The tides are caused by two things.

- The moon’s gravitational force pulling on the oceans.
- The sun’s gravitational force pulling on the oceans.

An eclipse does not cause the tides. It did not cause the so-called Supertide that enveloped Mont Saint Michel and London’s river Thames yesterday. The newspapers are getting it wrong, wrong, wrong, wrong, wrong.

However, the eclipse and the supertide do have a *common cause*, which is a new moon. What’s a common cause? Yellow teeth and lung cancer have a common cause: smoking. No one would say that yellow teeth causes lung cancer, right? Well, no one should say that an eclipse caused the super tide for exactly the same reason. (Read more about this in my Lecture Notes on Causation.)

Here is how the moon creates tides: it pulls on the oceans, which cause them to slosh into an American football-shape that points at the moon. As the earth spins the peak of the football sloshes across different locations, causing a high tide roughly once per day.

There is a stronger tide roughly twice per month, whenever the sun and the moon align, and in particular when there is a new moon.

Now, a new moon can cause a solar eclipse under much rarer circumstances, whenever the moon’s shadow happens to pass across your location. (For an explanation of why there isn’t an eclipse on every new moon, try this.) It is the monthly new moon, rather than the rare location of the shadow, is the cause of the tide — not the eclipse!

What *are* the physics that gave rise to the supertide? There are two main factors.

First, there is an especially strong tide when the sun and the moon align *and* the moon is closer than normal to the earth. This location is called the Perigee, and the stronger tide is called a Perigean tide, occurring a few times a year.

Second, there is an especially strong tide when the sun and the moon align *and* the earth is closer than normal to the sun. This is called a Spring tide, and happens every Spring, but is strongest nearest to the Equinox.

If all three things happen at once, then one gets an especially strong tide called a Perigean Spring tide.

That is the real cause of the Supertide. It is because the earth, moon and sun looked like this:

Here is the summary of what makes a tide stronger than normal.

**Strong tide:**New moon (sun and moon align — sometimes also causes eclipse)**Stronger tide:**New moon + Perigee (Perigean tide)**Strongest tide:**New moon + Perigee + Spring (Spring Perigean tide)

The Supertide was just a Spring Perigean tide that occurred the day before the equinox. That is the main reason that it was so strong. It is also possible that it was augmented by the effects of global wind.

In summary: the supertide in Western Europe was caused by the New Moon, Perigee and Spring tide occurring at once so close to the equinox. What is silly is that our friends over at the Daily Mail reported this occurrence, but still decided to report that the eclipse caused the high tides. The New Moon was also part of the cause of the eclipse. But the eclipse did not cause the supertide any more than yellow teeth cause lung cancer.

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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.

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