The existence of any elementary particle in universe requires the existence of some region of universe occupied by it. By taking the volume of this occupied region, the author will reformulate the relativistic quantum field theory using new 3-dimensional region-like idealization of elementary particles and hereinafter will call the total volume of all regions occupied by the elementary constituent particles of the quantum system the occupied volume. Also the author will call the set of all regions of universe filled by elementary constituent particles of the quantum system the occupied path. Always any quantum system is existed at a head of its occupied path. This path is growing by mutual filling and leaving regions of universe by its elementary constituent particles. The conservation of this elementary constituent particle requires the conservation of its occupied volume during this process. This requirement could be summarized by the following conditions: 1) the total volume of all regions of universe filled by the elementary constituent particles of the quantum system minus the total volume of all regions of universe left by these elementary constituent particles must be equal to the occupied volume of the quantum system; 2) the total increase in the occupied volume of the quantum system due to the absorption of another elementary particles from outside its occupied regions minus the total decreasing in its occupied volume due to the emission of another elementary particles outside its occupied regions must be equal to the occupied volume of it. The wave-particle duality of the elementary constituent particles implied accumulation of them as the finite set of interfered waves. This accumulation of elementary constituent particles causes the absolute probabilistic nature of event of finding the elementary consistent particle in specified interfered wave, and hence the mathematical representation of this interfered wave should take into account the value of probability amplitude of finding an elementary particle inside the region occupied specified interfered wave. In quantum theory this probability amplitude corresponds to complex amplitude of the wave function of interfered wave. Also in Hilbert’s representation of the quantum theory these wave functions are representing the components of the quantum state vector. In this paper the author will develop the transformation theory of the region-like quantum state of the quantum system.
The occupied volume of the elementary particle is the measurement of the volume of the region of universe occupied by it.
The occupied volume of the quantum system is the total occupied volume of its elementary constituent particles.
If we have a quantum system consist of N interference pattern then the occupied volume of the quantum systems is always given as following linear combination.
where
Up to now the shape of all elementary particles does not approved; so suppose that in general form the occupied volume of elementary particle is representing function of its wavelength
For example if the shape of boson
So the wavelength of the boson
By the same way if the shape of fermion fk is sphere then the occupied volume of fermion fk in this case is:
Also the inverse of
From the De-Broglie’s wave-particle duality the momentum of the elementary particle p is related to its wavelength
And hence by combination of De-Broglie’s formula
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However
And hence
where
Those two equations imply that the exchange of elementary particles between the quantum systems is bounded by speed of light c, and hence there is no action at distance on any quantum system in universe.
From the theory of special relativity the mass of the elementary matter particle m is representing function of its wavelength
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From the theory of special relativity the relativistic total energy E of the elementary matter particle is represent- ing function of its wavelength
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In the quantum mechanics the classical Newtonian equation of motion is replaced by another called Schrodinger’s equation. However the solution of Schrödinger equation is a wave function which is complex function of spatial and temporal coordinates with complex amplitude equal to probability of finding elementary particle at specified spatial and temporal coordinates, although the spatial coordinates is related to point-like idealization of elementary particle the temporal coordinate is not, so in region-like idealization of this waves we need to replace all spatial parameters of the wave function by occupied volumes of elementary constituent particles, so if we have quantum system consist of N interfered wave then the region-like quantum state of this quantum system at each instance of time t is representing the following ket vector:
where
1) The ith component of
2) The time derivation of
3)
The process of creating of the quantum system is the process in which the quantum system undergoes increasing in its occupied volume due to the absorption of elementary particles from outside its occupied region.
The process of annihilating of the quantum system is the process in which the quantum system undergoes decreasing in its occupied volume due to the emission of elementary particles outside its occupied region.
The process of animating of the quantum system is the process in which the quantum system mutually fills and leaves region of universe.
The occupied path of the quantum system is the 3-dimentional path followed by it during its creating, annihilating and animating process and consists of all regions of universe occupied by its elementary constituent particles.
The creating and annihilating interaction between the quantum systems is the creating and annihilating process of two or more quantum systems in which the total occupied volume of one quantum system changed due to transformation of elementary particles from or to another quantum system.
Now we could define the unified creating and animating law that govern the behavior of quantum system during its creating, annihilating and animating processes as following:
The transformation of the occupied volume of the quantum system form initial value
where:
And hence
to the final state:
is always governing by the following equations:
where
Such that
1)
2)
3)
1) The Equation (1.19) implied the equivalent between the deference i − d―where i is a dynamical variable of the process of creating and d is the dynamical variable of the process of annihilating―and the deference o ? l― where o and l are the dynamical variables of process of animating―so this equation implied the equivalent between the dynamical variables of creating, annihilating and animating processes.
2) Although the dynamical variables of the creating and annihilating process i, d could counted by the particle counters installed outside the region occupied by the elementary constituent particles of the quantum system, the animating dynamical variables immeasurable by this way, however the Equation (1.20) implied the complete equivalents between the dynamical variable of creating, annihilating and animating process and hence for each measurable dynamical variables of the process of creating and annihilating there exist an equivalent dynamical variables of animating satisfy the Equations (1.20) and (1.21).
1) The outer product
2)
If
Prove:
From the Theorem 2.1:
and
And hence from the Equation (1.21) the time evolution of the region-like quantum state of the system is given by the following equations:
1)
2) We could calculate the components of
(2.5) tell us the equivalents between the components of
3)
1) In
2) In
Using region-like idealization of the elementary constituent particles of the quantum system we could introduce a new formulation of the quantum theory independent from the energy scale, and hence free from hierarchal problem.
Thanks for my father who supported all of my educational levels and for my wife Ayaat Ahmed Osman for here incorporeal support to me on publication of this paper.
Elsadig Naseraddeen AhmedMohamed, (2015) Reformulation of Relativistic Quantum Field Theory Using Region-Like Idealization of the Elementary Particle. Journal of Modern Physics,06,1711-1720. doi: 10.4236/jmp.2015.611173