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.

An argument for hidden variables

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

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

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

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

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

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

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

Why we need superselection rules

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

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

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

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

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

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

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

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

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

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

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

* Added.

Wigner's elegant characterization of time reversal

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

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

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

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

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

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

This immediately implies that

But Taylor expanding the exponentials, we then have that:

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

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

David Albert on symmetries of motion in quantum mechanics

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

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

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

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

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

Adding these two equations, we now have that:

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

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

Two Theorems about Time Reversal

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

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

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

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

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

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

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




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

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

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

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

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

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

"An elementary particle 'is' an irreducible representation"

A well-known particle physicist's adage:

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

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

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

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

Unitary operators and spacetime symmetries

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

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

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

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

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

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

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

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

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

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

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

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

Special Relativity and the Bell Theorems

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

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

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

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

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

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

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

How to time-reverse a quantum system

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.

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.)

Group Structural Realism (Part 4)

Part 1 | Part 2 | Part 3 | Part 4
The Higher Structures Problem.

My worry about structural realism starts with the observation that a relation is a very general concept. Relations can describe not only objects, but also other relations. Consequently, it is a general fact about a structure (which is made up of relations) that it itself often admits a structure (made up of relations between the relations). The result, I argue, is this:

Structural realism is forced to either risk incoherence, or else adopt an overly extravagant and uninformative account of reality.

What does the 'structure of a structure' really mean? In the case of GSR, we have the following important sense in which symmetry groups are describable terms of their own symmetry group structure.

The ‘symmetry group structure’ describing a group G itself is called the automorphism group, Aut G. An automorphism of a group G is a mapping from G to itself that preserves group structure. The group Aut G is formed by collecting together the set of all such automorphisms, and taking the binary operation to be functional composition. Now, to see in what sense Aut G describes the ‘symmetries’ of G, consider the following analogy with the Wigner-approach to GSR.

As in our earlier post, take the specific example of angular momentum. On the old description, we had three tiers: an individual entity (like a Hydrogen atom), an invariant quantity j, and a group SO(3) whose action on the atom left j invariant. GSR announced that we should demote the individual entity, and promote the metaphysical status of the group. But now, consider by analogy the following three new tiers: an individual group SO(3), a representation H, and a group Aut SO(3) whose action on SO(3) leaves H invariant (up to isomorphism). Just as SO(3) described the symmetries of the Hydrogen atom, so Aut SO(3) describes the symmetries of SO(3). Shouldn’t the structural realist strategy demand that we now demote SO(3), and promote the status of Aut SO(3)?

To make this analogy more concrete, let’s think about what Aut SO(3) actually looks like. Begin by presenting SO(3) as the group of rotations of a sphere, where x, y and z are orthogonal axes of rotation. Then there is an automorphism of SO(3) formed by a smooth rotation of these axes, by mapping each rotation to another rotation about a new axis:

The class of all such automorphisms forms a subgroup of Aut SO(3), which is visibly isomorphic to SO(3) itself. The rest of the automorphisms involve an orthogonal transformation of the axes that is not accessible by a smooth rotation, and so the full automorphism group turns out to be given by the semi-direct product of SO(3) and {-1, 1}.

Moreover, all of the virtues of elevating SO(3) seem to carry over when we elevate the metaphysical status of Aut SO(3) instead. Note that in both cases, some important properties are left invariant under the action of the group (that is, both can be called ‘symmetry’ groups). In the case of the electron shell, they are the properties deriving from the total angular momentum j. In the case of SO(3) itself, they are the properties deriving from the group structure14. Note also that both can be taken as the basis for a construction in which the rest of quantum theory is recovered. The only difference is, the group Aut SO(3) is ‘one level more abstract,’ so that this construction begins by constructing an invariant group SO(3), and then proceeding as usual.

An infinite regress now threatens. In general, the group Aut G will also admit an automorphism group. This gives rise to what is known as an automorphism tower, given by

G, Aut G, Aut Aut G, ....As long as each successive automorphism group results in a distinct new group, we can continue producing new, ‘metaphysically fundamental’ structures all the way up. Since this tower can be very high, the result is a bloated, very abstract ontology. Indeed, there are even groups for which the tower can be continued transfinitely.

The Dilemma: Horn 1. The advocate of GSR would like to place the original group G at top of the metaphysical hierarchy. But there does not seem to be a well-motivated reason to choose G over Aut G. This pushes GSR to: Horn 2. We instead promote the highest automorphism group Aut G in the tower, or the ‘whole shebang.’ This introduces a tower’s worth of ‘lower down’ groups into our ontology, and risks the possibility that a ‘highest’ automorphism group doesn’t exist. Or, if it is possible to fix a structural foundation by promoting the ‘whole shebang’ of higher group structures, then it seems that our ontology is excessively extravagant and uninformative.

In summary: structural realism is forced to either risk incoherence (Horn 1), or else adopt an overly extravagant and uninformative account of reality (Horn 2).

For a summary of my comments on GSR, you can now read the paper! (PhilSci Archive)

Group Structural Realism (Part 3)

Part 1 | Part 2 | Part 3 | Part 4
Group Structure and Theory Change.

Can groups do a better job at surviving theory change than individual objects? Here's one observation that might make us think so: groups are often insensitive to a change in underlying set. So, it’s possible for the group structure of an early scientific theory to be preserved in a later theory, even if the descriptions of objects are not.

A toy example: Imagine that some theory leads us to propose the existence of a cube. Suppose that later, we discover that there is no cube, but rather an octahedron.This theory was wrong about what kinds of objects exist. However, it was right about the group-structure, since cubes and octahedrons have the same symmetry group (rotations of pi/2 about appropriate axes preserve the orientations of both objects; so do flips about an appropriate plane). So, if we were betting on which item would be preserved under theory change, a bet on groups would have won out over a bet on objects.

Two more relevant examples:
(1) Consider the change from the Galilei group to the Poincaré group, ushered in by special relativity. At first blush, this appears to be a discontinuity of group structure over theory change. As we discussed in the previous post, Wigner’s legacy allows for a theory of ‘Galilei particles.’ However, the group of Galilei transformations predicts the wrong kinds of particles (i.e., the wrong momentum eigenvalues), as well as the wrong commutation relations. Consequently, in the transition to the Poincaré group, the taxonomy of fundamental particles changed.

However, ‘Galilean particles’ do happen to have the right angular momentum quantum numbers – they allow for the existence of spin, for example. A realist about particles has little to say about this fact. But GSR can actually provide an explanation: it is because rotation group is what’s metaphysically fundamental about angular momentum, and the rotation group SO(3) was preserved in the transition from the Galilei to the Poincaré group -- as a subgroup of each. As for the Galilei group as a whole, one might say that it was also preserved in approximate form, in low-velocity regimes.

(2) This same rotation group SO(3) provides yet another example of preservation under theory change. With the discovery of spin, the traditional realist should seemingly admit that a new kind of particle was discovered, signifying a discontinuity over theory change. But, according to GSR, the important change was really the extension of the symmetry group SO(3) to a larger group, SU(2). The latter is the correct rotation group for a quantum theory of spin, because it admits j = 1/2-integer representations. However, SO(3) is not rejected in this correction – it is preserved as a subgroup of SU(2).

The point of these examples, for the budding structural realist, is to suggest that group structures – not individual objects, and not even algebras of observables – are the superior candidates for the survival of theory change. If this turns out to be right, then GSR not only provides a natural, precise example of structural realism; it also stands a promising chance of satisfying the original, ‘pessimistic meta-induction’ motivation for structural realism.

I hope to have argued (maybe a bit too long-windedly) that there is a fairly precise and compelling form of structural realism in quantum theory: Group Structural Realism. Unfortunately, I don't think it works. As we'll see in the next (and final) post, GSR actually illuminates a general objection to structural realism, which I call the `higher structures' objection.

Group Structural Realism (Part 2)

Part 1 | Part 2 | Part 3 | Part 4
Wigner's Legacy.

Yuval Ne’eman and Shlomo Sternberg have recorded an old 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, 327.)

This idea captures much of the physical basis of GSR. Let's discuss a bit about how one arrives at such a view.

Despite their abstractness, irreducible unitary representations do seem to satisfy our intuitions about elementary particles. Jonathan Bain points out two such intuitions: (1) an elementary particle should be uniquely labeled by a mass and a spin parameter (that is, by the eigenvalues of a total 4-momentum and a total 4-angular momentum operator); and (2) a particle should be invariant up to the group of spacetime symmetries, in order to satisfy “our intuitions concerning the continuity of particle identity through time” (Bain 2000, 402fn). One also wants that, (3) an elementary particle cannot be ‘decomposed’ into further particles; and (4) a particle should be associated with a set of observables that describe its possible states.

One can now observe: Wigner showed that the irreducible unitary representations of the Poincaré group do indeed satisfy (1) and (3) because of irreducibility; they satisfy (2) because they represent the Poincaré group; and (4) follows from the fact that they are unitary.

Although this metaphysical picture of ‘particles as representations’ is often attributed to Wigner, he does not seem to have advocated it in print. The famous 1939 paper (PDF) that Ne’emann and Sternberg refer to actually sets out only to identify the values of physical magnitudes (the so-called ‘quantum numbers’) with parameters labeling group representations, namely, which represent the group of spacetime symmetries. By classifying all the irreducible unitary representations of the Lorentz group, Wigner was able to identify all the possible labels of mass, spin and parity. This provided a deep connection between a symmetry group of nature, and the measurable properties of a quantum system.

A simple textbook example will help illustrate this connection. Take a familiar physical property like angular momentum. Quantum theory assigns a fixed value to some aspects of angular momentum, such as the total angular momentum of an isolated system. Others aspects, such as ‘angular momentum in the z-direction,’ are (prior to measurement) assigned a spectrum of values. How can concrete physical magnitudes like these be properties of a symmetry group?

In the case of angular momentum (ignoring spin, to simplify the example), one begins with the group SO(3) of continuous rotations about a point. The faithful irreducible representations of SO(3) turn out to be representable by groups of complex-valued matrices of odd dimension (2j+1), where j is a positive integer. If desired, a given representation can be thought of as acting on, say, the state space of an electron shell around a Hydrogen atom. However, the imagery of this individual object isn’t required for our construction. Instead, we can skip directly to defining the total angular momentum j = (n − 1)/2, in terms of the dimension n of the representation. The angular momentum operators can then be picked out as elements of the representation, and angular momentum in the z-direction can then be defined and shown to have the usual integer-valued spectrum, {−j, -j+1, ..., 0, ..., j-1, j}.

In summary: angular momentum is recovered, with all its expected properties, from facts about the symmetry group, and no assumptions about the state ψ of an individual object. Indeed, the construction seems to achieve precisely what Eddington hoped: “In fundamental investigations the conception of group-structure appears quite explicitly as the starting point; and nowhere in the subsequent development do we admit material not derived from group structure” (Eddington 1958, 147). That such a development is possible is a fact about the physics. But it is also what paves the way to a reasonable structuralist metaphysics. Wigner’s approach is just what is needed to allow the group structural realist to speak safely of properties like angular momentum, without recourse to an ontology of individual objects.

The Jump to Group Structural Realism.
Wigner's legacy provides us with a very interesting strategy. One can speak perfectly naturally about physical quantities, having begun the construction of quantum theory from a symmetry group. A measurable quantity like angular momentum j is of course derived from a representation space, and one can speak freely about its invariance under the action on that space. To the advocate of GSR, this is not a problem. GSR simply holds that the metaphysically most significant feature of this space is that it provides a copy of the rotation group – not that it refers to the possible states of an individual object.

Construed this way, GSR seems to lead to some surprisingly informative consequences:

Let’s think about what it would mean if spacetime had a symmetry group other than the Poincaré group. This new group would have different representations, and would thus allow for different properties of quantum systems. On Ne’eman and Sternberg’s definition, this would mean that there are different ‘particles.’ In fact, that is exactly what Bargmann (1954) and Lévy-Leblond (1967) were able to show: the Galilei group gives rise to a theory of ‘Galilei particles,’ which are different (in particular, with respect to the ‘mass’ parameter) than the usual ‘Poincaré particles.’

Let’s take another case: what would it mean for nature to admit more symmetries than just those of spacetime? According to GSR, this larger group would provide richer representations, and hence more properties for quantum particles. This is just what is suggested by the study of so-called ‘internal’ symmetries. For example, Gell-Mann’s (1961) adoption of the symmetry group SU(3) led him to organize a new taxonomy of hadrons (as they are now called) according to the irreducible representations of the new symmetry group.

Of course, in building up quantum theory as it gets used in practice, many other mathematical objects besides groups come into play: vector spaces, commutation relations, Hermitian forms, and on and on. What GSR postulates is that, out of all these tools, group structure is of central metaphysical importance. Other realists might propose a different foundation for the theory, perhaps by arguing (with Geoffrey Sewell) that, “theories of such systems should be based on the algebraic structure of their observables, rather than on particular representations thereof” (Sewell 2002, 18). So why choose GSR over all these other options? Here the two overarching aims of structural realism come into play: groups are thought to do a better job of providing a general programmatic account of science, or of solving specific problems in the interpretation of scientific theory.

We'll discuss why this is in the next post.

Group Structural Realism (Part 1)

Note: This post later turned into a paper, which turned into a forthcoming article in BJPS. You can read the full preprint here: PhilSci Archive.


Part 1 | Part 2 | Part 3 | Part 4
Introduction

Group theory plays a special role in the foundations of quantum theory. Today, I want to discuss a philosophical view that uses this role as its jumping-off point. I call the view, Group Structural Realism..

Structural Realism (A Brief Intro). The positive claim of structural realism differs greatly from author to author. However, there is one core consequence that follows from nearly every account of structural realism (as a metaphysical view):

The Structural Realist Hierarchy. The existing entities described by a scientific theory are organized into a hierarchy, in which ‘structure’ occupies the top, most fundamental position.

What it means to be ‘metaphysically fundamental’ is cashed out in various ways. For example, Ladyman and Ross characterize it using the notion of supervenience on properties:

Ontic Structural Realism (OSR) is the view that the world has an objective modal structure that is ontologically fundamental, in the sense of not supervening on the intrinsic properties of a set of individuals. According to OSR, even the identity and individuality of objects depends on the relational structure of the world. (Ladyman and Ross 2007, 130.)

An analogy: the droplets of paint on a canvas are perhaps more metaphysically fundamental than the images in the painting. Similarly: the atoms in a molecule are more fundamental than the molecule, which is more fundamental than the substance. Structural realists argue that at the very top, at the most most fundamental layer of this hierarchy, there is structure. Although there may be further concern about what kind of shadowy existence this structure might have, it is the metaphysical hierarchy that underwrites the view.

For this discussion, let's bracket the shadowy stuff, and focus on the hierarchy.

Broadly speaking, structural realism gets proposed with one of two main goals. First: structural realism aims to provide a general, programmatic account of science and scientific discovery. Worrall’s structural realist account of theory change is the canonical example. Second: structural realism aims to solve specific problems in the interpretation of a theory. For example, it has been proposed as a solution to the problem of identical particles (Ladyman and Ross 2007, §3.1) and to the problem of interpreting spacetime points (Dorato 2000).

Standing between structural realism and what it endeavors to achieve is the meaning of the word structure. Bas van Fraassen has described one broad agreement among critics: “As almost every commentator has somewhat sadly remarked, this key word has its own problems” (van Fraasen 2006, 303). Ladyman and French themselves note that “Because of the width of its embrace and its complex history, defining what is meant by structure and characterising the tendency in general, is problematic” (French and Ladyman forthcoming, §1). Ladyman and Ross similarly accept the criticism that structural realism may not be well ‘worked out.’ However, they retort that “it is far from clear that OSR’s rivals are ‘worked out’ in any sense that OSR isn’t (Ladyman and Ross 2007, 155).

Clearly, we ought to work something out. As a start, I will try to show that, using the resources of group theory and quantum mechanics, a precise characterization of ‘structure’ can be worked out in as much detail as you like. The specific view that I propose we work out is the following:

Group Structural Realism (GSR). The existing entities described by quantum theory are organized into a hierarchy, in which a particular symmetry group occupies the top, most fundamental position.

GSR has a good deal of precedent among structural realists. For example, Aharon Kantorovich argues for a conception of particle physics in which “internal symmetry is the deepest layer in the ontological hierarchy,” and in particular, that “flavor SU(3) symmetry was ontologically prior to hadrons... whereas SU(5) is ontologically prior only to baryons” (Kantorovich 2003, 673). Holger Lyre has suggested an account of objects that “takes the group structure as primarily given, group representations are then construed from this structure and have a mere derivative status” (Lyre 2004, 663). Similarly, Ladyman and Ross argue that, “elementary particles are hypostatizations of sets of quantities that are invariant under the symmetry groups of particle physics” (Ladyman and Ross 2007, 147).

In the next post, we will discuss some of the physical results that make GSR plausible. By identifying which structures are of interest in one particular context, GSR may stand a worthy chance of finding some connection to the physical quantities that we actually observe and measure in the lab. This step is absolutely essential. Bothersome though it may be, any acceptable account of reality must show some connection to the observable world.

Next post: Group Structural Realism (Part 2).

Quantum of Solace

What does quantum mean? With a new Hollywood blockbuster just out, and containing the word "quantum" in the title, it seems we'd better get to the bottom of this.

The word "quantum" was substantively introduced into atomic physics by Einstein, in his 1905 paper (PDF) on the thermodynamics of radiation. (Einstein won the 1921 Nobel Prize in physics for this work.) The meaning of "quantum" in this paper is clear: Einstein describes heat radiation as behaving thermodynamically as if it were made up of "energy quanta" -- discrete chunks of energy of exceedingly small size size hf (where f is frequency and h is Planck's constant, the latter being equal to about 0.0000000000000000000000000000000006). In other words:

    Meaning #1: quantum = very small chunk.

However, when someone refers to a "quantum theory" today, it almost always means much more than Meaning #1. As I've mentioned before, we can point to three categories of phenomena that can be tested in a laboratory, which in large part form the empirical basis of quantum theory. They are:

(1) particle diffraction;
(2) superposition; and
(3) discrete energy spectra.

Einstein's original use of the word "quantum" was an instance of item (3). But today, when one often refers to a "quantum particle," "quantum tunneling," "quantum teleportation," and the like, what is meant (broadly speaking) is:

    Meaning #2: quantum = exhibiting properties (1), (2) and (3).

Of course, this characterization is perhaps heavy on the side of the experimentalist. The theoretician might prefer to think of "quantum" as referring to the structure of a typical quantum theory. Unfortunately, it's not that easy to say precisely what that structure must be like. For example, it's certainly not a simple matter of casting one's theory in terms of bounded operators on Hilbert space, since Koopman and von Neumann showed that this is also possible for classical theories. But I think it's fair to ask that, in order for a theory to be "quantum," it must admit:

(5) a unitary representation of the Canonical Commutation Relations; or
(6) a unitary representation of the Canonical Anticommutation Relations.

Since the theories typically described as "quantum" all tend to admit at least one of these two properties, we now have available:

    Meaning #3: quantum = characterized by (5) or (6).

There's one last meaning for "quantum" that I should mention. It's a unique (and, for the moment, inescapable) feature of all our current quantum theories, but perhaps an undesirable one. Namely, quantum theories describe the world in terms of the following two kinds of processes. The first, called unitary (Schrodinger) evolution, is continuous through time. The second, called state reduction (or for von Neumann (1932), "wave function collapse"), is discontinuous. So, according to quantum theory, the typical lifetime of an entity in a particle physics lab consists in a sequence of successive evolutions and "jumps." A system evolves continuously, until a measurement interaction occurs. It then undergoes a discontinuous transition to a new point, and then begins evolving continuously again. (The nature of these jumps is a matter of debate among interpreters of quantum theory, but let's bracket that.) We thus have:

    Meaning #4: quantum = discontinuous transition.

This meaning seems most closely related to the use of the word in popular jargon. Indeed, my dear friend and fellow blogger Justin over at My Mind is Made Up thinks that the main way people use the word is in phrases like "a quantum leap." If this simply means a severe, discontinuous jump, then I'd say "the folk" have picked up on Meaning 4. Interestingly, Justin tells me that Hollywood has managed to pick up on something more like Meaning #1, which would be more historically accurate. But of course, I'm no expert on Hollywood, or the folk -- so check out what Justin has to say!

An Enormous Test of Bell's Inequalities

Nicolas Gisin (and colleagues) have executed an enormous test of the Bell-inequalities! Their experiment will span laboratories in the cities of Geneva, Satigny, and Jussy.

I believe that makes it the largest Bell-experiment ever done -- the idea being that gravitational effects might have some influence on the measurement problem when there's a lot of spacelike separation. Gisin first mentioned this idea to us at the New Directions conference in April, which I wrote about before.

And just so that you're not in suspense: the Bell inequalities were violated, as usual. Read the paper here.

How Far Out Is Quantum Theory Really?

The quantum world is commonly conceived of as (i) wildly strange, and (ii) far removed from everyday experience. But (i) certainly need not be true: many find quantum theory quite natural and intuitive (although others disagree, as I have discussed before). And it's similarly unclear whether or not (ii) is true. So I propose we try to figure it out.

For example, here's an experimental illustration of quantum diffraction (called "quantum eraser" by the authors) that you can do in your house. Admittedly, this particular experiment has a classical description. But is it possible to make the experiment just a little more sophisticated, and observe a truly non-classical effect? And are there other canonical "quantum phenomena" that are similarly accessible?

I know of no satisfying answer. So let me pose the following challenge:

Using only simple, every-day objects, design experiments illustrating all the canonical quantum effects.

There are different ways of breaking down what these effects are, but here's one way of doing it. Design "every-day" experiments of the following three phenomena:

1. Discrete energy values (e.g., a blackbody radiation experiment);


2. General diffraction (outside the scope of classical E&M); and


3. Coherent superposition.

These phenomena together are sufficient to give us much of the basis for ordinary non-relativistic quantum mechanics. Wouldn't it be nice to have some idea as to just how far out they really are?

Michelson-Morley meets Quantum Mechanics

The Michelson-Morley experiment (read the original paper here) is one of the first textbook experiments that you learn about in support of the light postulate. From this postulate, together with the principle of relativity, it is easy to derive the group of Lorentz transformations, which form the basis for special relativity theory. (Honesty note: some more mild-mannered principles are also required for this derivation, such as the homogeneity and isotropy of space.)Of course, it is possible to derive the Lorentz transformations without the light postulate (Torretti 1983, ch. 3). And famously, historian Gerald Holton revealed in an interview that Einstein was not influenced by this experiment when developing the theory of special relativity. (What experiments did influence him still remain a mystery, at least to me).

So when I read this article, somehow I just wasn't that shocked.

The guy is suggesting that the Michelson-Morley experiment could have been understood within Galilean kinematics, using the fact that in traditional quantum mechanics, phase-difference between transmittor and receiver is a velocity-independent quantity.

This now seems like a bit of a crank-trick to me. But the general idea that there might have been other ways to explain the experiment is interesting.