Issue 3 (July) PROGRESS IN PHYSICS Volume 11 (2015)A New Understanding of Particles by G-Flow Interpretationof Dierential EquationLinfan MaoChinese Academy of Mathematicsand System Science,Beijing 100190, P. R. China.E-mail:maolinfan@163.comApplying mathematics to the understanding of particles classically with an assumptionthat ifthevariablest andx1, x2, x3holdwithasystemofdynamical equations(1.4),thentheyareapoint(t, x1, x2, x3)inR4. However,ifweputothisassumption, howcan we interpret the solution space of equations?And are these resultants important forunderstanding the world?Recently, the author extended Banach and Hilbert spaces on atopological graph to introduce G-ows and showed that all such ows on a topologicalgraph Galso form a Banach or Hilbert space,which enables one to nd the multiversesolution of these equations on G. Applying this result, this paper discusses the G-owsolutions on Schr odinger equation, Klein-Gordon equation and Dirac equation, i.e., theeldequationsofparticles,bosonsorfermions,answerspreviousquestionsbyyes,andestablishesthemanyworldinterpretationofquantummechanicsofH. Everettbypurely mathematics in logic, i.e., mathematical combinatorics.1 IntroductionMatterconsistsofbosonswithintegerspinnandfermionswithhalf-integerspinn/2, n1 (mod2). Theelementaryparticles consist of leptons and hadrons, i.e.mesons, baryonsandtheirantiparticles, whicharecomposedofquarks[16].Thus, a hadron has an internal structure, which implies that allhadronsarenotelementarybutleptonsare, viewedaspointparticlesinelementaryphysics. Furthermore, thereisalsounmatterwhichisneithermatter nor antimatter, but some-thing 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 ischaracterized by solutions of dier-ential equationestablishedonitswavefunction(t, x). Innon-relativistic quantummechanics, the wave function (t, x)of a particle of mass m obeys the Schr odinger equationit=

22m2 + U, (1.1)where, =6.582 1022MeVsisthePlanckconstant,Uisthe potential energy of the particle in applied eld and=_ x,y,z_and 2=2x2+2y2+2z2.Consequently, afreeboson(t, x)holdwiththeKlein-Gordon equation_1c22t2 2_(x, t) +_mc

_2(x, t)= 0 (1.2)and a free fermion (t, x) satises the Dirac equation_imc

_(t, x)= 0 (1.3)in relativistic forms, where,=_0, 1, 2, 3_,=_1ct,x1,x2,x3_,c is the speed of light and0=_I2200 I22_, i=_0 ii0_with the usual Pauli matrices1=_0 11 0_, 2=_0 ii 0_,3=_1 00 1_.Itiswellknownthatthebehaviorofaparticleisonsu-perposition, i.e., in two or more possible states of being. Buthowtointerpretthisphenomenon inaccordance with(1.1)(1.3) ? Themanyworldsinterpretationonwavefunctionof(1.1)byH. Everett[2]in1957answeredthequestioninmachinery, i.e., viewed dierent worlds in dierent quantummechanicsandthesuperposition ofaparticlebelikedthoseseparate arms of a branching universe ([15], also see [1]). Infact, H. Everettsinterpretationclaimedthat thestatespaceofparticleisamultiverse,orparalleluniverse ([23,24]),anapplication of philosophical law that the integral always con-sists of its parts, or formally, the following.Denition1.1([6],[18]) Let(1; F1),(2; F2), ,(m; Fm)be m mathematical or physical systems, dierent two by two.A Smarandache multisystem is a unionm_i=1i with rules F=m_i=1Fion , denoted by_;F_.Linfan Mao.A New Understanding of Particles by G-Flow Interpretation 193Volume 11 (2015) PROGRESS IN PHYSICS Issue 3 (July)Furthermore, things are inherently related, not isolated inthe world.Thus, every particle in nature is a union of elemen-tary particlesunderlying agraph embedded inspace,where,agraph Gissaidtobeembeddable intoatopological spaceE ifthereisa1 1continuousmappingf : GE withf (p) f (q) ifpq for p, qG,i.e., edges only intersectat endverticesinE . Forexample, aplanargraphsuchasthose shown in Fig. 1.v1v2v3v4u1u2u3u4Fig. 1Denition1.2([6]) Foranyintegerm 1, let_;F_beaSmarandache multisystemconsistingofmathematicalsys-tems(1; F1), (2; F2), ,(m; Fm). Aninheritedtopologi-cal structures GL_;F_ on_;F_ is dened byV_GL_;F__=v1, v2, , vm,E_GL_;F__=(vi, vj)i_j, 1ij mwithalabelingL: viL(vi) =iandL: (vi, vj) L(vi, vj) =i_j, wherei_jdenotestheintersectionofspaces,oractionbetweensystemsiwithjforintegers1 i jm.For example, let =4_i=1iwith 1=a, b, c, 2=a, b,3=b, c, d, 4=c, d andFi=. Calculation shows that1_2=a, b, 1_3=b, c, 1_4=c, 2_3=b,2_4=,3_4=c, d. Such a graph GL_;F_is shown in Fig. 2.a, bbc, dcb, c1234Fig. 2Generally, a particle should be characterized by_;F_ intheory. However,wecanonlyverifyitbysomeofsystems(1; F1), (2; F2),, (m; Fm)forthelimitationofhumanbeingsbecauseheisalsoasystemin_;F_. Clearly, theunderlying graph in H. Everetts interpretation on wave func-tion is in fact a binary tree and there are many such traces inthe developing of physics.For example, a baryon is predomi-nantly formed fromthree quarks, and a meson is mainly com-posedofaquarkandanantiquarkinthemodelsofSakata,or Gell-Mann and Neeman on hadrons ([14]), such as thoseshowninFig. 3, where, qiu, d, c, s, t, bdenotesaquarkfor i= 1, 2, 3 and q2_u, d, c, s, t, b_, an antiquark. Thus, theunderlyinggraphs Gofameson, abaryonarerespectively K2and K3withactions. In fact,afree quark wasnot foundinexperiments untiltoday. Soitisonlyamachinery modelon hadrons. Even so, it characterizes well the known behaviorof particles.q1q2q3q1 q2.................. .. .. .. .. .. .. .. ... .....................................................Baryon MesonFig. 3It should be noted that the geometry on Denition 1.11.2canbealsousedtocharacterizeparticlesbycombinatorialelds ([7]), and there is a priori assumption for discussion inphysics, namely, the dynamical equation of a subparticle of aparticle is the same of that particle.For example, the dynam-icalequation ofquark isnothing elsebuttheDiracequation(1.3), acharacterizing on quark from themacroscopic tothemicroscopic, thequantumlevel inphysics. 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., thereare3parallel equations(1.3)indiscussion, aseemlyratio-nal interpretation in physics, but not perfect for mathematics.Why this happens is because the interpretation of solution ofequation. Usually, weidentifyaparticletothesolutionofitsequation,i.e., ifthevariablest andx1, x2, x3holdwithasystem of dynamical equationsFi_t, x1, x2, x3, ut, ux1, , ux1x2, _= 0,with 1im, (1.4)the particle in RR3is a point (t, x1, x2, x3), and if more thanone points (t, x1, x2, x3) hold with (1.4), the particle is nothingelse but consisting of all such points. However, the solutionsof (1.1)(1.3) are all denite on time t.Can this interpretationbeusedforparticlesinall times? Certainlynot becauseaparticle can be always decomposed into elementary particles,and it is a little ambiguous which is a point, the particle itselfor its one of elementary particles sometimes.194 Linfan Mao.A New Understanding of Particles by G-Flow InterpretationIssue 3 (July) PROGRESS IN PHYSICS Volume 11 (2015)Thisspeculationnaturallyleadstoaquestiononmathe-matics, i.e., what is the right interpretation on the solution ofdierential equation accompanying with particles?Recently,theauthor extended Banachspacesontopological graphs Gwithoperatoractionsin[13], andshownalloftheseexten-sionsarealsoBanachspace, particularly, theHilbert spacewithuniquecorrespondenceinelementsonlinear continu-ous functionals, which enables one to solve linear functionalequations in such extended space, particularly, solve dieren-tial equations on a topological graph, i.e., nd multiverse so-lutions for equations. This scheme also enables us to interpretthe superposition of particles in accordance with mathematicsin logic.The mainpurpose ofthispaper istopresent aninterpre-tationonsuperpositionof particlesby G-owsolutionsof(1.1)(1.3)inaccordancewithmathematics. Certainly, thegeometry on non-solvable dierential equations discussed in[9][12] brings us another general way for holding behaviorsof particles in mathematics. For terminologies and notationsnot mentioned here, we follow references [16] for elementaryparticles, [6]forgeometryandtopology, and[17][18]forSmarandache multi-spaces, and all equations are assumed tobe solvable in this paper.2 Extended Banach

G-ow space2.1 Conservation lawsAconservation law,suchasthoseonenergy, mass,momen-tum, angular momentum and electric charge states that a par-ticular measurablepropertyof anisolatedphysical systemdoesnotchangeasthesystemevolvesovertime,orsimply,constant ofbeing. Usually, alocal conservationlawisex-pressed mathematically as a continuity equation, which statesthattheamountofconservedquantityatapointorwithinavolume can only change by the amount of the quantity whichowsinoroutofthevolume. AccordingtoDenitions1.1and1.2,amatter inthenature isnothing elsebutaSmaran-dachesystem_;F_, oratopologicalgraph GL__;F__em-bedded in R3, hold with conservation laws

kF(v)k=

lF(v)+lon vV_GL__;F___, where, F(v)k , k 1 and F(v)+l, l 1denote respectively the input or output amounts on a particleor a volume v.2.2

G-ow spacesClassical operation systems can be easily extended on a graph Gconstraint on conditions for characterizing the unanimousbehaviors of groups in the nature, particularly, go along withthephysics. Forthisobjective, let Gbeanorientedgraphwith vertex set V(G) and arc set X(G) embedded in R3and let(A; )beanoperation systeminclassicalmathematics,i.e.,for a, bA, abA.Denoted by GLAall of those labeledgraphs GLwithlabelingL: X_ G_A. Then, wecanextend operation on elements in GAby a ruler following:R: For GL1, GL2 GLA, dene GL1 GL2= GL1L2,where L1 L2: eL1(e) L2(e) for eX_ G_.For example, such an extension on graph C4is shown inFig. 4, where, a3=a1a2, b3=b1b2, c3=c1c2, d3=d1d2.v1v2v3v4v1v2v3v4v1v2v3v4a1b1c1d1a2b2c2d2a3b3c3d3 Fig. 4Clearly, GL1 GL2 GLAby denition, i.e., GLAis also an op-eration system under ruler R, and it is commutative if (A, )is commutative,Furthermore, if(A, )isanalgebraic group, GLAisalsoan algebraic group because(1)_ GL1 GL2_ GL3= GL1_ GL2 GL3_ for GL1, GL2, GL3 GAbecause(L1(e) L2(e)) L3(e)=L1(e) (L2(e) L3(e))for eX_ G_, i.e., G(L1L2)L3= GL1(L2L3).(2)thereisanidentify GL1Ain GLA, whereL1A: e1Afor eX_ G_;(3) there is an uniquely element GL1for GL GLA.However,forcharacterizingtheunanimousbehaviorsofgroups inthenature,themostusefuloneistheextension ofvector space (V ; +, ) over eld/by dening the operations+and on elementsin GVsuchasthose shown inFig. 5 ongraph C4,wherea,b, c,d,ai,bi, ci,diV fori =1, 2, 3,x3=x1+x2for x=a, b, c or d and /.v1v2v3v4v1v2v3v4v1v2v3v4a1b1c1d1a2b2c2d2a3b3c3d3 v1 v2 v1v2v3v4v3v4abcdabcdFig. 5Linfan Mao.A New Understanding of Particles by G-Flow Interpretation 195Volume 11 (2015) PROGRESS IN PHYSICS Issue 3 (July)A G-ow on Gissuchanextension holdwithL(u, v)=L(v, u) and conservation laws

uNG(v)L (v, u)= 0forvV_ G_,where 0 isthe zero-vector inV . Thus, a G-ow is a subfamily of GLVlimited by conservation laws. Forexample, if G= C4, theremustbea=b=c=d, ai=bi=ci=difor i= 1, 2, 3 in Fig. 5.Clearly, all conservation G-ows on Galso form a vectorspace over /under operations + and with zero vector O= GL0, whereL0: e0foreX_ G_. Suchanextendedvector space on Gis denoted by GV.Furthermore, if (V ; +, ) is a Banach or Hilbert space withinner product (, ), we can also introduce the norm and innerproduct on GVby____ GL____=

(u,v)X_ G_jL(u, v)jor_ GL1, GL2_=

(u,v)X_ G_(L1(u, v), L2(u, v))for GL, GL1, GL2 GV, wherejL(u, v)j is thenormofL(u, v) inV .Then it can be veried that(1)____ GL____ 0 and____ GL____= 0 if and only if GL= O;(2)____ GL____= ____ GL____ for any scalar ;(3)____ GL1+ GL2________ GL1____+____ GL2____;(4)_ GL, GL_=_(u,v)X_ G_(L(uv), L(uv))0 and_ GL, GL_= 0 if and only if GL= O;(5)_ GL1, GL2_=_ GL2, GL1_ for GL1, GL2 GV;(6) For GL, GL1, GL2 GVand , /,_ GL1+ GL2, GL_= _ GL1, GL_+ _ GL2, GL_.The following result isobtained by showing that Cauchysequences in GVis converges hold with conservation laws.Theorem2.1([13]) Foranytopologicalgraph G, GVisaBanach space, and furthermore, ifVis a Hilbert space, GVis a Hilbert space also.AccordingtoTheorem2.1, theoperatorsactiononBa-nachorHilbertspace(V ; +, )canbeextendedon GV, forexample, the linear operator following.Denition 2.2 An operator T: GV GVis linear ifT_ GL1+ GL2_= T_ GL1_+ T_ GL2_for GL1, GL2 GVand, F, andiscontinuousata G-ow GL0if therealwaysexist anumber()for >0such that_____T_ GL_ T_ GL0______ < if____ GL GL0____ < ().ThefollowinginterestingresultgeneralizestheresultofFr echet andRieszonlinear continuousfunctionals, whichopens us mind for applying G-ows to hold on the nature.Theorem 2.3([13]) Let T: GVC be a linear continuousfunctional.Then there is a unique GL GVsuch thatT_ GL_=_ GL, GL_for GL GV.Particularly, if all ows L(u, v) on arcs (u, v) of Gare statefunction,weextendthedierentialoperatoron G-ows. Infact, a dierential operatortorxi: GV GVis denedbyt: GL GLt,xi: GL GLxifor integers 1i 3.Then, for , /,t_ GL1+ GL2_=t_ GL1+L2_= Gt (L1+L2)= Gt (L1)+t (L2)= Gt (L1)+ Gt (L2)=t G(L1)+t G(L2)= t GL1+ t GL2,i.e.,t_ GL1+ GL2_= t GL1+ t GL2.Similarly, we know also thatxi_ GL1+ GL2_= xi GL1+ xi GL2196 Linfan Mao.A New Understanding of Particles by G-Flow InterpretationIssue 3 (July) PROGRESS IN PHYSICS Volume 11 (2015)for integers 1i 3.Thus, operatorstandxi, 1i 3are all linear on GV.ttttetetetetetetetet1111tFig. 6Similarly, weintroduceintegraloperator_: GV GVby_: GL G_Ldt, GL G_Ldxifor integers 1i 3 and know that_ _ GL1+ GL2_= _ _ GL1_+ _ _ GL2_for , /,__t_and__xi_ : GL GL+ GLc,where Lcis such a labeling that Lc(u, v) is constant for (u, v)X_ G_.3 Particle equations in

G-ow spaceWeareeasilyndparticleequationswithnonrelativisticorrelativistic mechanics in GV.Notice thatit=E, i= p2andE=12m p2+ U,in classical mechanics, where is the state function, E, p, Uare respectively the energy, the momentum, the potential en-ergy and m the mass of the particle.Whence,O = G(E12m p2U)= GE G12m p2 GU= Git G

2m2 GU= i GLt+

2m2 GL GLU GL,where L: estate function and LU: epotential energyoneX_ G_. According totheconservation lawofenergy,there must be GU GV.We get the Schr odinger equation in GVfollowing.i GLt=

2m2 GL U GL, (3.1)where U= GLU GV. Similarly, by the relativistic energy-momentum relationE2=c2 p2+ m2c4for bosons andE=ck pk+ 0mc2forfermions, weget theKlein-GordonequationandDiracequation_1c22t2 2_ GL+_cm

_ GL= O (3.2)and_imc

_ GL= O, (3.3)ofparticlesin GVrespectively. Particularly, let Gbesuchatopological graphwithonevertexbut onlywithonearc.Then, (3.1)(3.3) are nothing else but (1.1)(1.3) respective-ly. However, (3.1)(3.3)concludethat wecannd G-owsolutions on (1.1)(1.3), which enables us to interpret mathe-matically the superposition of particles by multiverse.4

G-ows on particle equationsFormally, wecanestablishequationsin GVbyequations inBanach spaceVsuch as (3.1)(3.3). However, the importantthing isnot justonsuch establishing butnding G-owsonequations inV andtheninterpret thesuperposition ofparti-cles by G-ows.4.1

G-ow solutions on equationTheorem2.3concludesthat thereare G-owsolutionsforalinearequationsin GVforHilbertspaceV overeld/,includingalgebraicequations, linear dierential or integralequations withoutconsidering thetopological structure. Forexample, let ax=b.We are easily getting its G-ow solutionx= Ga1LifweviewanelementbV asb= GL, whereL(u, v)=b for(u, v)X_ G_ and 0a/, such as thoseshown in Fig. 7 for G= C4and a= 3, b= 5.Linfan Mao.A New Understanding of Particles by G-Flow Interpretation 197Volume 11 (2015) PROGRESS IN PHYSICS Issue 3 (July)53535353Fig. 7Generally, we know the following result:Theorem 4.1([13]) A linear system of equations___a11x1+ a12x2++ a1nxn=b1a21x1+ a22x2++ a2nxn=b2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .am1x1+ am2x2++ amnxn=bmwithai j, bj/forintegers1i n, 1 j mholdingwithrank_ai j_mn= rank_ai j_+m(n+1)has G-owsolutionsoninnitelymanytopologicalgraphs G, where_ai j_+m(n+1)=__a11a12 a1nL1a21a22 a2nL2. . . . . . . . . . . . .am1am2 amnLm__.We can also get G-ow solutions for linear partial dier-ential equations ([14]). For example, the Cauchy problems ondierential equationsXt=c2n

i=12Xx2iwith initial value Xt=t0= GL GVis also solvable in GVifL(u, v) is continuous and bou- nded in Rnfor (u, v) X_ G_and GL GV. In fact,X= GLFwithLF: (u, v)F(u, v)for (u, c) X_ G_, whereF (u, v) =1(4t)n2__+e(x1y1)2++(xnyn)24t L(u, v) (y1, , yn)dy1 dyn_is such a solution.Generally, if Gcanbedecomposedintocircuits C, thenextresult concludesthat wecanalwaysnd G-owsolu-tionsonequations, nomatterwhat theequationlookslike,linear or non-linear ([13]).Theorem 4.2If the topological graph Gis strong-connectedwith circuit decomposition G=l_i=1 CisuchthatL(u, v)=Li (x)for(u, v)X_ Ci_,1i landthe Cauchy problem_Fi_x, u, ux1, , uxn, ux1x2, _= 0ux0=Li(x)is solvable ina Hilbert spaceV on domain Rnfor inte-gers 1il, then the Cauchy problem_Fi_x, X, Xx1, , Xxn, Xx1x2, _= 0Xx0= GLsuchthatL(u, v)=Li(x)for(u, v)X_ Ci_issolvableforX GV.In fact,such asolution isconstructed byX= GLu(x)withLu(x) (u, v)=u(x) for (u, v) X_ G_ by applying the input andthe output at vertex v all being u(x) on C, which implies thatall ows at vertex vV_ G_ is conserved.4.2

G-ows on particle equationThe existence of G-owsolutions on particle equations (1.1)(1.3)isclearlyconcludedbyTheorem4.2, alsoimpliedby(3.1)(3.3) forany G. However,thesuperpositionofapar-ticleP showsthat there areN2statesof being associatedwith a particle P. Considering this fact, a convenient G-owmodel for elementary particle fermions, the lepton or quark Pis by abouquet BLN,and an antiparticleP ofP presented by BL1Nwith all inverse states on its loops, such as those shownin Fig. 8.PP12N11121N ParticleAntiparticleFig. 8An elementary unparticle is an intermediate formbetweenan elementary particle and its antiparticle, which can be pre-sentedby BLCN, whereLC: eL1 (e)ife Cbut LC:eL(e) ifeX_ BN_ \ Cfor asubset CX_ BN_,suchas those shown in Fig. 9,198 Linfan Mao.A New Understanding of Particles by G-Flow InterpretationIssue 3 (July) PROGRESS IN PHYSICS Volume 11 (2015)12N111121N2 Fig. 9 UnparticlewhereN1, N21areintegers. Thus,anelementary particlewith its antiparticles maybe annihilate or appears in pair at atime, which consists in an elementary unparticle by combina-tions of these state functions with their inverses.PPN2211NFig. 10DL0,2N,0For thoseofmediate interaction particlequanta,i.e.,bo-son,whichreectsinteraction between particles. Thus,theyare conveniently presented by dipoleDL0,2N,0but with dottedlines, suchasthoseinFig. 10, inwhichthevertexP, Pde-notesparticles, andarcswithstatefunctions1, 2, , Nare the Nstates ofP.Notice that BLNandDL0,2N,0both are aunion of Ncircuits.According toTheorem 4.2,weconsequently getthefol-lowing conclusion.Theorem 4.3For an integer N 1, there are indeedDL0,2N,0-ow solution on Klein-Gordon equation (1.2), and BLN-owsolution on Dirac equation (1.3).Generally, this model enables us to know that the G-owconstituents of a particle also.............. ..k21pq12 l Fig. 11 MesonThus, if aparticle Pisconsistedof l elementaryparti-clesP1, P1, , Plunderlyingagraph G_P_, its G-owisobtained byreplace eachvertex vby BLvNvandeach arcebyDLe0,2Ne,0in G_P_, denotedby GL_ Bv, De_. Forexample,the model of Sakata, or Gell-Mann and Neeman on hadronsclaimsthat themesonandthebaryonarerespectivelythedipoleDLek,2N,l-owshowninFig. 11andthetriplet G-owCLk,l,sshown in Fig. 12,....................... . . . . . . . . . . . . . . . . . . . .......................q1q2q31k12112l22213s3231Fig. 12 BaryonTheorem 4.4If P is a particle consisted of elementary parti-cles P1, P1, , Pl for an integer l 1, then GL_ Bv, De_ is a G-ow solution on the Schr odinger equation (1.1) wheneverGis nite or innite.Proof If Gis nite, the conclusion follows Theorem 4.2immediately. We only consider the case of G. In fact,if G, calculation shows thati limG_t_ GL_ Bv, De___= limG_it_ GL_ Bv, De___= limG_

22m2 GL_ Bv, De_+ GLU_=

22m2limG GL_ Bv, De_+ GLU,i.e.,i limG_t_ GL_ Bv, De___=

22m2limG GL_ Bv, De_+ GLU.In particular,ilimN__ BLNt__=

22m2limN BLN+ GLU,ilimNt_DL0,2N,0_=

22m2limNDL0,2N,0+ GLUfor bouquets and dipoles. Linfan Mao.A New Understanding of Particles by G-Flow Interpretation 199Volume 11 (2015) PROGRESS IN PHYSICS Issue 3 (July)5

G-ow interpretation on particle superpositionThesuperpositionof aparticlePis depictedbyaHilbertspace Vover complex eld C with orthogonal basis1) ,2) ,,n) ,inquantummechanics. Infact, thelinearityofSchr odingerequationconcludesthat all statesofparticlePareinsuchaspace. However, anobservercangrasponlyonestate, whichpromotedH. Everett devisedamultiverseconsisting ofstatesinsplittingprocess,i.e.,thequantum ef-fects spawn countless branches of the universe with dierenteventsoccurringineach, notinuenceoneanother,suchasthoseshowninFig. 13,andtheobserver selectsbyrandom-ness, where the multiverse is_i1ViwithVkl=Vfor integersk 1, 1 l 2kbut in dierent positions. 1V111V1112V1231V3132V3233V3334V34Fig. 13Whyit needsaninterpretationonparticlesuperpositionin physics lies in that we characterize the behavior of particlebydynamicequationonstatefunctionandinterpretittobethe solutions, and dierent quantum state holds with dierentsolution ofthatequation. However, wecanonly getone so-lution by solving the equation with given initial datum once,andholdonestateoftheparticleP, i.e., thesolutioncorre-spondent only to one position but the particle is in superposi-tion, which brought the H. Everett interpretation on superpo-sition. Itisonlyabiologicalmechanismbyinniteparallelspaces Vbut loses of conservations on energy or matter in thenature, whose independently runs also overlook the existenceof universal connection in things, a philosophical law.Evenso, it cannotblotouttheideologicalcontributionofH. Everett tosciencesashredbecauseall of thesemen-tions are produced by the interpretation on mathematical so-lutionswiththerealityofthings,i.e.,scanningonlocal,notthe global. However, if we extend the Hilbert space Vto BLN,DL0,2N,0or GL_ Bv, De_ingeneral, i.e., G-owspace GV,where GistheunderlingtopologicalgraphofP, thesitua-tion has been greatly changed because GVisitself aHilbertspace, and we can identify the G-ow on Gto particle P, i.e.,P= GL_ Bv, De_(5.1)for a globally understanding the behaviors of particle P what-ever Gor not byTheorem4.4. For example, letP= BLN, i.e., afreeparticlesuchasthoseofelectrone,muon ,tauon ,or their neutrinos e, , . Then the su-perposition of P is displayed by state functions onNloopsin BNhold on its each loop withinput i=ouput iat vertex Pfor integers 1iN.Consequently,input

iIi=ouput

iIiat vertex Pfor I1, 2, , N, the conservation law on vertex P.Fur-thermore, such a BLNis not only a disguise onP in form butalso a really mathematical element in Hilbert space BV, andcan be also used to characterize the behavior of particles suchas those of the decays or collisions of particles by graph oper-ations. For example, the -decay np+e+eis transferredto a decomposition formulaCLnk,l,s=CLpk1,l1,s1_ BLeN1_ BLN2,ongraph, where,CLpk1,l1,s1, BLeN1, BLN2areallsubgraphsofCLnk,l,s.Similarly, the - collision e+p n+e+is transferredto an equality BLeN1_CLpk1,l1,s1=CLnk2,l2,s2_ BLeN2.Eventhrough therelation(5.1)isestablishedonthelin-earity, itisin fact truly for the linear and non-liner cases be-causetheunderlying graphof GL_ Bv, De_-owcanbede-composed into bouquets and dipoles, hold with conditions ofTheorem 4.2. Thus, even if the dynamical equation of a par-ticleP is non-linear, we can also adopt the presentation (5.1)tocharacterizethesuperpositionandholdontheglobalbe-havior ofP. Whence, it isa presentation on superposition ofparticles, both on linear and non-linear.6 Further discussionsUsually, adynamicequationonaparticlecharacterizesitsbehaviors. Butisitssolutionthesameastheparticle? Cer-tainly not! Classically,adynamic equation isestablished oncharactersofparticles, anddierentcharactersresultindif-ferent equations. Thus the superposition of a particle shouldbecharacterizedbyat least 2dierential equations. How-ever, for a particle P, all these equations are the same one by200 Linfan Mao.A New Understanding of Particles by G-Flow InterpretationIssue 3 (July) PROGRESS IN PHYSICS Volume 11 (2015)chance,i.e.,oneoftheSchr odinger equation,Klein-Gordonequation or Dirac equation, which lead to the many world in-terpretation of H. Everett, i.e., put a same equation or Hilbertspace on dierent place for dierent solutions in Fig. 12. Asit isshowninTheorems4.14.4, wecaninterprettheso-lutionof(1.1)(1.3) tobea GL_ Bv, De_-ow, whichprop-erlycharacterizesthesuperpositionbehaviorofparticlesbypurely mathematics.The G-owinterpretation ondierentialequationopensanewwayforunderstanding thebehaviorofnature, partic-ularlyonsuperpositionofparticles. Generally,thedynamicequations on dierent characters maybe dierent, which willbringsabout contradictsequations, i.e., non-solvableequa-tions.For example, we characterize the behavior of meson orbaryon by Dirac equation (1.3). However, we never know thedynamic equation on quark. Although we can say it obeyingtheDiracequation butitisnotacompletepictureonquark.If we nd its equation some day, they must be contradicts be-cause it appear in dierent positions in space for a meson or abaryon at least. As a result,the G-solutions on non-solvabledierentialequationsdiscussedin[9][12]arevaluableforunderstanding the reality of the nature with G-ow solutionsa special one on particles.As it is well known for scientic community, any sciencepossessthe falsiability butwhich depends on known scien-ticknowledgeandtechnicalmeansatthattimes. Accord-ingly, it is very dicult to claim a subject or topic with logi-calconsistency istruth or falseon thenature sometimes,forinstancethemultiverseorparallel universesbecauseofthelimitation of knowing things in the nature for human beings.In that case, a more appreciated approach is not denied or ig-nored but tolerant, extends classical sciences and developingthoseofwell knowntechnicalmeans, andthenget abetterunderstandingonthenaturebecausethepointlessargumentwould not essentially promote the understanding of nature forhuman beings ([3,4,22]).Submitted on April 8, 2015 / Accepted on April 15, 2015References1. 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TegmarkM.Ourmathematicaluniverse: Myquestfortheul-timatenatureofreality. KnopfDoubledayPublishingGroup,2014.Linfan Mao.A New Understanding of Particles by G-Flow Interpretation 201