1. Introduction. Basics

Let us recall several definitions; for more details, see the book [10].

Let 𝔤 be a simple Lie algebra of rank l, let R₊ (resp. R₋) be the set of its positive (resp. negative) roots, and {α₁,...,α_l} be the set of simple roots. Let W be the Weyl group of the root system R acting in the space V = ℝ^l, let (·,·) be the non-degenerate W-invariant bilinear form in V, δ=∑1≤j≤lnjαj be the highest root, α₀ = −δ, and h=1+∑1≤j≤lnj be the Coxeter number.

The 2-dimensional Toda lattice is 2-dimensional relativistic field theory describing l interacting scalar fields. The 2-dimensional Toda lattice is related with 𝔰𝔩 (n).

In the paper [8], the 2-dimensional Toda lattice was generalized for the case of any simple finite-dimensional Lie algebra 𝔤; it was shown that the generalized construction has remarkable integrability properties. This is a relativistic system with Lagrangian

L=12∂μ∂μφ−U(φ),   where   μ=0,1  and   φ=φ(x0,x1)  is an l−dimensional vector., where μ = 0, 1 and ϕ = ϕ(x₀, x₁) is an l-dimensional vector.

The potential U(φ) is constructed using the set of simple roots {αj}j=0l of the simple Lie algebra 𝔤of rank l:

In [8], the mass spectrum of scalar fields was found for all simple Lie algebras, except for the most complicated case 𝔤 = E₈. For this algebra only numerical result was given.

In this note I describe two simple methods for obtaining the mass spectrum in the E₈ case. Note that both methods work also for any other finite-dimensional simple Lie algebra.

The numbering of simple roots of the Lie algebra E₈ is given on the Dynkin diagram:

The Dynkin diagram for the Lie algebra E₈.

For this numbering, the highest root δ has the form

Observe that in 1989 A.B. Zamolodchikov discovered, using conformal theory, that this system appears also in the Ising model with nonzero magnetic field and explicitly calculated the mass spectrum, see [13]. The four mass ratios are equal to the “golden ratio”

This remarkable property is related to the fact that the Coxeter number h = 30 of the Lie algebra E₈ is divisible by 5.

In 2010, Zamolodchikov’s theory was experimentally confirmed for 1-dimensional Ising ferro-magnet (cobalt niobate) near its critical point [2].

2. Method 1

As it was shown in papers [1, 5] the masses of particles are proportional to the components of a special eigenvector of the matrix A = 2I − C, where C is the Cartan matrix of 𝔤. This eigenvector is called the Perron–Frobenius vector, see [6, 12]. For 𝔤= E₈, we have

The characteristic equation of this matrix is

and its roots are the numbers a_j ∈ {1, 7, 11, 13, 17, 19, 23, 29} for 1 ≤ j ≤ 8 are called the exponents of E₈. Note that they have no common divisors with the Coxeter number.

Note also that x₅ = −x₄, x₆ = −x₃, x₇ = −x₂, x₈ = −x₁. Let us give the expressions of the xj in terms of radicals (these expressions might be used in calculations):

The matrix A has nonnegative elements and according to the Perron–Frobenius theorem [6, 12] it has a unique eigenvector (the Perron-Frobenius eigenvectors)

all coordinates of which are positive. This eigenvector corresponds to the maximal eigenvalue λ = 2cos(θ) and we have or, in more details,

Solving the system of these equations, and fixing u₁ = 2sin(θ), we obtain:

or, approximately,

Note that from eq. (2.1) it follows that (recall that θ=π30)

This is a very nice solution, because these expressions for uj can be written immediately just by looking at the Dynkin diagram of E₈.

Observe that for any simple Lie algebra the eigenvector corresponding to the maximal eigenvalue can also be written just by looking at the corresponding Dynkin diagram.

Let me also give expressions for some trigonometric quantities in terms of radicals (and use this occasion to correct a typo in the definition of H₃ on p. 382 of [9], where ε=2cos(π3) should be ε=2cos(π5)

3. Method 2

In the paper [8] it was shown that the squares of masses are eigenvalues of the 8 × 8 matrix whose elements are

and where quantities αja are coordinates of the vector α_j, and the n_j for j > 0 are coordinates of the vector δ=∑1≤j≤lnjαj.

For the Lie algebra E₈, the characteristic polynomial P of this matrix is

In the paper [1], it was observed that P = P₁ P₂, where

It is easy to check that the roots of polynomial P₁ (resp. P₂) are

Note that

The quantity M=23sin(6θ)sin(θ) can be found from the equation

So, formula (3.1) gives a relation between methods 1 and 2.

Let me give also the explicit expression for quantities mj2 in terms of radicals:

4. Conclusion

The remarkable property of the system under consideration is that the four mass ratios in (2.2) are equal to the “golden ratio”.

This is one more phenomenon of many in which the golden ratio appears. The golden ratio has a very long history, see e.g., the book [4, Ch. 11]. The first book on this topic, “Divina Proportione”, illustrated by Leonardo da Vinci, was published by Italian mathematician Luca Paccioli in 1509 [11].

Concluding, I would like to give here a quotation of the outstanding astronomer and mathematician Johannes Kepler [7]: “Geometry has two treasures: one of them is the Pythagorean theorem, and the other is dividing the segment in average and extreme respect ... The first can be compared to the measure of gold; the second is more like a gem”.

Acknowledgments

I am thankful to D. Leites who improved my English in this article.