Soul Physics

Beyond the CPT theorem

2013-05-17T20:28:00.000-04:00

Virtually all known laws of physics are invariant under the CPT transformation — that is, the combined operations of Charge conjugation (C), Parity or "mirror flipping" (P), and Time reversal (T). What that means is the following. Start with a trajectory $\psi(t)$ through state space, which represents some possible way for a system to change over time according to the known laws of physics. Now transform that trajectory, by reversing the order of all the states, and then applying C, P and T to each of them:

\[ \psi(t) \mapsto CPT\psi(-t). \]

$CPT$-invariance means that the resulting trajectory $CPT\psi(-t)$ will also be a possible according to the laws. One can check that this is equivalent to the statement that $CPT$ commutes with the Hamiltonian, $ [CPT,H]=0.$

Why is $CPT$ so often symmetry? There is a theorem that explains it: if we characterize quantum field theory in a very plausible and general way (such as by the Wightman axioms or Haag axioms), and in particular assume that it admits a unitary representation of the Poincaré group, then $CPT$-invariance is guaranteed. This result is called the CPT theorem. See Borchers & Yngvason for a very readable proof in the Haag framework.

Ok, that's the background for today. Now:

Question: as we move forward, and begin to adopt theories that go beyond the standard model of particle physics, will we continue to have a CPT theorem or something like it?

It is widely believe that the CPT may fail in generic extensions of quantum theory. In particular, the requirement of a unitary representation of the Poincaré group is pretty strong, and may not hold in the kind of general context of interest in quantum gravity. Just search for CPT-violation on the arxiv to see what I mean.

But there is a sense in which something "like" a CPT theorem probably will hold in physics beyond the standard model. That sense is this: every unitary dynamics admits infinitely many "time reversing transformations" (i.e., time reversal plus some other linear symmetries) under which the dynamics is invariant. Here's a more careful statement of this fact.

Fact. Let $\mathcal{H}$ be a separable Hilbert space with a unitary group $U_t = e^{-itH}$ describing the quantum dynamics, and let $T$ be the (antiunitary) time reversal operator. Then there exists a unitary operator $\Theta$ such that the dynamics is $\Theta T$-invariant, in that $[\Theta T,H]=0$.

Think of $\Theta$ as some kind of generalized symmetry transformation, similar to $CP$, but something else entirely. It is in this sense that this fact expresses something like the $CPT$-theorem, although unlike the $CPT$-theorem the mathematics is completely trivial.

There are two steps to seeing why this "Fact" is true. The first is to observe that, for every self-adjoint operator $H$, there is something called a conjugation operator $K_H$ such that $[K_H,H]=0$. Here's how it's defined. The self-adjoint operator $H$ comes with its own basis set for the Hilbert space, ${v_1,v_2,v_3,\dots}$. That's because of the spectral theorem. So for every vector $\psi$ in the Hilbert space there are complex constants $c_i$ that allow you to write that vector,

\[\psi = c_1v_1 + c_2v_2 + c_3v_3 + \cdots.\]

The conjugation operator $K_H$ is just the operator defined by conjugating all the complex constants of a vector written in the $H$-basis,

\[K_H\psi = c_1^*v_1 + c_2^*v_2 + c_3^*v_3 + \cdots.\]

So that's pretty easy. And it's easy to check that $K_H$ satisfies some special properties: it is antilinear $K_H(a\psi+b\phi)=a^*K_H\psi+b^*K_H\phi$, antiunitary $\langle K_H\psi,K_H\phi\rangle = \langle \psi,\phi \rangle^*$, and it commutes with the $H$ that we used to define it $[K_H,H]=0$. We will use all of these properties in the next step.

The second step to seeing why our "Fact" is true is to recognize that if $K_1$ and $K_2$ are any two antiunitary operators, then they are related by a unitary operator, $K_2 = UK_1$. It's a nice exercise to check for yourself that this is true, but if you get stuck, try here.

Since time reversal $T$ and the conjugation operator $K_H$ for the Hamiltonian are both anitunitary, this means that $K_H$ is related to $T$ by some unitary operator $\Theta$:

\[K_H = \Theta T.\]

So, there is always a unitary operator $\Theta$ such that $\Theta T$ commutes with the Hamiltonian $H$.

Above, I said there were actually infinitely many such operators. Puzzle: Can you work out why?

If you're impatient, here's the reason. Let $f:\mathbb{R}\rightarrow\mathbb{C}$ be a function (a Borel function if we're being pedantic), and let $f(H)$ be the corresponding Hilbert space operator as a function of the Hamiltonian $H$. (For example, if $f(x)=x^2$, then $f(H)=H^2=H\circ H$.) Every such function $f(H)$ commutes with $H$. And we already know that $K_H$ does as well. So their composition commutes with $H$ as well:

\[0 = [f(H)K_H,H] = [f(H)\Theta T,H] = [\Theta^\prime T, H],\]

where $\Theta^\prime = f(H)\Theta$. Since there are infinitely many such functions, this means that there are infinitely many such operators $\Theta^\prime$.

Four 1975 Lectures by Paul Dirac

2013-04-26T15:19:00.000-04:00

Slobodan Perovic (via Chris Joas) pointed this gem out to me. Four wonderful lectures given by Paul Dirac in 1975, on various topics in the foundations of quantum theory and cosmology.

"Quantum Mechanics"

"Quantum Electrodynamics"

"Magnetic Monopoles"

"Large Number Hypothesis"

Stop commercializing academic publishing

2012-03-07T09:37:00.000-05:00

Dear academic publishers: your business model runs completely counter to the aims of the academic community, for this reason: academic publishing is not like commercial publishing. Stop conflating the two.

I know of nary an academic that is publishing for the bling. So, stop thinking of us as obscure niche counterparts to J. K. Rowling. Scholarly authors would be crazy to write books for the tiny (or often non-existent) monetary compensation. They do it to disseminate information as widely as possible. So, stop treating academic work as if it were commercial. You're running completely off the rails.

Here's an example. You put a $229 USD price-tag on an important textbook, Souriau's (1970) Structure of Dynamical Systems. I'm sure you've done the calculation: how many people can be expected buy the textbook at that price? Not many. Not to mention that we could pick up two copies of J. K. Rowling's "complete works" for this royal sum. This is not dissemination of information. This is you failing the academic community.

Because of your silliness, Souriau's scholarship is not being widely shared in the way that the academic community needs. In this case, the author himself is taking steps to overcome your failing, by posting the French edition on his website. (Souriau's stated motto, translated from French: "I wanted this site to distribute my work as widely as possible.") Unfortunately, an English version of the book is not freely available. At least, not anymore.

It was freely available, on underground websites like Library.nu (rest in peace). Such websites came into existence because you did not meet the aims of the academic community. While we worked to share information, providing publishers with free content to put in their books and journals, you turned around and sold that content at commercially high prices. You actively prevented the dissemination of scholarly knowledge. On the other hand, by making half a million scholarly books publicly available, Library.nu actively enabled it. The end of this service amounted to a huge loss for the scholarly community.

Academics and publishers alike are beginning to recognize that we have a problem. Fortunately, there are many other publishing models on the table, many of which might go far to meet our aims. Take a long hard look in the mirror, academic publishers. It's time for a major change.

Philosophy of Science Journals and Elsevier

Philosophy of Science Journals and Elsevier

2012-02-21T18:02:00.000-05:00

Not Elsevier

Elsevier

Journal	Publisher
American Philosophical Quarterly	University of Illinois Press
Analysis	Oxford University Press
Australasian Journal of Philosophy	Taylor & Francis (Routledge)
Biology and Philosophy	Springer Verlag
British Journal for the History of Philosophy	Taylor & Francis (Routledge)
British Journal for the Philosophy of Science	Oxford University Press
Canadian Journal of Philosophy	University of Calgary Press
Dialectica	Wiley-Blackwell
Erkenntnis	Springer Verlag
European Journal of Philosophy	Blackwell Publishing
Foundations of Physics	Springer Verlag
General Relativity and Gravitation	Springer Verlag
Journal for General Philosophy of Science	Springer Verlag
Journal of Consciousness Studies	Imprint Academic
Journal of Applied Logic	Elsevier
Journal of Philosophical Logic	Springer Verlag
Journal of Philosophy	Journal of Philosophy, Columbia U.
Journal of Symbolic Logic	Association for Symbolic Logic
Journal of the History of Ideas	University of Pennsylvania Press
Journal of the History of Philosophy	Johns Hopkins University Press
Linguistics and Philosophy	Springer Verlag
Mind	Oxford University Press
Mind and Language	Blackwell Publishing
Minds and Machines	Springer Verlag
Monist	Hegeler Institute
Noûs	Wiley-Blackwell
Pacific Philosophical Quarterly	Blackwell Publishing
Philosophia Mathematica	Oxford University Press
Philosophical Quarterly	Wiley-Blackwell
Philosophical Review	Duke University Press
Philosophical Studies	Springer Verlag
Philosophy and Phenomenological Research	Wiley-Blackwell
Philosophy Compass	Blackwell Publishing
Philosophy of Science	University of Chicago Press
Proceedings of the Aristotelian Society	Wiley-Blackwell
Ratio	Blackwell Publishing
Review of Symbolic Logic	Cambridge University Press
Studia Logica	Springer Verlag
Studies in History & Philosophy of Science (A, B, C)	Elsevier
Synthese	Springer Verlag
Theoria	Wiley-Blackwell

about the boycott

The Cost of Knowledge: thecostofknowledge.com.
Chronicle: "As Journal Boycott Grows, Elsevier Defends Its Practices"
Azimuth: "Ban Elsevier"

Sellars and the Philosophy of Physics

2010-07-05T18:10:00.001-04:00

In a letter to Chisholm, Wilfred Sellars wrote:

Thus, while I agree with you that
'. . .' means - - -
is not constructable in Rylean terms ('Behaviorese,' I have called it), I also insist that it is not to be analyzed in terms of
'. . .' expresses t, and t is about - - -.
My solution is that "'. . .' means - - -" is the core of a unique mode of discourse which is as distinct from the description and explanation of empirical fact, as is the language of prescription and justification.

Although Sellars was concerned with the philosophy of mind, there is something important here for philosophers of physics to learn as well. A major activity of physics is the collection of empirical facts. Another is the prediction and justification of these facts. But the activity of investigating meaning is a distinct activity altogether. This last activity includes much of what concerns the philosophy of physics, when it is done well.

Whether it be causation, equivalence, gauge, prediction, or simultaneity -- among many examples -- I think much of what distinguishes philosophy of physics from physics is a central concern with the (particularly philosophical) activity of explicating meaning.

Elegant Desktop ToDo List

2010-06-25T11:20:00.002-04:00

A lot of people have been asking me about the Desktop ToDo list that appears in my last screencast. So, here's the scoop on this simple and elegant ToDo list system, which can be easily synchronized across multiple (Mac) computers.

Here's what you need to set this stuff up.

Mac OS X. I don't know of any good windows analogues for Quicksilver and Geektool. Let us know if you do!

Geektool. This beautiful little app is available for free from Tynsoe.org. For more on what it can do, try this tutorial. For our purposes, here's how to display a TXT file on your Desktop:

Create a new Shell geeklet in the Geektool preference pane
Enter the command: cat PathToYourList/YourList.txt
To synchronize multiple computers, make sure the file is in a Dropbox folder.

Quicksilver. This is why I own a mac. It's free from Blacktree.com. If you're new to it, try this beginner's guide. To set up the 'Append' and 'Completed' functions seen in the screencast:

On the Quicksilver > Plugins page, add the 'Text Manipulation Actions' plugin. Then make sure 'Append' box is checked on the Preferences > Actions page.
Now you can append text to any .TXT file -- but only files with that extension.
Download the Completed script.
Open it with ScriptEditor and set the path to your Completed.txt file.
Add the Completed.scpt script to the folder YourUsername/Library/Application Support/Quicksilver/Actions. Create this folder if it doesn't exist yet.

And that's it! As always, be sure to leave your thoughts, ideas or improvements in the comments below.

The three-way duel

2010-06-24T13:14:00.002-04:00

The late great Martin Gardner once posed this puzzle.

Suppose you're involved in a duel with two other people. You (Person A) shoot first, followed Person B, followed by Person C, then it goes back to you, and so on. Moreover, you know the following about everyone's shooting skills.

You (Person A) will hit your target with probability 1/3.
Person B will hit her target with probability 2/3.
Person C is a perfect marksman, will hit his target with probability 1.

You get to go first. Who would you shoot at, and why? Best solution gets a free sheep.

(Note: your options are Person B, or Person C, or neither.)

Update: Jonathan of Unshielded Colliders has been awarded a free sheep for his solution. Here you go, Jonathan:

CPT: The 'Intuitive' Approach

2010-06-14T15:50:00.053-04:00

Khriplovich and Lamoreaux (1997, §2) suggest a very interesting argument that CPT provides the correct notion of "complete reversal" in physics.

The background assumption is that "complete reversal" should have effect of reversing the sign of 4-vectors in spacetime. David Malament, for example, has suggested that time reversal in classical electrodynamics should have this effect on timelike vectors. The proposal here is that "complete" motion reversal to have this effect on all vectors (timelike, spacelike, and null).

Clearly, time reversal T on its own is not enough for this -- it doesn't reverse spacelike vectors. Parity reversal P isn't either -- it doesn't reverse timelike vectors.

What about PT? After all, flipping about two axes is equivalent to a rotation. Shouldn't that be enough to reverse all four vectors? As it turns out, it's not enough, at least when it comes to 4-current j^a. Since both P and T fix charge density and reverse current, we have:

PT j^a = PT (p, j) = P (p, -j) = (p, j).
To reverse current, we need an operator C that sends particles to antiparticles, and thus sending j^a to -j^a. Thus, to get "total" motion reversal in a world with current, we need the CPT operator.

What I like about this thinking is that it depends crucially on the kind of matter fields in play. It's only in the presence of 4-currents that PT is not enough to completely reverse motion. But similarly, the discovery of additional exotic matter fields might someday imply that CPT is not enough to reverse motion, either.

Update: Wolfgang reports news about evidence for CPT-violation in a recent Fermilab experiment.

Even more philosophy of physics conferences, Summer 2010

2010-06-04T07:56:00.000-04:00

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

2010-05-25T09:44:00.000-04:00

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

2010-05-16T21:40:00.000-04:00

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.

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.
The wave-function description of the measurement processes is *not* guaranteed to have determinate values at any given time.
All physical processes are governed by the same fundamental physical laws (and hence by equally deterministic equations of motion).
So, since Schrödinger evolution is governed by deterministic laws, the measurement process must be governed by deterministic laws as well.
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

2010-04-22T11:22:00.002-04:00

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:

a theory is (characterized by) the collection of its (Tarskian) models
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.

In Memory of Bacon

2010-04-09T06:00:00.002-04:00

Not the fatty breakfast-food, although you may remember that if you like. The great Sir Francis Bacon died 384 years ago today. And in memory of Bacon, here is your assignment:

THIS PASSAGE IS A FRANCIS BACON CIPHER WHERIN THE BOLD LETTERS REPRESENT B AND THE PLAIN LETTERS REPRESENT A

Cheers!

On Th' Electrodynamics O' Moving Bodies

2010-04-05T10:06:00.001-04:00

What do you do when you come across an excellent English-Pirate translator? You immediately translate Einstein's presentation of Special Relativity into Pirate-speak:

't might appear possible t' overcome all th' difficulties attendin' th' definition o' "time" by substitutin' "th' position o' th' wee hand o' me watch" fer "time." An' in fact such a definition be satisfactory when we be concerned wi' definin' a time exclusively fer th' place 'ere th' watch be located; but 'tis nay longer satisfactory when we be havin' t' connect in time series o' events occurrin' at different places, or--what comes t' th' same thing--t' evaluate th' times o' events occurrin' at places remote from th' watch. (More)

Now that's physics even a scurvy bilge-rat can understand. The entire Pirate-speak translation of "On the Electrodynamics of Moving Bodies" can be viewed here. Enjoy!

Improving the Peer Review Process

2010-04-04T09:43:00.000-04:00

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

2010-04-02T11:48:00.008-04:00

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.

Watch the LHC status updates live

2010-03-30T08:17:00.000-04:00

The Large Hadron Collider has finally begun collecting data for 7 TeV collisions. They've set up an interesting live feed, in which you can keep an eye on the collision rate in each detector:

The nodes that say "PHYSICS" are detectors that are now actively collecting data. Click each one to see its live collision rate feed. Hi-res 3D images of ATLAS detector events can be found at their project website:

Images from the ALICE experiment area available through their website:

And, in case all this is still making you nervous, here are a few rants about black holes at the LHC. Enjoy!

Constructing the Ultimate Machine

2010-03-29T09:13:00.001-04:00

The mathematician and engineer Claude Shannon kept a machine on his desk, which he called "The Ultimate Machine." As it turns out, there are a number of clever instantiations of Shannon's machine on the intertubes. Here are a few of the best ones.

The Classic Model. Large. Sumptuous. Classic.

The Elegant Two-Handed Version. Transparent so you can see the mechanism at work.

Lego Ultimate Machine. The ultimate application of your favorite building bricks.

Variant - The Unplugger. There seems to be a slippery slope from this to "The Exploder."

And now the craze over the Apple iUltimateMachine is just a matter of time. Not to mention the Ultimate iPhone App. You're welcome, Steve Jobs.

Wigner's elegant characterization of time reversal

2010-03-25T09:13:00.002-04:00

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

2010-03-22T07:31:00.005-04:00

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

2010-03-17T12:01:00.006-04:00

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

2010-03-15T18:19:00.000-04:00

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!

Reasons to love the "Dark Energy Task Force"

2010-03-12T05:07:00.006-05:00

It's official: the words 'dark,' 'energy,' 'task,' and 'force' have all been used in the title of a single scientific paper. The "Report of the Dark Energy Task Force" is available here on arxiv.org. Obviously, it's hard not to love a paper like this. The reasons appear to break down roughly as follows.

Of course, as I've noted before, there remain many alternatives to dark energy cosmology. But without a title like this, I'm afraid the competition is doomed.

1907 Crisis in Mathematical Physics According to Poincaré

2010-03-10T06:01:00.002-05:00

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:

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."
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?"
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.
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."
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.

Keep your caffeine tank at its optimum level

2010-03-08T06:06:00.005-05:00

Keeping your caffeine tank full is an essential part of history, philosophy and physics. Follow this easy chart to keep your caffeine intake at its optimum level. (Data from Wisebread.com.)

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