Mathematical Structure And Physical Reality

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-Avraham October 2009

N-dimensional Euclidean space 2 k (π ⋅ r ) n V ( r , n) = C n r = ; k = 2n k! And

k!= Γ( k + 1) = Γ( + 1) n 2

Yields

n 2

(π ⋅ r ) V ( r , n) = Γ(1 + n2 ) 2

Volume of an n-dimensional sphere 1. 2.

Radius = 1 V (r, n) V 5 4 3 2 1 5

10

15

20

n

1 0.75 0.5

r

0.25 0 8 6 V

4 2 0 0

2

4

6 n

8

10

• Acceleration introduces a velocity to a resting point, acceleration also needs to be introduced by a “jerk” j = da/dt. This would produce the following scenario: let us assume, |j| = 1, then a(t) = ∫01j dt = 1t =1, and v(t) = t2/2 with x(t) = t3/6. Vice versa, we need a mean jerk <j> of 6 to reach length one in unit time. • If the acceleration is known as one, the integral of dvdt equals ½. If Δx = 1 and Δv = ½, then their product will be half, with x = v2/2 from Fx = max = mv2/2 for starting from zero velocity and static zero position. Hence, Δx Δv = ½.

• Minimum mathematical uncertainty

1 ( ∫ ∆x | f ( x) | dx )( ∫ ∆v | f (v) | dv) ≥ −∞ −∞ 16π 2 ∞

2

2

∞

2

∧

2

Volume of Reciprocal Sphere in n Dimensions and Conditions at p = 1/2 •

Volume of momentum space in n dimensions, solved to p

2π

V p ( p, n) = − 2 2π

1 −n 2

2

(r )

−n 2

− 12 + n2

nπ |p | Γ(1 + n) sin( ) 2 Γ(1 + n2 ) −1− n

| p −1− n | Γ(1 + n) sin( n2π ) −1 = 0 n Γ(1 + 2 )

 π 2 − n (r 2 ) 2 csc( nπ )Γ(1 + n ) 2  2 2 ±    Γ(1 + n)   1

p ( r , n) = 2

3 2 ( 1+ n )

−n

1 −1− n

Solutions for p ≥ ½ in 6 Dimensions •

For r =

α

0.02

0.04

0.06 0.2 p 0.1 0 0

0.08 2 n

4 6

r

Fractional Charges • 1/3 and 2/3 of an elementar charge 0.0265

0.03

0.027

0.04 r

0.0275 r

0.028

0.02 p 0.01

0.05

p0.1 0.05

0

0 0.25

0.4 n

0.6

0.5

0.75

1 n

1.25

1.5

Square Root of 1/137 in n Dimensions •

Electric Charge Found in all Real Dimensions

0.04

0.06

p0.1 0.05

0.08

0 0 10 n

20

r

Momentum or velocity densities within a spherical ndimensional space element •

Jung’s smallest sphere that encloses an object with diameter 1

R=

n 2(n + 1)

•

With

•

We obtain for p (r, n) = ½ divided by V(R, n) and solved to alpha

p 2 2αq 2 / r = 2 r r2

 (π / 4)    n  Γ(1 + 2 )    n 2

α ( n) =

5

2

  1+ 3 1+ n  nπ   2(1+ n )  − 0 .5 + n 2 π Γ ( 1 + n ) sin     2    2 2  Γ(1 + n2 )   

       

−1 / n

Results for the Minima of Alpha • All fundamental constants lie on this curve log r

5 -5

-10

-15

-20

10

15

20

25

n

Acceleration and Jerk •

The volumes of acceleration and jerk space are Va = −

1 × Γ(1 + n2 )

 −1+ n2 n  nπ π | a | Γ ( − n ) Γ ( 1 + n ) sin    2 

Vj =

[

× π

(

1 2Γ(1 + n2 ) −3+ n 2

(

)

(

)

  nπ   nπ −2 n −2 n cos  sign (a ) sin   1 + (−1)  + i − 1 + (−1)   2   2

      

×

 nπ | j −1− n | Γ(− n)Γ(1 + n) 2 sin   2

)

(

)

(

(

)

  nπ   nπ −2n −2n cos sin  1 + (−1)  + i − 1 + (−1)   2   2

)

]

  nπ   nπ   2n ×  i − 1 + (−1) 2 n cos  sign( j ) + 1 + (−1) sin   2 2     

   × 

Velocity and Acceleration •

Elementar Charge, Velocity and Acceleration around 5 Dimensions

0.08

0.08

r 0.082

r 0.082

0.084

0.084

20000 0.03

15000 a 10000

0.02 p

5000

0.01 0 4.5

4.75

5 n

5.25

5.5

0 4.5

4.75

5 n

5.25

5.5

Analysis of Maximum Accelerations in n Dimensions •

For a (r, n) we obtain

π

a ( r , n) =

1 n

 n Γ 1 +   2

−2 n

×

n 2

n

 nπ   nπ   nπ  × [− 2π (r ) cos  Γ (− n)Γ (1 + n) sin  − 2 (− 1) − 2 n π n (r 2 ) 2 cos  Γ (− n)Γ (1 + n)  2   2   2  n

2

n

2

n

2

−1

 nπ   nπ   nπ  sin  + 2iπ n (r 2 ) 2 Γ (− n)Γ (1 + n) sin  − 2i (− 1) − 2 n π n (r 2 ) 2 Γ (− n) sin  Γ (1 + n)] n  2   2   2 

Acceleration in n Dimensions •

For a (r, n) ≥ 6 we obtain 1.2 1.4 r 1.6 1.8

2 1 0.75 0.5 a 0

0.25 20

40

60 n

80

0 100

Results •

Tables of Results where to find interaction constants Dimensions

Co-dimension

n for pmax

Min. Charge

Charge

0-2

1.4217

0.72

0.02685

1

1.0875

0.64

2/3

0.24

0.525

1/3

1.1061

4.96

4-6

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 √α r2 or αr

9.98

1/√137.036

8.3×10-4

1.3×10-10

5×10-16

4.18×10-23

Purely real dimensions

0
4
8
12
16
20
Conclusions • Is the presence of zitterbewegung a necessary requirement for time asymmetry? – If the answer is yes, this has far reaching consequences for how we need to look at the physics of our universe. Fractality and non-differentiability of time-related spaces that we represented as Fourier transforms can become a very simple explanation for time-asymmetry, uncertainty and similar features of the structure describing physics of the universe, but building such structure still requires observation and interpretation.

Conclusions •

We have demonstrated the dependence of a purely mathematical uncertainty on dimensionality. From geometrical considerations we have arrived at numerical values for minimum space for movement and movement densities in n dimensions. • There is no such thing as Tegmark’s “Reality independent of an observer”. We think we have shown a simplistic but viable example for a relatively naïve mathematical structure and minimal conceptual input, what richness lies in the structure’s (spherical space’s) transformations, if interpreted. Without such interpretation there is no way of recognizing such structure as a (simplified) physical reality, and such interpretation has to be made by an observer. So, we come back to Wheeler’s signposts and the space between them: only if all the space between them can be filled with certainty, we can say we have a mathematical universe that is determinable without an observer and his or her participation.