Post on 30-May-2018
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Can Mathematical Structure
and Physical Reality be the
Same Thing?An attempt to find the fine structure constant and other
fundamental constants in such a structure
Pinhas Ben-AvrahamOctober 2009
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N-dimensional Euclidean space
And
Yields
nkkrrCnrV
k
nn 2;
!)(),(
2
===
)1()1! 2 +=+(= nkk
)
)(),(
2
2 2
n
n
rnrV
+(1
=
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Volume of an n-dimensional sphere
1. Radius = 1
2. V (r, n)
5 10 15 20n
1
2
3
4
5
V
0 2 4 6 8 10
n
0
0.25
0.5
0.75
1
r
0
2
4
6
8
V
0.
0.5
0.75
1
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Acceleration introduces a velocity to a restingpoint, acceleration also needs to be introduced
by a jerkj = da/dt. This would produce thefollowing scenario: let us assume, |j| = 1, thena(t) =
0
1j dt= 1t=1, and v(t) = t2/2 with x(t) = t3/6.Vice versa, we need a mean jerk of 6 toreach length one in unit time.
If the acceleration is known as one, the integralof dvdtequals . If x = 1 and v = , then
their product will be half, with x = v2/2 from Fx =max = mv2/2 for starting from zero velocity andstatic zero position. Hence, x v = .
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Minimum mathematical uncertainty
2
2222
16
1)|)(|)(|)(|(
dvvfvdxxfx
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Volume of Reciprocal Sphere in n Dimensions andConditions at p = 1/2
Volume of momentum space in n dimensions, solved to p
)1(
)2
sin()1(||2
),(2
1221
n
n
p
nnp
npV
n
+
+=
+
01)1(
)sin()1(||)(22
2
2
12 221
=+
+
n
nnn nprn
nnnn
n
rnrp
n
n
++
=
+
11
2
22
2
)1(
)1()csc()(2),(
22
1
)1(2
3
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Solutions for p in 6 Dimensions
For r =
0.02
0.04
0.06
0.08
r
0
2
4
6n
0
0.1
0.2p
0
1
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Fractional Charges
1/3 and 2/3 of anelementar charge
0.0265
0.027
0.0275
0.028
r
0.4
0.6n
0
0.01
0.02p
0
0.03
0.04
0.05
r
0.250.5
0.751
1.25
1.5
n
0
0.05
0.1p
0
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Square Root of 1/137 in n Dimensions
Electric Charge Found in all RealDimensions
0.04
0.06
0.08
r
0
10
20n
0
0.05
0.1p
0
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Momentum or velocity densities within a spherical n-
dimensional space element Jungs smallest sphere that encloses an object with diameter 1
With
We obtain for p (r, n) = divided by V(R, n) and solved to alpha
)1(2 +=
n
nR
2
2
2
2 /2
r
rq
r
p
=
n
n
n
n
n
n
nn
n
n
/1
2
5.0
1
)1(2
31
25
2
)1(
2sin)1(2
22
)1(
)4/(
)(
2
+
+
++
+
+
+=
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Results for the Minima of Alpha
All fundamental constants lie on this curve
5 10 15 20 25
n
-20
-15
-10
-5
log r
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Acceleration and Jerk
The volumes of acceleration and jerk space are
( ) ( )
++
+
+
+
=
+
2
sin)()1(1
2
cos)1(1
2
sin)1()(||
)1(
1
2221
2
nasigni
nnnna
V
nnn
n
na
( ) ( )
( ) ( ) ][
2sin)1(1)(
2cos)1(1
2sin)1(1
2cos)1(1
2sin)1()(||
)1(2
1
22
22212
3
2
++
+
++
+
+
+
=
+
njsign
ni
ni
nnnnj
V
nn
nnn
n
nj
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Velocity and Acceleration
Elementar Charge, Velocity and Acceleration around 5 Dimensions
0.08
0.082
0.084
r
4.5 4.75 5 5.25 5.5
n
0
0.01
0.02
0.03
p
0
0.
0
0.08
0.082
0.084
r
4.5 4.75 5 5.25 5.5
n
0
5000
10000
15000
20000
a
0
50
1
1
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Analysis of Maximum Accelerations in nDimensions
For a (r, n) we obtain
n
n
nn
n
n
n
nn
n
n
n
n
nnnrinnnrin
nnn
rn
nnn
r
n
nra
12
222
2
22
22222
2
1
)]1(2
sin)()()1(22
sin)1()()(22
sin
)1()(2
cos)()1(22
sin)1()(2
cos)(2[
21
),(
+
++
+
+
+
=
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Acceleration in n Dimensions
For a (r, n) 6 we obtain
1.2
1.4
1.6
1.8
2
r
020
4060
80100
n
0
0.25
0.5
0.75
1
a
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Results
Tables of Results where to find interaction constantsDimensions Co-dimension n for p
max Min. Charge Charge
0 - 2 1.4217 0.72 0.02685 1
1.0875 0.64 2/3
0.24 0.525 1/3
4 - 6 1.1061 4.96 0.07826 1
Dimensional range 0 - 4 4 - 8 8 - 12 12 - 16 16 - 20 20 - 24
Interaction strong Electromag. weak spin spin gravitation
Numerical value r2 or r 9.98 1/137.036 8.310-4 1.310-10 510-16 4.1810-23
Purely real dimensions 0
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Conclusions
Is the presence ofzitterbewegunga necessaryrequirement for time asymmetry? If the answeris yes, this has far reaching consequences forhow we need to look at the physics of our
universe. Fractality and non-differentiability oftime-related spaces that we represented asFourier transforms can become a very simpleexplanation for time-asymmetry, uncertainty andsimilar features of the structure describing
physics of the universe, but building suchstructure still requires observation andinterpretation.
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Conclusions We have demonstrated the dependence of a purely mathematical
uncertainty on dimensionality. From geometrical considerations wehave arrived at numerical values for minimum space for movementand movement densities in n dimensions.
There is no such thing as Tegmarks Reality independent of anobserver.
We think we have shown a simplistic but viable example for a relativelynave mathematical structure and minimal conceptual input, what richnesslies in the structures (spherical spaces) transformations, if interpreted.Without such interpretation there is no way of recognizing such structure asa (simplified) physical reality, and such interpretation has to be made by an
observer. So, we come back to Wheelers signposts and the space betweenthem: only if all the space between them can be filled with certainty, we cansay we have a mathematical universe that is determinable without anobserver and his or her participation.