CPT: The 'Intuitive' Approach

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.

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.

How would negative mass behave in a gravitational field?

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!

Visualize a Wave Function

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.

Hyper-intelligent fish and black hole thermodynamics

Bill Unruh's recent collection on black hole analogues begins,

Deep beneath the great encircling seas of the Discworld lived a species of hyper-intelligent fish. (Unruh 2007, p.1)

Unusual, but inspiring: Unruh compares Hawking radiation -- the thermal heat bath emitted by black holes -- to a scenario he imagines in Terry Pratchett's Discworld. Pratchett's world is basically a big dish, with water flowing over the edges.

On Unruh's take, the dish-water is filled with little physicist fish, who are trying to determine the laws of physics. The fish are blind, but use sound waves to interpret their environment. And they are mostly successful. However, as water falls off the edge of the world, it reaches speeds faster than the speed of sound. Events beyond this "sound horizon" are thus inaccessible to the fish in the ocean.

One day, a graduate-student-fish goes flying off the edge while the professor-fish observes. (Professor Unruh apparently expects a lot of his students.) The graduate student yells "Help," while falling off. Then he plunges to his doom. But, from the professor's perspective, the sound of the graduate student's scream persists forever, getting ever more bass-shifted, as the student approaches the horizon.

The point is, the unlucky graduate-student-fish is directly analogous to an astronaut falling into a black hole. From the astronaut's perspective, nothing special happens as she crosses the event horizon. But from an outside observer's perspective, the astronaut appears to be forever approaching (but never crossing) the event horizon, and the light she emits getts ever more red-shifted.

Of course, the astronaut will get ripped to shreds by tidal forces, while the fish will not.

And so the "black hole analogue" debate begins. Black holes are widely believed to have a number of thermal properties -- for example, black holes have a temperature proportional to their surface gravity. Analogously, soundless "dumb-holes" (as Unruh calls them) in water can be shown to have interesting thermal properties as well. And -- tantalizingly -- it appears possible to carry out experiments that would actually test the properties of "dumb-holes," even though black holes remain outside our reach.

But does evidence for a sound-based analogue somehow provide us evidence about a real black hole?

I see no plausible way that it can. Although a black hole is mathematically similar to a "dumb hole," it is not the same thing. And history has something to teach us here: gas and fluid vortices are "mathematically similar" to Descartes' aether vortices. But experiments with the former do not provide evidence for the latter. After all, aether vortices don't exist! So, in spite of some interesting recent experiments (see here), we still don't have any new evidence that black holes have thermal properties.

Nevertheless, there might be one thing that sound-based experiments can still teach us about black holes, according to Unruh:

such successful experiments would greatly increase the confidence in the approximation which were being made in both the gravitational and the analogue situations. ... Certainly the suggestions from the sonic case are that Planckian physics is irrelevant to black hole evaporation, and that the radiation emitted by a black hole is due to low energy processes, processes on the length scale set by the black hole, and not by quantum gravity. (Unruh 2007, p.3.)

This to me seems very plausible: an analogy can tell us whether or not scale is relevant to the effect. According to Unruh, sound-based experiments are really teaching us that black hole thermodynamics is about essentially macroscopic effects. So, our prediction of thermal effects like Hawking radiation won't change when a new theory of quantum gravity comes along, and modifies our picture of the (microscopic, high-energy) Planck scale.

It's a bold and intriguing suggestion, but I'll wait for the iron hand of history to decide.
(If you have a Springer subscription, you can see a version of Unruh's article here.)

Can Time Unfold in the Wrong Direction?

The unfolding of time is typically described as a sequence of spatial regions: one region of space gets realized, and then another, and then another.

For example: consider a region of space in which Harold is crowned king. As events transpire, a number of further regions of space get realized. Finally, a region of space arrives in which Harold is slain by William. The unfolding of time here is described by a sequence spatial regions leading from Harold's crowning to Harold's death.

Although there does not seem to be any one-true-description of unfolding, it happens that intertial observers do agree about the order in which (time-like separated) events unfold.

In particular, in the weak-gravitational regime of Harold and William, spacetime is approximately Minkowski. So, William will identify one possible sequence of spatial regions leading up to the death of Harold. An astronaut moving away from William with velocity v = c/2 will identify a different sequence of spatial regions. However, William and the astronaut will agree about the order in which time-like separated events occur: first Harold is crowned, and then Harold is slain.

Figure 1: An observer on Earth and an astronaut traveling away with velocity c/2 will describe two different foliations of spacetime into space-like hypersurfaces of simultaneity.
On the other hand, an accelerating observer will not generally agree about the order in which these events unfold.

Imagine that at the moment of Harold's death, there is a second astronaut at rest relative to William, who begins to accelerate away at a constant rate. As the astronaut accelerates ever closer to the speed of light, her simultaneity hypersurfaces (the spatial regions that she uses to foliate spacetime) will tilt ever closer to 45 degrees, as dictated by the geometry of Minkowski spacetime. All these surfaces will intersect on some 2D surface I. Moreover, on the other side of the surface of I, the astronaut will judge the order of events to be the reverse of what William judges: first Harold is slain by William, and then Harold is crowned king. The astronaut will describe some events as unfolding in the wrong direction.

Figure 2: The simultaneity hypersurfaces of an accelerating astronaut. These surfaces tilt as the astronaut accelerates, so that events to the left of I are judged to unfold in the wrong direction.
So, the order in which time unfolds is not a fact that all observers agree about.

According to one common definition of objectivity, a claim is objectively true or false if and only if all observers agree about that claim. But in our example, the claim, 'time unfolds from Harold's crowning towards Harold's death' can only be valuated according to the subjective judgement of one observer. Other observers, such as the accelerating astronaut, are equally correct to valuate the claim differently.

Thus, in this sense, there is no objective fact of the matter as to the order in which time unfolds.

How to Create a Closed Timelike Curve

Time along the vertical axis; space along the horizontal....





Voila! And now, how to travel to the past....

Why does matter follow geodesics?

In the very first lessons on General Relativity, we learn that free particles follow geodesics -- the equivalent of straight lines in curved spacetimes. Why? Well, it's easy to show that point particles must follow geodesics, if Einstein's Field Equations are respected. But if you're like me, you might have had this lingering suspicion:

Real matter is not made of point particles. That might have lead you to ask: why does real matter follow geodesics?

Here's a very interesting answer, provided by Geroch and Jang (1975):

Theorem: Let (M, g_ab) be a space-time. Let γ be a worldline satisfying the following condition: For any neighborhood U of γ, there exists a nonzero, symmetric, conserved tensor field T^ab that satisfies the strong energy condition, and whose support is in U. Then γ is a timelike geodesic.

The idea is this: if there's a any kind of field of matter following a worldline through spacetime, which a) behaves like the matter we're used to, and b) is small compared to the curvature near the worldline, then that matter follows a geodesic.

Now, another lingering question: is there anything about this matter field that guaranteees the geodesic is unique?

Geneva Summer School in Philosophy of Physics

Read About It Here. The Geneva Summer School in Philosophy of Physics, 2008 was a 1-week intensive summer school in the middle of the Swiss alps. This year, scholars and graduate students met in Arolla's Hotel Mont-Collon for a conference on the nature of space and time. I was able to participate, thanks to a generous grant from the Wesley and Merilee Salmon Foundation. Here's my report on what happened. There is also a Google Picture Page for the event.

Next year's summer school is rumored to be about the foundations and interpretation of quantum theory. I highly recommend that any graduate student or new-Ph.D philosophers of physics consider going.

Update: 24.Sep.08, 8:45am. Justin Sytsma has posted this report, along with another summer school report by Jonathan Livengood. Justin's excellent blog is well worth checking out, for matters of phil-mind, x-phi, grad-life in Pittsburgh, and general entertainment.

Rotating Discs in GR: Part I

Introduction. Craig Callender (2001) and Jeremy Butterfield (2004) have recently suggested that the Rotating Disc Argument (RDA) fails in General Relativity. I argue that this conclusion is wrong: two recharged versions of the RDA do work in GR. In this post (Part I), we'll review the necessary background material (without GR):

1. The Original Rotating Disc Argument (RDA)
2. Butterfield's Sophisticated RDA.

In the next post (Part II), I'll explain the Callender-Butterfield complaint that the RDA fails in General Relativity, and then argue for two modifications of the RDA do work in GR.

1. The Original Rotating Disc Argument (RDA). The RDA is supposed to be a counterexample to one view about what an object's properties consist in. Fundamental objects can have intrinsic properties (such as the intrinsic angular momentum of an electron), as well as spatial location properties (described however you like). And there is a philosophical view, often called "Humean Supervenience" (HS), which says: those properties are all there are. Every property of every object is made up of fundamental objects (such as electrons), and so the properties of every object are just combinations of the intrinsic properties and spatial locations of its fundamental parts

The putative counterexample to this view: imagine two perfectly homogeneous, continuous discs; one is rotating and the other is not:


The idea is that these discs have different properties, but they they can't be distinguished by the intrinsic and spatial properties of their parts. Here's how this is supposed to work.

First, the continuous homogeneity is supposed to guarantee that you can't pick out the history of any individual point on the disc. So you can't follow the individual worldlines of the points to tell if they are all vertical (motionless through time), or spinning around like a cork-screw.

Second, you are supposed to ignore every other physical effect that angular momentum incurs on the parts of the disc. Then the claim is: these two cases can't be distinguished by the spatial locations and intrinsic properties of their parts, and "Humean Supervenience" is defeated.

2. A More Sophisticated Version. As Callender rightly points out: that second step was pretty bogus. Imagine a spinning bucket. What happens to the surface of the water inside? It becomes convex. So the spatial locations of the points on the rotating disc do distinguish it from the static one. Furthermore, phenomena like this always indicate whether or not an object is spinning in the real world. So if you imagine them away, you really don't have any reason to think the two discs are different.

However, Butterfield suggests a modification of the RDA that avoids this complaint. Instead of a rotating and non-rotating disc, consider these two cases:
In Case 1, two homogeneous discs rotate clockwise. In case 2, one rotates clockwise and the other counter-clockwise. Butterfield's more sophisticated RDA says: Cases 1 and 2 can't be distinguished by the intrinsic properties and spatial locations of the parts of the discs. The Newtonian effects of rotation equally appear on all four discs. But there is clearly a difference between the two cases. So Humean Supervenience is again defeated.

The point of all this has been to get clear on the RDA, not to advocate it. In fact, Butterfield (2004) has given a very original rebuttle of the RDA, by revising the Humean Supervenience claim in a way that allows him to sneak in velocity as an "intrinsic property." I will not be discussing this idea here. Instead, I'd like to argue against a claim that both Butterfield and Callender take for granted: that the RDA fails in GR.

This argument is to be continued on Wednesday.