Few remarks on the mass spectrum of two-dimensional Toda lattice of $E_8$ type

TL;DR

This paper presents a straightforward method for deriving the mass spectrum of the two-dimensional Toda lattice associated with the E8 Lie algebra, providing insights into its spectral structure.

Contribution

It introduces a simple procedure for calculating the mass spectrum of the E8 Toda lattice, enhancing understanding of its spectral properties.

Findings

Derived the mass spectrum for E8 Toda lattice

Provided a simplified computational approach

Enhanced understanding of spectral structure

Abstract

In this note the simple procedure for obtaining the mass spectrum of two-dimensional Toda lattice of $E_8$ type is given.

Equations

$$L=\frac{1}{2}{\partial }_{\mu }{\partial }^{\mu }\varphi -U(\varphi ),\mu =0,1,$$

$$U(\varphi )=\sum _{j=0}^l\exp (2{\alpha }_j,\varphi ).$$

$$\delta =2{\alpha }_1+3{\alpha }_2+4{\alpha }_3+5{\alpha }_4+6{\alpha }_5+4{\alpha }_6+2{\alpha }_7+3{\alpha }_8.$$

$$r=\frac{\sqrt{5}+1}{2}=2\cos (\frac{\pi }{5})=1,\mathrm{6180339887.....}$$

$${\rm A}\,=\left(\begin{array}[]{cccccccc}0&1&0&0&0&0&0&0\\ 1&0&1&0&0&0&0&0\\ 0&1&0&1&0&0&0&0\\ 0&0&1&0&1&0&0&0\\ 0&0&0&1&0&1&0&1\\ 0&0&0&0&1&0&1&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&1&0&0&0\end{array}\right).$$

$$x^8-7x^6+14x^4-8x^2+1=0,$$

$$x_j=2\cos (a_j\theta ),$$

$$\theta =\frac{\pi }{h},$$

$$\begin{array}{r}{x}_1=\frac{1}{2}\sqrt{(7+\sqrt{5}+\sqrt{30+6\sqrt{5}}},x_2=\frac{1}{2}\sqrt{(7+\sqrt{5}-\sqrt{30+6\sqrt{5}}},\\ x_3=\frac{1}{2}\sqrt{(7-\sqrt{5}+\sqrt{30-6\sqrt{5}}},x_4=\frac{1}{2}\sqrt{(7-\sqrt{5}-\sqrt{30-6\sqrt{5}}}.\end{array}$$

$$u=(u_1,u_2,u_3,u_4,u_5,u_6,u_7,u_8)$$

$$u\mathrm{A}=\lambda u,$$

$$\begin{array}{r}{u}_2=\lambda u_1,u_1+u_3=\lambda u_2,u_2+u_4=\lambda u_3,u_3+u_5=\lambda u_4,\\ u_4+u_6+u_8=\lambda u_5,u_5+u_7=\lambda u_6,u_6=\lambda u_7,u_5=\lambda u_8.\end{array}$$

$$\begin{gathered} u=\left(2\sin (\theta ),2\sin (2\theta ),2\sin (3\theta ),2\sin (4\theta ),2\sin (5\theta ), \\ \frac{\sin (2\theta )}{\sin (3\theta )},\frac{\sin (\theta )}{\sin (3\theta )},\frac{\sin (\theta )}{\sin (2\theta )}\right). \end{gathered}$$

$$u=(0,2091;0,4158;0,6180;0,8135;1;0,6728;0,3383;0,5028).$$

$$\frac{u_7}{u_1}=r,\frac{u_6}{u_2}=r,\frac{u_5}{u_3}=r,\frac{u_4}{u_8}=r,$$

$$r=\left(\frac{1}{2}\right)(1+\sqrt{5})=2\cos \left(\frac{\pi }{5}\right)$$

$$\begin{array}{r}2\cos (\frac{\pi }{5})=\frac{1}{2}(1+\sqrt{5})=r,2\sin (\frac{\pi }{5})=\sqrt{\frac{5-\sqrt{5}}{2}},\\ 2\cos (\frac{\pi }{10})=\sqrt{\frac{5+\sqrt{5}}{2}},2\sin (\frac{\pi }{10})=\sqrt{\frac{3-\sqrt{5}}{2}},\\ 2\cos (\frac{\pi }{15})=\frac{1}{2}\sqrt{9+\sqrt{5}+2\sqrt{3}\sqrt{\frac{5-\sqrt{5}}{2}}},2\sin (\frac{\pi }{15})=\frac{1}{2}\sqrt{7-\sqrt{5}-2\sqrt{3}\sqrt{\frac{5-\sqrt{5}}{2}}},\\ 2\cos (\frac{\pi }{30})=\frac{1}{2}\sqrt{7+\sqrt{5},+2\sqrt{3}\sqrt{\frac{5+\sqrt{5}}{2}}},2\sin (\frac{\pi }{30})=\frac{1}{2}\sqrt{9-\sqrt{5}-2\sqrt{3}\sqrt{\frac{5+\sqrt{5}}{2}}}\end{array}$$

$$B_{a,b}=\sum _{j=0}^l{n}_j{{\alpha }_j}^a{{\alpha }_j}^b{n}_0=1$$

$$P=x^8-60x^7+1440x^6-18000x^5+127440x^4-518400x^3+1166400x^2-1296000x+518400.$$

$$P=P_1{P}_2,P_1=x^4-30x^3+240x^2-720x+720,P_2=x^4-30x^3+300x^2-1080x+720.$$

$${m_1}^2,{m_3}^2,{m_4}^2,{m_6}^2$$

$${m_2}^2,{m_5}^2,{m_7}^2,{m_8}^2.$$

$$u_2{u}_5{u}_7{u}_8=u_1{u}_3{u}_4{u}_6,$$

$${m_j}^2=M{u_j}^2.$$

$$M^4(u_2{u}_5{u}_7{u}_8{)}^2=720$$

$$M=2\sqrt{3}\frac{\sin (6\theta )}{\sin (\theta )}.$$

$$\begin{array}{r}{m_5}^2=\frac{1}{2}\sqrt{15+3\sqrt{5}+\sqrt{6}\sqrt{25+11\sqrt{5}}},\\ {m_7}^2=\frac{1}{2}\sqrt{15+3\sqrt{5}-\sqrt{6}\sqrt{25+11\sqrt{5}}},\\ {m_8}^2=\frac{1}{2}\sqrt{15-3\sqrt{5}+\sqrt{6}\sqrt{25-11\sqrt{5}}},\\ {m_2}^2=\frac{1}{2}\sqrt{15-3\sqrt{5}-\sqrt{6}\sqrt{25-11\sqrt{5}}},\\ {m_4}^2=\frac{1}{2}\sqrt{15+3\sqrt{5}+\sqrt{6}\sqrt{5-\sqrt{5}}},\\ {m_6}^2=\frac{1}{2}\sqrt{15+3\sqrt{5}-\sqrt{6}\sqrt{5-\sqrt{5}}},\\ {m_3}^2=\frac{1}{2}\sqrt{15-3\sqrt{5}+\sqrt{6}\sqrt{5+\sqrt{5}}},\\ {m_1}^2=\frac{1}{2}\sqrt{15-3\sqrt{5}-\sqrt{6}\sqrt{5+\sqrt{5}}}.\end{array}$$

$$r=\frac{\sqrt{5}+1}{2}=2\cos (\frac{\pi }{5})=1,\mathrm{6180339887.....}$$

Full text

1. Introduction. Basic notations

Two-dimensional Toda lattice is two-dimensional relativistic field theory describing $l$ interacting scalar fields. In paper [MOP 1981] it was generalized for the case of arbitrary simple Lie algebra ${\mathcal{G}}$ and it has remarkable integrability properties.

This is the relativistic system with Lagrangian

[TABLE]

where $\phi=\phi(x_{0},\,x_{1})$ is $l$-dimensional vector.

The potential $U(\phi)$  is constructed using some finite set of vectors $\{\alpha_{j}\}$,

${j=0,\,1,\,...,\,l}$ in $l$-dimensional Euclidean space related to the simple Lie algebra ${\mathcal{G}}$ of rank $l$ :

[TABLE]

Let us give some formal definitions, more details may be found in the book [OV 1990].

Let $\mathcal{G}$ be a compact simple Lie algebra of rank $l$, $R_{+},(R_{-})$ be the set of positive (negative) roots, and $\{\alpha_{1},\ldots,\alpha_{l}\}$ be the set of simple roots. Let also $W$ be the Weyl group of root system acting in the space $V={{\mathbb{R}}^{l}}$, $(\ ,\ )$ be the $W$-invariant bilinear form in $V$, $\delta=\sum_{j=1}^{l}n_{j}\alpha_{j}$ be the highest root, $\alpha_{0}=-\delta$, $h={\sum_{j=1}^{l}n_{j}}+1$ be the Coxeter number.

In [MOP 1981] the mass spectrum of of scalar fields was found for all simple Lie algebras except the most complicated case ${\mathcal{G}}=E_{8}$. For this algebra only numerical result was given.

In this note we describe two simple methods to obtain the mass spectrum for the $E_{8}$ case. Note that both methods are valid also for the cases of other simple Lie algebras.

The enumeration of simple roots of the Lie algebra $E_{8}$ is given on Dynkin diagram.

\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\alpha_{4}$$\alpha_{5}$$\alpha_{8}$$\alpha_{6}$$\alpha_{7}

The Dynkin diagram for the Lie algebra $E_{8}$.

For this enumeration the highest root $\delta$ has the form :

[TABLE]

Note then that in 1989 A.B. Zamolodchikov using conformal theory discover that such system appears also at consideration of Ising model in nonzero magnetic field and he calculated mass spectrum explicitely [Za 1989].

It appears that four mass ratios are equal to the ”golden ratio”

[TABLE]

This remarkable property is related to the fact that for Lie algebra $E_{8}$ the Coxeter number $h=30$ has factor 5.

In 2010 this theory was confirmed experimentally for quasi one - dimensional Ising ferromagnet (cobalt niobate) near its critical point [Co 2010].

2. Method 1

As it was shown in papers [BCDS 1990], [Fr 1991] masses of paricles are proprtional to the components of special eigenvector of matrix $\rm A$ – Perron - Frobenius vector ( [Pe 1907], [Fr 1912] ). Here $\rm{A}=2\,\rm{I}-\rm{C},\,\rm{C}$ is the Cartan matrix of Lie algebra $\mathcal{G}$. For the case $E_{8}$ we have

[TABLE]

The characteristic equation of this matrix is

[TABLE]

and his roots are

[TABLE]

where

[TABLE]

$h\,=\,30$  is Coxeter number.

The numbers $\{a_{j}\}\,=\,(\,1,\,7,\,11,\,13,\,17,\,19,\,23,\,29)$ are so called exponents of Lie algebra $E_{8}$.

Note that they have not common divisors with Coxeter number $30$.

Note also that $x_{5}=-x_{4},\,x_{6}=-x_{3},\,x_{7}=-x_{2},\,x_{8}=-x_{1}$ and let us give the expressions for $x_{j}$ in terms of radicals.

[TABLE]

The matrix $\rm A$ has nonnegative elements and according to Perron - Frobenius theorem [Pe 1907], [Fr 1912] it has unique eigenvector

[TABLE]

all components of which are positive. It corresponds to the maximal eigenvalue $\lambda=2\cos(\theta)$ and we have

[TABLE]

[TABLE]

Solving these equations, fixing $u_{1}=2\sin(\,\theta\,)$, we obtain:

[TABLE]

[TABLE]

Note that from this it follows relationes

[TABLE]

where

[TABLE]

is the so called golden section.

This is very nice solution, because these expressions for $u_{j}$ may be written immediately just looking to the Dynkin diagram for $E_{8}$.

We would like to underline that the situation for arbitrary simple Lie algebra is the same, i. e. the solution may be written just to looking to corresponding Dynkin diagram.

Let us give also the expressions for some trigonometric quantities in terms of radicals

[TABLE]

3. Method 2

In the paper [MOP 1981] it was shown that squares of masses are eigenvalues of matrix

[TABLE]

where quantities ${\alpha_{j}}^{a}$ are components of vector $\alpha_{j}$.

For the case of Lie algebra $E_{8}$ characteristic polynomial $P$ of this matrix is

[TABLE]

In paper [BCDS 1990] was noted factorisation of the characteristic polynomial.

[TABLE]

It is easy to check that the roots of polynomial $P_{1}$ are

[TABLE]

and the roots of polynomial $P_{2}$ are

[TABLE]

Note that

[TABLE]

and

[TABLE]

The quantity $M$ may be found from the equation

[TABLE]

and it has the form

[TABLE]

So the formulae (3.7) and (3.9) give the relation between methods 1 and 2.

Let us give also the explicit expression for quantities ${m_{j}}^{2}$ in terms of radicals:

[TABLE]

4. Conclusion

The remarkable property of system under consideration is that four mass ratios in (2.10) are equal to the ”golden ratio” (”golden section”)

[TABLE]

This is only one more fenomenon in which golden ratio appeared. Note that golden ratio has very long history, see for example book [Co 1961], Ch. 11. The first book on this topic ”Divina Proportione”, illustrated by Leonardo da Vinci, was published by Italian mathematician Luca Paccioli in 1509 [Pa 1509]. In conclusion I would like to give here the quotation of outstanding astronomer and mathematicien Johannes Kepler [Ke 1596]: ”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”.