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JMP> Vol.9 No.14, December 2018
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Fluid State of Dirac Quantum Particles

Abstract Full-Text HTML XML Download Download as PDF (Size:600KB) PP. 2402-2419
DOI: 10.4236/jmp.2018.914154    279 Downloads   504 Views   Citations
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Vu B. Ho

Affiliation(s)

Advanced Study, 9 Adela Court, Mulgrave, Victoria, Australia.

ABSTRACT

In our previous works, we suggest that quantum particles are composite physical objects endowed with the geometric and topological structures of their corresponding differentiable manifolds that would allow them to imitate and adapt to physical environments. In this work, we show that Dirac equation in fact describes quantum particles as composite structures that are in a fluid state in which the components of the wavefunction can be identified with the stream function and the velocity potential of a potential flow formulated in the theory of classical fluids. We also show that Dirac quantum particles can manifest as standing waves which are the result of the superposition of two fluid flows moving in opposite directions. However, for a steady motion a Dirac quantum particle does not exhibit a wave motion even though it has the potential to establish a wave within its physical structure, therefore, without an external disturbance a Dirac quantum particle may be considered as a classical particle defined in classical physics. And furthermore, from the fact that there are two identical fluid flows in opposite directions within their physical structures, the fluid state model of Dirac quantum particles can be used to explain why fermions are spin-half particles.

KEYWORDS

Dirac Equation, Wave Mechanics, Stan, Fluid Mechanics, Stream Function, Velocity Potential, Potential Flow, General Relativity, Maxwell Field Equations, CW Complexes, Differential Geometry, Topology, Differentiable Manifolds, Topological Transformation

Cite this paper

Ho, V. (2018) Fluid State of Dirac Quantum Particles. Journal of Modern Physics, 9, 2402-2419. doi: 10.4236/jmp.2018.914154.

1. Introductory Summary

In our previous works on spacetime structures of quantum particles, we suggest that quantum particles should be endowed with geometric and topological structures of differentiable manifolds and their motion should be described as isometric embeddings in higher Euclidean space. We also suggest that all quantum particles are formed from mass points which are joined together by contact forces as a consequence of viewing quantum particles as CW-complexes [1] [2] [3]. Fundamentally, we show that the three main dynamical descriptions of physical events in classical physics, namely Newton mechanics, Maxwell electromagnetism and Einstein gravitation, can be formulated in the same general covariant form and they can be represented by the general equation

β M = k J (1)

where M is a mathematical object that represents the corresponding physical system and β is a covariant derivative. For Newton mechanics,

M = 1 2 m μ = 1 3 ( d x μ / d t ) 2 + V and J = 0 . For Maxwell electromagnetism,

M = F α β = μ A ν ν A μ , with the four-vector potential A μ ( V , A ) and J can be identified with the electric and magnetic currents. And for Einstein gravitation, M = R α β and J can be defined in terms of a metric g α β and the Ricci

scalar curvature using the Bianchi identities β R α β = 1 2 g α β β R , that is,

J = 1 2 g α β β R . If we use the Bianchi identities as field equations for the gravitational field then Einstein field equations T μ ν = k ( R μ ν 1 2 R g μ ν + Λ g μ ν ) , as in

the case of the electromagnetic field, should be regarded as a definition for the energy-momentum tensor T μ ν for the gravitational field [4]. An interesting feature that emerges from Equation (1) for the gravitational field is that we can derive the Ricci flow g α β / t = κ R α β for a vacuum field J = 0 . Mathematically, the Ricci flow is a geometric process that can be employed to smooth out irregularities of a Riemannian manifold [5]. From the definition of the four-current j α = ( ρ , j i ) = 1 2 g α β β R for the gravitational field, by comparing with the Poisson equation for a potential V in classical physics, 2 V = 4 π ρ , we can identify the scalar potential V with the Ricci scalar curvature R and then obtain a diffusion equation t R = k 2 R whose solutions can be found to take the form R ( x , y , z , t ) = ( M / ( 4 π k t ) 3 ) e ( x 2 + y 2 + z 2 ) / 4 k t , which determines the probabilistic distribution of an amount of geometrical substance M which is defined via the Ricci scalar curvature R and manifests as observable matter [6]. It is worth mentioning that in fact a similar diffusion equation can also be derived from the Ricci flow g α β / t = κ R α β of the form R / t = Δ R + 2 | Ric | 2 , where Δ is the Laplacian defined as Δ = g α β α β and | Ric | is a shorthand for a mathematical expression that we will not be concerned with in this work [7]. Therefore, the Bianchi field equations of general relativity in the covariant form given in Equation (1) can be used to formulate quantum particles as differentiable manifolds. For example, we showed that the Ricci scalar curvature R associated with a differentiable manifold that represents a quantum system, such as the hydrogen atom, can be expressed in terms of the Schrödinger wavefunction ψ in quantum mechanics as R = k ( μ = 1 3 ( d x μ / d t ) 2 ( / m ) ( t ψ + μ = 1 3 μ ψ ( d x μ / d t ) / ψ ) ) .

On the other hand, we have also shown that Maxwell field equations of electromagnetism and Dirac relativistic equation of quantum mechanics can be formulated covariantly from a general system of linear first order partial differential equations [8] [9]. An explicit form of a system of linear first order partial differential equations can be written as follows [10] [11]

i = 1 n j = 1 n a i j r ψ i x j = k 1 l = 1 n b l r ψ l + k 2 c r , r = 1 , 2 , , n (2)

The system of equations given in Equation (2) can be rewritten in a matrix form as

( i = 1 n A i x i ) ψ = k 1 σ ψ + k 2 J (3)

where ψ = ( ψ 1 , ψ 2 , , ψ n ) T , ψ / x i = ( ψ 1 / x i , ψ 2 / x i , , ψ n / x i ) T , A i , σ and J are matrices representing the quantities a i j k , b l r and c r , and k 1 and k 2 are undetermined constants. Now, if we apply the operator i = 1 n A i / x i on the left on both sides of Equation (3) then we obtain

( i = 1 n A i 2 2 x i 2 + i = 1 n j > i n ( A i A j + A j A i ) 2 x i x j ) ψ = k 1 2 σ 2 ψ + k 1 k 2 σ J + k 2 i = 1 n A i J x i (4)

In order for the above systems of partial differential equations to be used to describe physical phenomena, the matrices A i must be determined. We have shown that for both Dirac and Maxwell field equations, the matrices A i must take a form so that Equation (4) reduces to the following equation

( i = 1 n A i 2 2 x i 2 ) ψ = k 1 2 σ 2 ψ + k 1 k 2 σ J + k 2 i = 1 n A i J x i (5)

To obtain Dirac equation we simply set A i A j + A j A i = 0 with A i 2 = ± 1 , and in this case the matrices A i are the matrices given as [12]

γ 1 = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) , γ 2 = ( 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 ) , γ 3 = ( 0 0 0 i 0 0 i 0 0 i 0 0 i 0 0 0 ) , γ 4 = ( Fluid State of Dirac Quantum Particles
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