Solitons and Particles^{∗}

Vieri Benci^{∗}, Donato Fortunato^{∗∗} ∗Dipartimento di Matematica Applicata ”U. Dini” Università di Pisa, via Bonanno 25/b 56126 PISA (Italy) ∗∗Dipartimento di Matematica Università di Bari, via Orabona 4 70125 BARI (Italy)

January 31, 1997

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The Sine-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Solitons in a 3-dimensional space . . . . . . . . . . . . . . . . . . . . . . 3 4 The topological invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Particles and radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6 Mass and energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 7 Some remarks on the dynamics . . . . . . . . . . . . . . . . . . . . . . . 17

1. Introduction

In this conference we deal with solitons of Lorentz invariant equations and we want to show that they behave pretty much as relativistic particles. The solitons occur in many questions of mathematical physics, such as classical and quantum field theory, non linear optics, fluid mechanics, plasma physics (see [8], [10], [14], [16]).

∗Conference held by the first author at the ”International Conference on Nonlinear Differential Equations and Applications”, Tata Institute of Fundamental Research, Bangalore, India, August 19-23 , 1996. Supported by M.U.R.S.T. (40% and 60% funds); the second author was supported also by E.E.C., Program Human Capital Mobility (Contract ERBCHRXCT 940494)

Roughly speaking a soliton is a solution of a field equation whose energy travels as a localized packet and which preserves its form under perturbations. In this respect solitons have a particle-like behavior; however, if the equation is relativistic invariant, the similarity between solitons and particles is very surprising.

In general the solitonic behavior rises when one of the following circumstances occurs:

• existence of infinitely many first integrals of motion (e.g. KdV equation);

• existence of topological constraints which characterize the solutions.

In this conference we consider only the second case, namely we shall study topological solitons which are solutions of Lorentz invariant equations in more than one space dimensions.

2. The Sine-Gordon equation

A classical interesting one dimensional model is the sine-Gordon equation

u_{tt} − c^{2}u_{xx} + ω_{0}^{2} sin u = 0  (2.1)
lim_{x→−∞} u(x, t) = 0,  (2.2)
lim_{x→+∞} u(x, t) = 2kπ,  (2.3)

being c the light velocity, k ∈ Z and ω_{0} ∈ R. The asymptotic conditions allow to have solutions with finite energy

E(u) = ∫_{R} [\frac{1}{2} (u_{t}^{2} + c^{2}u_{x}^{2}) + ω_{0}^{2} (1 − cos u)] dx.

Any static solution u(x, t) = φ(x) of 2.1 solves the O. D. E.

−c^{2}φ′′ + ω_{0}^{2} sin φ = 0,  (2.4)
lim_{x→−∞} φ(x) = 0,  (2.5)
lim_{x→+∞} φ(x) = 2kπ.  (2.6)

These static solutions give rise to the travelling solutions of 2.1 with velocity v ∈ R, |v| < c

u(x, t) = φ (\frac{x − vt}{\sqrt{1 − (\frac{v}{c} )^{2}}} ) .

Observe that the solution of 2.4 are the critical points of the energy functional

E(φ) = ∫_{R} [\frac{c^{2}}{2} (φ′(x))^{2} + V (φ)] dx

with

V (φ) = ω_{0}^{2} (1 − cos φ)

The function space Λ on which E is defined can be divided into infinitely many connected components according to the asymptotic conditions 2.5, 2.6

Λ = ⋃_{k∈Z} Λ_{k}
Λ_{k} = {φ | lim_{x→−∞} φ(x) = 0, lim_{x→+∞} φ(x) = 2kπ}

The existence of local minima of E in the connected components Λ_{±1} can be proved. These solutions possess a solitonic behavior.

Moreover, observe that the linearized equation at u = 0 of the 2.1 is the one-dimensional Kline-Gordon equation (with m = ω_{0}):

u_{tt} − c^{2}u_{xx} + ω_{0}^{2}u = 0

3. Solitons in a 3-dimensional space

The goal of this section is to determine an equation defined in a four dimensional space-time which preserves the main peculiarity of the sine-Gordon equation namely:

• it has soliton solutions;

• it is Lorentz invariant;

• the linearized equation at u = 0 is the Kline-Gordon equation

u_{tt} − c^{2}Δu + ω_{0}^{2}u = 0

Now consider the analogous case of a scalar equation with 3 space dimensions:

u_{tt} − c^{2}Δu + V′(u) = 0.

Then the energy functional for the static solutions is

E(φ) = ∫_{R^{3}} [\frac{c^{2}}{2} |∇φ|^{2} + V (φ)] dx.  (3.1)

If V is bounded from below, it is not difficult to show, by a rescaling argument, that any φ minimizing 3.1 is necessarily trivial, i.e. it takes a constant value which is a minimum point of V (Derrick’s Theorem, [8]).

On the other hand, if we consider non positive potential, we are forced to seek saddle points, instead of minima, and for these static solutions we have lack of stability. As an example we recall that, if we take

V (ξ) = \frac{1}{2}ξ^{2} − \frac{1}{4}ξ^{4},

critical points of the energy functional

E(φ) = ∫_{R^{3}} [\frac{c^{2}}{2} |∇φ|^{2} + \frac{1}{2}φ^{2} − \frac{1}{4}φ^{4}] dx.

have been found in [7] and [13] and for more general potentials in [6], [15]; but in [1] and [5] it has been proved that these static solution are not stable.

So we are forced to study systems of non linear wave equation with a suitable correction. In the following we derive, by means of some heuristic arguments, one model equation.

We are interested in maps

u : R^{3+1}→ R^{n}, u = (u^{1}, ...., u^{n}).

We refer to the target space R^{n} as an internal parameters space. Since we require the Lorentz invariance, we shall consider Lagrangian densities of the form

L = L(u, ρ)

where ρ = (ρ^{1}, ...., ρ^{n}) and

ρ^{j} = c^{2} |∇u^{j}|^{2} − (u_{t}^{j})^{2} .

We shall consider

L(u, ρ) = −\frac{1}{2}α(ρ) − V (u),  (3.2)

where α : R^{n}→ R and the potential function V is defined in an open set Ω ⊂ R^{n}. The action functional related to 3.2 is

S(u) = ∫_{R^{3+1}} L(u, ρ) dxdt = ∫_{R^{3+1}} [−\frac{1}{2}α(ρ) − V (u)] dxdt.

So the Euler-Lagrange equations are

\frac{∂}{∂t} (\frac{∂α}{∂ξ_{j}} (ρ) u_{t}^{j}) − c^{2}div (\frac{∂α}{∂ξ_{j}} (ρ) ∇u^{j}) + \frac{∂V}{∂ξ_{j}} (u) = 0,  (3.3)
(1 ⩽ j ⩽ n).

When

α(ξ_{1}, ...., ξ_{n}) = ξ_{1} + .... + ξ_{n}  (3.4)

the equations 3.3 reduce to a classical system of n nonlinear wave equations

u_{tt}^{j} − c^{2}Δu^{j} + \frac{∂V}{∂ξ_{j}} (u) = 0,
(1 ⩽ j ⩽ n).

The static solutions of the 3.3 solve the equation

−c^{2}div (\frac{∂α}{∂ξ_{j}} (c^{2} |∇φ|^{2}) ∇φ^{j}) + \frac{∂V}{∂ξ_{j}} (φ) = 0,  (3.5)
(1 ⩽ j ⩽ n).

If φ = φ(x_{1}, x_{2}, x_{3}) is a solution of the 3.5 then, for any vector v with |v| < c,

φ_{v}(x_{1}, x_{2}, x_{3}, t) = φ (\frac{x − vt}{\sqrt{1 − \frac{v^{2}}{c^{2}}}} )  (3.6)

is a solution of the equations 3.3.

In particular, if v = (v, 0, 0), we have the following solution:

φ_{v}(x_{1}, x_{2}, x_{3}, t) = φ (\frac{x_{1} − vt}{\sqrt{1 − (\frac{v}{c} )^{2}}} , x_{2}, x_{3}) .  (3.7)

Notice that the function φ experiences a contraction of a factor

γ = \frac{1}{\sqrt{1 − (\frac{v}{c} )^{2}}}

in the direction of the motion: and this fact is consistent with the fact that 3.3 is Lorentz invariant.

Observe that the 3.5 are the Euler-Lagrange equation relative to the energy for functions independent of t. In fact

E(u) = ∫ T_{00} dx

where (cf. e.g. [9] pg. 184 or [11])

T_{00} = ∑_{j=1}^{n} \frac{∂L}{∂ (\frac{∂u_{j}}{∂t} )} \frac{∂u_{j}}{∂t} − L

= −\frac{1}{2} ∑_{j=1}^{n} α′ (ρ) \frac{∂ρ}{∂ (\frac{∂u_{j}}{∂t} )} \frac{∂u_{j}}{∂t} + \frac{1}{2}α (ρ) + V (u)

= α′ (ρ) ∑_{j=1}^{n} (\frac{∂u_{j}}{∂t} )^{2} + \frac{1}{2}α (ρ) + V (u)

= α′ (ρ) (\frac{∂u}{∂t})^{2} + \frac{1}{2}α (ρ) + V (u)

Thus, if u is independent of t, we have

E(u) = ∫ [\frac{1}{2}α (c^{2} |∇u|^{2}) + V (u)] dx.  (3.8)

Moreover, if we make the following assumptions on α and V :

α′(0) = 1
V′′(0) [ξ, ξ] = ω_{0}^{2}|ξ|^{2}

the linearized equation at u = 0 of the 3.3 reduce to the Kline-Gordon equation:

u_{tt}^{j} − c^{2}Δu^{j} + ω_{0}^{2}u^{j} = 0,
(1 ⩽ j ⩽ n).

Now, we consider a simple correction of 3.4, namely

α(ξ_{1}, ..., ξ_{n}) = ∑_{j=1}^{n} (ξ_{j} + \frac{ε}{3}ξ_{j}^{3}) ,

being ε > 0. Then 3.3 can be written

\frac{∂}{∂t} ((1 + ε(ρ^{j})^{2}) u_{t}^{j}) − c^{2}div ((1 + ε(ρ^{j})^{2}) ∇u^{j}) + \frac{∂V}{∂ξ_{j}} (u) = 0,
(1 ⩽ j ⩽ n).

or

□u^{j} + ε□_{6}u^{j} + \frac{∂V}{∂ξ_{j}} (u) = 0  (3.9)
(1 ⩽ j ⩽ n)

where

□_{6}u^{j} = \frac{∂}{∂t} [(c^{2} |∇u^{j}|^{2} − (u_{t}^{j})^{2})^{2} u_{t}^{j}] − c^{2}div [(c^{2} |∇u^{j}|^{2} − (u_{t}^{j})^{2})^{2} ∇u^{j}] .

So the static solutions φ solve the system of equations

−c^{2}Δφ^{j} − εc^{6}Δ_{6}φ^{j} + \frac{∂V}{∂ξ_{j}} (φ) = 0,  (3.10)
(1 ⩽ j ⩽ n).

where

Δ_{6}φ^{j} = div (|∇φ^{j}|^{4} ∇φ^{j}) .

Then the energy functional becomes

E(φ) = ∫_{R^{3}} (\frac{c^{2}}{2} |∇φ|^{2} + ε\frac{c^{6}}{6} |∇φ|^{6} + V (φ)) dx  (3.11)

As to the potential V and the open set

Ω = R^{n} \ Σ

where it is defined, we make the following assumptions:

• Ω is connected and

min_{ξ∈Σ} |ξ| = 1

• V ∈ C^{2}(Ω, R);

• V (ξ) ⩾ V (0) = 0 for every ξ ∈ Ω; • there exist k, r > 0 such that

dist(ξ, Σ) < r ⇒ V (ξ) > \frac{k}{dist(ξ, Σ)^{6}}  (3.12)

• V′′(0) [ξ, ξ] = ω_{0}^{2}|ξ|^{2}

Of course we can consider our evolution equation 3.9 as a dynamical system. The configurations space for this system is given by

Λ = W^{1,6}(R^{3}, Ω) ∩ W^{1,2}(R^{3}, Ω)

equipped with the norm

∥φ∥_{Λ} = ∥φ∥_{W^{1,6}} + ∥φ∥_{W^{1,2}} .

Since the functions in Λ are continuous and decay to 0 at infinity, by means of a suitable topological invariant, the algebraic structure of π_{3}(Ω) is reflected into a decomposition of the configurations space:

Λ = ⋃_{α∈π_{3}(Ω)} Λ_{α}.

About the structure of π_{3}(Ω) we can say that, in many cases (e.g. when it is finitely generated and torsion free) it has the structure

π_{3}(Ω) = Z^{k}

with k ∈ N. For example this situation occurs when

Ω = R^{4} \ {ξ_{1}, …, ξ_{k}}.

In [3] it has been proved the following theorem.

Theorem 3.1. Let

A = {α ∈ π_{3}(Ω) | E attains its minimum in Λ_{α}}

then the subgroup of π_{3}(Ω) generated by A coincides with π_{3}(Ω) itself.

Since every local minimum of E gives rise to a weak solution of system 3.10, if π_{3}(Ω) is not trivial, then there exists at least a non trivial solution of 3.10 in the class

Λ^{∗} = ⋃_{α≠0} Λ_{α}

i.e. in the class of configurations which are not homotopic to zero.

On the other hand, in the case π_{3}(Ω) = Z^{k}, Theorem 1.1 implies that there exist at least k homotopically distinct static solutions of 3.9.

4. The topological invariants

First of all we define a ”global homotopic invariant”. To this end we consider an homeomorphism

π : S^{3} → R^{3} ∪ {∞}

such that π(∗) = ∞, where ∗ is the base point. Now, for every function φ ∈ Λ, we can define the continuous map

φ ∘ π : S^{3} → Ω.

We denote with φ^{#} the homotopy class of φ ∘ π, that is

φ^{#} = [φ ∘ π] ∈ π_{3}(Ω).

Clearly, this invariant is stable under uniform convergence.

In order to define a local topological invariant more work is necessary.

Definition 4.1. For every φ ∈ Λ we call support of φ the compact set defined as follows

K(φ) = \overline{\{x : |φ(x)| > 1\}}.

It can be proved that (see. [3])

φ^{#} ≠ 0 ⇒ K(φ) ≠ ∅;

if we have two functions φ, ψ ∈ Λ such that

K(φ) = K(ψ) = K

and for every x ∈ K

φ(x) = ψ(x),

then

φ^{#} = ψ^{#};

indeed the functions φ and ψ are homotopic in Ω, so φ∘π and ψ∘π are homotopic too.

Now we want to ”localize” in a more precise fashion the topological invariant, by evaluating the contribution from a fixed compact set in R^{3}; in other words, if φ ∈ Λ and K ⊂ R^{3}, we would like to define

(φ_{|K})^{#} ∈ π_{3}(Ω)

with suitable properties (finite additivity and so on). Having in mind, as model case, the topological degree, we notice that this kind of object can be defined only for a class of admissible subsets K.

Fix φ ∈ Λ and consider the following class of admissible sets

K(φ) = {K ⊂ R^{3} closed | |φ(x)| ⩽ 1 on ∂K}.

This set is obviously not empty; indeed it contains K(φ), each connected component of K(φ) and every ball with radius sufficiently large.

Before introducing the local invariant we need some notation; let

Ω_{1} = {ξ ∈ Ω | |ξ| ⩽ 1}.

and

U = ⋃_{ξ∈Ω_{1}} B_{ρ_{ξ}}(ξ).

where ρ_{ξ} is such that

∀η ∈ B_{ρ_{ξ}}(ξ), ∀s ∈ [0, 1] , sη ∈ Ω.

Clearly U is open and star-shaped.

If K ∈ K(φ), by the continuity of φ, there exists an open neighborhood N of ∂K such that φ(x) ∈ U for every x ∈ N.

By the Uryson lemma, there exists a continuous function

c : R^{3} → [0, 1]

such that

c(x) = \{ \begin{matrix}1  on K, \\ 0  on R^{3}\Ñ. \end{matrix}

where

Ñ = K ∪ N.

is an open neighborhood of K Now we consider φ̃ : R^{3}→ R^{n} defined as follows

φ̃(x) = c(x)φ(x).

Definition 4.2. We set

(φ_{|K})^{#} = φ̃^{#}.

This definition is well posed, i.e. it does not depend on φ̃ (see [3]).

The natural connection between the global and the local homotopic invariant is the following

(φ_{|K(φ)})^{#} = φ^{#},  (4.1)

where in the left hand and in the right hand side we are using respectively the local and the global invariant.

Indeed, for any choice of φ̃ in the previous definition, since φ and φ̃ coincide in their common support, we have φ̃^{#} = φ^{#}.

The local invariant satisfy the finite additivity property. Let K_{1}, K_{2} ∈ K(φ)

with K_{1}^{∘} ∩ K_{2}^{∘}= ∅, then we have

(φ_{|K_{1}∪K_{2}})^{#} = (φ_{|K_{1}})^{#} + (φ_{|K_{2}})^{#}

(the sum is meant, of course, in the group π_{3}(Ω)).

5. Particles and radiation

Now we use some terminology (quantum numbers, particles, ....) which can be useful to interpret the notions and the model introduced. However this terminology does not refer to any specific model in field theory.

Suppose that π_{3}(Ω) is isomorphic Z^{k}, and fix a set of generators {α_{1}, ...., α_{k}}; then every configuration φ ∈ Λ is characterized by a k-tuple of integer numbers, which we call quantum numbers of φ: if φ^{#} = m_{1}α_{1} + .... + m_{k}α_{k}, then we set

ν(φ) = (m_{1}, ...., m_{k}).

If C is a connected component of K(φ), C ∈ K(φ) and hence (φ_{|C})^{#} is well defined. If (φ_{|C})^{#} ≠ 0,

σ := φ_{|C},

is called a particle.

So each particle has its quantum numbers, which are the coefficients of (φ_{|C})^{#} for a fixed systems of generators.

The state φ will be said a radiation field if φ^{#} = 0.

On the other hand, every configuration can be written, in some sense, as the disjoint union of a set of particles and a radiation field. Indeed we can write

φ = (⋁ σ_{i}) ∨ ρ

being σ_{i} and ρ the restrictions of φ respectively to the admissible sets C_{i} and R (with disjoint interiors) such that

R^{3} = (⋃ C_{i}) ∪ R,
(φ_{|C_{i}})^{#} ≠ 0, (φ_{|R})^{#} = 0.

6. Mass and energy

Now we want to define the mass of a soliton. The most natural thing to do is to define the inertial mass which can be defined as the ratio between the momentum and the speed of a particle.

Now let u(x, t) = u(x_{1}, x_{2}, x_{3}, t) be a solution of the 3.3 (where for simplicity we set c = 1). It is well known (see e.g. [9]) that the momentum of u is given by

p_{k} = ∫ T_{0,k} dx, k = 1, 2, 3  (6.1)

where T_{i,k} is the energy-momentum tensor defined by

T_{ik} = ∑_{j=1}^{n} \frac{∂L}{∂ (\frac{∂u_{j}}{∂x_{i}} )} \frac{∂u_{j}}{∂x_{k}} − δ_{ik}L; with x_{0} := t

In our case we have

T_{0,k} = −α′(ρ) u_{x}u_{t}  (6.2)

so we can easily compute the momentum of the soliton 3.7:

Lemma 6.1. Let φ_{v} is the solution of 3.3 (with c = 1) given by the 3.7, then the momentum is given by

p_{1}(φ_{v}) = vγ ∫ α′(|∇φ|^{2}) (\frac{∂φ}{∂x_{1}} )^{2} dx

p_{2}(φ_{v}) = v ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂x_{1}} \frac{∂φ_{v}}{∂x_{2}} dx

p_{3}(φ_{v}) = v ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂x_{1}} \frac{∂φ_{v}}{∂x_{3}} dx

where

γ = \frac{1}{\sqrt{1 − v^{2}}} .

Proof. By the 6.2 we have

p_{1}(φ_{v}) = − ∫ α′ (|∇φ_{v}|^{2} − (\frac{∂φ_{v}}{∂t} )^{2}) \frac{∂φ_{v}}{∂x_{1}} \frac{∂φ_{v}}{∂t} dx;

exploiting the Lorentz invariance of |∇φ_{v}|^{2} − (\frac{∂φ_{v}}{∂t} )^{2}, we get

[|∇_{x}φ_{v}|^{2} − (\frac{∂φ_{v}}{∂t} )^{2}] (γ (x_{1} − vt) , x_{2}, x_{3}) = |∇_{ξ}φ|^{2} (ξ_{1}, ξ_{2}, ξ_{3})

where

\{ \begin{matrix}ξ_{1} = γ (x_{1} − vt) \\ ξ_{2} = x_{2} \\ ξ_{3} = x_{3} \end{matrix}

Thus

p_{1}(φ_{v}) = − ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂x_{1}} \frac{∂φ_{v}}{∂t} dx_{1} dx_{2} dx_{3}

= − ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂ξ_{1}} \frac{∂ξ_{1}}{∂x_{1}} \frac{∂φ_{v}}{∂ξ_{1}} \frac{∂ξ_{1}}{∂t} \frac{1}{γ} dξ_{1} dξ_{2} dξ_{3}

= γv ∫ α′ (|∇_{ξ}φ|^{2}) (\frac{∂φ_{v}}{∂ξ_{1}} )^{2} dξ_{1} dξ_{2} dξ_{3}.

Now let us compute p_{2} :

p_{2}(φ_{v}) = − ∫ α′ (|∇φ_{v}|^{2} − (\frac{∂φ_{v}}{∂t} )^{2}) \frac{∂φ_{v}}{∂x_{2}} \frac{∂φ_{v}}{∂t} dx

= − ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂x_{2}} \frac{∂φ_{v}}{∂t} dx_{1} dx_{2} dx_{3}

= ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂ξ_{2}} \frac{∂φ_{v}}{∂ξ_{1}} γv \frac{1}{γ} dξ_{1} dξ_{2} dξ_{3}

= v ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂ξ_{2}} \frac{∂φ_{v}}{∂ξ_{1}} dξ_{1} dξ_{2} dξ_{3}.

The mass matriix {m_{ij}} of φ_{v} is defined by the following relaation:

p_{i}(φ_{v}) = m_{ij}(φ_{v})v_{j},

then, by the previous lemma we get:

m_{11}(φ_{v}) = γ ∫ α′(|∇φ|^{2}) (\frac{∂φ}{∂x_{1}} )^{2} dx

m_{12}(φ_{v}) = ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂x_{1}} \frac{∂φ_{v}}{∂x_{2}} dx

m_{13}(φ_{v}) = ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂x_{1}} \frac{∂φ_{v}}{∂x_{3}} dx.

and by simetry arguments we get:

m_{ii}(φ_{v}) = γ ∫ α′(|∇φ|^{2}) (\frac{∂φ}{∂x_{i}} )^{2} dx

m_{ij}(φ_{v}) = ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂x_{i}} \frac{∂φ_{v}}{∂x_{j}} dx, for i ≠ j

The rest mass matrix, then is defined as follows:

m_{ij}(φ_{v}) = ∫ α′ (|∇_{ξ}φ|^{2}) \frac{∂φ_{v}}{∂x_{i}} \frac{∂φ_{v}}{∂x_{j}} dx,

Now let us compute the energy of a soliton. By definition we have

E(φ_{v}) = ∫ T_{0,0} dx  (6.3)

Since the direct computation of T_{0,0} is quite involved, we start to compute the energy of a static solution; in this case (see 3.8) we have that

E(φ) = ∫ [\frac{1}{2}α(|∇φ|^{2}) + V (φ)] dx  (6.4)

At this point we need the following lemma

Lemma 6.2. Let φ be a solution of 3.5 with c = 1; then

∫ α′(|∇φ|^{2}) (\frac{∂φ_{v}}{∂x_{i}} )^{2} dx = \frac{1}{2} ∫ α(|∇φ|^{2}) dx + ∫ V (φ) dx, i = 1, 2, 3

Proof. Let

φ_{δ}(x) = φ(δx_{1}, x_{2}, x_{3});

then, setting y = (δx_{1}, x_{2}, x_{3})

E(φ_{δ}) = \frac{1}{2} ∫ α(|∇φ_{δ}|^{2}) dx + ∫ V (φ_{δ}) dx

= \frac{δ^{−1}}{2} ∫ α (δ^{2} (\frac{∂φ_{v}}{∂x_{1}} )^{2} + (\frac{∂φ_{v}}{∂x_{2}} )^{2} + (\frac{∂φ_{v}}{∂x_{3}} )^{2}) dy + δ^{−1} ∫ V (φ) dy

Since φ minimizes the energy

\frac{d}{dδ} E(φ_{δ})|_{δ=1} = 0

and since

\frac{d}{dδ} E(φ_{δ}) = −\frac{δ^{−2}}{2} ∫ α (δ^{2} (\frac{∂φ_{v}}{∂x_{1}} )^{2} + (\frac{∂φ_{v}}{∂x_{2}} )^{2} + (\frac{∂φ_{v}}{∂x_{3}} )^{2}) dy

+ ∫ α′ (δ^{2} (\frac{∂φ_{v}}{∂x_{1}} )^{2} + (\frac{∂φ_{v}}{∂x_{2}} )^{2} + (\frac{∂φ_{v}}{∂x_{3}} )^{2}) (\frac{∂φ_{v}}{∂x_{1}} )^{2} dy

−δ^{−2} ∫ V (φ) dy

we get

−\frac{1}{2} ∫ α(|∇φ|^{2}) dy + ∫ α′(|∇φ|^{2}) (\frac{∂φ_{v}}{∂x_{1}} )^{2} dy − ∫ V (φ) dy = 0

By the ?? the 6.4 and the above lemma the following result follows straightforward.

Theorem 6.3. If φ is a solution of 3.5 with c = 1; then

m_{ii}(φ) = E(φ)
m_{ij}(φ) = 0 if i ≠ j

Proof. Since the matrix m_{ij}(φ) is symmetric, we can put ourselves in a reference frame in which it is diagonal. In this referenceframe, the conclusions of lemma 6.2 holds; thus

m_{ij}(φ) = E(φ)δ_{ij}

where δ_{ij} is the Kroneker symbol. This implies that m_{ij}(φ) = E(φ)δ_{ij} in any reference frame.

By tha above lemma we get the two unexpected facts:

(i) since the mass matrix is a scalar times the identity, the mass is in fact a scalar;

(ii) the mass is equal to the energy.

Notice that

(p(φ_{v}), E(φ_{v})) = (∫ T_{0,1} dx, ∫ T_{0,2} dx, ∫ T_{0,3} dx, ∫ T_{0,0} dx)

is a 4-vector since the equation 3.3 is Lorentz invariant; so, its Lorentzian norm

|p(φ_{v})|^{2} − E(φ_{v})^{2}  (6.5)

is independent of v. Using this fact we get:

Theorem 6.4. If φ is a solution of 3.5 (with c = 1) which satisfies the ?? and φ_{v} is the function ??, then

E(φ_{v}) = γE(φ)  (6.6)

Proof. Since the 6.5 is independent of v, |p(φ_{v})|^{2} − E(φ_{v})^{2} = −E(φ)^{2}; then by ?? and Th. 6.3

E(φ_{v})^{2} = E(φ)^{2} + |p(φ_{v})|^{2} = E(φ)^{2} + m(φ)^{2}γ^{2}v^{2}

= E(φ)^{2} + E(φ)^{2}γ^{2}v^{2} = (1 + \frac{v^{2}}{1 − v^{2}} ) E(φ)^{2} = γ^{2}E(φ)^{2}.

Comparing the 6.6 with the ?? and using the Th. 6.3 we deduce the equality between mass and energy:

E(φ_{v}) = m(φ_{v})  (6.7)

If c ≠ 1, the above formula gives the celebrated Einstein equation

E(φ_{v}) = m(φ_{v})c^{2}

It is interesting to note that this formula is a consequence not only of the Lorentz invariance of the Langrangian L but also of the fact that φ_{v} is a topological soliton (and hence lemma 6.2 holds).

7. Some remarks on the dynamics

We now make some remarks on the solutions of the Cauchy problem relative to the equation 3.3 assuming that they are sufficiently regular. Of course the existence and the regularity of these solutions depends on α and V and it has not yet been investigated.

Now let (u_{0}(x), v_{0}(x)) = (u(x, 0), u_{t}(x, 0)) be the initial conditions. By the previous sections the following quantities are well defined:

• the global energy

• the global momentum

• the global angular momentum defined by

∫ M_{0kl} dx, (k < l)  (7.1)

where M_{ikl} is the angular momentum tensor (cf. e. g. [9] pg. 185)

• the global quantum numbers.

All these quantities are integrals of the motion. The fact that also the quantum numbers are integrals of the motion is a consequence of the fact that the function

t ↦ u( · , t)^{♯}

is constant, i.e. u( · , t) does not change homotopy class in Λ.

It is interesting to observe that all these quantities can be defined for single particles as defined in section 5. In fact, for every t ∈ R, the support K(u( · , t)) of u( · , t), its connected components C(t) and the particles u( · , t)_{|C(t)} are well defined. This allows to define mass, momentum, and angular momentum of a particle, by restricting the integrals 6.1, 6.3 and 7.1 to C(t). The quantum numbers of the particle are well defined by the discussion of section 5.

If a particle is far away from the other particles and it minimizes the energy in the homotopy class determined by its quantum numbers, then, essentially, it is a soliton, and its energy is equal to the mass according to the 6.7.

If two solitons ”collide” and generate other solitons, the usual laws of the Relativistic Mechanics (cf. [11] pg.47-59) are satisfied and moreover the quantum numbers are preserved.

Acknowledgments. It is a pleasure to thank Leone Fronzoni for useful discussions on these topics.

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