Twenty years ago now, Jenann Ismael wrote a fascinating little article called "Curie's Principle" (1997 *Synthese*), which discussed a symmetry principle due to Pierre Curie:

“When certain effects show a certain asymmetry, this asymmetry must be found in the causes which gaves rise to them” (Curie 1894).

It’s an intriguing principle — for example, it has applications in particle physics! — and there are lots of great ideas in Ismael’s article. The article also did a wonderful service in helping to bring Curie’s Principle to the attention of philosophers. However, one of the central claims of Ismael (1997) is to have formulated Curie’s Principle as a mathematical theorem, argued for in Section 3 of the paper, “The Proof”.

In fact, Ismael’s statement is not a theorem. The formal argument given in the article is invalid, and in particular contains an erroneous first premise. I’d like to explain here in clear terms what went wrong, and why even Ismael’s statement of Curie’s principle is false.

**Note:** my intention is really not to beat up on this lovely little article, which I very much enjoy! But I would like to set the record straight on matters of fact, especially for students interested in this subject.

There are lots of statements of Curie’s principle out there. I am interested in Ismael’s (1997) mathematical formulation: begin with a set (a set of “causes”) and a set (a set of “effects”). Assume that every cause has no more than one effect, by asserting that the causes and effects are related by a function . Ismael views this (possibly many-to-one) function equivalently as a subset such that, if and are both in , then .

Finally, let a “symmetry transformation” be a pair of bijections: one on , and the other on . These are formally different functions, since they have different domains. But I will follow Ismael in a slight abuse of notation, by dropping the subscripts and just writing for both and .

The statement of Curie’s principle that Ismael formulates is then the following.

Statement.Given a function between sets, and two bijections and , if it is the case that (symmetry of the cause), then it follows that , where (symmetry of the effect).

Some readers will immediately see that this statement is false. The bijections and the function are totally arbitrary and unrelated, so this surely cannot be true! But before getting to that, let me go through Ismael’s argument, as it illustrates a key point that I would like to make.

Ismael’s argument in Section 3 (pg.170) has only a few components, which can be summarised as follows.

*(1) Let be in (the set of pairs or “solutions” characterising the function ). Then is also in . [Ismael refers to these together as “a pair of solutions” without further argument (p.170).]*

*(2) Let (symmetry of the cause).*

*(3) Combining 1 and 2, we get that and are both in .*

*(4) But is a function, so (symmetry of the effect). [Ismael writes, “any Curie-cause has only one physically possible effect among the B’s and it follows… that ” (p.170).]*

**The error in this argument is in the very first premise.** There is certainly no guarantee that an arbitrary bijection will act invariantly on an arbitrary subset . For example, assume has just one element, and has many. Let be a singleton, and let map this singleton to any other singleton; then . A similar example: let map every element of to the same element , and let map this to a different element. Then (where ), so . Thus, Ismael’s first assumption fails: does not always mean that .

There are more physical counterexamples, too: let’s give these symbols a physical interpretation. Suppose we’re describing a falling rock near the surface of the earth, and that the pairs represent the initial and final states. Suppose further that is an ordinary symmetry transformation, like rotation in space. Then it is certainly not true that the symmetry-transformed initial and final state will always be in the solution set.

To see this, suppose the rock begins at rest at a position above the ground, and ends by colliding with the ground at a few moments later. In Ismael’s notation, this “solution” to the law of freefall is the pair . Let our symmetry transformation be “rotation 90 degrees anti-clockwise about the initial state “. The initial state is obviously invariant under this rotation, while the final state gets rotated to a point on the horizontal. Thus, it certainly doesn’t follow that is a solution to the laws of freefall as Ismael suggests: the rock falls down, not sideways! The counter-example is illustrated in the animation below.

This is not just a counterexample to Ismael’s first premise, but to Curie’s principle more generally: a symmetry of the initial “cause” is not necessarily a symmetry of the final “effect”, in that but . John D. Norton (in 2016 / preprint) has identified a slew of similar counterexamples, including one quite similar to the example above.

Castellani and Ismael (2016 / preprint) have repeated Ismael’s (1997) statement above, quoting that formulation of the principle verbatim:

“Ismael (1997, 169) writes: ‘What the principle says depends crucially on how the terms ’cause’ and ‘effect’ are understood…. Let and be families and respectively, of mutually exclusive and jointly exhaustive event types, and let the statement that is a Curie-cause and its Curie-effect mean that the physical laws provide a many-one mapping of into .’ So construed, the principle is quite general. There is no restriction on what the terms of the relation can be. … Someone who wants to apply the principle simply has to find a functional relationship.” (p.1009)

I hope it is now clear that, precisely because there is no restriction on what the terms of the relation can be, Ismael’s formulation of Curie’s Principle is not generally true at all. Someone who applies it simply upon finding “a functional relationship” rather does so at their hazard.

One of the lessons of the above discussion should be that there must be some systematic relationship between the symmetry transformation and the causal law if something like Curie’s principle is going to be true. Many have recognised this, and managed to formulate correct statements that they refer to as Curie’s principle. Here, I will just point to my own version, which appears in my (2016 / preprint) take on Curie’s principle in electromagnetism; there, I use the phrase “Curie’s Hazard” to refer to what Ismael calls “Curie’s Principle”.

Regardless of how one calls it, here is a proposition in the neighbourhood of Curie’s Principle that is actually true.

Proposition.Let and be sets, and let and be bijections. If is a function such that,(symmetry preservation) for all ,

then Curie’s principle is true, in the sense that (symmetry of a cause) only if .

The proof is an easy exercise (and found in the 2016 paper I cite above).

It also indicates a straightforward lesson: Curie’s principle is only true under a judicious interpretation of “causes”, “effects”, and “symmetries”, in addition to a strict limitation on their relationship (which I call “symmetry preservation” above). And, as Norton (2016 / preprint) points out, the suggestion that such a limitation is an “analytic truth” is often a dubious exercise in causal metaphysics.

Katherine Brading once asked me, following a chat about Curie’s Principle, what remains for researchers and students in the history and philosophy of physics to do on this topic. I wanted to end with a few thoughts about that question.

**What not to do.** As you might discern from the above, I don’t see much use in further attempts to state an “analytic formulation” of Curie’s Principle for general causal metaphysics. Norton’s article above argued, conclusively in my view, that this programme is fruitless. And the facts regarding simple mathematical statements resembling Curie’s Principle seem to be more or less understood.

However, there remain at least three topics of interest, where I suspect a clever young researcher could make interesting progress in the field.

**What to do in the history and philosophy of physics.** Curie originally made use of his principle in order to calculate the electric field that is produced when a crystal is compressed. This curious phenomenon, which Curie and his brother Jean discovered, is called the Piezoelectric Effect, and has interesting applications for renewable energy. But Curie’s original use and understanding of the principle has still not been thoroughly explored. Most treatments have focused on his (1894) reflections, cited above. But the principle also appeared in one of the founding articles on the Piezoelectric Effect, which Curie wrote with his brother in 1882 (see pg.247). Very little historical work has been done on this article that I know of, and remains ripe for the picking for a (ideally French-reading) researcher or student.

Moreover, a symmetry principle resembling Curie’s continues to be used in the modern field of plasmonics, such as in this (2017) usage as a selection rule for photonic crystal defect modes. Only some very early philosophical work has been done in understanding the conceptual basis and generality of such principles (e.g. in my 2016 article above), and it would be helpful to have an account that clarifies and explain the justification of such symmetry principles in this (and in Curie and Curie’s 1882) application (and especially if they turn out to be false in general!).

**What to do about dualities in physics.** Another important symmetry principle of intense interest in recent physics and philosophy is the notion of a duality. A classic example is the duality that exchanges the electric and magnetic fields in electromagnetism with magnetic monopoles. One schema for philosophical analysis of dualities in physics as simple “isomorphisms of theories” was recently suggested in a remarkable series of papers by de Haro and Butterfield (“A Schema for Duality, Illustrated by Bosonization“), de Haro, Teh and Butterfield (“Comparing Dualities and Gauge Symmetries“) and de Haro, Mayerson and Butterfield (“Conceptual Aspects of Gauge/Gravity Duality“).

A consequence of this (often merely conjectured) relationship between dual theories is that *“a symmetry of a theory is a symmetry of its dual”*. This idea, though strongly reminiscent of Curie’s Principle, clearly goes beyond a context where the language of “cause and effect” stands a chance of being clarifying. However, a more careful conceptual analysis of how such principles are being used in physics may prove fruitful, for example as a heuristic for interesting conjectures, or as a foundation for understanding dualities. These wonderful papers by de Haro, Butterfield et al. are an excellent place to start!

**What to do in philosophy of science more generally.** Curie’s original use of his principle, though widely explored in the philosophy of physics, has received no attention at all in the philosophy of sciences outside physics. Nevertheless, the claim that “A symmetry of the cause is a symmetry of the effect” may turn out to play an interesting role in the philosophical foundations of other sciences, such as biology, medicine or economics, were a creative philosopher of science to take up the search.

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.

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