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.

]]>A promising proof of the so-called Kadison-Singer conjecture was announced yesterday. Mathematicians are excited, because this conjecture is equivalent to a remarkable number of open problems in other fields. And mathematicians just love to make connections between wildly disparate fields.

Here is another reason to be excited: the Kadison-Singer conjecture has important consequences for the foundations of physics! Settling this conjecture shows an important way in which our experiments are enough to provide a complete description of a quantum system. Here’s why.

The Kadison-Singer question comes out of thinking about how to describe a quantum system. Here is a simplified picture of how that works. When we describe matter in quantum mechanics, there are two main things we are concerned with:

*Observables:*Each observable represents a property whose values we can measure, and*States:*Each state is a “probability measure,” which determines the probability of measuring each value of each observable, given our experimental setup.

This is a lot like the situation with a coin: a property that we can measure is the side that is facing up; it can have value “heads” or “tails.” When the coin is in a state of flipping in the air under normal circumstances, the probability of measuring each value is 1/2. So, the state “flipping normally” determines a probability measure. And that measure assigns probability 1/2 to each value that the observable “which side of the coin is facing up” can take.

Quantum mechanics is a lot like that, except we’re measuring the fundamental properties of matter. Our observable might be the spin of an electron, which can take the values “up” or “down.” And our starting state might be an electron flying out of an oven and through a magnetic field. But the principle is the same, and in this case, the state happens to assign probability 1/2 to each value.

Now, here is a crucial distinction for understanding the Kadison-Singer conjecture. One thing that is special about quantum observables is that there are some that can be measured simultaneously, and some others that can’t. Two observables that can be simultaneously measured are called “compatible,” and are otherwise called “incompatible.”

For example, you can simultaneously measure position and spin. But you cannot simultaneously measure position and momentum; the latter is known as Heisenberg’s uncertainty principle. Position and spin are compatible observables, but position and momentum are not. This is a simple fact of quantum life. Indeed, according to many, it is the crucial distinction between the quantum and the classical descriptions of the world.

Dirac suggested a simple formula for taking experiments and using them to describe a quantum system. The following is a slight improvisation on what Dirac actually wrote; the historically-precise may prefer to think of it as a “Dirac-inspired” picture.

We want to know the properties of a quantum system, meaning the probabilities (the states) that get assigned to all the measurable quantities (the observables). How can we use experiments to learn about this? Here is one way.

*Write down what you can measure simultaneously*. Collect together a set of compatible observables, and keep adding to it until it is as large as it can be. For example, you could take a finite set of observables (say, position and spin ), and then just add in all the functions of those observables (like , , etc.). The resulting set is called a “maximal abelian algebra.” This set represents the most we can learn from a single experiment.*Determine the basic probabilities (i.e., pure states)*. Arrange your lab in some way, and then repeat an experiment that determines the values of those observables. Then do it again. After enough repetitions, you’ll be able to choose a probability distribution for the experiment. A structurally “basic” probability distribution is called a*pure state*.*Generalize this information to the entire system*. We now want to extend this probability distribution to all the other observables, those that are not compatible with our set. That is, we want to choose a probability distribution that applies to*all*observables, and which is the same as our probability distribution when restricted to our set of compatible observables. And, if we may ask so much, we’d like this extension to be*unique*, so that we can be sure we’ve chosen the right one.

The last step is what’s interesting. Dirac’s procedure requires us to extend what we learned about the basic quantum probabilities of compatible observables to all the other observables as well, and to ensure that there is only one way to do so. Is this always possible? *That* is the Kadison-Singer conjecture.

The Kadison-Singer conjecture says we have at least one way to completely characterize a quantum system on the basis of what we can learn from experiments. We can take what we know about the quantum probabilities of simultaneously measurable quantities, and uniquely extend this to all the other measurable quantities as well.

In particular, if we have determined a “basic” probability distribution on a set of compatible observables that is as large as it can be, then this probability distribution extends uniquely to all the other observables.

Here is the precise statement of this claim.

Kadison-Singer Conjecture.Let be a discrete maximal abelian subalgebra of , the algebra of bounded linear operators on a separable Hilbert space. Let be a pure state on that subalgebra. Then there exists a pure extension of to all of , and that extension is unique.

Proof of this statement provides a very nice assurance, that our experiments really are enough to describe quantum systems as we understand them. (Note for those familiar with quantum theory: I have included some notes on the terminology of this statement in the last section of this post.)

Unfortunately, the result is not without caveats, and an important one is the “discreteness” caveat. Although a great many physical descriptions satisfy discreteness, there are also some continuous ones (some of which are relevant to quantum field theory) that are not discrete, and for which the conjecture does not apply. Indeed, Kadison and Singer showed in the second theorem of their original paper that the conjecture fails for such continuous geometries!

Kadison and Singer wrote a number of wonderful papers on the mathematical foundations of quantum theory in the 1950′s. The conjecture that is named after them comes from the “Related Questions” section of their 1959 paper on extensions of pure states. On the question of uniqueness, they say that “we incline to the view that such an extension is non-unique.” Kadison and Singer seem to have thought that their conjecture is probably false! You might have expected this too, if you had just proven that uniqueness fails in the continuous case. This makes it all the more surprising that the conjecture turns out to be true.

In settling the Kadison-Singer conjecture, a number of other equivalent propositions are have also been settled. Those familiar with the mathematics of quantum theory may enjoy reading this paper by Casazza and Tremain on some equivalent formulations, which is very accessible from that perspective.

For the statement of the Kadison-Singer conjecture above, here are a few definitions.

We say that an algebra is *discrete* if it contains “atoms,” i.e. minimal projections such that only if . This is the familiar situation for most. For example, a 1-dimensional projections onto a ray in a Hilbert space is a minimal projection for .

A *state* is a probability distribution that is -additive on families of mutually orthogonal projections. (Often, one takes a state to be just a positive normalized functional, and defines a state to be *normal* if it is -additive.) One of the cornerstones of quantum theory, called *Gleason’s Theorem*, says that a (normal) state can always be implemented by a density matrix, in that there always exists a density matrix such that for all .

A state is *pure* if it cannot be expressed as a non-trivial convex combination of other states, i.e., for only if . In this sense, pure states are “basic.” The pure states on can always be implemented by Hilbert space vectors, in that there exists some such that for all .

*21 June 2013.* Orr Shalit at Noncommutative Analysis some very nice comments on the proof of the K-S Conjecture, and Gil Kalai has a nice collection of references.

*11 March 2014.* Danny points out an interesting example in the comments, which clarifies the importance of requiring that all states be “normal” in the last sentence of this post.