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Contents
1 Intro duction. 2
2 Feynman Checkerb oards. 3
3 Hyp erDiamond Lattices. 10
3.1 8-dimensional Hyp erDiamond Lattice. :: :: ::: :: :: :: 12
3.2 4-dimensional Hyp erDiamond Lattice. :: :: ::: :: :: :: 14
3.3 From 8 to 4 Dimensions. : ::: :: :: :: :: ::: :: :: :: 19
4 Internal Symmetry Space. 21
5 Protons, Pions, and Physical Gravitons. 25
5.1 3-Quark Protons. : :: :: ::: :: :: :: :: ::: :: :: :: 29
5.2 Quark-AntiQuark Pions. : ::: :: :: :: :: ::: :: :: :: 32
5.3 Spin-2 Physical Gravitons. ::: :: :: :: :: ::: :: :: :: 35 1
1 Intro duction.
The 1994 Georgia Tech Ph. D. thesis of Michael Gibbs under David Finkel-
stein [11] constructed a discrete 4-dimensional spacetime Hyp erDiamond lat-
tice, here denoted a 4HD lattice, in the course of building a physics mo del.
The 1992 MIT Ph. D. thesis of Hrvo je Hrgovcic under Tommasso To oli
[12] constructed a discrete 4-dimensional spacetime Minkowski lattice in the
course of building a simulation of solutions to the Dirac equation and other
equations.
Hrgovcic's lattice is closely related to the 4HD lattice.
Both the Gibbs mo del and the Hrgovcic simulation di er in signi cant
ways from the D D E mo del constructed from 3 3 o ctonion matrices
4 5 6
in hep-ph/9501252 and quant-ph/9503009 .
However, b oth theses have b een very helpful with resp ect to this pap er,
which generalizes the 2-dimensional Feynman checkerb oard [8, 9] to a phys-
ically realistic 4-dimensional spacetime.
In this 4-dimensional 4HD lattice generalized Feynman
Hyp erDiamond checkerb oard, the prop erties of the particles that move
around on the checkerb oard are determined by the physically
realistic D D E mo del constructed from 3 3 o ctonion matrices
4 5 6
in hep-ph/9501252 and quant-ph/9503009 .
The 4-dimensional Hyp erDiamond 4HD checkerb oard is related to
the 8-dimensional Hyp erDiamond 8HD = E in the same way
8
the 4-dimensional physical asso ciative spacetime is related to
the 8-dimensional o ctonionic spacetime in the D D E mo del,
4 5 6
so the Hyp erDiamond 4HD Feynman checkerb oard is fundamentally
the discrete version of the D D E mo del.
4 5 6
For more ab out o ctonions and lattices, see the works of Go e rey Dixon
[3, 4, 5, 6]. 2
2 Feynman Checkerb oards.
The 2-dimensional Feynman checkerb oard [8, 9] is a notably successful and
useful representation of the Dirac equation in 2-dimensional spacetime.
To build a Feynman 2-dimensional checkerb oard,
start with a 2-dimensional Diamond checkerb oard with
two future lightcone links and two past lightcone links at eachvertex.
The future lightcone then lo oks like
@I
@
If the 2-dimensional Feynman checkerb oard is
co ordinatized by the complex plane C :
the real axis 1 is identi ed with the time axis t;
the imaginary axis i is identi ed with the space axis x; and
p p
the two future lightcone links are (1= 2 )(1 + i) and (1= 2)(1 i).
2 2
In cylindrical co ordinates t; r with r = x ,
2 2 2 2
the Euclidian metric is t + r = t + x and
the Wick-Rotated Minkowski metric with sp eed of light c is
2 2 2 2
(ct) r =(ct) x .
For the future lightcone links to lie on
the 2-dimensional Minkowski lightcone, c =1.
Either link is taken into the other link by complex multiplication by i.
Now, consider a path in the Feynman checkerb oard.
At a given vertex in the path, denote the future lightcone link in
the same direction as the past path link by1,and
the future lightcone link in the (only p ossible) changed direction by i. 3
1
i
@I
@
The Feynman checkerb oard rule is that
if the future step at a vertex p oint of a given path is in a di erent direction
from the immediately preceding step from the past,
then the path at the p ointofchange gets a weightof im,
where m is the mass (only massive particles can change directions),and
is the length of a path segment.
Here I have used the Gersch [10] convention
of weighting each turn by im
rather than the Feynman [8, 9] convention
of weighting by+im, b ecause Gersch's convention gives a b etter nonrela-
tivistic limit in the isomorphic 2-dimensional Ising mo del [10].
HOW SHOULD THIS BE GENERALIZED
TO HIGHER DIMENSIONS?
2 2 0
The 2-dim future light-cone is the 0-sphere S = S = fi; 1g ,
with 1 representing a path step to the future in
the same direction as the path step from the past, and
i representing a path step to the future in a
(only 1 in the 2-dimensional Feynman checkerb oard lattice)
di erent direction from the path step from the past.
The 2-dimensional Feynman checkerb oard lattice spacetime can b e
represented by the complex numb ers C ,
with 1;i representing the two future lightcone directions and
1; i representing the two past lightcone directions.
Consider a given path in
the Feynman checkerb oard lattice 2-dimensional spacetime. 4
Atany given vertex on the path in the lattice 2-dimensional spacetime,
the future lightcone direction representing the continuation of the path
in the same direction can b e represented by1,and
the future lightcone direction representing the (only 1 p ossible)
change of direction can b e represented by i since either
of the 2 future lightcone directions can b e taken into the other
bymultiplication by i,
+ for a left turn and for a right turn.
If the path do es change direction at the vertex,
then the path at the p ointofchange gets a weightof im,
where i is the complex imaginary,
m is the mass (only massive particles can change directions), and
is the timelike length of a path segment, where the 2-dimensional sp eed of
light is taken to b e 1.
Here I have used the Gersch [10] convention
of weighting each turn by im
rather than the Feynman [8, 9] convention
of weighting by+im, b ecause Gersch's convention gives a b etter nonrela-
tivistic limit in the isomorphic 2-dimensional Ising mo del [10].
For a given path, let
C b e the total numb er of direction changes, and
c b e the cth change of direction, and
i b e the complex imaginary representing the cth change of direction.
C can b e no greater than the timelikecheckerb oard distance D
between the initial and nal p oints.
The total weight for the given path is then
Y Y
C C
im =(m) ( i)= ( im) (1)
0cC 0cC
The pro duct is a vector in the direction 1ori. 5
Let N (C ) b e the numb er of paths with C changes in direction.
The propagator amplitude for the particle to go from the initial vertex
to the nal vertex is the sum over all paths of the weights, that is the path
integral sum over all weighted paths:
X
C
N (C )( im) (2)
0C D
The propagator phase is the angle b etween
the amplitude vector in the complex plane and the complex real axis.
Conventional attempts to generalize the Feynman checkerb oard from
2-dimensional spacetime to k -dimensional spacetime are based on
the fact that the 2-dimensional future light-cone directions are
2 2 0
the 0-sphere S = S = fi; 1g.
The k -dimensional continuous spacetime lightcone directions are
k 2
the (k 2)-sphere S .
2
In 4-dimensional continuous spacetime, the lightcone directions are S .
Instead of lo oking for a 4-dimensional lattice spacetime, Feynman
0 2
and other generalizers went from discrete S to continuous S
for lightcone directions, and then tried to construct a weighting
2
using changes of directions as rotations in the continuous S , and
never (as far as I know) got any generalization that worked. 6
THE D D E MODEL 4HD HYPERDIAMOND
4 5 6
GENERALIZATION HAS DISCRETE LIGHTCONE DIRECTIONS.
If the 4-dimensional Feynman checkerb oard is co ordinatized by
the quaternions Q:
the real axis 1 is identi ed with the time axis t;
the imaginary axes i; j; k are identi ed with the space axes x; y ; z ; and
the four future lightcone links are
(1=2)(1 + i + j + k ),
(1=2)(1 + i j k ),
(1=2)(1 i + j k ), and
(1=2)(1 i j + k ).
In cylindrical co ordinates t; r
2 2 2 2
with r = x + y + z ,
2 2 2 2 2 2
the Euclidian metric is t + r = t + x + y + z and
the Wick-Rotated Minkowski metric with sp eed of light c is
2 2 2 2 2 2
(ct) r =(ct) x y z .
For the future lightcone links to lie on
p
the 4-dimensional Minkowski lightcone, c = 3.
Any future lightcone link is taken into any other future lightcone
link by quaternion multiplication by i, j ,ork.
For a given vertex on a given path,
continuation in the same direction can b e represented by the link 1, and
changing direction can b e represented by the
imaginary quaternion i; j; k corresp onding to
the link transformation that makes the change of direction.
Therefore, at a vertex where a path changes direction,
a path can b e weighted by quaternion imaginaries
just as it is weighted by the complex imaginary in the 2-dimensional case.
If the path do es change direction at a vertex, then
the path at the p ointofchange gets a weightof im, jm,or km
where i; j; k is the quaternion imaginary representing the change of direction, 7
m is the mass (only massive particles can change directions), and
p
3 is the timelike length of a path segment,
p
where the 4-dimensional sp eed of light is taken to b e 3.
For a given path,
let C b e the total numb er of direction changes,
c b e the cth change of direction, and
e b e the quaternion imaginary i; j; k representing the cth change of direction.
c
C can b e no greater than the timelikecheckerb oard distance D
between the initial and nal p oints.
The total weight for the given path is then
p p
Y Y
C
e m 3 =(m 3) ( e ) (3)
c c
0cC 0cC
Note that since the quaternions are not commutative,
the pro duct must b e taken in the correct order.
The pro duct is a vector in the direction 1, i, j ,or k. and
Let N (C ) b e the numb er of paths with C changes in direction.
The propagator amplitude for the particle to go
from the initial vertex to the nal vertex is
the sum over all paths of the weights,
that is the path integral sum over all weighted paths:
p
X Y
C
N (C )(m 3 ) ( e ) (4)
c
0C D 0cC
The propagator phase is the angle b etween
the amplitude vector in quaternionic 4-space and
the quaternionic real axis. 8
The plane in quaternionic 4-space de ned by
the amplitude vector and the quaternionic real axis
can b e regarded as the complex plane of the propagator phase. 9
3 Hyp erDiamond Lattices.
The name "Hyp erDiamond" was rst used byDavid Finkelstein in our dis-
cussions of these structures.
n-dimensional Hyp erDiamond structures nH D are
constructed from D lattices.
n
An n-dimensional Hyp erDiamond structures nH D is a lattice
if and only if n is even.
If n is o dd, the nH D structure is
only a "packing", not a "lattice", b ecause a nearest neighb or link
from an origin vertex to a destination vertex
cannot b e extended in the same direction
to get another nearest neighb or link.
n-dimensional Hyp erDiamond structures nH D are
constructed from D lattices.
n
The lattices of typ e D are n-dimensional checkerb oard lattices, that is,
n
n
the alternate vertices of a Z hyp ercubic lattice. A general reference on
lattices is Conway and Sloane [1].
For the n-dimensional Hyp erDiamond lattice construction from D ,
n
Conway and Sloane use an n-dimensional glue vector [1] = (0:5; :::; 0:5)
(with n 0:5's).
Consider the 3-dimensional structure 3HD.
Start with D , the fcc close packing in 3-space.
3
Make a second D shifted by the glue vector (0:5; 0:5; 0:5).
3
Then form the union D [ ([1] + D ).
3 3
That is a 3-dimensional Diamond crystal structure,
the familiar 3-dimensional thing 10
for which Hyp erDiamond lattices are named. 11
3.1 8-dimensional Hyp erDiamond Lattice.
When you construct an 8-dimensional Hyp erDiamond 8HD lattice,
you get D [([1]+D )=E , the fundamental lattice of the o ctonion structures
8 8 8
in the D D E mo del describ ed in hep-ph/9501252 [14] and quant-
4 5 6
ph/9503009. [15 ]
The 240 nearest neighb ors to the origin in the E lattice can b e written
8
in 7 di erentways using
o ctonion co ordinates with basis
f1;e ;e ;e ;e ;e ;e ;e g (5)
1 2 3 4 5 6 7
One way is:
16 vertices:
1; e ; e ; e ; e ; e ; e ; e (6)
1 2 3 4 5 6 7
96 vertices:
(1 e e e )=2
1 2 3
(1 e e e )=2
2 5 7
(1 e e e )=2
2 4 6
(7)
(e e e e )=2
4 5 6 7
(e e e e )=2
1 3 4 6
(e e e e )=2
1 3 5 7
128 vertices:
(1 e e e )=2
3 4 7
(1 e e e )=2
1 5 6
(1 e e e )=2
3 6 7
(1 e e e )=2
1 4 7
(8)
(e e e e )=2
1 2 6 7
(e e e e )=2
2 3 4 7
(e e e e )=2
1 2 4 5
(e e e e )=2
2 3 5 6
That the E lattice is, in a sense, fundamentally 4-dimensional can b e
8
seen from several p oints of view: 12
the E lattice nearest neighbor vertices have only 4 non-zero co ordinates,
8
p
like 4-dimensional spacetime with sp eed of light c = 3 , rather than 8 non-
p
zero co ordinates, like 8-dimensional spacetime with sp eed of light c = 7, so
the E lattice light-cone structure app ears to b e 4-dimensional rather than
8
8-dimensional;
the representation of the E lattice by quaternionic icosians, as describ ed
8
by Conway and Sloane [1];
the Golden ratio construction of the E lattice from the D lattice, which
8 4
has a 24-cell nearest neighb or p olytop e (The construction starts with the
24 vertices of a 24-cell, then adds Golden ratio p oints on each of the 96
edges of the 24-cell, then extends the space to 8 dimensions by considering
p
the algebraicaly indep endent 5 part of the co ordinates to b e geometrically
indep endent, and nally doubling the resulting 120 vertices in 8-dimensional
space (by considering b oth the D lattice and its dual D ) to get the 240
4
4
vertices of the E lattice nearest neighb or p olytop e (the Witting p olytop e);
8
and
the fact that the 240-vertex Witting p olytop e, the E lattice nearest
8
neighb or p olytop e, most naturally lives in 4 complex dimensions, where it is
self-dual, rather than in 8 real dimensions.
Some more material on such things can b e found at
WWW URL http://www.gatech.edu/tsmith/home.html [13].
In referring to Conway and Sloane [1], b ear in mind that they use the
convention (usual in working with lattices) that the norm of a lattice distance
is the square of the length of the lattice distance. 13
3.2 4-dimensional Hyp erDiamond Lattice.
The 4-dimensional Hyp erDiamond lattice 4HD is
4HD = D [ ([1] + D ).
4 4
4
The 4-dimensional Hyp erDiamond 4HD = D [ ([1] + D )istheZ
4 4
hyp ercubic lattice with null edges.
It is the lattice that Michael Gibbs [11] uses in his Ph. D. thesis advised
byDavid Finkelstein.
The 8 nearest neighb ors to the origin in the 4-dimensional Hyp erDiamond
4HD lattice can b e written in o ctonion co ordinates as:
(1 + i + j + k )=2
(1 + i j k )=2
(1 i + j k )=2
(1 i j + k )=2
(9)
( 1 i + j + k )=2
( 1+i j +k)=2
( 1+i+j k)=2
( 1 i j k)=2
Here is an explicit construction of the 4-dimensional Hyp erDiamond 4HD
lattice nearest neighb ors to the origin. 14
START WITH THE 24 VERTICES OF A 24-CELL D :
4
+1 +1 0 0
+1 0 +1 0
+1 0 0 +1
+1 1 0 0
+1 0 1 0
+1 0 0 1
1 +1 0 0
1 0 +1 0
1 0 0 +1
1 1 0 0
1 0 1 0
1 0 0 1
(10)
0 +1 +1 0
0 +1 0 +1
0 +1 1 0
0 +1 0 1
0 1 +1 0
0 1 0 +1
0 1 1 0
0 1 0 1
0 0 +1 +1
0 0 +1 1
0 0 1 +1
0 0 1 1 15
SHIFT THE LATTICE BY A GLUE VECTOR,
BY ADDING
0:5 0:5 0:5 0:5 (11)
TO GET 24 MORE VERTICES [1] + D :
4
+1:5 +1:5 0:5 0:5
+1:5 0:5 +1:5 0:5
+1:5 0:5 0:5 +1:5
+1:5 0:5 0:5 0:5
+1:5 0:5 0:5 0:5
+1:5 0:5 0:5 0:5
0:5 +1:5 0:5 0:5
0:5 0:5 +1:5 0:5
0:5 0:5 0:5 +1:5
0:5 0:5 0:5 0:5
0:5 0:5 0:5 0:5
0:5 0:5 0:5 0:5
(12)
0:5 +1:5 +1:5 0:5
0:5 +1:5 0:5 +1:5
0:5 +1:5 0:5 0:5
0:5 +1:5 0:5 0:5
0:5 0:5 +1:5 0:5
0:5 0:5 0:5 +1:5
0:5 0:5 0:5 0:5
0:5 0:5 0:5 0:5
0:5 0:5 +1:5 +1:5
0:5 0:5 +1:5 0:5
0:5 0:5 0:5 +1:5
0:5 0:5 0:5 0:5 16
FOR THE NEW COMBINED LATTICE D [ ([1] + D ),
4 4
THESE ARE 6 OF THE NEAREST NEIGHBORS
TO THE ORIGIN:
0:5 0:5 0:5 0:5
0:5 0:5 0:5 0:5
0:5 0:5 0:5 0:5
(13)
0:5 0:5 0:5 0:5
0:5 0:5 0:5 0:5
0:5 0:5 0:5 0:5
HERE ARE 2 MORE THAT COME FROM
ADDING THE GLUE VECTOR TO LATTICE VECTORS
THAT ARE NOT NEAREST NEIGHBORS OF THE ORIGIN:
0:5 0:5 0:5 0:5
(14)
0:5 0:5 0:5 0:5
THEY COME, RESPECTIVELY, FROM ADDING
THE GLUE VECTOR TO:
THE ORIGIN
0 0 0 0 (15)
ITSELF;
AND 17
THE LATTICE POINT
1 1 1 1 (16)
WHICH IS SECOND ORDER, FROM
1 1 0 0
pl us (17)
0 0 0 0
FROM
1 0 1 0
pl us (18)
0 1 0 1
OR FROM
1 0 0 1
pl us
(19)
0 1 1 0 18
3.3 From 8 to 4 Dimensions.
Dimensional reduction of the 8HD = E lattice spacetime to 4-dimensional
8
spacetime reduces eachoftheD lattices in the
8
E =8HD = D [ ([1] + D )
8 8 8
lattice to D lattices.
4
Therefore, we should get a 4-dimensional Hyp erDiamond
4HD = D [ ([1] + D )
4 4
lattice.
To see this, start with E =8HD = D [ ([1] + D ).
8 8 8
We can write:
D = f(D ; 0; 0; 0; 0)g[f(0; 0; 0; 0;D )g[ f(1; 0; 0; 0; 1; 0; 0; 0) + (D ;D )g
8 4 4 4 4
(20)
The third term is the diagonal term of
an orthogonal decomp osition of D , and
8
the rst two terms are the orthogonal
asso ciativephysical 4-dimensional spacetime
and
coasso ciative 4-dimensional internal symmetry space
as describ ed in
StandardModel plus Gravity from
Octonion Creators and Annihilators,
quant-th/9503009 [15].
Now, we see that the orthogonal decomp osition of 8-dimensional space-
time into 4-dimensional asso ciativephysical spacetime plus 4-dimensional
internal symmetry space gives a decomp osition of D into D D .
8 4 4 19
Since E = D [ ([1] + D ), and
8 8 8
since [1] = (0:5; 0:5; 0:5; 0:5; 0:5; 0:5; 0:5; 0:5) can
b e decomp osed by
[1] = (0:5; 0:5; 0:5; 0:5; 0; 0; 0; 0) (0; 0; 0; 0; 0:5; 0:5; 0:5; 0:5) (21)
wehave
E = D [([1] + D ) (22)
8 8 8
= ((D ; 0; 0; 0; 0) (0; 0; 0; 0;D ))
4 4
[
(((0:5; 0:5; 0:5; 0:5; 0; 0; 0; 0) (0; 0; 0; 0; 0:5; 0:5; 0:5; 0:5))
+
((D ; 0; 0; 0; 0) (0; 0; 0; 0;D )))
4 4
= ((D ; 0; 0; 0; 0) [ ((0:5; 0:5; 0:5; 0:5; 0; 0; 0; 0) + (D ; 0; 0; 0; 0)))
4 4
((0; 0; 0; 0;D ) [ ((0; 0; 0; 0; 0:5; 0:5; 0:5; 0:5) + (0; 0; 0; 0;D )))
4 4
Since 4HD is D [ ([1] + D ),
4 4
E =8HD =4HD 4HD (23)
8 a ca
where 4HD is the 4-dimensional asso ciativephysical spacetime and
a
4HD is the 4-dimensional coasso ciativeinternal symmetry space.
ca 20
4 Internal Symmetry Space.
4HD is the 4-dimensional coasso ciativeInternal Symmetry Space of the 4-
ca
dimensional Hyp erDiamond Feynman checkerb oard version of the D D
4 5
E mo del.
6
Physically, the 4HD Internal Space should b e thought of as a space
ca
"inside" eachvertex of the 4HD Feynman checkerb oard spacetime, sort of
a
like a Kaluza-Klein structure.
The 4 dimensions of the 4HD Internal Symmetry Space are:
ca
electric charge;
red color charge;
green color charge; and
blue color charge.
Eachvertex of the 4HD lattice has 8 nearest neighb ors, connected by
ca
lightcone links. They have the algebraic structure of the 8-element quaternion
group < 2; 2; 2 >. [2]
Eachvertex of the 4HD lattice has 24 next-to-nearest neighb ors, con-
ca
nected bytwo lightcone links. They have the algebraic structure of the
24-element binary tetrahedral group < 3; 3; 2 > that is asso ciated with the
24-cell and the D lattice. [2]
4 21
The 1-time and 3-space dimensions of the 4HD spacetime can b e repre-
a
sented by the 4 future lightcone links and the 4 past lightcone links as in the
following pair of "Square Diagrams" of the 4 lines connecting the future ends
of the 4 future lightcone links and of the 4 lines connecting the past ends of
the 4 past lightcone links:
Z 1 i j +k Y 1 i+j k
T 1+i+j +k X 1+i j k
X 1 i+j +k T 1 i j k
Y 1+i j +k Z 1+i+j k
The 8 links fT ; X; Y ; Z; T; X; Y; Zg corresp ond to the 8 ro ot vec-
tors of the S pin(5) de Sitter gravitation gauge group, which has an 8-element
2
Weyl group S S .
2
2
The symmetry group of the 4 links of the future lightcone is S , the Weyl
4
group of the 15-dimensional Conformal group SU (4) = S pin(6).
10 of the 15 dimensions make up the de Sitter S pin(5) subgroup, and
the other 5 x the "symmetry-breaking direction" and scale of the Higgs
mechanism (for more on this, see [14] and
WWW URL http://www.gatech.edu/tsmith/cnfGrHg.html). 22
Similarly, the E-electric and RGB-color dimensions of the 4HD Internal
ca
Symmetry Space can b e represented by the 4 future lightcone links and the
4 past lightcone links.
However, in the 4HD Internal Symmetry Space the E Electric Charge
ca
should b e treated as indep endent of the RGB Color Charges.
As a result the following pair of "Square Diagrams" lo ok more like"Tri-
angle plus Point Diagrams".
B 1 i j +k G 1 i+j k
@I
@
@
@
@
@
@
@
@
@
E 1+i+j +k R 1+i j k
R 1 i+j +k E 1 i j k
@
@
@
@
@
@
@
@
@
@
G 1+i j +k B 1+i+j k 23
The 2+6 links fE; E; R; G; B ; R; G; B g corresp ond to:
the2rootvectors of the weak force SU (2), which has a 2-elementWeyl
group S ; and
2
the 6 ro ot vectors of the color force SU (3), which has a 6-elementWeyl
group S .
3 24
5 Protons, Pions, and Physical Gravitons.
In his 1994 Georgia Tech Ph. D. thesis under David Finkelstein, Spacetime as
a Quantum Graph, Michael Gibbs [11 ] describ es some 4-dimensional Hyp er-
Diamond lattice structures, that he considers likely candidates to represent
physical particles.
The terminology used by Michael Gibbs in his thesis [11] is useful with
resp ect to the mo del he constructs. Since his mo del is substantially di erent
from my Hyp erDiamond Feynman checkerb oard in some resp ects, I use a
di erent terminology here. However, I wanttomake it clear that I have
b orrowed these particular structures from his thesis.
Three useful Hyp erDiamond structures are:
3-link Rotating Propagator, useful for building a proton out of 3 quarks;
2-link Exchange Propagator, useful for building a pion out of a quark and
an antiquark; and
4-link Propagator, useful for building a physical spin-2 physical graviton
out of S pin(5) Gauge b osons..
In the 2-dimensional Feynman checkerb oard, there is only one massive
particle, the electron.
What ab out the D D E mo del, or any other mo del that has di erent
4 5 6
particles with di erent masses?
In the context of Feynman checkerb oards, mass is just the amplitude for
a particle to haveachange of direction in its path.
More massive particles will change direction more often.
In the D D E mo del, the 4HD Feynman checkerb oard fundamental
4 5 6
path segment length of any particle the Planck length L .
PL
However, in the sum over paths for a particle of mass m, 25
it is a useful approximation to consider the path segment length to b e the
Compton wavelength L of the mass m,
m
L = h=mc
m
That is b ecause the distances b etween direction changes in the vast bulk
of the paths will b e at least L , and
m
those distances will b e approximately integral multiples of L ,
m
so that L can b e used as the e ective path segment length.
m
This is an imp ortant approximation b ecause the Planck length L is
PL
33
ab out 10 cm,
while the e ective length L for a particle of mass 100 GeV is ab out
100GeV
16
10 cm.
In this section, the 4HD lattice is given quaternionic co ordinates.
The orgin 0 designates the b eginning of the path.
The 4 future lightcone links from the origin are given the co ordinates
1+i+j +k,1+i j k,1 i+j k,1 i j+k
The path of a "Particle" at rest in space, moving 7 steps in time, is
denoted by
P ar ticl e
0
1
2
3
4
5
6
7
p p
Note that since the 4HD sp eed of lightis 3, the path length is 7 3. 26
The path of a "Particle" moving along a lightcone path in the 1 + i + j + k
direction for 7 steps with no change of direction is
P ar ticl e
0
1+i+j +k
2+2i+2j +2k
3+3i+3j +3k
4+4i+4j +4k
5+5i+5j +5k
6+6i+6j +6k
7+7i+7j +7k
At each step in either path, the future lightcone can b e represented bya
"Square Diagram" of lines connecting the future ends of the 4 future lightcone
links leading from the vertex at which the step b egins.
1 i j +k 1 i+j k
1+i+j +k 1+i j k
In the following subsections, protons, pions, and physical gravitons will
b e represented bymultiparticle paths.
The multiple particles representing protons, pions, and physical gravitons will
b e shown on sequences of such Square Diagrams, as well as by a sequence of co ordinates.
The co ordinate sequences will b e given only for a representative sequence of
timelike steps, with no space movement, b ecause
the notation for a timelike sequence is clearer and
it is easy to transform a sequence of timelike steps into a sequence of lightcone
link steps, as shown ab ove.
Only in the case of gravitons will it b e useful to explicitly discuss a path
that moves in space as well as time. 28
5.1 3-Quark Protons.
In this 4HD Feynman checkerb oard version of the D D E mo del,
4 5 6
protons are made up of 3 valence rst generation quarks, (two up quarks and
one down quark), with one Red, one Green, and one Blue in color.
The proton b ound state of 3 valence quarks has a soliton structure (see
WWW URL http://www.gatech.edu/tsmith/SolProton.html [13 ]).
The complicated structure of the sea quarks and binding gluons can b e
ignored in a 4HD Feynman checkerb oard approximation that uses the R, G,
and B valence quarks and path length L that is the Compton wave-
313GeV
length of the rst generation quark constituent mass.
The 4HD structure used to approximate the proton is the 3-link Rotating
Propagator of the thesis of Michael Gibbs [11]. 29
Here is a co ordinate sequence representation of the approximate 4HD
Feynman checkerb oard path of a proton:
R Quar k G Quar k B Quar k
1+i j k 1 i+j k 1 i j +k
2 i+j k 2 i j +k 2+i j k
3 i j +k 3+i j k 2 i+j k
4+i j k 4 i+j k 4 i j +k
5 i+j k 5 i j +k 5+i j k
6 i j +k 6+i j k 6 i+j k
7+i j k 7 i+j k 7 i j +k
The following page contains a Square Diagram representation of the ap-
proximate 4HD Feynman checkerb oard path of a proton: 30
R 1 i j +k G 1 i+j k
1+i+j +k B 1+i j k
B 1 i j +k R 1 i+j k
1+i+j +k G 1+i j k
G 1 i j +k B 1 i+j k
1+i+j +k R 1+i j k
R 1 i j +k G 1 i+j k
31
1+i+j +k B 1+i j k
5.2 Quark-AntiQuark Pions.
In this 4HD Feynman checkerb oard version of the D D E mo del, pions
4 5 6
are made up of rst generation valence Quark-AntiQuark pairs.
The pion b ound state of valence Quark-AntiQuark pairs has a soliton struc-
ture that would, pro jected onto a 2-dimensional spacetime, b e a Sine-Gordon
breather (see WWW URL http://www.gatech.edu/tsmith/SnGdnPion.html
[13]).
The complicated structure of the sea quarks and binding gluons can b e
ignored in a 4HD Feynman checkerb oard approximation that uses the va-
lence Quark and AntiQuark and path length L that is the Compton
313GeV
wavelength of the rst generation quark constituent mass.
The 4HD structure used to approximate the pion is the 2-link Exchange
Propagator. 32
Here is a co ordinate sequence representation of the approximate 4HD
Feynman checkerb oard path of a pion:
Quar k AntiQuar k
0 0
1+i+j +k 1 i+j k
2 2
3+i j k 3 i j +k
4 4
5 i+j k 5+i+j +k
6 6
7 i j +k 7+i j k
8 8
9+i+j +k 9 i+j k
10 10
11 + i j k 11 i j + k
12 12
13 i + j k 13 + i + j + k
14 14
15 i j + k 15 + i j k
16 16
The following page contains a Square Diagram representation of the ap-
proximate 4HD Feynman checkerb oard path of a pion: 33
1 i j +k Q 1 i+j k
Q 1+i+j +k 1+i j k
Q 1 i j +k 1 i+j k
1+i+j +k Q 1+i j k
1 i j +k Q 1 i+j k
Q 1+i+j +k 1+i j k
Q 1 i j +k 1 i+j k
34
1+i+j +k Q 1+i j k
5.3 Spin-2 Physical Gravitons.
In this 4HD Feynman checkerb oard version of the D D E mo del, spin-2
4 5 6
physical gravitons are made up of the 4 translation spin-1 gauge b osons of
the 10-dimensional S pin(5) de Sitter subgroup of the 15-dimensional S pin(6)
Conformal group used to construct Einstein-Hilb ert gravityintheD D
4 5
E mo del describ ed in hep-ph/9501252 and quant-ph/9503009 .
6
These spin-2 physical gravitons are massless, but they can have energy
up to and including the Planck mass.
The Planck energy spin-2 physical gravitons are really fundamental struc-
tures with 4HD Feynman checkerb oard path length L .
P lanck 35
Here is a co ordinate sequence representation of
the 4HD Feynman checkerb oard path of a fundamental
Planck-mass spin-2 physical graviton,
where T; X; Y; Z represent the 4 translation
generator of the S pin(5) de Sitter group:
T X Y Z
0 0 0 0
1+i+j +k 1+i j k 1 i+j k 1 i j +k
2 2 2 2
3+i+j +k 3+i j k 3 i+j k 3 i j +k
4 4 4 4
5+i+j +k 5+i j k 5 i+j k 5 i j +k
6 6 6 6
7+i+j +k 7+i j k 7 i+j k 7 i j +k
8 8 8 8
The following is a Square Diagram representation of the 4HD Feynman
checkerb oard path of a fundamental Planck-mass spin-2 physical graviton:
Z 1 i j +k Y 1 i+j k
T 1+i+j +k X 1+i j k
The representation ab ove is for a timelike path at rest in space. 36
With resp ect to gravitons, we can see something new and di erentby
letting the path move in space as well.
Let T and Y represent time and longitudinal space, and
X and Z represent transverse space.
Then, as discussed in Feynman's Lectures on Gravitation, pp. 41-42 [7],
the Square Diagram representation shows that our spin-2 physical graviton
is indeed a spin-2 particle.
transverse Z 1 i j + k longitudinal Y 1 i + j k
@
@
@
@
@R
@I
@
@
@
@
time T 1+i+j +k transverse X 1+i j k
Spin-2 physical gravitons of energy less than the Planck mass are more
complicated comp osite gauge b oson structures with approximate 4HD Feyn-
man checkerb oard path length L .
g r av itonener g y
They can b e deformed from a square shap e, but retain their spin-2 nature
as describ ed byFeynman [7]. 37
References
[1] J. Conway and N. Sloane, Sphere Packings, Lattices, and Groups, 2nd
ed, Springer-Verlag (1993).
[2] H. Coxeter, Regular Complex Polytopes, 2nd ed, Cambridge (1991).
[3] G. Dixon, Division algebras: octonions, quaternions, complex numbers,
and the algebraic design of physics, Kluwer (1994).
[4] G. Dixon, Octonion X-product Orbits, hep-th/9410202 .
[5] G. Dixon, Octonion X-product and
Octonion E Lattices, hep-th/9411063.
8
[6] G. Dixon, Octonions: E Latticeto , hep-th/9501007.
8 16
[7] R. Feynman, Lectures on Gravitation, 1962-63, Caltech (1971).
[8] R. Feynman, Notes on the Dirac Equation, Caltech archives, b ox 13,
folder 6 (1946).
[9] R. Feynman and Hibbs, Quantum Mechanics and Path Integrals,
McGraw-Hill (1965).
[10] H. Gersch, Int. J. Theor. Phys. 20 (1981) 491.
[11] M. Gibbs, Spacetime as a Quantum Graph, Ph. D. thesis of J. Michael
Gibbs, with advisor David Finkelstein, Georgia Tech (1994).
[12] H. Hrgovcic, Quantum mechanics on space-time lattices using path in-
tegrals in a Minkowski metric, MIT Ph. D. thesis under T. To oli, July
1992.
[13] F. Smith, WWW URL http://www.gatech.edu/tsmith/home.html.
[14] F. Smith, Gravity and the StandardModel with 130 GeV Truth Quark
from D D E Model using 3 3 Octonion Matrices, preprint:
4 5 6
THEP-95-1; hep-ph/9501252.
[15] F. Smith, StandardModel plus Gravity from Octonion Creators and An-
nihilators, preprint: THEP-95-2; quant-th/9503009. 38