- Email: [email protected]
Particle-like behavior
ANNALS
OF PHYSICS:
59,
19-26 (1970)
Particle-Like M. MURASKIN
Behavior AND T. CLARK
University of North Dakota, Grand Forks, North Dakota 58201 Received September 29, 1969
We study the field theory previously introduced by one of the authors (M. M.) by means of a finite difference approximation using a computer. We focus attention on particle-like behavior centered at an arbitrary origin. We find in the two examples under consideration that one kind of particle-like property induces other particle-like properties to also appear as a consequence of the dynamics.
1. INTRODUCTION
There are many criteria that may be used to characterize an elementary particle. For one thing, we may think of an elementary particle as denserthan the region around it. But this is clearly not the only property of an elementary particle. For example, an electron can be considered to be a spinning body. Thus, in any attempt to formulate a model of a particle it would be important to show that a condition on density of the particle also causes other particle properties to appear as well. In this paper, we are not at the point where we can talk of real particles like electrons. The study of particle structure is still rather in a primitive state. What we shall do is investigate a nonlinear field equation that has the desirable property of allowing for particle-like behavior. The equation we consider is I’$;, = 0. This equation was introduced in a previous paper [l] where some of its properties were studied. We found this equation can be motivated in a simple way. Furthermore, an extremum in the determinant of gij can be made to appear at an arbitrary origin. An extremum in a field component also characterizes the particle-like behavior obtained by Rosen and Born-Infeld [2] (in these cases, the field component is maximum). In this present work, we study this particle-like object in greater detail making use of a finite difference approximation to the field equations. This technique enables us to map out a solution to the equations using a computer. We are able, in this way, to gather some new results which are difficult to obtain by analytic 19
20
MURASKIN
AND
CLARK
methods due to the nonlinear complexities of the field equations. We find that the conditions on g induce other particle-like features to appear. Our present work is restricted to the neighborhood of the origin because of errors coming in from the approximation scheme. We are also restricted by the limitations of computer time. Thus, we have not entirely mapped out the particlelike object. It is the ultimate goal of the program to map out completely the particlelike object and, then, follow its development in time. Only then will we be in a position to gauge how close to a real particle we have come.
11. FINITE
DIFFERENCE
APPROXIMATION
The field equations that we study are
(1) and agik -
a2
p -
r&k
-
ri%h
=
O.
(2)
Thus, from
dg,,++”
and
drji, = %
dxz
we get rjk(Q)
= rjk(P)
+ {r;,r;+
riimr:
- r;rA,jdxZ
(3)
and
where dx” is the displacement between Q and P. The bracketed terms on the right side are evaluated in lowest order at the origin point P. Thus, the field at Q is obtained if we know the field at P. Once we have determined the field at Q from (3) and (4), we then procede to the next point in a similar fashion. In this way we may map out the field throughout space. The imput data is the field at the origin point. The scheme gives only an approximate solution to the field equations since the displacement between points is taken to be a finite number and not an infinitesimal. The smaller the grid size, the more accurate the results. It is also necessary to require that I’jk(P) obey consistency relations. If we consider the field at some point in space, it is necessary that the field be unique,
PARTICLE-LIKE
21
BEHAVIOR
independent of the path chosen from the origin point. In our previous work we found that this condition can be met by introducing 16eiu so that at P r,tk = eaieEjeykrEa,
(3
with r& obeying riArin - r;r;?
+ r,",r;
- r&rFo = 0.
We have shown that solutions to (6) exist defining a nontrivial III.
PARTICLE-LIKE
(6)
field theory.
BEHAVIOR
We can require that g (the determinent of gJ be an extremum at the origin point P by requiring that r& at P, obey rjt, = 0, k = 1, 2, 3. We have shown in our previous work that this condition can be met. We have considered two different particle-like structures which we have labeled particle A and particle B. The term particle is used loosely since, at this point, a g particle has not been shown to correspond to a real particle. We shall use the following values for the nonvanishing r;,, : rtl
= r;L2 = 0.5,
r,l, = riz = 0.2,
(7)
ri3 = rio = 0.3, r& = rio = 0.1. This choice enables us to satisfy the consistency conditions described by e’, = e21 = e31 = eel =
-0.62 0.8 0.3 0.6
(6). Particle A is
el, = e22 = e32 = eo2 =
-0.76 0.4 0.7 0.9
e13 = e23 = e33 = eo3 =
-0.7 0.5 0.8 0.1
el, = e20 = e30 = coo =
0.1 0.2 0.3 2.0.
(8)
el, = ez2 = e32 = e20 =
-0.6 0.6 0.3 0.9
e13 = ez3 = e33 = eo3 =
-0.5 0.1 0.5 0.8
el, = e20 = e30 = coo =
0.1 0.2 0.3 2.0.
(9)
Particle B is described by el, = e21 = e31 = eel =
-0.82 0.8 0.7 0.4
22
MURASKIN
AND
CLARK
These values obey Tit, = 0, k = 1,2,3. We have made runs down the x, y, z, -x, -y, --z axes. The results are tabulated in Tables I and II. We see in the tables that when the coordinates have, say, the value 0.19, then the value of g is smaller than the value of g at the origin point. The grid sizes are taken to be either 0.005 or 0.01. g itself is rather slowly varying close to the origin. We have made longer runs down the axes (x = 0.39, -x = 0.39), and the trends shown continue. We note, from the data, a reflection symmetry exists along the axes for the values TABLE
I
Behavior of g for Particle g at origin = -0.09948
Axis
x --x Y -Y z -2
Value
of g
Value
-0.09959 -0.09957 -0.10000 -0.09958 -0.10003 -0.10007 -0.10008 -0.09988 -0.09988
X
-x Y -Y z -2
of g
-0.06410 -0.06409 -0.06408 -0.06428 -0.06425 -0.06516 -0.06510 -0.06428 -0.06416 -0.06416 -0.06426 -0.06426
Grid
Size
0.005 0.01 0.01 0.01 0.01 0.005 0.005 0.005 0.005
II
Behavior of g for Particle g at origin = -0.06405
Value
of Coordinate 0.19 0.19 0.39 0.19 0.39 0.19 0.19 0.19 0.19
TABLE
Axis
A
Value
B
of Coordinate 0.10 0.10 0.10 0.19 0.19 0.39 0.39 0.19 0.19 0.19 0.19 0.19
Grid
Size
0.0025 0.005 0.01 0.005 0.01 0.005 0.01 0.005 0.005 0.005 0.005 0.005
PARTICLE-LIKE
23
BEHAVIOR
of g (within the errors involved in the program). That is, g at x = 0.19 and x = -0.19 are the same. This holds for the y and z axes, also. This property is not true for all .l’ik . For example, r& at P for B is 0.37411. At x = 0.19 for grid size 0.005, it is 0.38683. At x = -0.19 for the same grid size, it is 0.35303. Even the increment here is not reflection symmetric. On the other hand, there are other components of rjfi (which we shall soon discuss) that have the same reflection symmetry as g has.
IV. EXISTENCE OF ADDITIONAL
PARTICLE
FEATURES
In seeking as extremum in g, we have chosen a function that is easy to treat analytically. We can use gii , rjdlc also at each point to define a geometric representation of our theory (here, the location of points becomes a dynamical variable since ds2 is a function of x). In this representation, 2/-g dxl dx2 dx3 dx” is a volume element. Thus, we can think of g as a kind of density function. ag/ax” = 0, k = 1,2, 3 would then be a reasonable condition to impose if we want a particlelike object. At this point, we discuss an unexpected result that has emerged from study of the data coming out of the computer. This result appears for both particle A and B. We have found that 36 of the l-‘ji, have the same properties as g when one compares the results down the six axes with that of the origin. In all cases the magnitude of the values at the origin is less than the values for runs down the six axes. The 36r,“, are
cl
c2 c3 co
cl
c2
c3
CO
cl
r;,
r;,
CO
cl rt2 co co rzol rzoz rzo3 r,3, r301 r302 r303 co. The same l?& have this property for both particle A and B. (See Tables III and IV for details for the function I’i3 , which we have taken as representative of the 36.)
24
MURASKIN
AND
TABLE
CLARK
III
Behavior of I’& for Particle A r.& at origin = -0.53361
Axis x
-x Y -Y z -2
Value of ris
Value of Coordinate
-0.53403 -0.53402 -0.53539 -0.53402 -0.53539 -0.53530 -0.53530 -0.53478 -0.53418
0.19 0.19 0.39 0.19 0.39 0.19 0.19 0.19 0.19
TABLE
Grid Size 0.005
0.01 0.01 0.01 0.01 0.005 0.005 0.005 0.005
IV
Behavior of 7,$ for Particle B I’,‘, at origin = 0.22660
Axis X
-x
Y -Y 2 -2,
Value of r:, 0.22614 0.22673 0.22673 0.22710 0.22708 0.22874 0.22812 0.22710 0.22686 0.22686 0.22702 0.22702
Value of Coordinate
Grid Size
0.10 0.10 0.10 0.19 0.19 0.39 0.39 0.19 0.19 0.19 0.19 0.19
0.0025 0.005
0.01 0.005
0.01 0.005
0.01 0.005 0.005 0.005 0.005 0.005
The I’!k that do not have this property can be characterized as follows: those I’.& having zero for the first bottom index; those having the top index the same as the second bottom index (this group includes I$ , I’& , r& , I’k). We do not have, as yet, an understanding of this result, although we can suggest some significance for r;,kl . Let us consider a field of velocity vectors (for example,
PARTICLE-LIKE
BEHAVIOR
25
the normals to g = constant contour surfaces). We may suppose that the change of ui is described by in the same manner as other tensor functions [l]. Then
Thus, we have (2r;d,1 = I’:k - rLi)
aui --T= a9
au, axa
2ujqar;, .
(12)
The curl of the ui which measures rotation properties is given by l&1 . We note that a g particle existence implies existence of a I& particle as well. We have also found that the I’jk in (IO) are reflection symmetric. The significance of this is not clear. Thus, we have a model of a particle in which one kind of particle feature associated with g induces other particle-like properties to appear as a consequence of dynamics. (The derivatives of rji, are determined by the field equations.) We have not proved this result in generality since we have only considered the two examples, A and B. However, we feel confident that this holds for many choices of parameters at P, since our particular choice of parameters was largely arbitrary. We have evaluated aI’&/&‘, which is a representative example, analytically. We have found by means of this tedious direct calculation that this quantity is zero when ag/ax” = 0, k = 1,2, 3. The calculation involved the cancellation of hundreds of terms so we did not repeat the calculation for other components and for other derivatives.
V.
DISCUSSION
We have still not discussed the problem of particle size. This may be a difficult task due to the lack of scale in the field equations. Thus, the particle we have could be billions of points big, for all we know. Awaiting future work on the size problem, we have not made any concerted attempt to map out the particle in any detail. Eventually, we would like to study the particle as it evolves in time. Thus, we still have a long way to go before we can assesswhether the particle we have is realistic. Nevertheless, it is encouraging to find that a density particle induces other particle characteristics as a consequence of the dynamics. This result was rather unexpected since ag/a.xk = 0, k = 1,2, 3 gives but three conditions on the theory. We have found that 36 of the I’jk have behavior similar to g. Thus, a highly cooperative effect is clearly taking place.
26
MURASKIN AND CLARK ACKNOWLEDGMENT
One of the authors (M. M.) would like to thank Professor Max Dresden for his encouragement of our work.
REFERENCES 1. M.
MURASKIN,
A reexamination of unified field theory, Ann. Phys. (1970),27.
2. G. ROSEN, J. Math. Phys.
144(1934),425.
6 (1965),
1269;
M.
BORN
AND L. INFELD,
Proc. Roy. Sot. Ser. A
OF PHYSICS:
59,
19-26 (1970)
Particle-Like M. MURASKIN
Behavior AND T. CLARK
University of North Dakota, Grand Forks, North Dakota 58201 Received September 29, 1969
We study the field theory previously introduced by one of the authors (M. M.) by means of a finite difference approximation using a computer. We focus attention on particle-like behavior centered at an arbitrary origin. We find in the two examples under consideration that one kind of particle-like property induces other particle-like properties to also appear as a consequence of the dynamics.
1. INTRODUCTION
There are many criteria that may be used to characterize an elementary particle. For one thing, we may think of an elementary particle as denserthan the region around it. But this is clearly not the only property of an elementary particle. For example, an electron can be considered to be a spinning body. Thus, in any attempt to formulate a model of a particle it would be important to show that a condition on density of the particle also causes other particle properties to appear as well. In this paper, we are not at the point where we can talk of real particles like electrons. The study of particle structure is still rather in a primitive state. What we shall do is investigate a nonlinear field equation that has the desirable property of allowing for particle-like behavior. The equation we consider is I’$;, = 0. This equation was introduced in a previous paper [l] where some of its properties were studied. We found this equation can be motivated in a simple way. Furthermore, an extremum in the determinant of gij can be made to appear at an arbitrary origin. An extremum in a field component also characterizes the particle-like behavior obtained by Rosen and Born-Infeld [2] (in these cases, the field component is maximum). In this present work, we study this particle-like object in greater detail making use of a finite difference approximation to the field equations. This technique enables us to map out a solution to the equations using a computer. We are able, in this way, to gather some new results which are difficult to obtain by analytic 19
20
MURASKIN
AND
CLARK
methods due to the nonlinear complexities of the field equations. We find that the conditions on g induce other particle-like features to appear. Our present work is restricted to the neighborhood of the origin because of errors coming in from the approximation scheme. We are also restricted by the limitations of computer time. Thus, we have not entirely mapped out the particlelike object. It is the ultimate goal of the program to map out completely the particlelike object and, then, follow its development in time. Only then will we be in a position to gauge how close to a real particle we have come.
11. FINITE
DIFFERENCE
APPROXIMATION
The field equations that we study are
(1) and agik -
a2
p -
r&k
-
ri%h
=
O.
(2)
Thus, from
dg,,++”
and
drji, = %
dxz
we get rjk(Q)
= rjk(P)
+ {r;,r;+
riimr:
- r;rA,jdxZ
(3)
and
where dx” is the displacement between Q and P. The bracketed terms on the right side are evaluated in lowest order at the origin point P. Thus, the field at Q is obtained if we know the field at P. Once we have determined the field at Q from (3) and (4), we then procede to the next point in a similar fashion. In this way we may map out the field throughout space. The imput data is the field at the origin point. The scheme gives only an approximate solution to the field equations since the displacement between points is taken to be a finite number and not an infinitesimal. The smaller the grid size, the more accurate the results. It is also necessary to require that I’jk(P) obey consistency relations. If we consider the field at some point in space, it is necessary that the field be unique,
PARTICLE-LIKE
21
BEHAVIOR
independent of the path chosen from the origin point. In our previous work we found that this condition can be met by introducing 16eiu so that at P r,tk = eaieEjeykrEa,
(3
with r& obeying riArin - r;r;?
+ r,",r;
- r&rFo = 0.
We have shown that solutions to (6) exist defining a nontrivial III.
PARTICLE-LIKE
(6)
field theory.
BEHAVIOR
We can require that g (the determinent of gJ be an extremum at the origin point P by requiring that r& at P, obey rjt, = 0, k = 1, 2, 3. We have shown in our previous work that this condition can be met. We have considered two different particle-like structures which we have labeled particle A and particle B. The term particle is used loosely since, at this point, a g particle has not been shown to correspond to a real particle. We shall use the following values for the nonvanishing r;,, : rtl
= r;L2 = 0.5,
r,l, = riz = 0.2,
(7)
ri3 = rio = 0.3, r& = rio = 0.1. This choice enables us to satisfy the consistency conditions described by e’, = e21 = e31 = eel =
-0.62 0.8 0.3 0.6
(6). Particle A is
el, = e22 = e32 = eo2 =
-0.76 0.4 0.7 0.9
e13 = e23 = e33 = eo3 =
-0.7 0.5 0.8 0.1
el, = e20 = e30 = coo =
0.1 0.2 0.3 2.0.
(8)
el, = ez2 = e32 = e20 =
-0.6 0.6 0.3 0.9
e13 = ez3 = e33 = eo3 =
-0.5 0.1 0.5 0.8
el, = e20 = e30 = coo =
0.1 0.2 0.3 2.0.
(9)
Particle B is described by el, = e21 = e31 = eel =
-0.82 0.8 0.7 0.4
22
MURASKIN
AND
CLARK
These values obey Tit, = 0, k = 1,2,3. We have made runs down the x, y, z, -x, -y, --z axes. The results are tabulated in Tables I and II. We see in the tables that when the coordinates have, say, the value 0.19, then the value of g is smaller than the value of g at the origin point. The grid sizes are taken to be either 0.005 or 0.01. g itself is rather slowly varying close to the origin. We have made longer runs down the axes (x = 0.39, -x = 0.39), and the trends shown continue. We note, from the data, a reflection symmetry exists along the axes for the values TABLE
I
Behavior of g for Particle g at origin = -0.09948
Axis
x --x Y -Y z -2
Value
of g
Value
-0.09959 -0.09957 -0.10000 -0.09958 -0.10003 -0.10007 -0.10008 -0.09988 -0.09988
X
-x Y -Y z -2
of g
-0.06410 -0.06409 -0.06408 -0.06428 -0.06425 -0.06516 -0.06510 -0.06428 -0.06416 -0.06416 -0.06426 -0.06426
Grid
Size
0.005 0.01 0.01 0.01 0.01 0.005 0.005 0.005 0.005
II
Behavior of g for Particle g at origin = -0.06405
Value
of Coordinate 0.19 0.19 0.39 0.19 0.39 0.19 0.19 0.19 0.19
TABLE
Axis
A
Value
B
of Coordinate 0.10 0.10 0.10 0.19 0.19 0.39 0.39 0.19 0.19 0.19 0.19 0.19
Grid
Size
0.0025 0.005 0.01 0.005 0.01 0.005 0.01 0.005 0.005 0.005 0.005 0.005
PARTICLE-LIKE
23
BEHAVIOR
of g (within the errors involved in the program). That is, g at x = 0.19 and x = -0.19 are the same. This holds for the y and z axes, also. This property is not true for all .l’ik . For example, r& at P for B is 0.37411. At x = 0.19 for grid size 0.005, it is 0.38683. At x = -0.19 for the same grid size, it is 0.35303. Even the increment here is not reflection symmetric. On the other hand, there are other components of rjfi (which we shall soon discuss) that have the same reflection symmetry as g has.
IV. EXISTENCE OF ADDITIONAL
PARTICLE
FEATURES
In seeking as extremum in g, we have chosen a function that is easy to treat analytically. We can use gii , rjdlc also at each point to define a geometric representation of our theory (here, the location of points becomes a dynamical variable since ds2 is a function of x). In this representation, 2/-g dxl dx2 dx3 dx” is a volume element. Thus, we can think of g as a kind of density function. ag/ax” = 0, k = 1,2, 3 would then be a reasonable condition to impose if we want a particlelike object. At this point, we discuss an unexpected result that has emerged from study of the data coming out of the computer. This result appears for both particle A and B. We have found that 36 of the l-‘ji, have the same properties as g when one compares the results down the six axes with that of the origin. In all cases the magnitude of the values at the origin is less than the values for runs down the six axes. The 36r,“, are
cl
c2 c3 co
cl
c2
c3
CO
cl
r;,
r;,
CO
cl rt2 co co rzol rzoz rzo3 r,3, r301 r302 r303 co. The same l?& have this property for both particle A and B. (See Tables III and IV for details for the function I’i3 , which we have taken as representative of the 36.)
24
MURASKIN
AND
TABLE
CLARK
III
Behavior of I’& for Particle A r.& at origin = -0.53361
Axis x
-x Y -Y z -2
Value of ris
Value of Coordinate
-0.53403 -0.53402 -0.53539 -0.53402 -0.53539 -0.53530 -0.53530 -0.53478 -0.53418
0.19 0.19 0.39 0.19 0.39 0.19 0.19 0.19 0.19
TABLE
Grid Size 0.005
0.01 0.01 0.01 0.01 0.005 0.005 0.005 0.005
IV
Behavior of 7,$ for Particle B I’,‘, at origin = 0.22660
Axis X
-x
Y -Y 2 -2,
Value of r:, 0.22614 0.22673 0.22673 0.22710 0.22708 0.22874 0.22812 0.22710 0.22686 0.22686 0.22702 0.22702
Value of Coordinate
Grid Size
0.10 0.10 0.10 0.19 0.19 0.39 0.39 0.19 0.19 0.19 0.19 0.19
0.0025 0.005
0.01 0.005
0.01 0.005
0.01 0.005 0.005 0.005 0.005 0.005
The I’!k that do not have this property can be characterized as follows: those I’.& having zero for the first bottom index; those having the top index the same as the second bottom index (this group includes I$ , I’& , r& , I’k). We do not have, as yet, an understanding of this result, although we can suggest some significance for r;,kl . Let us consider a field of velocity vectors (for example,
PARTICLE-LIKE
BEHAVIOR
25
the normals to g = constant contour surfaces). We may suppose that the change of ui is described by in the same manner as other tensor functions [l]. Then
Thus, we have (2r;d,1 = I’:k - rLi)
aui --T= a9
au, axa
2ujqar;, .
(12)
The curl of the ui which measures rotation properties is given by l&1 . We note that a g particle existence implies existence of a I& particle as well. We have also found that the I’jk in (IO) are reflection symmetric. The significance of this is not clear. Thus, we have a model of a particle in which one kind of particle feature associated with g induces other particle-like properties to appear as a consequence of dynamics. (The derivatives of rji, are determined by the field equations.) We have not proved this result in generality since we have only considered the two examples, A and B. However, we feel confident that this holds for many choices of parameters at P, since our particular choice of parameters was largely arbitrary. We have evaluated aI’&/&‘, which is a representative example, analytically. We have found by means of this tedious direct calculation that this quantity is zero when ag/ax” = 0, k = 1,2, 3. The calculation involved the cancellation of hundreds of terms so we did not repeat the calculation for other components and for other derivatives.
V.
DISCUSSION
We have still not discussed the problem of particle size. This may be a difficult task due to the lack of scale in the field equations. Thus, the particle we have could be billions of points big, for all we know. Awaiting future work on the size problem, we have not made any concerted attempt to map out the particle in any detail. Eventually, we would like to study the particle as it evolves in time. Thus, we still have a long way to go before we can assesswhether the particle we have is realistic. Nevertheless, it is encouraging to find that a density particle induces other particle characteristics as a consequence of the dynamics. This result was rather unexpected since ag/a.xk = 0, k = 1,2, 3 gives but three conditions on the theory. We have found that 36 of the I’jk have behavior similar to g. Thus, a highly cooperative effect is clearly taking place.
26
MURASKIN AND CLARK ACKNOWLEDGMENT
One of the authors (M. M.) would like to thank Professor Max Dresden for his encouragement of our work.
REFERENCES 1. M.
MURASKIN,
A reexamination of unified field theory, Ann. Phys. (1970),27.
2. G. ROSEN, J. Math. Phys.
144(1934),425.
6 (1965),
1269;
M.
BORN
AND L. INFELD,
Proc. Roy. Sot. Ser. A