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A New Understanding ofParticles by G-Flow Interpretation of Differential EquationLinfan MAOChinese Academy of Mathematics and System Science, Beijing 100190, P.R.ChinaE-mail: maolinfan@163.com

Applying mathematics to understanding of particles classically enables one with an assumption, i.e., if the variables t and x1 , x2 , x3 hold with a system of dynamical equations

Fi t, x1 , x2 , x3 , ut , ux1 , , ux1 x2 , = 0, 1 i m,(DEq4m )they are a point (t, x1 , x2 , x3 ) in R4 . However, if we put off this assumption, how canwe interpret the solution space of equations? And are these resultants important forunderstanding the world? Recently, the author extended Banach and Hilbert spaces ona topological graph to introduce G-flows and showed that all such flows on a topologicalgraph G also form a Banach or Hilbert space, which enables one to find the multiversesolution of equations (DEq4m ) on G. Applying this result, This paper discusses the Gflow solutions on Schrodinger equation, Klein-Gordon equation and Dirac equation,i.e., the field equations of particles, bosons or fermions, answers previous questions byYES, and establishes the many world interpretation of quantum mechanics of H.Everettby purely mathematics in logic, i.e., mathematical combinatorics.

Gordon equation

Linfan Mao. A New Understanding of Particles by G-Flow Interpretation

! mc 21 22(x,t)+(x, t) = 0c2 t2~

1. IntroductionAll matters consist of two classes particles, i.e., bosons withinteger spin n, fermions with spin n/2, n 1(mod2), andby a widely held view, the elementary particles consists ofquarks and leptons with interaction quanta including photonsand other particles of mediated interactions ([16]), which constitute hadrons, i.e., mesons, baryons and their antiparticles.Thus, a hadron has an internal structure, which implies that allhadrons are not elementary but leptons are, viewed as pointparticles in elementary physics. Furthermore, there are alsounmatter which is neither matter nor antimatter, but something in between ([19-21]). For example, an atom of unmatteris formed either by electrons, protons, and antineutrons, or byantielectrons, antiprotons, and neutrons.Usually, a particle is characterized by solutions of differential equation established on its wave function (t, x). Innon-relativistic quantum mechanics, the wave function (t, x)of a particle of mass m obeys the Schrodinger equation~2i~= 2 + U,t2m

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

mc i (t, x) = 0~

(1.3)

in relativistic forms, where,

= 0 , 1 , 2 , 3 ,

=

!1 ,,,,c t x1 x2 x3

c is the speed of light and0 =

I220

0I22

!

3 =

1 00 1

!

i0

0i

, i =

with the usual Pauli matrices!0 11 =, 2 =1 0

(1.1)

where, ~ = 6.582 1022 MeV s is the Planck constant, U isthe potential energy of the particle in applied field and! 222=, ,and 2 = 2 + 2 + 2 .x y zxyz

(1.2)

0i

i0

!

!

,

.

It is well known that the behavior of a particle is on superposition, i.e., in two or more possible states of being. Buthow to interpret this phenomenon in according with equation(1.1)-(1.3) ? The many worlds interpretation on wave funcConsequently, a free boson (t, x) hold with the Klein- tion of equation (1.1) by H.Everett [2] in 1957 answered the

Linfan Mao. A New Understanding of Particles by G-Flow Interpretation

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question in machinery, i.e., viewed different worlds in different quantum mechanics and the superposition of a particlebe liked those separate arms of a branching universe ([15],also see [1]). In fact, H.Everetts interpretation claimed thatthe 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.

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1

mS

i=1

{c}

{b}

3

{c, d}Fig.2

i=1

Ri on e, denoted by e; eR.

Furthermore, things are inherently related, not isolated inthe 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 spaceE if there is a 1 1 continuous mapping f : G E withf (p) , f (q) if p , q for p, q G, i.e., edges only intersectat end vertices in E . For example, a planar graph such asthose shown in Fig.1.v1

v2u1

u2

u4

u3

e inGenerally, a particle should be characterized by e; Rtheory. However, we can only verify it by some of systems(1 ; R1 ), (2 ; R2 ), , (m ; Rm ) for the limitationof human

e . Clearly, thebeings because he is also a system in e; Runderlying graph in H.Everetts interpretation on wave function is in fact a binary tree and there are many such traces inthe 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 Neeman on hadrons ([14]), such as thoseshown in Fig.3, where,n qi {u, d, c,o s, t, b} denotes a quarkfor i = 1, 2, 3 and q2 u, d, c, s, t, b , an antiquark. Thus, theunderlying graphs G of a meson, a baryon are respectivelyK 2 and K 3 with actions. In fact, a free quark was not foundin experiments until today. So it is only a machinery modelon hadrons. Even so, it characterizes well the known behaviorof particles.

v3

v4

2

{b, c}

4Definition 1.1([6],[18]) Let (1 ; R1 ), (2 ; R2 ), , (m ; Rm )be m mathematical or physical systems, different two by two.mSe=A Smarandache multisystem e is a union i with rules R

{a, b}

Fig.1

e beDefinition 1.2([6]) For any integer m 1, let e; Ra Smarandache multisystem consisting of mathematical systems (1 ; R1 ), (2h; R2 ),i , (m ; Rm ). An inherited topologie is defined bycal structures G L e; eR on e; R

hiV GL e; eR = {v1 , v2 , , vm }, hie = {(vi , v j )|i T j , , 1 i , j m}E GL e; Rwith a labeling L : vi L(vi ) = i and L : (vi , v j ) TTL(vi , v j ) = i j , where i j denotes the intersectionof spaces, or action between systems i with j for integers1 i , j m.

q2

.. q1........................................... q3

q1 ................................. q2

Baryon

Meson

Fig.3

It should be noted that the geometry on Definition 1.11.2can be also used to characterize particles by combinatorialfields ([7]), and there is a priori assumption for discussion inphysics, namely, the dynamical equation of a subparticle ofa particle is the same of that particle. For example, the dynamical equation of quark is nothing else but the Dirac equa4tion (1.3), a characterizing on quark from the macroscopicSFor example, let e = i with 1 = {a, b, c}, 2 = {a, b}, to the microscopic, the quantum level in physics. However,i=13 = {b, c, d}, 4 = {c, d} and Ri = . Calculation shows that the equation (1.3) can not provides such a solution on the beTTTT1 2 = {a, b}, 1 3 = {b, c}, 1 4 = {c}, 2h 3i haviors of 3 quarks. We can only interpret it similar to thatTTe of H.Everett, i.e., there are 3 parallel equations (1.3) in dis; R= {b}, 2 4 = , 3 4 = {c, d}. Such a graph G L ecussion, a seemly rational interpretation in physics, but notis shown in Fig.2.2

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perfect for mathematics. Why this happens is because the interpretation of solution of equation. Usually, we identify aparticle to the solution of its equation, i.e., if the variables tand x1 , x2 , x3 hold with a system of dynamical equations

Fi t, x1 , x2 , x3 , ut , u x1 , , u x1 x2 , = 0,1 i m,(DEq4m )

Volume 10 (2014)

flows in or out of the volume. According to Definitions 1.1and 1.2, a matter in the nature is nothing else buth a Smaranie , or a topological graph G L ee emdache system e; R; Rbedded in R3 , hold with conservation lawsXXF(v)k =F(v)+lk

l

hithe particle in R R3 is a point (t, x1 , x2 , x3 ), and if more on v V G L ee , where, F(v) , k 1 and F(v)+ , l 1; Rklthan one points (t, x1 , x2 , x3 ) hold with (ES q4m ), the particle denote respectively the input or output amounts on a particleis nothing else but consisting of all such points. However, or a volume v.the solutions of equations (1.1)-(1.3) are all definite on timet. Is this interpretation can be used for particles in all times? 2.2 G-Flow SpacesCertainly not because a particle can be always decomposedinto elementary particles, and it is a little ambiguous which Classical operation systems can be easily extended on a graphis a point, the particle itself or its one of elementary particles G constraint on conditions for characterizing the unanimoussometimes.behaviors of groups in the nature, particularly, go along withThis speculation naturally leads to a question on mathe- the physics. For this objective, let G be an oriented graphmatics, i.e., what is the right interpretation on the solution of with vertex set V(G) and arc set X(G) embedded in R3 and letdifferential equation accompanying with particles? Recently, (A ; ) be an operation system in classical mathematics, i.e.,Lthe author extended Banach spaces on topological graphs G for a, b A , ab A . Denoted by G A all of those labeledwith operator actions in [13], and shown all of these extenLsions are also Banach space, particularly, the Hilbert space graphs G with labeling L : X G A . Then, we canAwith unique correspondence in elements on linear continu- extend operation on elements in G by a ruler following:ous functionals, which enables one to solve linear functionalLLL12 L1 L2 L1 L2R: For G , G G A , define G G = G,equations in such extended space, particularly, solve differential equations on a topological graph, i.e., find multiverse so- where L1 L2 : e L1 (e) L2 (e) for e X G .lutions for equations. This scheme also enables us to interpretFor example, such an extension on graph C 4 is shown inthe superposition of particles in accordance with mathematicsFig.4, where, a3 =a1 a2 , b3 =b1 b2 , c3 =c1 c2 , d3 =d1 d2 .in logic.The main purpose of this paper is to present an interprev1 av2v1 av2v1 av2312tation on superposition of particles by G-flow solutions ofb1d2b2d3b3equations (1.1)-(1.3) in accordance with mathematics. Cer- d1tainly, the geometry on non-solvable differential equationsv 4 c1 v 3v 4 c2 v 3v 4 c3 v 3discussed in [9]-[12] brings us another general way for holding behaviors of particles in mathematics. For terminologiesFig.4and notations not mentioned here, we follow references [16]LLL 1 2 Lfor elementary particles, [6] for geometry and topology, and Clearly, G G G A by definition, i.e., G A is also an op[17]-[18] for Smarandache multi-spaces, and all equations are eration system under ruler R, and it is commutative if (A , )assumed to be solvable in this paper.is commutative,LFurthermore, if (A , ) is an algebraic group, G A is alsoan algebraic group because2. Extended Banach G-Flow Space

L

L 1 L2 L3 L1 2 L3 L1 L2(1) G G G = G G G for G , G ,2.1 Conservation Laws L3 AG G becauseA conservation law, such as those on energy, mass, momen(L1 (e) L2 (e)) L3 (e) = L1 (e) (L2 (e) L3 (e))tum, angular momentum and electric charge states that a particular measurable property of an isolated physical system (L1 L2 )L3 L1 (L2 L3 )=G.does not change as the system evolves over time, or simply, for e X G , i.e., GL1ALconstant of being. Usually, a local conservation law is ex(2) there is anidentify Gin G A , where L1A : e pressed mathematically as a continuity equation, which states 1 for e X G ;Athat the amount of conserved quantity at a point or within a L1L Lvolume can only change by the amount of the quantity which(3) there is an uniquely element G for G G A .

-

6

Linfan Mao. A New Understanding of Particles by G-Flow Interpretation

?

6 ?

6 ?

3

Volume 10 (2014)

PROGRESS IN PHYSICS

Issue 1 (January)

L1 L2 L1 L2 However, for characterizing the unanimous behaviors of(3)

G + G

G

+

G

;groups in the nature, the most useful one is the extension of L L L LP vvvector space (V ; +, ) over field F by defining the operations(4) G , G =hL(u),L(u)i0andG ,G!V+ and on elements in G such as those shown in Fig.5 on(u,v)X Ggraph C 4 , where a, b, c, d, ai , bi , ci , di V for i = 1, 2, 3,L= 0 if and only if G = O;x3 =x1 +x2 for x=a, b, c or d and F . L L

1 L2 2 L1 L1 L2 V(5) G , G= G ,Gfor G , G G ;v1 a 1 v2v1 a 2 v2v1 a 3 v2L L1 L2 V(6) For G , G , G G and , F ,d1b1d2b2d3b3 L

1 L2 LG+G,Gccc123v3v3v3v4v4v4 L

L

1L 2Lv1 a v2v1 a v2= G ,G + G ,G .bbddThe following result is obtained by showing that CauchyVv4 c v3v4 c v3sequences in G is converges hold with conservation laws.

-

6 ??6

6 ?

6 ?6 ?

Fig.5A G-flow on G is such an extension hold with L (u, v) =L (v, u) and conservation lawsXL (v, u) = 0uNG (v)

for v V G , where 0 is the zero-vector in V . Thus, a GLflow is a subfamily of G V limited by conservation laws. Forexample, if G = C 4 , there must be a=b=c=d, ai =bi =ci =difor i = 1, 2, 3 in Fig.5.Clearly, all conservation G-flows on G also form a vectorspace over F under operations + and with zero vector O = L0G , where L0 : e 0 for e X G . Such an extendedVvector space on G is denoted by G .Furthermore, if (V ; +, ) is a Banach or Hilbert space withinner product h, i, we can also introduce the norm and innerVproduct on G by

X L

kL(u, v)k

G

=!(u,v)X Gor

L

1 L2G ,G=

X

(u,v)X

!G

hL1 (u, v), L2 (u, v)i

VTheorem 2.1([13]) For any topological graph G, G is aVBanach space, and furthermore, if V is a Hilbert space, Gis a Hilbert space also.According to Theorem 2.1, the operators action on BaVnach or Hilbert space (V ; +, ) can be extended on G , forexample, the linear operator following.VVDefinition 2.2 An operator T : G G is linear if L

L L 1 L2 1 2T G + G = T G + T G L1 L2 Vfor G , G G and , F , and is continuous at a L0G-flow G if there always exist a number () for > 0such that

L

L 0

T GTG