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A new understanding of particles by [??]-flow interpretation of differential equation.

1 Introduction

Matter consists of bosons with integer spin n and fermions with half-integer spin n/2, n [equivalent to] 1 (mod 2). The elementary particles consist of leptons and hadrons, i.e. mesons, baryons and their antiparticles, which are composed of quarks [16]. Thus, a hadron has an internal structure, which implies that all hadrons are not elementary but leptons are, viewed as point particles in elementary physics. Furthermore, there is also unmatter which is neither matter nor antimatter, but something in between [19-21]. For example, an atom of unmatter is formed either by electrons, protons, and antineutrons, or by antielectrons, antiprotons, and neutrons.

Usually, a particle is characterized by solutions of differential equation established on its wave function [psi](t, x). In non-relativistic quantum mechanics, the wave function [psi](t, x) of a particle of mass m obeys the Schrodinger equation

i[??] [partial derivative][psi]/[partial derivative]t = [[??].sup.2]/2m [[nabla].sup.2] [psi]+ U, (1.1)

where, [??] = 6.582 x [10.sup.-22] MeVs is the Planck constant, U is the potential energy of the particle in applied field and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, a free boson [psi](t, x) hold with the Klein-Gordon equation

(1/[c.sub.2] [[partial derivative].sup.2]/[partial derivative][t.sup.2] - [[nabla].sup.2]) [psi](x, t) + [(mc/[??]).sup.2] [psi](x, t) = 0 (1.2)

and a free fermion [psi](t, x) satisfies the Dirac equation

(i[[gamma].sup.[mu]] [[partial derivative].sub.[mu]])[psi](t, x) = 0 (1.3)

in relativistic forms, where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

c is the speed of light and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with the usual Pauli matrices

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is well known that the behavior of a particle is on superposition, i.e., in two or more possible states of being. But how to interpret this phenomenon in accordance with (1.1)-(1.3)? The many worlds interpretation on wave function of (1.1) by H.Everett [2] in 1957 answered the question in machinery, i.e., viewed different worlds in different quantum mechanics and the superposition of a particle be liked those separate arms of a branching universe ([15], also see [1]). In fact, H. Everett's interpretation claimed that the state space of particle is a multiverse, or parallel universe ([23, 24]), an application of philosophical law that the integral always consists of its parts, or formally, the following.

Definition 1.1([6],[18]) Let ([[summation].sub.1]; [R.sub.1]), ([[summation].sub.2]; [R.sub.2]), ... ([[summation].sub.m]; [R.sub.m]) be m mathematical or physical systems, different two by two. A Smarandache multisystem [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Furthermore, things are inherently related, not isolated in the world. Thus, every particle in nature is a union of elementary particles underlying a graph embedded in space, where, a graph G is said to be embeddable into a topological space E if there is a 1-1 continuous mapping f : G [right arrow] E with f (p) [not equal to] f(q) if p [not equal to] q for [for all]p, q [member of] G, i.e., edges only intersect at end vertices in E. For example, a planar graph such as those shown in Fig. 1.

[FIGURE 1 OMITTED]

Definition 1.2([6]) For any integer m [greater than or equal to] 1, let ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) be a Smarandache multisystem consisting of mathematical systems ([[summation]; [R.sub.1]), ([[summation].sub.2]; [R.sub.2]), ..., ([[summation].sub.m]; [R.sub.m]). An inherited topological structures [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the intersection of spaces, or action between systems [[summation].sub.i] with [[summation].sub.j] for integers 1 [less than or equal to] i [not equal to] j [less than or equal to] m.

For example, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is shown in Fig. 2.

[FIGURE 2 OMITTED]

Generally, a particle should be characterized by ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) in theory. However, we can only verify it by some of systems ([[summation].sub.1]; [R.sub.1]), ([[summation].sub.2]; [R.sub.2]), ..., ([[summation].sub.m]; [R.sub.m]) for the limitation of human beings because he is also a system in ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Clearly, the underlying graph in H. Everett's interpretation on wave function is in fact a binary tree and there are many such traces in the developing of physics. For example, a baryon is predominantly formed from three quarks, and a meson is mainly composed of a quark and an antiquark in the models of Sakata, or Gell-Mann and Ne'eman on hadrons ([14]), such as those shown in Fig. 3, where, [q.sub.i] [member of] {u, d, c, s, t, b} denotes a quark for i = 1, 2, 3 and [[bar.q].sub.2] [member of] {[bar.u], [bar.d], [bar.c], [bar.s], [bar.t], [bar.b]}, an antiquark. Thus, the underlying graphs [??] of a meson, a baryon are respectively [[??].sub.2] and [[??].sub.3] with actions. In fact, a free quark was not found in experiments until today. So it is only a machinery model on hadrons. Even so, it characterizes well the known behavior of particles.

[FIGURE 3 OMITTED]

It should be noted that the geometry on Definition 1.1-1.2 can be also used to characterize particles by combinatorial fields ([7]), and there is a priori assumption for discussion in physics, namely, the dynamical equation of a subparticle of a particle is the same of that particle. For example, the dynamical equation of quark is nothing else but the Dirac equation (1.3), a characterizing on quark from the macroscopic to the microscopic, the quantum level in physics. However, (1.3) cannot provide such a solution on the behaviors of 3 quarks. We can only interpret it similar to that of H. Everett, i.e., there are 3 parallel equations (1.3) in discussion, a seemly rational interpretation in physics, but not perfect for mathematics. Why this happens is because the interpretation of solution of equation. Usually, we identify a particle to the solution of its equation, i.e., if the variables t and [x.sub.1], [x.sub.2], [x.sub.3] hold with a system of dynamical equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.4)

the particle in R x [R.sup.3] is a point (t, [x.sub.1], [x.sub.2], [x.sub.3]), and if more than one points (t, [x.sub.1], [x.sub.2], [x.sub.3]) hold with (1.4), the particle is nothing else but consisting of all such points. However, the solutions of (1.1)-(1.3) are all definite on time t. Can this interpretation be used for particles in all times? Certainly not because a particle can be always decomposed into elementary particles, and it is a little ambiguous which is a point, the particle itself or its one of elementary particles sometimes.

This speculation naturally leads to a question on mathematics, i.e., what is the right interpretation on the solution of differential equation accompanying with particles? Recently, the author extended Banach spaces on topological graphs [??] with operator actions in [13], and shown all of these extensions are also Banach space, particularly, the Hilbert space with unique correspondence in elements on linear continuous functionals, which enables one to solve linear functional equations in such extended space, particularly, solve differential equations on a topological graph, i.e., find multiverse solutions for equations. This scheme also enables us to interpret the superposition of particles in accordance with mathematics in logic.

The main purpose of this paper is to present an interpretation on superposition of particles by G-flow solutions of (1.1)-(1.3) in accordance with mathematics. Certainly, the geometry on non-solvable differential equations discussed in [9]-[12] brings us another general way for holding behaviors of particles in mathematics. For terminologies and notations not mentioned here, we follow references [16] for elementary particles, [6] for geometry and topology, and [17]-[18] for Smarandache multi-spaces, and all equations are assumed to be solvable in this paper.

2 Extended Banach [??]-flow space

2.1 Conservation laws

A conservation law, such as those on energy, mass, momentum, angular momentum and electric charge states that a particular measurable property of an isolated physical system does not change as the system evolves over time, or simply, constant of being. Usually, a local conservation law is expressed mathematically as a continuity equation, which states that the amount of conserved quantity at a point or within a volume can only change by the amount of the quantity which flows in or out of the volume. According to Definitions 1.1 and 1.2, a matter in the nature is nothing else but a Smarandache system ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) embedded in [R.sup.3], hold with conservation laws

[summation over (k)] [F(v).sup.-.sub.k] = [summation over (l)] [F(v).sup.+.sub.l]

on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where, [F(v).sup.-.sub.k], k [greater than or equal to] 1 and [F(v).sup.+.sub.l], l [greater than or equal to] 1 denote respectively the input or output amounts on a particle or a volume v.

2.2 [??]-flow spaces

Classical operation systems can be easily extended on a graph [??] constraint on conditions for characterizing the unanimous behaviors of groups in the nature, particularly, go along with the physics. For this objective, let [??] be an oriented graph with vertex set V(G) and arc set X(G) embedded in [R.sup.3] and let (A; o) be an operation system in classical mathematics, i.e., for [for all] a, b [member of] A, a o b [member of] A. Denoted by [[??].sup.L.sub.A] all of those labeled graphs [[??].sup.L] with labeling L : X ([??]) [right arrow] A. Then, we can extend operation o on elements in [[??].sup.A] by a ruler following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For example, such an extension on graph [[??].sub.4] is shown in Fig.4, where, [a.sub.3]=o[a.sub.2], [b.sub.3] =[b.sub.1] o[b.sub.2], [c.sub.3]=[c.sub.1]o[c.sub.2], [d.sub.3] =[d.sub.1]o[d.sub.2].

[FIGURE 4 OMITTED]

Clearly, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by definition, i.e., [[??].sup.L.sub.A] is also an operation system under ruler R, and it is commutative if (A, o) is commutative,

Furthermore, if (A, o) is an algebraic group, [[??].sup.L.sub.A] is also an algebraic group because

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(2) there is an identify [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(3) there is an uniquely element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

However, for characterizing the unanimous behaviors of groups in the nature, the most useful one is the extension of vector space (V; +, *) over field F by defining the operations + and * on elements in [[??].sup.V] such as those shown in Fig. 5 on graph [[??].sub.4], where a, b, c, d, [a.sub.i], [b.sub.i], [c.sub.i], [d.sub.i] [member of] V for i = 1,2,3, [x.sub.3]=[x.sub.1] + [x.sub.2] for x=a, b, c or d and [alpha] [member of] F.

[FIGURE 5 OMITTED]

A [??]-flow on [??] is such an extension hold with L(u, v) = -L(v, u) and conservation laws

[summation over (u[member of][N.sub.G(v)])] L(v, u) = 0

for [for all]v [member of] V ([??]), where 0 is the zero-vector in V. Thus, a [??]- flow is a subfamily of [[??].sup.L.sub.V] limited by conservation laws. For example, if [??] = [[??].sub.4], there must be a=b=c=d, [a.sub.i]=[b.sub.i]=[c.sub.i]=[d.sub.i] for i = 1,2,3 in Fig. 5.

Clearly, all conservation [??]-flows on [??] also form a vector space over F under operations + and * with zero vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Such an extended vector space on [??] is denoted by [[??].sup.V].

Furthermore, if (V; +, *) is a Banach or Hilbert space with inner product <*, *>, we can also introduce the norm and inner product on [[??].sup.V] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [parallel] L(u, v) [parallel] is the norm of L(u, v) in V. Then it can be verified that

(1) [parallel] [[??].sup.L] [parallel] [greater than or equal to] 0 and [parallel] [[??].sup.L] [parallel] = 0 if and only if [[??].sup.L] = O;

(2) [parallel] [[??].sup.[xi]L] [parallel] = [xi] [parallel] [[??].sup.L] [parallel] for any scalar [xi];

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(4) <[[??].sup.L], [[??].sup.L]> = [summation over ((u, v))[member of]X([??])] <L([u.sup.v]), L([u.sup.v])> [greater than or equal to] 0 and <[[??].sup.L], [[??].sup.L]> = 0 if and only if [[??].sup.L] = O;

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The following result is obtained by showing that Cauchy sequences in [[??].sup.V] is converges hold with conservation laws.

Theorem 2.1([13]) For any topological graph [??], [[??].sup.V] is a Banach space, andfurthermore, if V is a Hilbert space, [[??].sup.V] is a Hilbert space also.

According to Theorem 2.1, the operators action on Banach or Hilbert space (V; +, *) can be extended on [[??].sup.V], for example, the linear operator following.

Definition 2.2 An operator T : [[??].sup.V] [right arrow] [[??].sup.V] is linear if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if there always exist a number [delta]([epsilon]) for [for all][epsilon] > 0 such that

if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The following interesting result generalizes the result of Frechet and Riesz on linear continuous functionals, which opens us mind for applying [??]-flows to hold on the nature.

Theorem 2.3([13]) Let T : [[??].sup.V] [right arrow] C be a linear continuous functional. Then there is a unique [??].sup.L] [member of] [[??].sup.V] such that

T([[??].sup.L]) = <[[??].sup.L], [[??].sup.L]>

for [for all][[??].sup.L] [member of] [[??].sup.V].

Particularly, if all flows L(u, v) on arcs (u, v) of [??] are state function, we extend the differential operator on [??]-flows. In fact, a differential operator [partial derivative]/[partial derivative]t or [partial derivative]/[partial derivative] [x.sub.i] : [[??].sup.V] [right arrow] [[??].sup.V] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for integers 1 [less than or equal to] i [less than or equal to] 3. Then, for V[mu], [lambda] [member of] F,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for integers 1 [less than or equal to] i [less than or equal to] 3. Thus, operators [partial derivative]/[partial derivative]t and -[partial derivative]/[partial derivative][x.sub.i], 1 [less than or equal to] i [less than or equal to] 3 are all linear on [[??].sup.V].

[FIGURE 6 OMITTED]

Similarly, we introduce integral operator [integral] : [[??].sup.V] [right arrow] [[??].sup.V] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [L.sub.c] is such a labeling that [L.sub.c](u, v) is constant for [for all](u, v) [member of] x([??]).

3 Particle equations in [??]-flow space

We are easily find particle equations with nonrelativistic or relativistic mechanics in [[??].sup.V]. Notice that

ih [partial derivative][psi]/[partial derivative]t = E[psi], -ih[nabla][psi] = [[??].sup.2] [psi]

and

E = 1/2m [[??].sup.2] + U,

in classical mechanics, where [psi] is the state function, E, [psi], U are respectively the energy, the momentum, the potential energy and m the mass of the particle. Whence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [L.sub.[psi]] : e [right arrow] state function and [L.sub.U] : e [right arrow] potential energy on e [member of] X ([??]). According to the conservation law of energy, there must be [[??].sup.U] [member of] [[??].sup.V]. We get the Schrodinger equation in [[??].sup.V] following.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly, by the relativistic energy-momentum relation

[E.sup.2] = [c.sup.2] [[??].sup.2] + [m.sup.2] [c.sup.4]

for bosons and

E = c [[alpha].sub.k] [[??].sub.k] + [[alpha].sub.0][mc.sup.2]

for fermions, we get the Klein-Gordon equation and Dirac equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.3)

of particles in [[??].sup.V] respectively. Particularly, let [??] be such a topological graph with one vertex but only with one arc. Then, (3.1)-(3.3) are nothing else but (1.1)-(1.3) respectively. However, (3.1)-(3.3) conclude that we can find G-flow solutions on (1.1)-(1.3), which enables us to interpret mathematically the superposition of particles by multiverse.

4 [??]-flows on particle equations

Formally, we can establish equations in [[??].sup.V] by equations in Banach space V such as (3.1)-(3.3). However, the important thing is not just on such establishing but finding [??]-flows on equations in V and then interpret the superposition of particles by [??]-flows.

4.1 [??]-flow solutions on equation

Theorem 2.3 concludes that there are [??]-flow solutions for a linear equations in [[??].sup.V] for Hilbert space V over field F, including algebraic equations, linear differential or integral equations without considering the topological structure. For example, let ax = b. We are easily getting its [??]-flow solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if we view an element b [member of] V as b = [[??].sup.L], where L(u, v) = b for [for all](u, v) [member of] X ([??]) and 0 [not equal to] a [member of] F, such as those shown in Fig. 7 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and a = 3, b = 5.

[FIGURE 7 OMITTED]

Generally, we know the following result:

Theorem 4.1([13]) A linear system of equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [a.sub.ij], [b.sub.j] [member of] F for integers 1 [less than or equal to] i [less than or equal to] n, 1 [less than or equal to] j [less than or equal to] m holding with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

has [??]-flow solutions on infinitely many topological graphs [??], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can also get [??]-flow solutions for linear partial differential equations ([14]). For example, the Cauchy problems on differential equations

[partial derivative]X/[partial derivative]t = [c.sup.2] [n.summation over (i=1)] [[partial derivative].sup.2]X/ [partial derivative][x.sup.2.sub.i]

with initial value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also solvable in [[??].sup.V] if L'(u, v) is continuous and bou- nded in [R.sup.n] for [for all](u, v) [member of] X ([??]) and [for all][[??].sup.L]' [member of] [G.sup.V]. In fact, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is such a solution.

Generally, if [??] can be decomposed into circuits [??], the next result concludes that we can always find [??]-flow solutions on equations, no matter what the equation looks like, linear or non-linear ([13]).

Theorem 4.2 If the topological graph [??] is strong-connected with circuit decomposition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that L(u, v) = [L.sub.i] (x) for [for all](u, v) [member of] X ([[??].sub.i]), 1 [less than or equal to] i [less than or equal to] l and the Cauchy problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is solvable in a Hilbert space V on domain [DELTA] [subset] [R.sup.n] for integers 1 [less than or equal to] i [less than or equal to] l, then the Cauchy problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that L(u, v) = [L.sub.i](x) for [for all](u, v) [member of] X ([[??].sub.i]) is solvable for X [member of] [[??].sup.V].

In fact, such a solution is constructed by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [L.sub.u(x)] (u, v) = u(x) for (u, v) [member of] X ([??]) by applying the input and the output at vertex v all being u(x) on [??], which implies that all flows at vertex v [member of] V ([??]) is conserved.

4.2 [??]-flows on particle equation

The existence of [??]-flow solutions on particle equations (1.1)-(1.3) is clearly concluded by Theorem 4.2, also implied by (3.1)-(3.3) for any [??]. However, the superposition of a particle P shows that there are N [greater than or equal to] 2 states of being associated with a particle P. Considering this fact, a convenient [??]-flow model for elementary particle fermions, the lepton or quark P is by a bouquet [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and an antiparticle [bar.P] of P presented by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with all inverse states on its loops, such as those shown in Fig. 8.

[FIGURE 8 OMITTED]

An elementary unparticle is an intermediate form between an elementary particle and its antiparticle, which can be presented by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for a subset C [subset] X ([[??].sub.N]), such as those shown in Fig. 9, where [N.sub.1], [N.sub.2] [greater than or equal to] 1 are integers. Thus, an elementary particle with its antiparticles maybe annihilate or appears in pair at a time, which consists in an elementary unparticle by combinations of these state functions with their inverses.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

For those of mediate interaction particle quanta, i.e., boson, which reflects interaction between particles. Thus, they are conveniently presented by dipole [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] but with dotted lines, such as those in Fig. 10, in which the vertex P, P' denotes particles, and arcs with state functions [[psi].sub.1], [[psi].sub.2], ..., [[psi].sub.N] are the N states of P. Notice that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] both are a union of N circuits.

According to Theorem 4.2, we consequently get the following conclusion.

Theorem 4.3 For an integer N [greater than or equal to] 1, there are indeed [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-flow solution on Klein-Gordon equation (1.2), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-flow solution on Dirac equation (1.3).

Generally, this model enables us to know that the [??]-flow constituents of a particle also.

[FIGURE 11 OMITTED]

Thus, if a particle [??] is consisted of l elementary particles [P.sub.1], [P.sub.1], ..., [P.sub.l] underlying a graph [??] [[??]], its [??]-flow is obtained by replace each vertex v by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For example, the model of Sakata, or Gell-Mann and Ne'eman on hadrons claims that the meson and the baryon are respectively the dipole [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] shown in Fig. 12,

[FIGURE 12 OMITTED]

Theorem 4.4 If [??] is a particle consisted of elementary particles [P.sub.1], [P.sub.1], ..., [P.sub.l] for an integer l [greater than or equal to] 1, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a [??]-flow solution on the Schrodinger equation (1.1) whenever AG is finite or infinite.

Proof If [[lambda].sub.G] is finite, the conclusion follows Theorem 4.2 immediately. We only consider the case of [[lambda].sub.G] [right arrow] [infinity]. In fact, if [[lambda].sub.G] [right arrow] [infinity], calculation shows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for bouquets and dipoles.

5 [??]-flow interpretation on particle superposition

The superposition of a particle P is depicted by a Hilbert space V over complex field C with orthogonal basis |1>, |2>, ..., |n>, ... in quantum mechanics. In fact, the linearity of Schrodinger equation concludes that all states of particle P are in such a space. However, an observer can grasp only one state, which promoted H. Everett devised a multiverse consisting of states in splitting process, i.e., the quantum effects spawn countless branches of the universe with different events occurring in each, not influence one another, such as those shown in Fig. 13, and the observer selects by randomness, where the multiverse is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for integers k [greater than or equal to] 1, 1 [less than or equal to] l [less than or equal to] [2.sup.k] but in different positions.

[FIGURE 13 OMITTED]

Why it needs an interpretation on particle superposition in physics lies in that we characterize the behavior of particle by dynamic equation on state function and interpret it to be the solutions, and different quantum state holds with different solution of that equation. However, we can only get one solution by solving the equation with given initial datum once, and hold one state of the particle P, i.e., the solution correspondent only to one position but the particle is in superposition, which brought the H. Everett interpretation on superposition. It is only a biological mechanism by infinite parallel spaces V but loses of conservations on energy or matter in the nature, whose independently runs also overlook the existence of universal connection in things, a philosophical law.

Even so, it can not blot out the ideological contribution of H. Everett to sciences a shred because all of these mentions are produced by the interpretation on mathematical solutions with the reality of things, i.e., scanning on local, not the global. However, if we extend the Hilbert space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in general, i.e., [??]-flow space [[??].sup.V], where [??] is the underling topological graph of P, the situation has been greatly changed because [[??].sup.V] is itself a Hilbert space, and we can identify the [??]-flow on [??] to particle P, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.1)

for a globally understanding the behaviors of particle P whatever [[lambda].sub.G] [right arrow] [infinity] or not by Theorem 4.4. For example, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e., a free particle such as those of electron [e.sup.-], muon [[mu].sup.-], tauon [[tau].sup.-], or their neutrinos [v.sub.e], [v.sub.[mu]], [v.sub.[tau]]. Then the superposition of P is displayed by state functions [psi] on N loops in [[??].sub.N] hold on its each loop with

input [[psi].sub.i] = ouput [[psi].sub.i] at vertex P

for integers 1 [less than or equal to] i [less than or equal to] N. Consequently,

input [summation over (i[member pf]I)] = ouput input [summation over (i[member pf]I)] [[psi].sub.i] at vertex P

for [for all]I [subset] {1,2, ..., N}, the conservation law on vertex P. Furthermore, such a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not only a disguise on P in form but also a really mathematical element in Hilbert space [[??].sup.V], and can be also used to characterize the behavior of particles such as those of the decays or collisions of particles by graph operations. For example, the [beta]-decay n [right arrow] p + [e.sup.-] + [[mu].sup.-.sub.e] is transferred to a decomposition formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

on graph, where, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly, the [beta]- collision [v.sub.e] + p [right arrow] n + [e.sup.+] is transferred to an equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Even through the relation (5.1) is established on the linearity, it is in fact truly for the linear and non-liner cases because the underlying graph of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-flow can be decomposed into bouquets and dipoles, hold with conditions of Theorem 4.2. Thus, even if the dynamical equation of a particle P is non-linear, we can also adopt the presentation (5.1) to characterize the superposition and hold on the global behavior of P. Whence, it is a presentation on superposition of particles, both on linear and non-linear.

6 Further discussions

Usually, a dynamic equation on a particle characterizes its behaviors. But is its solution the same as the particle? Certainly not! Classically, a dynamic equation is established on characters of particles, and different characters result in different equations. Thus the superposition of a particle should be characterized by at least 2 differential equations. However, for a particle P, all these equations are the same one by chance, i.e., one of the Schrodinger equation, Klein-Gordon equation or Dirac equation, which lead to the many world interpretation of H. Everett, i.e., put a same equation or Hilbert space on different place for different solutions in Fig. 12. As it is shown in Theorems 4.1-4.4, we can interpret the solution of (1.1)-(1.3) to be a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-flow, which properly characterizes the superposition behavior of particles by purely mathematics.

The [??]-flow interpretation on differential equation opens a new way for understanding the behavior of nature, particularly on superposition of particles. Generally, the dynamic equations on different characters maybe different, which will brings about contradicts equations, i.e., non-solvable equations. For example, we characterize the behavior of meson or baryon by Dirac equation (1.3). However, we never know the dynamic equation on quark. Although we can say it obeying the Dirac equation but it is not a complete picture on quark. If we find its equation some day, they must be contradicts because it appear in different positions in space for a meson or a baryon at least. As a result, the [??]-solutions on non-solvable differential equations discussed in [9]-[12] are valuable for understanding the reality of the nature with [??]-flow solutions a special one on particles.

As it is well known for scientific community, any science possess the falsifiability but which depends on known scientific knowledge and technical means at that times. Accordingly, it is very difficult to claim a subject or topic with logical consistency is truth or false on the nature sometimes, for instance the multiverse or parallel universes because of the limitation of knowing things in the nature for human beings. In that case, a more appreciated approach is not denied or ignored but tolerant, extends classical sciences and developing those of well known technical means, and then get a better understanding on the nature because the pointless argument would not essentially promote the understanding of nature for human beings ([3,4,22]).

Submitted on April 8, 2015 / Accepted on April 15, 2015

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Linfan Mao

Chinese Academy of Mathematics and System Science, Beijing 100190, P. R. China.

E-mail: maolinfan@163.com
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