## Examples where physical heuristics led to incorrect answers?

I have always been impressed by the number of results conjectured by physicist, based on mathematically non-rigorous reasoning, then (much) later proved correct by mathematicians. A recent example is the $\sqrt{2+\sqrt{2}}$ connective constant of the honeycomb lattice, derived non rigorously by the physicist B. Nienhuis in 1982 and rigorously proved this year by S. Smirnov and H. Duminil-Copin.

I would be interested in knowing examples of results conjectured by physicists and later proved wrong by mathematicians. Furthermore it would be interesting to understand why physical heuristics can go wrong, and how wrong they can go (for example, were the physicists simply missing an important technical assumption or was the conjecture unsalvagable).

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Closed. I do not think MathOverflow should be a place where we indiscriminately collect the failures of others. – S. Carnahan Jul 1 2010 at 8:31
Hm. Replacing "wrong" with something like "not fully right" could be helpful... – Wadim Zudilin Jul 1 2010 at 8:44
my idea was not to create a "collection of failures" ! Physicists sometimes base their argumentation on divergent series, fishy exchange of limits,replica methods and other nightmares for mathematicians. Maybe surprisingly, the results obtained this way are more often correct than not - it could be interesting, I think, to have examples that show how the mathematical community can be helpful in these kind of situations. A short description of the non rigorous argumentation and some thoughts on the reason(s) why this cannot be true could be helpful, indeed. – Alekk Jul 1 2010 at 9:22
I made an edit in an attempt to make the question appear less argumentative. I hope that I did not change the meaning of the question, and Alekk you should feel free to undo my changes if you think they've made the question worse. – Noah Snyder Jul 1 2010 at 15:50

I'll describe below a controversy in statistical mechanics in the 1980's: the case of the lower critical dimension of the Ising model with an applied random magnetic field.

## Background

Let me give a little background, though you might want to read Terry Tao's discussion of basic statistical mechanics instead. The Ising model is a statistical mechanical model of "spins" on a hypercubic lattice. The energy functional is: $E=\sum_{\langle ij\rangle}\frac{1}{2}(1-S_iS_j)-\sum h_iS_i$ where the first sum is taken over nearest neighbor pairs on the lattice and the second is taken over all sites, and $S_i$ is a $\pm1$ valued variable on each site called the spin and $h_i$ is the real-valued "externally applied magnetic field" applied to each site. Each possible configuration of spins on the lattice is assigned a probability proportional to its Boltzmann weight $e^{-\beta E}$ where $\beta>0$ is a parameter that is interpreted physically as the inverse temperature $T$.

Given such a model, one question is to determine the "phase behavior", or what are the typical properties of the ensemble of configuration at a given $\beta$, and how does this change with $\beta$.

Considering at the moment just the Ising model with $h_i=0$, one might expect that for large $\beta$, the typical configuration will tend to have lower energy, and hence have all its spins aligned to either all $+1$ or all $-1$. At small $\beta$, all the Boltzmann factors tend to 1 and the typical configuration will have random spins. This rough argument is just meant to guide the intuition that there might be a phase transition between "mostly aligned" configurations to "mostly random" configurations at some special value of $\beta$.

As it turns out, what happens is highly dependent on the dimensionality of the lattice.

The lower critical dimension $d_L$ of a model is the dimension below which no phase transitions can occur because even as $\beta\rightarrow\infty$, there is not enough of an energy gain from ordering to create a phase with long-range correlations. In the ordinary Ising model (with all $h_i=0$), the lower critical dimension is 1, and hence at any finite $\beta$, the average $\langle S_iS_{j}\rangle$ over configurations weighted with the Boltzmann distribution will approach zero (exponentially fast, even) as the distance between sites $i$ and $j$ approaches $\infty$. For two dimensions and above, it can be shown that above a certain $\beta_c$ (depending on dimension) this average will be finite in that long-distance limit.

## Controversy

In the 1980's there was a controversy in the physics literature over the value of $d_L$ for the Random Field Ising model, a model where the $h_i$ are independent Gaussian random variables with zero mean and constant variance $\epsilon^2$.

I'm not in a position to describe the history accurately, but I believe that there were physical arguments by Imry and Ma originally that $d_L\leq 2$, which were disputed when an amazing connection between random systems in $d$ dimensions and their pure counterparts in $d-2$ dimensions was found, known as the "Parisi-Sourlas correspondence". My understanding of Parisi-Sourlas is that it is based on a hidden supersymmetry in some series representation of the model which yields order-by-order agreement in the "epsilon expansions" of the two systems. Their argument was also made rigorous by Klein, Landau and Perez (MR). Based on this, since the Ising model has $d_L=1$, the RFIM was argued to have $d_L=3$ by various authors, though this was never a consensus view.

This controversy was settled by work of John Imbrie (MR) and later work of Bricmont and Kupianen (MR) building off his results that proved rigorously that $d_L\leq2$ in this system. Apparently terms like $e^{-1/\epsilon}$ become important and the epsilon expansion breaks down in low dimensions, though I'm not sure if this has been made precise, and even today the RFIM is far from being completely understood.

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I do not count the example below as failure but a remarkable insight with strong mathematical consequences.

One of the most famous examples of explicit constructions in Mirror Symmetry was introduced in the epoch-making paper by the physicists Candelas et al. Starting from the family ${\mathbf M}$ of quintic hypersurfaces in $\mathbb{P}^4(\mathbb{C})$ defined by $\sum_ {k=1}^5x_ k^5-5z \prod_ {k=1}^5x_ k=0$ ($z$ being a complex parameter), Candelas et al. naturally associate another family $\mathbf{W}$ of manifolds (the "mirror of ${\mathbf M}$") which turn out to be Calabi--Yau. To $\mathbf{W}$, one can naturally associate a vector of periods (depending on $z$) which are solutions of the same differential equation (namely, the Picard--Fuchs equation of $\mathbf{W}$). This equation is simply a hypergeometric differential equation satisfied by a hypergeometric function $F(z)$ and its "neighbour" $G(z)+\log(z)F(z)$. Then they observed the non-trivial property that the Taylor coefficients of $q(z)=\exp(G(z)/F(z))$ are integers. Furthermore, let us define the Yukawa coupling $$K(q):= \frac{5}{1-5^5z(q)}\cdot \frac{1}{F(z(q))^2}\cdot \bigg(\frac{qz'(q)}{z(q)}\bigg)^3 \in \mathbb{Q}[[q]],$$ where $z(q)$ is the compositional inverse of $q(z)$, and write it as $$K(q) = 5+ \sum_ {d=1}^{\infty} c_ d \frac{q^d}{1-q^d},$$ which is formally possible. Candelas et al. observed that the instanton number $n_ d=c_ d/d^3$ is an integer for all $d\ge 1$, which is already a non-trivial fact, but that furthermore $n_ d$ seems to be the number of rational curves of degree $d$ lying on the initial quintic ${\mathbf M}$, thereby providing an effective algorithm to compute these numbers. These striking observations generated much interest amongst algebraic geometers, and this culminated in the further math work (of Givental, Lian et al.) where it is proved that, if for a given $d$ the curves of degree $d$ are all rigid, then there are $n_ d$ of them. In fact, the coincidence was proved to be true for $d\le 9$, and the first difference occurs at $d=10$ (see [E. Cotterill, Comm. Algebra 33 (2005) 1833--1872]).

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Fascinating! Where does one go to learn about such things? I am actually most interested in the analytic aspects in the first half of your message. – Jacques Carette Jul 1 2010 at 12:22
Thanks, Jacques! I escaped refs as there should be too many. There are a lot of math papers in the arXiv discussing "mirror symmetry" from quite different points of view, there is a book on mirror symmetry and algebraic geometry (on google) by David Cox and Sheldon Katz. – Wadim Zudilin Jul 1 2010 at 12:32
@Jacques, try Sheldon Katz's book on "Enumerative Geometry and String theory". books.google.com/… – SandeepJ Jul 1 2010 at 15:50
This is a good example of what seems to be quite common with results based on physical intuition: they're true in many cases but ultimately rely on an additional technical assumption. This is the tradeoff inherent in having a lower standard of rigor, you get better results faster but often then have to go back and rethink the exceptional cases. – Noah Snyder Jul 1 2010 at 18:14

This question seems overly broad. I also do not think that one should single out physicists here, because all sorts of heuristic arguments can go awry, not to mention that, like mathematicians, physicists correct themselves and each other all the time.

Having said that, very much along the same lines as OP's example, but wrong: Kelvin conjecture about the minimal area honeycombs that was disproved by Weaire and Phellan. Also, Close-packing and froth by H.S.M. Coxeter offers ample evidence in a closely related context of statistical honeycombs. Coxeter himself gave two different answers for the average number of faces ($13.39$ and $13.56$)! In spite of the recent work of MacPherson and Srolovitz, the mathematical side of the story is still far from complete.

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 Beautiful stuff, though the work by MacPherson and Srolovitz stands slightly apart from the work on "minimal honeycombs". They provided the higher-dimensional analogue to the "von Neumann coarsening law" that is a statement about dynamics of foams under mean curvature evolution. – jc Jul 1 2010 at 17:30

I believe knot theory got started by 19th-century physicists (Lord Kelvin was one of the initiators) based on the assumption that atoms have to be knots in the ether. They started tabulating knots in the hope to shed light on the periodic table.

The physics was rubbish but not the maths.

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I don't think this is a very good example of what the question is looking for as there's no mathematical conjecture here coming out of physical ideas. – Noah Snyder Jul 1 2010 at 18:15
The physics wasn't rubbish. It was merely wrong. – Dan Piponi Jul 1 2010 at 22:18
I agree with Noah. This is better as an answer to a question like, "what are examples of wrong physics inspired by elegant mathematics?" It fits in well with Kepler's idea about the correspondence between planetary orbits and platonic solids. – S. Carnahan Jul 2 2010 at 13:44

Somewhat related to the ergodic hypothesis mentioned in another answer is the assumption that generic non-linearities leads to thermalization and equipartition of energy. To be more precise, start with a linear, completely integrable, finite dimensional Hamiltonian system (say de-coupled system of many harmonic oscillators). The system has independent excitation modes that, if the initial data is set to be one of the modes, the evolution will stay on the mode. The assumption from physics is that by addition a non-linear coupling, this would let modes interact and in the long run, the system will settle down to a thermalized state where each mode contributes the same amount to the total energy.

This, of course, is now known to be false, in view of the KAM theorem.

But an interesting side development is that Fermi, Pasta, and Ulam were convinced that the thermalization should take place (in fact Fermi had published a "proof" to that effect), so they ran a computer simulation (way back when in Los Alamos on one of the first computers built) for a vibrating string, taking in account of the second order effects (the first order effects are just the linear wave equation, which in finite grid approximation is completely integrable ODE), and tried to numerically compute the rate at which thermalization will occur. What they observed, however, is that the system is quasi-periodic. This discovery gave birth to the modern study of solitons. See an account of this in Palais' article in the Bulletin http://dx.doi.org/10.1090/S0273-0979-97-00732-5

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Many physicists believed that quantum entanglement would (eventually) be understood in terms of some simple-to-compute measure. The surprising mathematical finding that the quantum separability problem is NP-hard proved these expectations to be wrong.

See, for example, Lawrence M. Ioannou's review, "Computational complexity of the quantum separability problem", (arXiv:quant-ph/0603199).

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Physicists originating statistical mechanics asserted that the path in phase space of the state of the system will visit every point on the surface of constant energy. Bolzmann's ergodic hypothesis, as formulated by Ehrenfest, 1911, states exactly that. Later mathematicians found it to be in error for topological reasons: a differentiable curve cannot cover a surface of dimension higher than 1. You can then go on to study this...the curve is dense in the surface, and there are "ergodic" conditions much stronger than that which are (presumably) obeyed by physical systems.

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The ergodic hypothesis as such has been discredited among physicists since the work of Krylov showed that the stronger notion of mixing is the relevant property. But suitably amended, it is still a very powerful POV that has informed the fluctuation theorem for Anosov systems (statistical-physical systems were famously conjectured by Gallavotti and Cohen to be "essentially" of this type). The ergodic hypothesis is also used wrongly but quite effectively in signal processing. – Steve Huntsman Jul 1 2010 at 14:59
Also I don't believe that anyone ever seriously assumed a curve would visit every point in phase space (though I could be wrong). – Steve Huntsman Jul 1 2010 at 15:28
I know the ergodic hypothesis as "time averages = space averages", which is 'true'. ( en.wikipedia.org/wiki/Ergodic_hypothesis ) – Helge Jul 1 2010 at 19:38
Edited to be more complete. – Gerald Edgar Jul 1 2010 at 22:32
Though I can't access this paper by Jan von Plato springerlink.com/content/p53862520r345141 , the abstract claims "If Boltzmann's work is read with any attention, it becomes impossible to ascribe to him the claim that one single trajectory would fill the whole of state space." – jc Jul 1 2010 at 22:56
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