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Note on the golden mean, nonlocality in quantum mechanics and fractal cantorian Space time


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Abstract

Given the inverse of the Golden Mean <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <semantics>
  <mrow>
   <msup>
    <mi>&#x03C4;</mi>
    <mrow>
     <mo>&#x2212;</mo><mn>1</mn></mrow>
   </msup>
   <mo>=</mo><mi>&#x03D5;</mi><mo>=</mo><mfrac>
    <mn>1</mn>
    <mn>2</mn>
   </mfrac>
   <mrow><mo>(</mo>
    <mrow>
     <msqrt>
      <mn>5</mn>
     </msqrt>
     <mo>&#x2212;</mo><mn>1</mn></mrow>
   <mo>)</mo></mrow><mo>,</mo></mrow>
  <annotation encoding='MathType-MTEF'>MathType@MTEF@5@5@+=
  feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
  hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
  4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb
  a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
  0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiXdq
  NcdaahaaWcbeqaaiabgkHiTiaaigdaaaqcLbsacqGH9aqpcqaHvpGz
  cqGH9aqpkmaalaaabaqcLbsacaaIXaaakeaajugibiaaikdaaaGcda
  qadaqaamaakaaabaqcLbsacaaI1aaaleqaaKqzGeGaeyOeI0IaaGym
  aaGccaGLOaGaayzkaaGaaiilaaaa@4967@
  </annotation>
 </semantics>
</math> t is known that the continuous fraction expansion of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <semantics>
  <mrow>
   <msup>
    <mi>&#x03D5;</mi>
    <mrow>
     <mo>&#x2212;</mo><mn>1</mn></mrow>
   </msup>
   <mo>=</mo><mn>1</mn><mo>+</mo><mi>&#x03D5;</mi><mo>=</mo><mi>&#x03C4;</mi></mrow>
  <annotation encoding='MathType-MTEF'>MathType@MTEF@5@5@+=
  feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
  hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
  4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb
  a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
  0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqy1dy
  McdaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGH9aqpcaaIXaGaey4k
  aSscLbsacqaHvpGzcqGH9aqpcqaHepaDaaa@4430@
  </annotation>
 </semantics>
</math> is (1; 1; 1;…). Integer solutions for the pairs of numbers
 

 are found obeying the equation   The latter equation was inspired from El Naschie's formulation of fractal Cantorian space time   and such that it furnishes the continuous fraction expansion of   generalizing the original expression for the Golden mean. Hardy showed that is possible to demonstrate nonlocality without using Bell inequalities for two particles prepared in nonmaximally entangled states. The maximal probability of obtaining his nonlocality proof was found to be precisely   Zheng showed that three-particle nonmaximally entangled states revealed quantum nonlocality without using inequalities, and the maximal probability of obtaining the nonlocality proof was found to be   Given that the two-parameter   quantum-calculus deformations of the integers   coincide precisely with the Fibonacci numbers, as a result of Binet's formula when   we explore further the implications of these results in the quantum entanglement of two-particle spin-s states. We finalize with some remarks on the generalized Binet's formula corresponding to generalized Fibonacci sequences, and their role in the physics of quasicrystals with   rotational symmetry.

Keywords

cantorian fractal spacetime, quantum calculus, golden mean, noncommutative geometry, quantum mechanics, nonlocality

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