht, has now produced an English translation:

I met Mr Picot here, in whom I recognize a man of good sense, and to whom I am much obliged. I believe he will arrive at Leiden today and has the intention to stay. In his company is a nobleman from Touraine who brought me the greetings from Father Bourdin, whose student he is; he also spoke of Mr Petit in such terms that I am obliged to tone down what I wrote on him in the Preface to the reader, which I send you now to be printed, if you please, at the beginning of the book, after the dedicatory letter to the Gentlemen of the Sorbonne. Neither the fourth part of the Discours de la méthode, nor the little preface I put in next, nor the one preceding the theologian’s objections, must be printed, but only the Synopsis. Finally, rest assured that there is nothing in Mr Gassendi’s objections with which I have problems; the only thing I shall have to attend to is the style. Indeed, he expressed himself with so much elegance, that I should attempt to reply in the same way. I am


Your much obliged and affectionate
servant Des Cartes


27 May 1641


The two highlighted passages are sure to make a lasting impression on Descartes scholarship. The first (in blue) shows that the original draft of the Meditations on First Philosophy had more chapters, which Descartes later omitted. The second (in red) indicates that Descartes didn't originally mind the criticisms of Gassendi -- this is strange, because Descartes later came to loathe Gassendi's objections, and responded with harsh personal attacks. One wonders what happened in between.

Two Theorems about Time Reversal

Time reversal is an important philosophical topic, but not because of any whacky metaphysics. It's just a transformation from the space of possible motions to itself. For example, the motion of a ball rolling down an inclined plane is a possible motion according to classical mechanics.

Time reversal (as it's normally understood) transforms this motion to another possible motion, that of a ball rolling up an inclined plane until coming to rest.

In other words, this picture of time reversal reverses the time-ordering, preserves position, and reverses momentum.

The philosophical question for us is: what does time reversal generally mean? In particular, we'd like to know which transformations count as time reversal, as opposed to any other transformations. Well, here's one interesting way to produce an answer. Suppose the following principle is true of the world.

Free Motion Symmetry. In the absence of forces of interactions, when a system contains only free particles or fields, the laws of nature are time reversal invariant.

Nature need not be time reversal invariant in general. But adopting this principle means that when we turn off all interactions, there is no question -- if a free motion is allowed by nature, then so is its time reverse. I think there is good reason to believe this principle. But whether there is or not, it's interesting to observe that it provides a unified way to recover the standard time reversal operators. For example:

Theorem 1. Suppose that Hamilton's equations are time-reversal invariant for the free-particle Hamiltonian, and that T is a linear involution (T²=1). Then T is characterized by one of the following:




Tq = q and Tp = -p
or
Tq = -q and Tp = p
where q and p are the canonical position and momentum variables, respectively.

In other words, free motion symmetry (together with the assumption that T is an involution or reversal) is enough to pick out the two time-reversal operators of classical mechanics. The first is the standard time reversal operator. The second is the space-and-time reversal operator.

As it turns out, this same method recovers the standard (but more unusual looking) time-reversal operator in quantum mechanics as well. In particular, we have the following.

Theorem 2. Suppose that T is a Hilbert space symmetry that commutes with the free-particle Hamiltonian. If the Schrödinger equation with this Hamiltonian is time-reversal invariant, then T is antiunitary.

In other words, assuming free motion symmetry in a system characterized by a time-independent Hamiltonian is enough to guarantee that T is antiunitary -- which is the standard characterization of the quantum time reversal operator.

For more on this approach (and the very elementary proofs of the theorems), see my recent draft: How to time-reverse a quantum system, or my post on how to visualize why time-reversal involves conjugation in quantum mechanics.

Rescher Prize for Contributions to Systematic Philosophy

Photo Credit: John D. Norton

The University of Pittsburgh has announced the establishment of the Nicholas Rescher Prize for Contributions to Systematic Philosophy. The details:

• Rescher Prize gold medal for work in philosophy
• $25,000 award
• Awarded every two years

The prize is being offered in part to establish an award in philosophy hoped to become comparable to the Field Medal, Pulitzer Prize, and Nobel Prizes in other fields. It is also being offered to combat the fracturing specialization of the field:

"The philosopher's key job is to integrate philosophy and to provide a systematic picture of the whole field: Systematic thinking across frontiers is not fashionable but nevertheless crucial. Virtually all major contributors to philosophy have been systematic thinkers." (Nicholas Rescher, press release)

Pittsburgh is honoring Rescher for his own contributions, which include over 100 books, 1000 articles, and half a century of systematic contributions from the perspective of American pragmatism. In return, Rescher is donating his extensive library to the University of Pittsburgh, which includes 40,000 pages of correspondence and volumes of original manuscripts by 20th century philosophers.

Thanks to Jonah at Choice & Inference for bringing this to my attention.

More Philosophy of Physics in the Blogosphere

Readers of this blog may be interested in Chris Wüthrich's new blog on the philosophy of physics, Taking up Spacetime.

Chris is a philosopher of physics at UCSD, and fellow Pittsburgh graduate. Philosophy of physics has such a scarce presence on the bloggersphere, this is certainly a welcome addition! For more philosophy of physics on the intertubes, you might also try:

• The Statistical Mechanic / Mostly Harmless Hoofless
• The Truth Makes Me Fret
• Backreaction

And of course, let us know in the comments if you know about more philosophy of physics blogs!

"An elementary particle 'is' an irreducible representation"

A well-known particle physicist's adage:

Ever since the fundamental paper of Wigner on the irreducible representations of the Poincaré group, it has been a (perhaps implicit) definition in physics that an elementary particle ‘is’ an irreducible representation of the group, G, of ‘symmetries of nature’ (Ne’eman and Sternberg 1991, pp. 327.) 

But what exactly does that mean? Many would agree that Wigner's seminal work on the Poincaré group has some deep metaphysical implication. But what? Wigner himself provided very little indication as to what it might be, even in his later work in the philosophy of physics.

However, there seems to be what Arthur Fine would call a "core position" about Wigner's result. That is: there is a tight mathematical connection between the symmetry groups of nature, and the measurable quantities of quantum theory. You can even diagram it:

However, as I've argued before (and in a forthcoming article), a certain realist addition to this core position just doesn't make sense. So what, if anything, can be said about Wigner's result beyond the core position?

Unitary operators and spacetime symmetries

In quantum mechanics, certain unitary operators have been understood since the time of Wigner in terms of spacetime symmetries. Why?

The foundation for this kind of thinking has an interpretive and a mathematical aspect. The interpretive aspect has to do with the way we connect certain observables to experience; the mathematical aspect has to do with the way unitary operators look under this interpretation.

First, on interpreting observables. We’ll take an observable to be a self-adjoint operator acting on a Hilbert space. It’s well known that quantum mechanics must make some assumption about how to connect this operator to measurement; one common such assumption is the eigenvalue-eigenstate link. However, we make an additional interpretive assumption about some (though not all) observables, which is the
following.

Assumption. The expected or average value ⟨O⟩ of an observable O can be identified with a vector in spacetime.

For example, an eigenstate of the position operator in a single-particle Hilbert space assigns the property ‘there is a particle located here’ to a vector in 3-dimensional space. Average position can thus be identified with the average of these vectors. Similarly, an eigenstate of an angular momentum operator assigns a property like ‘spin +1’ to a direction (say, a unit vector) in space. Average angular momentum can thus be identified with the average of these vectors.

That’s our interpretive connection between quantum theory and spacetime. Now, here’s the mathematical part: such unitary operators turn out to implement spacetime symmetries under this interpretation. In short, it can be proved that these unitary transformations are equivalent to symmetry transformations of the corresponding spacetime structures.

The details of how this works of course depends on the situation. But it’s useful to see one example. Consider the angular momentum operators S_x, S_y, S_z on the Hilbert space of a spin-1/2 system. When we interpret these observables, the expected values (⟨S_x⟩, ⟨S_y⟩, ⟨S_z⟩) form a vector, at a point in the background spacetime.

Now, a unitary operator U is a symmetry transformation on vectors in Hilbert space: ψ → Uψ. It can also be viewed as transforming observables O → U⁻¹OU. That’s because:

     ⟨Uψ, S_xUψ⟩ = ⟨ψ, U⁻¹S_xUψ⟩ = ⟨U⁻¹S_xU⟩

Moreover, there is an operator R_z(θ), which can be shown (see, e.g., Sakurai 1994, 3.2) to transform the expected value of the angular momentum observables as follows:

     ⟨R_z⁻¹(θ) S_x R_z (θ)⟩ = ⟨S_x ⟩cosθ − ⟨S_y ⟩sinθ
     ⟨R_z⁻¹(θ) S_y R_z (θ)⟩ = ⟨S_x⟩sinθ + ⟨S_y⟩cosθ
     ⟨R_z⁻¹(θ) S_z R_z(θ)⟩ =⟨S_z⟩.

In other words, transforming Hilbert space by the unitary operator R_z(θ) has the same effect as applying a rotation matrix through a degree θ about the z-axis, to vectors in spacetime. And of course, such a transformation can equivalently be implemented by just rotating the background spacetime, instead of the vector itself. Thus, such unitary transformations can be equivalently understood as implementing spacetime symmetries.

Get Started Reading Recent Classics on the Philosophy of Physics

Which philosophy of physics books are relatively recent (say, post-1980), but still clear classics that every graduate student in the field should at least paw through? Here's a preliminary list, ordered alphabetically.

• Albert, David: QM & Experience
• Albert, David: Time & Chance
• Barrett, Jeff: QM of minds & worlds
• Bell, John: Speakable & Unspeakable in QM
• Bub, Jeff: Interpreting the Quantum World
• Cartwright, Nancy: How the Laws of Physics Lie
• Earman, John: Bangs, Crunches, Whimpers & Shrieks
• Earman, John: Primer on Determinism
• Earman, John: World Enough & Spacetime
• Fine, Arthur: The Shaky Game
• Friedman, Michael: Foundations of Spacetime Theories
• Hughes, RIG: Structure & Interpretation of QM
• Maudlin, Tim: Metaphysics Within Physics
• Maudlin, Tim: Quantum Non-locality & Relativity
• Penrose, Roger: The Emperor's New Mind
• Price, Hugh: Time's Arrow & Archimedes' Point*
  
• Redhead, Michael: Incompleteness, Non-locality & Realism
• Redhead, Michael: From Physics to Metaphysics
• Sklar, Lawrence: Philosophy of Physics*
• Sklar, Lawrence: Physics & Chance*
  
• Teller, Paul: Interpretive Introduction to QFT
• van Fraassen, Bas: QM An Empiricist View

Also, some classic unpublished texts:

• David Malament, Notes on Geometry & Spacetime
• Rob Clifton, Introductory Notes on QM

Also, quickly becoming classics:

• Brown, Harvey: Physical Relativity
• Healey, Richard: Gauging What's Real
• Lange, Marc: Introduction to the Philosophy of Physics
• Monographs contained in the Handbook of Philosophy of Physics, Earman & Butterfield (eds)

One important book on the list, John Earman's (1986) Primer on Determinism, has unfortunately reached "rare" status, and is fairly difficult get ahold of for less than $200. Nevertheless, let it be known that an electronic copy of this book does is circulating on the inter-tubes. If you look around a bit, you'll likely be able to find and download a copy for free. Just throwing that out there.

See also my advice about reading on the cheap, and about learning GR online. This list was inspired by a recent post over at It's Only a Theory -- further suggestions are more than welcome. Happy reading!

^(*) Added Feb. 11 - Thanks commenters!

Special Relativity and the Bell Theorems

The Bell Theorems, together with a collection of experimental results (such as those of Aspect et al.), provide good statistical evidence that quantum theory is "non-local." Roughly, this means that the interaction between two bodies in quantum theory doesn't necessarily get weaker as those bodies become spatially separated.

Is this a problem for Special Relativity? That depends on what you think Special Relativity means. Here's a simple flow-chart illustrating some of what's at stake.

For a very accessible view on how we should navigate many of these options, I highly recommend Tim Maudlin's excellent book on the subject. But here are a few thoughts on each of the steps.

• Bell-inequality violation. There is a sect of conspiracy theorists who aren't convinced that the Bell-inequalities are violated by experiment. If that's you, then there's no reason to worry about Special Relativity.
  
• Minkowski Geometry. If Special Relativity requires only that the background spacetime be Minkowski spacetime, then there is no problem for non-local quantum effects. After all, we have plenty of matter theories (quantum field theories) that take place on such a background, and even respect its symmetries to a certain extent. Non-locality is not a problem here.
• Upper limit on the speed of mass-energy transfer. We would normally like to add that matter-energy cannot be transferred faster than the speed of light. But this is not a problem for quantum non-locality, either -- unless you adopt a pretty unusual view of matter-energy transfer. Then what matters is statistical correlation -- see below.
• Signal/Information Transfer. These terms are a bit vague, and people disagree about how to explicate them. However, as the chart suggests, I think that what's really important is whether or not you think there are consequences for the statistical behavior of distant regions.
• Statistical Correlation. This seems to be the heart of the problem. If you think that Special Relativity implies an upper limit on the "speed" at which statistical correlation can occur, then you'll think the Bell-type results violate this. What I mean by that is: interactions in one region can have near-immediate consequences for the statistical behavior of another region, no matter how far apart the two regions are.

But why would someone answer "yes" to the last choice in the chart? Why should we think that Special Relativity implies anything at all about the statistical behavior of matter?

There is no probability measure in SR. Of course, matter satisfying the assumption of local realism appears consistent with Special Relativity, and the Bell inequalities hold for such matter. But I see no reason to think that such matter is required by Special Relativity. If it isn't, then Special Relativity isn't enough to derive the Bell inequalities, and doesn't contradict non-locality.

And that's exactly how we all like it. Right?

PSA Submissions Are a Go

The Philosophy of Science Association is holding its biennial conference in November, in Montréal, QC. Many of you were frantically preparing your manuscript for submission before midnight last night. Phew! Done and done.

Now you can start looking forward to your upcoming trip to Montréal! A few facts of interest:

• Montréal is the second largest city in Canada. It's slightly smaller than Phoenix, AZ and slightly larger than Marseille, Fr.
• Nearly 2 out of 3 people in the city are native French speakers.
• The weather in November is usually 28° to 50° F (-2° to 13° C) and rainy.
• The conference is at the Hyatt Regency in Old Montréal, which dates back to the 17th century. The nearby hill, called "Mont Royal" (hence Montréal) was previously inhabited by the Mohawk Nation.

For a quick view of the city, try this "Montréal in 2 Minutes" video.

Hopefully I'll see you there. And don't forget to share your freshly prepared manuscript on PhilSci Archive before you go!