Category Archives: Bell’s Theorem

Bohr-van Leeuwen Theorem and Micro/Macro Disconnect

A couple weeks ago, I wrote a post about the Gibbs paradox and how it represented a case where, if particle indistinguishability was not taken into account, led to some bizarre consequences on the macroscopic scale. In particular, it suggested that entropy should increase when partitioning a monatomic gas into two volumes. This paradox therefore contained within it the seeds of quantum mechanics (through particle indistinguishability), unbeknownst to Gibbs and his contemporaries.

Another historic case where a logical disconnect between the micro- and macroscale arose was in the context of the Bohr-van Leeuwen theorem. Colloquially, the theorem says that magnetism of any form (ferro-, dia-, paramagnetism, etc.) cannot exist within the realm of classical mechanics in equilibrium. It is quite easy to prove actually, so I’ll quickly sketch the main ideas. Firstly, the Hamiltonian with any electromagnetic field can be written in the form:

H = \sum_i \frac{1}{2m_i}(\textbf{p}_i - e\textbf{A}_i)^2 + U_i(\textbf{r}_i)

Now, because the classical partition function is of the form:

Z \propto \int_{-\infty}^\infty d^3\textbf{r}_1...d^3\textbf{r}_N\int_{-\infty}^\infty d^3\textbf{p}_1...d^3\textbf{p}_N e^{-\beta\sum_i \frac{1}{2m_i}(\textbf{p}_i - e\textbf{A}_i)^2 + U_i(\textbf{r}_i)}

we can just make the substitution:

\textbf{p}'_i = \textbf{p}_i - e\textbf{A}_i

without having to change the limits of the integral. Therefore, with this substitution, the partition function ends up looking like one without the presence of the vector potential (i.e. the partition function is independent of the vector potential and therefore cannot exhibit any magnetism!).

This theorem suggests, like in the Gibbs paradox case, that there is a logical inconsistency when one tries to apply macroscale physics (classical mechanics) to the microscale and attempts to build up from there (by applying statistical mechanics). The impressive thing about this kind of reasoning is that it requires little experimental input but nonetheless exhibits far-reaching consequences regarding a prevailing paradigm (in this case, classical mechanics).

Since the quantum mechanical revolution, it seems like we have the opposite problem, however. Quantum mechanics resolves both the Gibbs paradox and the Bohr-van Leeuwen theorem, but presents us with issues when we try to apply the microscale ideas to the macroscale!

What I mean is that while quantum mechanics is the rule of law on the microscale, we arrive at problems like the Schrodinger cat when we try to apply such reasoning on the macroscale. Furthermore, Bell’s theorem seems to disappear when we look at the world on the macroscale. One wonders whether such ideas, similar to the Gibbs paradox and the Bohr-van Leeuwen theorem, are subtle precursors suggesting where the limits of quantum mechanics may actually lie.

The Unswattable

One of the deepest results in all of quantum mechanics is Bell’s inequality. While it is remarkably profound, it is also an irksome, ever-present, unswattable fly. Nature’s violation of this inequality implies that nature is intrinsically non-local. Until the day Bell published his theorem, it was thought that everything in nature could be understood under the locality assumption.

While I continue with my research life day-to-day without thinking too much about it, every once in a while that pesky fly reappears, seemingly out of thin air. In this particular instance, it emerged while I was re-reading the excellent piece by N. David Mermin in Physics Today entitled Is the moon there when nobody looks? (pdf!) Alright, so not exactly thin air, perhaps this was self-inflicted.

Regardless, this article by Mermin is particularly pedagogical and explains Bell’s theorem in the simplest manner that I have found. It describes how in a sea of seemingly random data, there are correlations that cannot be explained without considering the existence of an entangled state. Any “local hidden variable” theories cannot explain the data.

What is so bothersome about Bell’s theorem

At some point in the article, Mermin challenges the reader to try to come up with scheme to explain all the observed results using a purely local and deterministic picture. This appears, at first sight, not to be an impossibly difficult task. However (at least for me), one’s schemes are quickly exhausted (very frustratingly!), and one has to face the reality that this may not be possible. In fact, Bell unequivocally showed that this was not possible.

However, there seems to me to be a (rather pathological) way out of Bell’s constraints. It is possible that embedded in Bell’s theorem is an assumption that perhaps we unaware that we are making. This would be analogous to the implicit assumption in Newton’s formulation of gravity of the infinite speed of light — an assumption that when just looking at Newton’s equations, we would not know that we were making. If we are making such an assumption in the Bell experiments, it may be possible to salvage locality in some extremely contrived manner. While this situation seems unlikely even to me, I sincerely hope that there is such an assumption lurking somewhere rather than face up to the more probable idea that nature is intrinsically non-local.

One of the most unfortunate historical circumstances surrounding the publication of Bell’s inequality was that Einstein was not alive to see its formulation. One wonders what his reaction would have been and whether he would have taken The Unswattable to his grave.

The truth we all know but agree not to talk about

I am currently in the process of reading an enlightening short book entitled Quantum Chance by Nicolas Gisin, an authority on the foundations of quantum mechanics. You can actually download it for free here if your institution has a Springer publishing subscription. The book stands out as being accessible to the educated lay reader without sacrificing much in the form of profundity.

The main topics of this book are the implications of Bell’s theorem. Prior to Bell’s paper (pdf!), a possible view of quantum mechanics was that it predicted the statistical distribution of many events. However, back then, it was plausible that there existed an underlying theory, one we had yet to discover, that was described by a set of “hidden variables”. The idea was that these hidden variables would allow us to calculate the trajectory of a single particle deterministically, but that we just didn’t know the equations obeyed by the hidden variables. Quantum mechanics was just an approximate theory allowing us to calculate probability distributions of many events.

Behind the hidden variable theory was a philosophical stance called realism, strongly espoused by Einstein. Simply stated, realism is the belief that reality exists independent of the observers. This is counter to the orthodox view of quantum mechanics, which was emphasized by Bohr. The orthodox view is that the measurement of quantum systems causes a “wave function collapse” and that observables have no meaning until they are measured. The implication of the orthodox view is that reality is in the eyes of the observer and does not exist independent of the observer. There are other interpretations of quantum mechanics out there as well, but it is my understanding that these were the two prevailing views before Bell worked out his theorem.

Even though Bohr found it quite easy to give up the notion of realism, I find it quite difficult to abandon. In the very least, one should be able to describe the mechanism giving rise to “wave function collapse” if this indeed even occurs. Regardless, Bell’s theorem, when it was published in 1964 (pdf!), showed that local realism was untenable in quantum mechanics.

What does this mean? Well, I’ve described what realism means, so let me now take on locality, which is implicit in the hidden variables idea in the way Einstein originally conceived of it. According to Wikipedia “the principle of locality states that an object is only directly influenced by its immediate surroundings”. This sounds quite vague, but Bell was able to show in a rigorous mathematical sense that if Bell’s inequality was violated, that an event on one side of the universe can instantaneously affect another event on the other side of the universe. Stunningly, experiments suggest that quantum mechanics does indeed appear to violate Bell’s inequality.

For a realist (and for adherents to most other interpretations of quantum mechanics), Bell’s theorem then suggests that the universe is inherently nonlocal. This notion of nonlocality, the idea that two things are somehow connected over vast empty space on an instantaneous time scale, bothered both Einstein and Newton greatly. Newton, whose theory of gravity is also nonlocal said:

It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact…That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro’ a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain laws; but whether this Agent be material or immaterial, I have left to the Consideration of my readers.

I suspect that the solution to the nonlocality problem in quantum mechanics may end up needing a large conceptual overhaul. It is going to take a work of great insight to preserve locality, if it indeed can at all be preserved in some contrived way. Whatever the solution to this problem, I hope that I am alive to see it. I won’t be betting on it though.