Last year I had a back-and-forth Abhay Ashtekar. It was during a lively conference organised by Emily Grosholz and hosted by the Center for Gravitaiton and the Cosmos where Ashtekar is director. Our discussion was about the arguments underpinning the evidence for time asymmetry in fundamental physics.

Our discussion has finally come out in a special issue of *Studies in History and Philosophy of Modern Physics:*

- Roberts (2014) Three merry roads to T-violation (philsci-archive)
- Ashtekar (2014) Response to Bryan Roberts: A new perspective on T violation (arxiv)
- Roberts (2014) Comment on Ashtekar: Generalization of Wigner׳s principle (arxiv)

Here’s a cheerful little essay on what our discussion was about.

We all experience asymmetry in time. We’re not getting any younger. We smash glasses, but don’t un-smash them. We notice cigarette smoke dissipating but never compressing back into the cigarette.

However, those time asymmetries only appear in systems arranged with special initial conditions. They occur when a system begins in a “highly organised” low entropy state, and therefore must evolve into a “disorganised” high entropy state by the second law of thermodynamics. These asymmetries do not occur when a system begins in an equilibrium state, where all the familiar time asymmetries tend to disappear.

However, in the mid-20th century, a radical new kind of time asymmetry was discovered. This new time asymmetry did not depend on initial conditions. It was built right into the laws of physics themselves. And it came as an incredible surprise to physicists when it was first discovered by Cronin and Fitch in 1964. Leading theories were overturned. Nobel prizes were awarded. The phenomenon was recorded in the textbooks, and came to be referred to as *T-violation*.

But here is something curious about this episode: at the time, we didn’t actually know what the laws of physics were. Somehow managed to determine that the laws of physics are T-violating without actually knowing the precise laws of physics.

In particular, in 1964 there was no standard model. We did not understand the Hamiltonian or Lagrangian for these systems, which is what determines the precise form of the laws governing their motion. And yet, through some beautifully clever reasoning, experimentalists managed to show that the laws are T-violating.

So, what kinds of arguments made this possible? And how robust were those arguments? There were basically three kinds of answers.

In “Three merry roads to T-violation“, I argued that if you draw out the basic skeletal arguments, you see that there are three roads to T-violation currently being explored. Each makes use of a symmetry principle in order to establish that the laws of physics are T-violating. And each works even when we don’t have a very clear picture of the laws themselves.

*T-violation by Curie’s principle*. Pierre Curie declared that there is never an asymmetric effect without an asymmetric cause. This idea, together with the CPT theorem, provided the road to the very first detection of T-violation in the 20th century. (It is also itself the subject of some recent debate in philosophy of physics, e.g. here, here and here.)*T-violation by Kabir’s principle*. Pasha Kabir pointed that, whenever the probability of an ordinary particle decay A → B differs from that of the time-reversed decay B′ → A′, then we have T-violation. This second road provides a very direct test for T-violation, which was successfully carried out by the CPLEAR experiment at CERN in 1998.*T-violation by Wigner’s principle*. If certain kinds of exotic matter turn out to exist, such as an elementary electric dipole, then this would lead immediately to T-violation. This provides the final road, although it has not yet led to a successful detection of T-violation.

But how robust are these principles? The standard model of particle physics will certainly be adjusted as physics continues to progress. Will the arguments for T-violation be lost when we proceed beyond the standard model? Or, are they robust enough to stay with us even as our theories change?

Ashtekar pointed out in his response that in fact Curie’s principle and Kabir’s principle are both surprisingly robust.

His approach introduced a helpful tool that he calls *general mechanics.* It is a framework that allows one to peel back much of the special structure of quantum theory that distinguishes it from other theories, and focus on a few core structures that are shared by many other theories as well. This includes many alternatives to the standard model.

What Ashtekar showed was that the first two roads to T-violation, Curie’s Principle and Kabir’s Principle, are valid even in the stripped-down framework of general mechanics. These principles rely on very little of the basic structures that characterise quantum theory:

- Unitary evolution (or “Schrödinger” evolution) is not presumed.
- The superposition principle is not presumed (nor is any linear structure)
- The notion of an observable is not presumed.

And yet the core arguments that our world is T-violating are still correct.

What about the Wigner’s Principle approach? In my short comment on Ashtekar written after the conference, I showed that the it too is valid in general mechanics.

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Bohmian mechanics is not just an “interpretation” of quantum mechanics. It is a radical revision. In this note, I’d like to point out one reason that it’s an *implausible* revision: Bohmian mechanics is rampantly indeterministic in a way that quantum mechanics is not.

In Bohmian mechanics, the locations of particles are described by a point in a real manifold , called the configuration space. The trajectory of a system of particles is a curve through that manifold. The theory also includes a set of square-integrable functions on this space called wavefunctions.

A physical system in Bohmian mechanics can be characterized by a configuration space , a wavefunction space , and also a self-adjoint linear operator called the Hamiltonian. This Hamiltonian generates a one-parameter group of wavefunctions that solves the Schrödinger equation,

For a given Bohmian system , an *initial condition* is a pair , with and . The fundamental law of Bohmian mechanics then says that, given an initial , the trajectory of the system is a solution to the Bohmian *Guidance Equation*,

where is the solution to the Schrödinger equation with initial condition .

Take as simple a Bohmian system as you can imagine: a free particle confined to a finite space. It turns out that the Bohmian description is rampantly indeterministic.

Let the configuration space be , which describes the possible locations of a particle on a string of finite length. Let be the set of differentiable square integrable functions . And let , the Hamiltonian for a particle free of any forces or interactions, and where .

As a dirt-simple example of indeterminism, choose the initial condition , and given by,

This wavefunction is square integrable and differentiable. (Square-integrability follows from the fact that , and differentiability is obvious.) But let’s calculate what the Guidance Equation looks like for this initial wavefunction. Since , the unitary propagator for our Hamiltonian satisfies . Therefore,

The differential equation is well-known to be indeterministic, following a much-discussed example of John Norton (2008 / animated summary). But let me make it explicit: a Bohmian particle with this initial configuration is compatible with a continuum of future trajectories all satisfying the Guidance Equation. Namely,

for any arbitrary time . (We restrict our attention to times during which the particle is in the interval , namely .) These solutions correspond to a Bohmian particle that sits at up until an arbitrary time , when it randomly begins moving.

As a point of comparison, recall that many have complained about the “unphysical” features of the surface of Norton’s dome, such as the infinite Gaussian curvature at the apex (e.g. here and here). No such complaining need be tolerated in the case of Bohmian mechanics. There is no surface to complain about. There is only the wavefunction , which is a perfectly boring, deterministic wavefunction from the perspective of orthodox quantum mechanics. In particular, it is the initial condition for a unique solution to the Schrödinger equation, which is defined for all times . It is only with the addition of the Bohmian Guidance Equation that a pathology occurs.

In order to avoid such pathologies, Bohmian mechanics must somehow excise this class of wavefunctions from the theory. But it’s not clear how to motivate this excision in a non-ad hoc way. And it’s even less clear whether it can be done in a way that avoids doing damage to the ordinary quantum dynamics.

]]>I remember that when I first learned the Canonical Commutation Relations in quantum mechanics, they seemed mysterious:

I knew I was supposed to view this as a law of nature, and that it could be used in some contexts to explain important observations like position-momentum uncertainty. But I remember it being a huge revelation to me when I realized that *the canonical commutation relations are just the local expression of spatial translations when space is homogeneous.*

This is well-known by experts. But since I couldn’t find an obvious source for it on the interwebs, I thought I’d share the story here for others.

We want to interpret a self-adjoint operator as representing position in space. To keep it simple, let’s say it represents position along an infinite length of string, which is easy because it’s 1-dimensional.

Then we can interpret as position that has been translated by a distance in space.

Now, if space is homogeneous, then no point in space is any different than any other. So, the self-adjoint operators and are equally good representatives of space. Setting up the same experiment in two laboratories that differ only a distance will produce the same results.

In quantum mechanics, experimental results are probabilistic, and the transformations that preserve probabilities are the unitary ones. So, we can capture this homogeneity precisely by say that the spatially translated position operators are related by a unitary transformation,

More can be said about these translation operators . If we think of the infinite string as continuous, then we’ll want to have a continuous collection of operators , one for each real number . We’ll also want to capture the additive relations between distances on the string, .

Whenever this is the case, Stone’s theorem says that there exists a self-adjoint operator such that . (Of course we’ve chosen the letter suggestively — but wait for it.) So, our statement of homogeneity above can be expressed,

First consequence: this equation implies a special form of the canonical commutation relations known as the Weyl CCRs, . It only takes one line to check this, so do give it a try. In fact, this equation is *equivalent* to the Weyl CCRs.

Second consequence: when we take the derivative of both sides with respect to , we get the normal canonical commutation relations. This is also a nice exercise, which only uses the product rule and the definition of the derivative for exponentials, so I’ll let you give it a go.

What this means is that the canonical commutation relations in quantum mechanics are the local expression of translations in space — where “local” is in the sense of a derivative, as above. But this should warn you that the derivation needn’t go the other way — in fact, you can’t derive translations in space (or the Weyl CCRs) from the canonical commutation relations.

This is because there is a lot of information in our statement of spatial homogeneity above that is not needed for the CCRs.

For example, we assumed that a dimension of space is described by the entire real line . But it may be of interest to restrict space to a finite interval of , or a loop, or even a discrete set of points. As long there space is homogeneous in the sense of there being unitary operators relating the points (and some notion of a derivative can be defined) we can often still construct the commutation relations.

]]>There is a quirk in the literature on time-energy uncertainty. It might have started as a little sloppiness. But it has grown into an error that seems to have spread all over the place.

The problem comes from a footnote in Wolfgang Pauli’s Quantum Mechanics textbook, where he wrote (pg.63, fn.2):

In the older literature on quantum mechanics, we often find the operator equation

… . It is generally not possible, however, to construct a Hermitian operator (e.g. as function of and ) which satisfies this equation. This is so because, from the C.R. written above, it follows that possess continuously all eigenvalues fro to … whereas on the other hand, discrete eigenvalues of can be present. We, therefore, conclude that the introduction of an operator is basically forbidden and the time must necessarily be considered an ordinary number (“-number”) in Quantum Mechanics

The conclusion that a time operator is “basically forbidden” has a strong following. Here’s just a small sampling:

- “In quantum mechanics there is in fact no self-adjoint time observable of any suitable sort” (Dürr, Goldstein and Zanghi, pg. 12)
- “In , only the energy is a physical quantity like and ; , on the other hand,
*is a parameter*, with which no quantum mechanical operator is associated” (Cohen-Tannoudji, Diu and Laloë, vol.1 pg. 251) - “time enters into Schrödinger’s equation, not as an operator (i.e., and “observable”) but rather, as a parameter” (Aharanov and Bohm).

These statements are strictly false. Although people often repeat Pauli’s conclusion that there can be no time operator in quantum mechanics, they are not stating a theorem.

Here is a simple counterexample, which was pointed out by Busch, Grabaowski and Lahti (1994), but which apparently had little impact on the broader physics community.

Let , and consider the Hamiltonian . Define . Then by a trivial calculation,

This is a time observable by Pauli’s own definition. It is not “forbidden.” It would be best if everyone stops saying that it is.

Before the erroneous conclusion, Pauli’s original passage says something like, “systems with discrete energy do not admit a time observable.” This isn’t a very powerful thing to say, since many interesting physical systems (like the free particle) have a continuous energy spectrum. But more importantly, this statement is also incorrect. Garrison and Wong proved back in 1970 that the harmonic oscillator (which has a discrete energy spectrum) does allow for a self-adjoint operator satisfying the commutation relation above.

A more interesting (and correct) thing to say is that *if is bounded from below*, then there can be no self-adjoint operator that satisfies the relation,

This was proved rigorously by Srinivas and Vijayalakshmi (1981). Equivalently, it says that there is no self-adjoint operator conjugate to in the sense of the Weyl commutation relations.

But perhaps most intuitively, the result says that if energy is bounded from below, then there can be no self-adjoint operator that continuously tracks the value of the time parameter describing the system’s unitary evolution.

All known physical systems have energy that is bounded from below. In fact, most of them have energy that is always positive! So, a more correct slogan for the community to adopt would be this:

In all physical systems in which energy is bounded below, there is no self-adjoint observable that tracks the time parameter .

This slogan excludes the counter-example above, because that Hamiltonian does not describe any known physical system, as its energy is unbounded above and below.

I first learned about many of these results from my friend Tom Pashby, who has written a very interesting dissertation on this topic. Some of his work has been summarized here.

]]>A superselection rule is a special limitation on what is observable in the quantum world. For example, we can never seem to observe interference from the relative phases between a boson and a fermion. This can be seen as a consequence of a superselection rule.

However, there are lots of ways to make the limitation of a superselection rule precise, and the literature is sometimes confusing as a result. To help keep things straight, John Earman (2008) started off a very rich article with five ways to express the presence of a superselection rule. I find John’s classification helpful, and so I thought I’d share a little summary. I’ll then briefly mention the fermion-boson superselection rule as an example, and end with some warnings about common mistakes.

John describes five basic characterizations of a superselection rule that are all mathematically equivalent. They’re expressed in the language of operator algebras, the rigorous mathematical language of modern quantum theory.

Let be an algebra of bounded linear operators acting on a Hilbert space . Note that it may in general be a proper subalgebra of the set of all bounded operators . The pair is said to have a superselection rule if any of the following are true.

- SSRI. acts reducibly on , meaning that its action leaves a non-null proper subspace of invariant.
- SSRII. The commutant , which is the set of all bounded linear operators that commute with everything in , contains operators besides the constant multiples of the identity.
- SSRIII. For any normalized vector states and any unitary operator such that and , the superposition (with ) is a mixed state.
- SSRIV. The Hilbert space has subspaces such that no observable has non-zero matrix elements in both subspaces; i.e., for all and , .
- SSRV. The Hilbert space has subspaces such that the relative phases between the two subspaces are unobservable; i.e., if and , then for all observables .

An example of a superselection rule, mentioned above, is the so-called Boson-Fermion Superselection Rule. This was the first known superselection rule, discovered in a famous (1952) paper by Wick, Wightman and Wigner.

If you have a single boson and a single fermion, these particles occupy a Hilbert space of the form . Let be a vector state of the boson, and let be a vector state of the fermion.

As you may know, if you time reverse a boson twice, it goes back to where it started. But if you time reverse a fermion twice, it picks up a negative sign. That is,

That’s an inevitable fact of quantum life. But it still seems to be the case that no measurable quantities are changed under the transformation , and so we seem to have physical grounds to believe that for all observables . This gives rise to a superselection rule, in all of the senses above. is the relevant non-constant operator for SSRII and SSRIII, and are the relevant subspaces for SSRIV and SSRV. It is a good exercise to check that all these superselection rules follow. But if you get stuck, many of the calculations can be found in textbooks like Ballentine (1998).

It’s easy to slip up with superselection rules, especially when time reversal is involved. Here are two words of warning to keep an eye out for.

*1. There’s sloppy language out there.* Notice that none of the expressions of a superselection rule above imply that a superposition of a boson and a fermion is “impossible” or “not realizable.” Experts often say this when they’re being sloppy, but it has a tendency to confuse students. The best thing to say is that we are limited in what we can observe about the superposition of a boson and a fermion, in the sense of SSRI-V above. Nothing more.

*2. Time reversal invariance has nothing to do with it.* Following Wick, Wightman and Wigner, I said above that the boson-fermion superselection rule follows from the way that two successive applications of the time reversal operator transform fermions and bosons, and the fact that for all observables . I did not assume time reversal invariance, which says that where is the Hamiltonian, and this assumption is not needed. That’s good, because time reversal invariance has been known to fail since the 1964 discovery of T-violation by James Cronin and Val Fitch.

And yet, in a very strange historical episode, Hegerfeldt, Kraus and Wigner (1968) declared that the Wick-Wightman-Wigner argument is invalid because it relies on time reversal invariance. Earman follows them in this mistake (pg.379), as do many others.

But this declaration is simply wrong. Time reversal invariance has nothing to do with it. Just do the derivation to see what I mean, or look at the original Wick, Wightman and Wigner. Or, more intuitively, notice that superselection rules are properties of the “kinematics” or observable structure of a quantum system. Invariance under time reversal is a property of the “dynamics” or time evolution of that system. The observation that can be a perfectly good constraint on the kinematics, whether or not is a property of the dynamics.

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