Examples where physical heuristics led to incorrect answers?

Question

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 this year (2010) by S. Smirnov and H. Duminil-Copin in The connective constant of the honeycomb lattice equals $\sqrt{2 + \sqrt2}$.

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).

Answer 1

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.

Answer 2

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 $$ K(q):= \frac{5}{1-5^5z(q)}\cdot \frac{1}{F(z(q))^2}\cdot \biggl(\frac{qz'(q)}{z(q)}\biggr)^3 \in \mathbb{Q}[[q]], $$ 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, Rational curves of degree 10 on a general quintic threefold, Comm. Algebra 33 (2005) 1833–1872]).

Answer 3

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 The symmetries of solitons.

Answer 4

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 The von Neumann relation generalized to coarsening of three-dimensional microstructures of MacPherson and Srolovitz, the mathematical side of the story is still far from complete.

Answer 5

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).

Answer 6

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.

Answer 7

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.

Answer 8

In the context of the Abelian sandpile model, in the paper “Absorbing-state phase transitions in fixed-energy sandpiles” the physicists Vespignani, Dickman, Muñoz, and Zapperi predicted that the threshold density of the fixed energy sandpile should be the same as the stationary density of this model. But in fact this heuristic was refuted in some graphs by Fey, Levine and Wilson (The approach to criticality in sandpiles and Driving sandpiles to criticality and beyond). In particular while the two quantities in question appear to be close to one another in the cases of interest to physicists (torus/grid graphs), they aren't the same. The issue is that the threshold density depends on the initial configuration one starts from: the system remembers some of its pre-critical past at criticality. One then suspects that if the system is allowed enough time to mix, the threshold density will be the same as the stationary density. Hence, some more physicists (Poghosyan et al., Numerical Study of the Correspondence Between the Dissipative and Fixed Energy Abelian Sandpile Models) conjectured that if the total number of grains of sand of the initial configuration goes to negative infinity, in the limit the threshold density and stationary density agree. Levine proved this is the case (”Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle”). Levine's paper is very well written and he goes over all this history in the introduction.

Answer 9

I realize that this question was posed some time ago, but perhaps I may contribute with this (humble) post.

I'd have guessed that plenty of examples would have been written, but apart from the posts on the lower critical dimension, the Kelvin conjecture, and the KAM theorem, not many "incorrect physical ansätze" (the plural of "ansatz" in German, which btw should also start with a capital letter in German, but since this is English...) have been pointed out. Though the parsing of such a phrase might be somehow ambiguous, I would understand it as follows: supposing that some axiomatic (physical) grounds are accepted (i.e. those constituting the building blocks of the corresponding physics model), an incorrect physical ansatz would be the result of an heuristic argument, which though it seems compelling, it is not valid (in the "mathematical" sense of the word). I will refrain from mentioning the (typical) opaque arguments (if one may call them in such a manner) that physicists provide on their basic textbooks, and which most of the time deprive any rigor whatsoever (e.g. the derivation of the Liénard-Wiechert formula of EM potentials given in the book of L. Landau and E. Lifshitz, "The classical theory of fields. Vol 2", Ch. 8, 62, to mention a classic one, which may be correctly proved just using the "obvious" convolution arguments on fundamental solutions for the wave equation), for these can be put into solid mathematical grounds, even though these solid mathematical arguments are lacking (most of the time) in the physics literature.

The example I want to mention is the Kirchhoff formula of (scalar) diffraction (of light) by a planar screen. This is just the expression of any scalar component of a nonvanishing (monochromatic) electric and/or magnetic field wave passing through a planar screen with an aperture. The initial data is the wave at the aperture and the Kirchhoff formula is the expression of the wave after the aperture in terms of the wave at the aperture. However, Kirchhoff imposed two conditions: over the plane screen (excepting its aperture) the scalar field and its normal derivative vanish. This formula gives an expression which was used among physicists. As was later pointed out by A. Sommerfeld, imposing the Kirchhoff boundary conditions would imply that the scalar field should vanish everywhere. This implies that Kirchhoff formula could not be applied to fields nonvanishing at the aperture, which implies its inconsistent use. Moreover, it was "physically inconsistent" (something perhaps of much interest to physicists), i.e. contrary to measurements, for they could indeed measure light (i.e. not dark!) at those situations. Fortunately, Sommerfeld did also provide a consistent formulation of the problem by replacing the Kirchhoff boundary condition by the so-called Sommerfeld radiation condition. The expression of (Rayleigh-)Sommerfeld formula seems however quite similar to that of Kirchhoff. As a reference one may check the book of J. Goodman "Introduction to Fourier optics", Sections 3-4 to 3-6. Needless to say that this "subtlety" is (usually) not properly treated -as far as I know- in the standard physics literature (e.g. the book "Optics" by E. Hecht, see the end of 10.1 and 10.5).