Angularly Deformed Special Relativity and its Results for Quantum ...

arXiv:1510.01603v1 [physics.gen-ph] 15 Sep 2015

Angularly Deformed Special Relativity and Its Results for Quantum Mechanics Lukasz Andrzej Glinka October 7, 2015

Abstract In this paper, the deformed Special Relativity, which leads to an essentially new theoretical context of quantum mechanics, is presented. The formulation of the theory arises from a straightforward analogy with the Special Relativity, but its foundations are laid through the hypothesis on breakdown of the velocity-momentum parallelism which affects onto the Einstein equivalence principle between mass and energy of a relativistic particle. Furthermore, the derivation is based on the technique of an eikonal equation whose well-confirmed physical role lays the foundations of both optics and quantum mechanics. As a result, we receive the angular deformation of Special Relativity which clearly depicts the new deformation-based theoretical foundations of physics, and, moreover, offers both constructive and consistent phenomenological discussion of the theoretical issues such like imaginary mass and formal superluminal motion predicted in Special Relativity for this case. In the context of the relativistic theory, presence of deformation does not break the Poincar´e invariance, in particular the Lorentz symmetry, and provides essential modifications of both bosons described through the Klein-Gordon equation and fermions satisfying the Dirac equation. On the other hand, on the level of discussion of quantum theory, there arises the concept of emergent deformed spacetime, wherein the presence of angular deformation elucidates a certain new insight into the nature of spin, as well as both the Heisenberg uncertainty principle and the Schr¨odinger wave equation.

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1

Introduction

In Special Relativity, Cf. the Ref. [1], the Einstein equivalence principle E 2 = m2 c4 + p2 c2 ,

(1)

where pc is the speed of light in vacuum, relates kinetic energy E to magnitude p = pi pi of a momentum vector pi of a particle equipped with a mass m. In general, a velocity vector vi and speed v of a particle are determined as ∂E pc2 = . (2) ∂p E p In particular, for a massive particle one has E = γmc2 , p = γ 2 − 1mc, and   2 −1/2 v = p/(γm) where γ = 1 − vc2 is the Lorentz factor, for a massless particle E = pc and v = c, for an imaginary mass one has to deal with superluminal particles known as tachyons. The space-time coordinates fourvectors are either contravariant xµ = [ct, xi ] or covariant xµ = ηµν xν , and the energy-momentum four-vectors are pµ = [E, pi c] or pµ = ηµν pν , both space-time and energy-momentum space are equipped with the Minkowski metric ηµν = diag[1, −1, −1, −1]. Locally, the energy-momentum interval ds2 = ηµν dpµ dpν = dE 2 −c2 dpi dpi is symmetric, that is ds2 = ds′2 , with respect to the action of the Poincar´e transformations p′µ = Λµ ν pν + pµ0 with a constant energy-momentum fourvector pµ0 , while the space-time interval ds2 = ηµν dxµ dxν = c2 dt2 − dxi dxi displays invariance with respect to the action of the Poincar´e transformations x′µ = Λµ ν xν + xµ0 with a constant space-time four-vector xµ0 , whenever the Lorentz matrices obey (Λ−1 )µ κ = η µλ Λλ ν ηνκ and det Λµ ν = ±1. Relativistic invariance includes symmetry with respect to action of the Lorentz transformations p′µ = Λµ ν pν and x′µ = Λµ ν xν . Globally, the Lorentz symmetry holds for either s2 = ηµν pµ pν = E 2 −c2 p2 or s2 = ηµν xµ xν = c2 t2 −x2 , that is s2 = s′2 , whereas preservation of the Poincar´e symmetry demands vi =

∂E , ∂pi

v i pi = vp,

v = (v i vi )1/2 =

ηµν (pµ0 pν0 + {Λµ κ pκ , pν0 }) = 0,

(3)

ηµν (xµ0 xν0

(4)

µ

+ {Λ

κx

κ

, xν0 })

= 0,

for every xµ and pµ . For a particle motion, the well-known representation is # " i γ −γRji vc µ   , (5) Λ ν= vk v v −γ cj Rki δjk + (γ − 1) v2 j

where a matrix of rotation Rji ∈ SO(3) in three-dimensional Euclidean space R3 with metric δji = diag[1, 1, 1], can be defined through either the Euler angles (φ, ϕ, θ) h i h i h i Rji (θ, ϕ, φ) = exp (rji )(3) φ exp (rji )(2) ϕ exp (rji )(3) θ , (6) 2

or the Tait-Bryan angles (φ, ϕ, θ) h i h i h i Rji (θ, ϕ, φ) = exp (rji )(3) θ exp (rji )(2) ϕ exp (rji )(1) φ ,

with the Rodriques formula h i exp (rji )(p) α = δji + (rji )(p) sin α + (rki )(p) (rjk )(p) (1 − cos α),

(7)

(8)

where (rji )(p) are infinitesimal rotations matrices around a p-axis 

     0 0 0 0 0 1 0 −1 0 (rji )(1) =  0 0 −1  , (rji )(2) =  0 0 0  , (rji )(3) =  1 0 0  , 0 1 0 −1 0 0 0 0 0 (9) o n i (1) i (2) i (3) of SO(3) and form the Lie algebra so(3) = span (rj ) , (rj ) , (rj ) group Y sgn(aj − ai ), (10) [(rki )(a1 ) , (rjk )(a2 ) ] = εa1 a2 a3 (rji )(a3 ) , εa1 a2 a3 = 1≤i