Soul Physics

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.)

Translation of the long-lost Descartes letter

2010-03-06T11:43:00.001-05:00

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

2010-03-03T15:32:00.003-05:00

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.

Is philosophy fracturing, and what should we do about it?

2010-02-27T16:22:00.000-05:00

I was once a teaching assistant for John Earman's "Introduction to Philosophy of Science" undergraduate course. On the very first day of class, John put this image on the overhead projector:

John's worry was that philosophy of science is becoming so specialized that we can hardly communicate our work to others in the field -- let alone to the philosophical community at large. He made a similar remark in his 2002 PSA Presidential Address (PhilSci-Archive). What John didn't say is what we should do about it. I suppose he didn't know.

To investigate, I decided to participate in a conference consisting mostly of philosophers in very different fields than my own, at this year's NCPS meeting in Charlotte. (The conference was wonderful, by the way). I tried my very best to pitch a paper on time reversal to a general audience (Draft PDF). But all things said and done, I don't think I connected with more than a third of my audience. Discussion with a few people afterward suggested that I was thwarted by the Problem of Babel, after all.

I find it hard to believe that our concerns are that different across analytic philosophy. If so, then why is the Problem of Babel so pervasive? And what should be done to overcome it?

Rescher Prize for Contributions to Systematic Philosophy

2010-02-24T06:39:00.004-05:00

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.

A peek inside the mind of a mathematician

2010-02-23T06:01:00.000-05:00

This is not new, but it's one of the very best math videos on the inter-tubes. It's designed by Bill Thurston, who's known not only for his mathematical talent (winning him a Fields medal), but for his incredible ability to teach.

The video is about Thurston's technique for everting a sphere. But the real gems are the little visual techniques (complete with sound effects), so important for this kind of mathematics. Thurston's video not only makes them accessible, but provides a rare peek into the mind of one of the greatest mathematicians of our time.

See also my note on the history of sphere eversion, and on a more efficient method; you might also try this software from the Geometry Center, which provides an even more hands on experience of turning a sphere inside out. Enjoy!

More Philosophy of Physics in the Blogosphere

2010-02-21T06:06:00.002-05:00

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:

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"

2010-02-20T06:06:00.002-05:00

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

2010-02-17T16:35:00.002-05:00

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)⟩ = cos⟨S_x ⟩ − sin⟨S_y ⟩
   ⟨R_z⁻¹(θ) S_y R_z (θ)⟩ = cos⟨S_y⟩ + sin⟨S_z⟩
   ⟨R_z⁻¹(θ) S_zR_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

2010-02-10T13:14:00.005-05:00

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!

Galilean Freefall Confirmed

2010-02-06T06:00:00.001-05:00

It wasn't centuries of successful theories of mechanics, planetary dynamics, and interplanetary travel that confirmed Galilean freefall. Oh no. It was this.

Special Relativity and the Bell Theorems

2010-02-03T18:02:00.001-05:00

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

2010-02-02T09:03:00.001-05:00

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!

Announcement: Meet Me in North Carolina

2010-01-31T16:50:00.003-05:00

*Beeeeeeeeep* (Is this thing on?) Ahem.

Are there any philosophers of science out there who, on the weekend of February 27th, will be either (a) attending the NCPS/SCSP annual meeting, or (b) interested in skydiving near Charlotte? Weather permitting, I'll be doing both, and would welcome company. The conference and the dropzone are about an hour apart:

View Larger Map
If this means you, feel free to contact me so we can meet up!

I'm always interested in checking out new corners of philosophy departments I haven't seen before. And February seems like a nice time to migrate South for a few jumps. Charlotte just seemed like the perfect opportunity to kill both birds with one stone! Maybe I'll see some of you there.

AMFPVNP9M895

How would negative mass behave in a gravitational field?

2010-01-10T11:32:00.002-05:00

Answer: exactly the same way positive mass would.

That's unusual, because if you try to push a negative mass, it behaves in surprising ways. While a positive mass will accelerate in the direction you push it in, in accordance with Newton's second law,

x'' = F/m,

a negative mass will accelerate in the opposite direction of the applied force. Strange. But now, what if F were a gravitational force? The force of gravity acting on a normal positive mass is

F = -mMG/r²,

where F is directed toward the central mass M. For a negative mass, this force would be in the opposite direction -- the central mass M would repel the negative mass m.

Since (1) the gravitational force on a negative mass m is directed away from the (positive) central mass M, and (2) the negative mass accelerates in the direction opposite to the applied force, these two strange effects cancel each other out. A negative mass in a gravitational field will behave in exactly the same way as a regular mass.

It seems that this is just another interesting consequence of the equality of inertial and gravitational mass. Doing some searching around, it seems that Hermann Bondi was the first to write about it, in the context of General Relativity -- Bondi plays around with various scenarios involving positive and negative masses, and find some surprising results.

Of course, the evidence for negative-mass matter remains zilch. But it seems to be an interesting theoretical plaything!

Soul Physics: Best of 2009

2010-01-03T17:07:00.001-05:00

If you're new to Soul Physics, try some of our classic posts! The following posts received either a lot of traffic or a lot of comments (or both) in 2009.

Where the material conditional gets its truth conditions. A simple explanation to aid students of classical predicate logic.
Hyper-intelligent fish and black hole thermodynamics. A thought on the well-known analogy between hydrodynamics and black hole thermodynamics.
Can Time Unfold in the Wrong Direction? An argument that yes, it can, according to Special Relativiy.
How Time Really Passes. A response to John Norton's argument that there's some sort of problem here.
Another unexpectedly simple failure of determinism. An ill-posed initial value problem of the Norton-dome kind, constructed from a simple mass-on-a-spring system.
Possible Positions on the Passage of Time. An illustrated catalogue of your options.
Group Structural Realism (Part 4). The last of a 4-part argument that there's little hope for this kind of view.

Best wishes for 2010!
-- Bryan

Get Started Handling Academic Citations Like a Pro

2009-12-31T10:49:00.005-05:00

Using a Mac to do your academic work? Here's a brief tutorial on how to optimize your day-to-day dealings with academic citations, by integrating Bibdesk, Textmate, Quicksilver and Scholar. Below, you'll find instructions on how to set up all these neat little tricks.

Quicksilver Web Search. After you download Quicksilver (free) and open the preferences window, go to Plug-ins > All Plug-ins, and check the box next to "Web Search Module." Restart Quicksilver. Go to Catalog, click the "+" at the bottom-left, and select "Web Search List." A new pane should appear (if not, click the little i), in which you can add any number of web-search shortcuts by clicking the other "+" appearing above "Source Options" on the new pane. Just go to any search box (such as Google Scholar) and search for "***", and copy resulting page's URL. Paste it in the web-search shortcut you just created in Quicksilver under "URL." Under "Name," write something useful like "GoogleScholar." Finally, rescan your catalog by clicking the circular-arrow in the bottom right. That's it! To search for stuff, just invoke Quicksilver, hit "." (period) to enter text, hit Tab, start typing "Find with", hit Tab again, and start typing "GoogleScholar."

Even more useful -- if you want to use a secure remote access service (like VPN) to search for articles from home, just log into VPN before going searching for "***" in Google Scholar. When you first search for something, you'll have to enter your username and password. But in every search after that, you'll have your regular access to secure articles.

Bibtex Records on Google Scholar. You won't see these unless you actually set them up. Log into your Google account, go to Google Scholar, and click the "Scholar Preferences" link next to the search box. Click the dial that says, "Show links to import citations" and select "BibTex" from the drop-down menu. Now you'll see that useful BibTex link below all your search results.

Bibdesk. You can download this beautiful little app for free from Sourceforge. If you copy a bibtex citation record to the clipboard, you can add it automatically to Bibdesk by typing Command-Option-L.

Textmate Drop-down Menu. Make sure you've installed Textmate (trial available from Macromates) and Bidesk. To set up that neat little drop-down menu in Textmate, first download and unzip the "Completion.zip" package (available here or here). Second, double-click each of the .tmComand files. Next, copy the binary file "BibDeskTMCompletions" somewhere convenient, like a folder called "bin" in your home directory. Finally, open Textmate and go to Bundles > Bundle Editor > Show Bundle Editor. Click the newly-added bundle, "Build Cite With BibDesk." Find the line that begins "CMD = '"$HOME"...," and set it to the path of your binary file. If you chose the "bin" folder in your home directory, just change this line to the following:

CMD = '"$HOME"/bin/BibDeskTMCompletions

Do the same thing in the bundle, "Bibdesk DO Completion." While still in this second bundle, set a useful Activation key, such as the Tab-Trigger "cite". Close the editor and restart Textmate. Now, to make the drop-down menu work, you just need to open a Latex document with a bibliography set at the end -- for example, mine reads \bibliography{~/Documents/MasterBibliography.bib}. Now, whenever you type "cite" followed by the Tab key, you'll get a drop-down menu displaying all the references in you .bib-file.

Ice water on closed timelike curves

2009-12-05T19:59:00.001-05:00

Today, let's kick off our shoes and relax our standards for what counts as a reasonable spacetime. Let's talk about spacetimes allowing for a certain kind of time travel, called a closed timelike curve. Being stuck on one would be like being in Nietzsche's "eternally recurring" world -- or more recently, like Bill Murray in the movie Groundhog Day.

Although worlds containing closed timelike curves arise from classical general relativity, one can worry about whether such worlds obey the laws of statistical physics. For example, what would happen to a glass of ice water on a closed timelike curve?

The state of the ice water would have to be cyclic. So, if the ice were able to melt at all, then it would also have to "unmelt," and coagulate as ice again when it came back around. Apparently, entropy in our time-traveling ice water has to regularly and dramatically decrease. But that's supposed to be impossible.

Does that make closed timelike curves incompatible with the principles of statistical physics? Not necessarily. Boltzmann would argue that the reason the ice should always melt is that, among all the possible trajectories that the particles in the ice water might travel, those in which the ice melts form the overwhelming majority.

So, assuming each possible trajectory is equally likely, it follows that the ice is overwhelmingly likely to melt.

But suppose we restrict what counts as a "possible trajectory" for a particle in our time-traveling ice water. After all, we're on a closed timelike curve. The possible trajectories are different, because they are required to be cyclic -- every particle has to end up back where it started. This constraint guarantees that, if the ice melts at all, it always also unmelts. Boltzmann's counting argument then just determines what most commonly happens in between, among these trajectories.

Of course, everything has to happen the same way every time around an individual closed timelike curve. So, "most commonly" would have to be made sense of in terms of an array closed timelike curves, each with a glass of ice water on it. But apparently, no fundamental principles are violated. And so statistical physics might yet live in harmony with closed timelike curves.

Where the material conditional gets its truth conditions

2009-12-01T12:57:00.003-05:00

Oh, the material conditional. Some love it, some hate it. But can we all agree that explaining it to the uninitiated is a perennial headache? If you've taught baby-logic, you know how this goes. There you are, giving a just lucid shpeel on deductive systems, until you get to this part:

A	B	A → B
T	T	T
T	F	F
F	T	T
F	F	T

and the tires screech to a halt. Why are those bottom two values True? -- they demand. The first two rows don't bother them. But if A is false, why should it be that A → B is true, regardless of the truth of B?

You could just say it's a convention, get over it. But why is this the convention adopted in classical logic? My colleague Jonathan Livengood and I discussed this, and came up with a better answer:

Suppose we agree on the first two rows of the above truth table. If implication (→) is both non-trivial and asymmetric, then this its only possible truth table.

Here's why. Start by writing down all the possibilities for these bottom two rows. There are only four, and A → B has to be one of them.

A	B	1	2	3	4
T	T	T	T	T	T
T	F	F	F	F	F
F	T	T	F	F	T
F	F	F	T	F	T

Column 1 is trivial, because it has the same values as B. If this were the correct column, then saying A → B would mean the same thing as just saying B. So, assuming → is not trivial, we can throw this column out.

Column 2 has a symmetry property that implication doesn't. Namely, it stays the same if we reverse the A and B cells. If this were material implication, then A → B would be true if and only if B → A is true. So, assuming → is asymmetric, we can throw this column out too. (This column is actually the usual truth table for A ↔ B.)

Column 3 has exactly the same problem: it stays the same when we reverse the cells containing A and B. So we can throw it out for the same reason. (This column is actually the usual truth table for A&B. So, plausibly, we can also observe that "implies" should mean something different than "and.")

And that's it! If → is non-trivial and asymmetric, then Column 4 is the only option left: the standard, not-just-conventional truth table for material implication.

How to time-reverse a quantum system

2009-11-25T14:06:00.002-05:00

Time-reversing a classical Newtonian trajectory is simple. If q(t) and p(t) are the positions and momenta of a particle on the trajectory, then time reversal flips that trajectory as follows:

q(t) → q(-t) = q(t)
p(t) → p(-t) = -p(t)

For example, a particle traveling along some path with velocity to the left becomes a particle traveling along that path with velocity to the right -- just like when we play a movie in reverse. Very simple.

In quantum mechanics, time-reversal looks comparatively strange, because it involves complex conjugation. (More precisely, it is implemented by an antiunitary Hilbert space operator.) Why? Here's my answer today (more answers later):

One reason that time reversal in quantum mechanics requires complex conjugation is that time-reversing a wave function requires time-reversing its phase.

I recently pointed out an oversimplified way to visualize certain wave functions. Here's a better way to put the idea: The phase of a simple plane wave can be visualized as the assignment of a "dial value" to little regions in spacetime.

In particular, if the plane wave (in the position representation) has the form:

ψ(x, t) = exp(ipx - it),

then exp(-it) -- the phase -- is just a point on a circle, lying on the complex plane. So, we can think of ψ(x, t) as assigning dial-values to points -- each corresponding to a different location on the circle. Moving smoothly forward through time gives rise to a changing dial value. This just represents the changing phase as the plane wave propagates through space:

(Click to enlarge)

Now, how should we time-reverse such a system? Well, minimally, it seems we'd want our dial to run in reverse. That's exactly what conjugation does. Notice that by sending ψ to its conjugate ψ*, we flip the arrow about the real axis of the dial:

The result is an arrow moving in the opposite direction. Moreover, the arrow cannot be reversed by any symmetry operator that does not conjugate (i.e., a unitary operator). That's because the wavefunction ψ(x, t) is given by an inner product, and unitary operators preserve inner products.

So there you have it: one way to see why quantum time-reversal requires conjugation.

As a final note: a wave's phase really just describes its relationship to the origin of a coordinate system. So, one might complain that phase isn't a physical feature of a wave, any more than a coordinate system is. However, differences in phase, and in particular changes in phase, are physical features of a wave. So, our account of time reversal must be sure to reverse these quantities as well.

2009 Nobel Laureates in Physics

2009-10-06T08:56:00.002-04:00

The 2009 Nobel Laureates in Physics have been announced! This year the award honors pioneers in information technology. Half the prize is going to Charles Kao (Standard Tel Labs), and the other half to Willard Boyle and George Smith (Bell Labs).

Charles Kao (left) - Willard Boyle and George Smith (right)

Charles Kao is being recognized for his foundational work in the development of optical fibers. Those fibers are now the veins and arteries of our electronic lives, allowing us to transmit data efficiently across long distances. They function on a very simple principle: light signals can be transmitted by bouncing light bouncing through a hair-thin glass tube.

However, the first optical fibers couldn't even transmit data across a football field -- the signal attenuated too quickly as it passed through the fibers. In the 60's, Kao was able to show that this attenuation was the result of the absorption and scattering of light, caused by metal ions in the glass. The problem was thus to figure out how to develop more pure glass fibers. Kao correctly argued that this could be done using silica. His research on the problem later led to the development of the super-efficient optical fibers we know today.

Boyle and Smith are being recognized for their invention of the Charged Coupled Device (CCD). That's the device that takes the place of photo-film in your digital camera. From the mid-60's up through the 70's, Bell Labs tried to develop and market an early video conferencing tool, called the Picturephone. The idea never took. But the CCD's developed along the way became central in our ability to take and transmit digital images.

In the semi-conductor department of Bell Labs, Boyle and Smith were asked to develop a semi-conductor memory device, on pain of losing funding in their department. After an hour's discussion, the two got the basic idea sketched on a chalkboard: to store a charge in a confined region using a metal-oxide semiconductor. The surface of the device is a matrix of little capacitors, or pixels. When light hits a pixel on the surface, it knocks out electrons that are stored in the capacitor. The lucky feature of this process is that the number of electrons is proportional to the intensity of light. (That's an essentially quantum phenomenon called the photoelectric effect, explained by Einstein in 1905.) So by recording the number of electrons ejected, you can get a reading on the intensity of light in each little region, and use this to produce a photo image.

Congratulations to Kao, Boyle and Smith!

Visualize a Wave Function

2009-10-04T22:29:00.024-04:00

Can you visualize a normalized wave function on spacetime?

Let's try with a simple example. The role of a wave function is to assign a complex number to each point (x, t) in spacetime. This is central to a quantum description of the world. The complex number at each point is interpreted as an amplitude, which determines a probability -- the probability of measuring some physical quantity (like position or momentum) at that point.

But in the end, it's just a complex number, of unit length.

Now, a complex number lives on the complex plane -- a plane with the vertical axis representing a complex value, and the horizontal axis representing a real value. And the complex numbers of unit length live on a circle around the origin. So you can think of these numbers as readings on a circular meter -- like a speedometer or an altimeter -- except that the meter reads amplitudes instead of speeds or altitudes.

That means you can visualize the wave function ψ(x, t) as assigning a meter-reading to each point in spacetime. And if I fix a point in space -- like a spot on my kitchen floor -- then I can trace through the history of this wave function over time. The result will be a smoothly changing meter reading. For example, the meter arrow might just spin around clockwise over time.

Then it would look something like the following.

(Click to enlarge)

Challenge Question: How would you characterize the "time-reverse" of this description of the world? Tune in next post for a discussion...

Edit: The above account is not quite right -- see the post comments for more.

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