A New Understanding of Particles by G - Flow Interpretation of Differential Equation
A New Understanding of Particles by G - Flow Interpretation of Differential Equation
A New Understanding of Particles by G - Flow Interpretation of Differential Equation
PROGRESS IN PHYSICS
Volume 10 (2014)
A New Understanding of
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 can
we interpret the solution space of equations? And are these resultants important for
understanding the world? Recently, the author extended Banach and Hilbert spaces on
a topological graph to introduce G-flows and showed that all such flows on a topological
graph G also form a Banach or Hilbert space, which enables one to find the multiverse
solution 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 by
YES, and establishes the many world interpretation of quantum mechanics of H.Everett
by purely mathematics in logic, i.e., mathematical combinatorics.
Gordon equation
!
mc 2
1 2
2
(x,
t)
+
(x, t) = 0
c2 t2
~
1. Introduction
All matters consist of two classes particles, i.e., bosons with
integer spin n, fermions with spin n/2, n 1(mod2), and
by a widely held view, the elementary particles consists of
quarks and leptons with interaction quanta including photons
and 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 all
hadrons are not elementary but leptons are, viewed as point
particles in elementary physics. Furthermore, there are 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 (t, x). In
non-relativistic quantum mechanics, the wave function (t, x)
of a particle of mass m obeys the Schrodinger equation
~2
i~
= 2 + U,
t
2m
(1.3)
1
,
,
,
,
c t x1 x2 x3
I22
0
0
I22
3 =
1 0
0 1
i
0
0
i
, i =
(1.1)
(1.2)
0
i
i
0
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 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
Volume 10 (2014)
PROGRESS IN PHYSICS
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.Everetts 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.
Issue 1 (January)
m
S
i=1
{c}
{b}
{c, d}
Fig.2
i=1
Ri on e
, denoted by e
; e
R.
v2
u1
u2
u4
u3
e in
Generally, a particle should be characterized by e
; R
theory. However, we can only verify it by some of systems
(1 ; R1 ), (2 ; R2 ), , (m ; Rm ) for the limitation
of human
e . Clearly, the
beings because he is also a system in e
; R
underlying graph in H.Everetts 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 Neeman on hadrons ([14]), such as those
shown in Fig.3, where,n qi {u, d, c,
o s, t, b} denotes a quark
for i = 1, 2, 3 and q2 u, d, c, s, t, b , an antiquark. Thus, the
v3
v4
{b, c}
4
Definition 1.1([6],[18]) Let (1 ; R1 ), (2 ; R2 ), , (m ; Rm )
be m mathematical or physical systems, different two by two.
m
S
e=
A Smarandache multisystem e
is a union i with rules R
{a, b}
Fig.1
e be
Definition 1.2([6]) For any integer m 1, let e
; R
a Smarandache multisystem consisting of mathematical systems (1 ; R1 ), (2h; R2 ),i , (m ; Rm ). An inherited topologie is defined by
cal structures G L e
; e
R on e
; R
h
i
V GL e
; e
R = {v1 , v2 , , vm },
h
i
e = {(vi , v j )|i T j , , 1 i , j m}
E GL e
; R
with a labeling L : vi L(vi ) = i and L : (vi , v j )
T
T
L(vi , v j ) = i j , where i j denotes the intersection
of spaces, or action between systems i with j for integers
1 i , j m.
q2
.. q1
..
..
..
...........
.
.
.
.
.
.....
.....
.....
.....
. q3
q1 ................................. q2
Baryon
Meson
Fig.3
Issue 1 (January)
PROGRESS IN PHYSICS
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 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)
h
i
the particle in R R3 is a point (t, x1 , x2 , x3 ), and if more on v V G L e
e , where, F(v) , k 1 and F(v)+ , l 1
; R
k
l
than one points (t, x1 , x2 , x3 ) hold with (ES q4m ), the particle denote respectively the input or output amounts on a particle
is 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 time
is a point, the particle itself or its one of elementary particles G constraint on conditions for characterizing the unanimous
sometimes.
behaviors of groups in the nature, particularly, go along with
This speculation naturally leads to a question on mathe- the physics. For this objective, let
G be an oriented graph
matics, 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 let
differential equation accompanying with particles? Recently, (A ; ) be an operation system in classical mathematics, i.e.,
L
the author extended Banach spaces on topological graphs G for a, b A , ab A . Denoted by
G A all of those labeled
with operator actions in [13], and shown all of these exten
L
sions are also Banach space, particularly, the Hilbert space graphs G with labeling L : X G A . Then, we can
A
with 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 functional
L
L
L
1
2
L1
L2
L1 L2
R: For G , G G A , define G G = G
,
equations in such extended space, particularly, solve differen
tial 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 interpret
3
1
2
tation on superposition of particles by G-flow solutions of
b1
d2
b2
d3
b3
equations (1.1)-(1.3) in accordance with mathematics. Cer- d1
tainly, the geometry on non-solvable differential equations
v 4 c1 v 3
v 4 c2 v 3
v 4 c3 v 3
discussed in [9]-[12] brings us another general way for holding behaviors of particles in mathematics. For terminologies
Fig.4
and notations not mentioned here, we follow references [16]
L
L
L
1
2
L
for 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,
L
Furthermore, if (A , ) is an algebraic group, G A is also
an algebraic group because
1
L2
L3
L1
2
L3
L1
L2
(1) G G G = G G G for G , G ,
2.1 Conservation Laws
L3
A
G G because
A 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 par
ticular 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., G
L1A
L
L1
L
L
volume can only change by the amount of the quantity which
(3) there is an uniquely element G for G G A .
6
6 ?
6 ?
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 L
P
v
v
vector space (V ; +, ) over field F by defining the operations
(4) G , G =
hL(u
),
L(u
)i
0
and
G ,G
!
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 ,G
for G , G G ;
v1 a 1 v2
v1 a 2 v2
v1 a 3 v2
L
L1
L2
V
(6) For G , G , G G and , F ,
d1
b1
d2
b2
d3
b3
L
L2
L
G
+
G
,
G
c
c
c
1
2
3
v3
v3
v3
v4
v4
v4
L
L
1
L
2
L
v1 a v2
v1 a v2
= G ,G + G ,G .
b
b
d
d
The following result is obtained by showing that Cauchy
V
v4 c v3
v4 c v3
sequences in G is converges hold with conservation laws.
6 ?
?
6
6 ?
6 ?
6 ?
Fig.5
L0
G , where L0 : e 0 for e X G . Such an extended
V
vector space on G is denoted by G .
Furthermore, if (V ; +, ) is a Banach or Hilbert space with
inner product h, i, we can also introduce the norm and inner
V
product on G by
X
L
kL(u, v)k
G
=
!
(u,v)X G
or
L
1
L2
G ,G
=
(u,v)X
!
G
V
Theorem 2.1([13]) For any topological graph G, G is a
V
Banach space, and furthermore, if V is a Hilbert space, G
is a Hilbert space also.
According to Theorem 2.1, the operators action on Ba
V
nach or Hilbert space (V ; +, ) can be extended on G , for
example, the linear operator following.
V
Definition 2.2 An operator T : G G is linear if
L
L
L
L2
2
T G + G = T G + T G
L1
L2
V
for G , G G and , F , and is continuous at a
L0
G-flow G if there always exist a number () for > 0
such that
L
L
0
T
G
T
G
<
if
L
L0
G
G
< ().
V
Theorem 2.3([13]) Let T : G C be a linear continuous
bL
V
functional. Then there is a unique G G such that
L * L bL +
T G = G ,G
L
L1
L2
V
for G , G , G G , where kL(u, v)k is the norm of
L(u, v) in V . Then it can be verified that
L
V
L
L
L
for G G .
(1)
G
0 and
G
= 0 if and only if G = O;
Particularly, if all flows L(u, v) on arcs (u, v) of G are state
L
L
Issue 1 (January)
PROGRESS IN PHYSICS
i.e.,
V
or
: G G is defined
t
xi
L
Lt
L
xL
: G G ,
: G G i
t
xi
for integers 1 i 3. Then, for , F ,
L1
L2
G + G
t
L1 +L2
t (L1 +L2 )
G
=G
=
t
t (L1 )+ t (L2 )
t (L1 )
t (L2 )
=G
=G
+G
(L1 )
(L2 )
+ G
= G
t
t
L1
L2
= G + G ,
t
t
Volume 10 (2014)
V
relativistic mechanics in G . Notice that
i~
= E,
t
p
i~ =
2
and
1
p 2 + U,
2m
p , U
in classical mechanics, where is the state function, E,
are respectively the energy, the momentum, the potential energy and m the mass of the particle. Whence,
E=
=
=
L1
L2
L1
L2
G + G = G + G .
t
t
t
Similarly, we know also that
L1
L2
L1
L2
G + G =
G +
G
xi
xi
xi
=
=
2
1
E 2m p U
G
E
2m1 p
U
G G
G
i~ t
2m~ 2
U
G
G
G
L
~ 2
G
L
LU
L
i~
+
G G G ,
t
2m
U
V
for integers 1 i 3. Thus, operators
and
, 1i3
there must be G G . We get the Schrodinger equation in
t
xi
V
G following.
are all linear on G .
t
t
e
e
L
G
~ 2
L b
L
i~
=
G U
G ,
(3.1)
t
1 t
t
2m
t
t
e
e
e
et
t
t
LU
V
t
1 where U
1
b =
G G . Similarly, by the relativistic energymomentum relation
t
1
p 2 + m2 c4
E 2 = c2
t
t
e
e
for bosons and
Fig.6
p + 0 mc2
E = ck
k
Z
V
for
fermions,
we
get
the
Klein-Gordon
equation and Dirac
Similarly, we introduce integral operator
: G
equation
V
G by
!
R
R
Z
cm L
1 2
L
L
Ldt
L
Ldxi
2
G
+
G =O
(3.2)
: G G
, G G
2
2
~
c t
6 }=
-
6 =} ?
-
L1
L2
L1
L2
G + G =
G +
G
and
mc
L
i
G = O,
~
(3.3)
L
L
Lc
and
: G G +G ,
Volume 10 (2014)
PROGRESS IN PHYSICS
Issue 1 (January)
differential equations
n
X
X
2 X
V
2
=
c
Formally, we can establish equations in G by equations in
t
x2i
i=1
Banach space V such as those equations (3.1)-(3.3). However, the important thing is not just on such establishing but
L
V
finding G-flows on equations in V and then interpret the su- with initial value X|t=t0 = G G is also solvable in G if
L
V
LF
and G G . In
fact, X = G with LF : (u, v) F(u, v)
Z +
Theorem 2.3 concludes that there are G-flow solutions for
(x1 y1 )2 ++(xn yn )2
1
V
4t
F (u, v) =
e
n
a linear equations in G for Hilbert space V over field F ,
(4t) 2
including algebraic equations, linear differential or integral
L (u, v) (y1 , , yn )dy1 dyn
equations without considering the topological structure. For
a L
L
x = G
if we view an element
b V as b = G , where
Generally, if G can be decomposed into circuits C , the
L(u, v) = b for (u, v) X G and 0 , a F , such as those
5
3
?
5
3
5
3
Fig.7
Generally, we know the following result .
Theorem 4.1([13]) A linear system of equations
..............................
G=
Ci
l
i=1
such that L(u, v) = Li (x) for (u, v) X C i , 1 i l and
the Cauchy problem
(
Fi x, u, u x1 , , u xn , u x1 x2 , = 0
u|x0 = Li (x)
is solvable in a Hilbert space V on domain Rn for integers 1 i l, then the Cauchy problem
Fi x, X, X x1 , , X xn , X x1 x2 , = 0
L
X|x0 =
G
with ai j , b j F for integers 1 i n, 1 j m holding such that L (u, v) = Li (x) for (u, v) X
C i is solvable for
with
V
h i
h i+
XG .
rank ai j
= rank ai j
mn
m(n+1)
Lu(x)
a
a
a
L
11
12
1n
1
.
ai j
=
m(n+1)
. . . . . . . . . . . . .
We can also get G-flow solutions for linear partial differ- (1.3) is clearly concluded by Theorem 4.2, also implied by
ential equations ([14]). For example, the Cauchy problems on equations (3.1)-(3.3) for any G. However, the superposition
Issue 1 (January)
PROGRESS IN PHYSICS
of a particle P shows that there are N 2 states of being associated with a particle P. Considering this fact, a convenient
L
quark P is by a bouquet B N , and an antiparticle P of P pre
L1
sented by B N with all inverse states on its loops, such as
those shown in Fig.8.
N
2 1
1
1
1
2
1
N
Antiparticle
Particle
N1
L
are conveniently presented by dipole D 0,2N,0 but with dotted
lines, such as those in Fig.10, in which the vertex P, P denotes particles, and arcs with state functions 1 , 2 , , N
L
L
are the N states of P. Notice that B N and D 0,2N,0 both are a
union of N circuits.
According to Theorem 4.2, we consequently get the following conclusion.
L
Theorem 4.3 For an integer N 1, there are indeed D 0,2N,0
L
-flow solution on Klein-Gordon equation (1.2), and B N -flow
solution on Dirac equation (1.3).
Fig.8
Volume 10 (2014)
1
1
1
2
2 1
p
q
..............
Meson
Fig.11
1
N2
h ei
Unparticle
cles P1 , P1 , , Pl underlying a graph G P
, its G-flow is
Lv
obtained by replace each vertex v by B Nv and each arc e by
Le
h ei
L h
i
Fig.9
D 0,2Ne ,0 in G P
, denoted by G
B v , D e . For example,
where N1 , N2 1 are integers. Thus, an elementary particle the model of Sakata, or Gell-Mann and Neeman on hadrons
the meson and the baryon are respectively the
with its antiparticles maybe annihilate or appears in pair at a claims that
L
e
time, which consists in an elementary unparticle by combina- diploe D k,2N,l -flow shown in Fig.11 and the triplet G-flow
L
tions of these state functions with their inverses.
C k,l,s shown in Fig.12,
-
-
2
l
2
1
D 0,2N,0
1k
-
1 1
2 1
2
2
. q
........ 3
.
.
.. ....
...
...
.
.
.
...
.
.
.
...
.
.
..... . .. .. . .. .. . .. . .. ....
q1
q2
Fig.10
Baryon
For those of mediate interaction particle quanta, i.e., boson, which reflects interaction between particles. Thus, they
Fig.12
3
1
3
2
3s
Volume 10 (2014)
PROGRESS IN PHYSICS
L h
i
cles P1 , P1 , , Pl for an integer l 1, then G
B v , D e is a
Issue 1 (January)
31 V31
o
32 V32
33 V33
o 7
34 V34
Proof If G is finite, the conclusion follows Theorem 4.2
immediately. We only consider the case of G . In fact,
11 V11
12 V12
if G , calculation shows that
!
L h
i
1 V1
i~ lim
G
B v , De
G t
!
L h
i
= lim i~
G
B v , De
G
t
!
Fig.13
~2 2
L h
i
LU
B v , De + G
= lim G
G
2m
Why it needs an interpretation on particle superposition
~2 2
in physics lies in that we characterize the behavior of particle
L h
i
LU
= lim G
B v , De + G ,
by dynamic equation on state function and interpret it to be
2m G
the solutions, and different quantum state holds with different
i.e.,
solution of that equation. However, we can only get one so!
lution by solving the equation with given initial datum once,
h
i
L
G
B v , De
i~ lim
and hold one state of the particle P, i.e., the solution correG t
spondent only to one position but the particle is in superposi~2 2
L h
i
LU
tion, which brought the H.Everett interpretation on superpo= lim G
B v, De + G .
2m G
sition. It is only a biological mechanism by infinite parallel
spaces V but loses of conservations on energy or matter in the
Particularly,
nature, whose independently runs also overlook the existence
L
2
of universal connection in things, a philosophical law.
B
~
L
LU
i~ lim N = 2 lim B N + G ,
Even so, it can not blot out the ideological contribution
N
t
2m N
of H.Everett to sciences a shred because all of these mentions are produced by the interpretation on mathematical so!
L
lutions with the reality of things, i.e., scanning on local, not
i~ lim
D 0,2N,0
L
N t
the global. However, if we extend the Hilbert space V to B N ,
L
L h
i
V
L
~2 2
LU
D
or
G
B
,
D e in general, i.e., G-flow space G ,
v
0,2N,0
= lim D 0,2N,0 + G
2m N
where G is the underling topological graph of P, the situaV
for bouquets and dipoles.
tion has been greatly changed because
G is itself a Hilbert
L h
i
P=G
B v , De
(5.1)
Superposition
for a globally understanding the behaviors of particle P whatever G or not by Theorem 4.4. For example, let
L
P = B N , i.e., a free particle such as those of electron e ,
muon , tauon , or their neutrinos e , , . Then the superposition of P is displayed by state functions on N loops
iI
Issue 1 (January)
PROGRESS IN PHYSICS
V
also a really mathematical element in Hilbert space B , 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 -decay n p+e +e is transferred
to a decomposition formula
Ln
L p [
Le [
L
C k,l,s = C k1 ,l1 ,s1
B N1
B N2 ,
L p
Le
L
on graph, where, C k1 ,l1 ,s1 , B N1 , B N2 are all subgraphs of
Ln
C k,l,s. Similarly, the - collision e + p n+e+ is transferred
to an equality
L p
Ln [
Le [
Le
B N1
C k1 ,l1 ,s1 = C k2 ,l2 ,s2
B N2 .
Even through the relation (5.1) is established on the linearity, it is in fact truly for the linear and non-liner cases be
L h
i
cause the underlying graph of G
B v , D e -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
characterized by at lest 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
L h
i
equation (1.1)-(1.3) to be a G
B v , D e -flow, which properly characterizes the superposition behavior of particles by
purely mathematics.
Volume 10 (2014)
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PROGRESS IN PHYSICS
Issue 1 (January)
10