mathematical structure and physical reality

8/14/2019 Mathematical Structure and Physical Reality

1/18

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


2/18

N-dimensional Euclidean space

And

Yields

nkkrrCnrV

k

nn 2;

!)(),(

2

===

)1()1! 2 +=+(= nkk

)

)(),(

2

2 2

n

n

rnrV

+(1

=


3/18

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


4/18

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 = .


5/18

Minimum mathematical uncertainty

2

2222

16

1)|)(|)(|)(|(

dvvfvdxxfx


6/18

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


7/18

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


8/18

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


9/18

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


10/18

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

+

+

++

+

+

+=


11/18

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


12/18

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


13/18

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


14/18

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

),(

+

++

+

+

+

=


15/18

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


16/18

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


17/18

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.


18/18

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.

mathematical structure and physical reality

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