The Dirac Quantum Field I
Diaa A Ahmed
e-mail: diahmed@yahoo.com
(Amsterdam August 2001, Reviewed Washington DC October 2005)
According to P A M Dirac (1966) the divergences arise from the running out of countability of the space of the states due to the violent fluctuations in the quantum field! The global topological property of the functional space does not depend on the non-countability of the space the states. I attempt here at a correct representation of the dynamical operator of the Dirac quantum field in quantum space. We will consider representation of the linear dynamical operator as a complex variable for the reasons stated here and see the result of this consideration on our understanding of the fluctuations and the divergences. Quantum space is in one to one correspondence with the representation space of quantum theory, and the quantum dynamical operator will be represented by a complex variable like the state vector of the representation space and therefore it will represent purely quantum dynamics in the quantum space. Hermitian operators oscillate in a complex plane around their real values. The mass term in the Hamiltonian represents an oscillatory component in the complex plane and contributes spin current. Spinors are anticommutative extension of complex numbers. Charge conjugation is a symmetry between the positive and negative solutions along the imaginary axes. The new field propagator we study has formal symmetry and more sensible behavior, then, we will apply this new field propagator to study a finite field theory (this article will be revised and completed in a later date).
The Linear Dynamical Operator
"We cannot, however, simply assert that the negative-energy solutions represent positrons, as this would make the dynamical relations all wrong." P A M Dirac.
The arguments for the derivation of this equation is not in the spirit of quantum theory, they are classical arguments. While in classical dynamics we deal with disjoint real positive-energy and negative-energy solutions and we can eleminate the negative-energy solutions on the ground of reality condition that conveniently chooses the positive one, but this condition does not apply in quantum theory where the positive-energy and the negative-energy solutions are not disjoint due to quantum tunelling.
Quantum space is in one to one correspondence with the representation space and linear dynamical operators such as the Hamiltonian operator or the Dirac operator has to be considered complex like the state vectors. The negative and the positive solutions reflect an underlying symmetry of the theory between the positive and negative imaginary axes in the complex plane. The ground state of the quantum field represents oscillations around the zero point in the complex plane around the positive and negative halves of the imaginary axes.
The m term in the Hamiltonian is not manifestly Hermitian and this is reminiscent of the idol role played by mass in classical dynamics. We may pass into the complex Hamiltonian by replacing it by im. The physical condition that justify this replacement is that mass represents a component of the momentum
in the relativistic theory and a component of oscillation in quantum theory.
[The whole real line solution of the operator from
to , takes ground state as the bottom of the real line at , and this is nonsensible because there exists no symmetry between the positive and the negative solutions which is required by charge conjugation.]
Spinors in Quantum Space
Hermiticity
Hermitian operators represent oscillations in the complex plane around their real value. This implies physical conditions for mass in quantum theory as a component of a linear complex variable. Mass represents a component of the momentum pm and component of wm oscillation in quantum theory.
Modulus of a complex
means a positive definite H.
The Quantum Hamiltonian
The mathematical condition is that the quantum Hamiltonian H is a linear function of the y's.
Being linear function in the complex 4-vectors y,
H might be a complex linear operator, an image of y, its modulus is positive definite.
The Dirac Hamiltonian
We proceed by introducing the Dirac gm matrices as anticommutative extensions of the complex number. The Dirac Hamiltonian is also complex, therefore,
The Dirac matrices have 5 components (0,1,2,3,m),
Bilinears
Subject to reality condition,
The classical Hamiltonian is a modulus of the bilinear variable ( pr2 + m2 ). This does not imply that the operator ( pr + m ) must be real operator by itself. There is no priori reason that ( pr + m ) should be a real operator since the measurable variables are functions of the modulus of the bilinear variable,
Dirac Spinors
The Dirac matrices gm are non-commutative extension of complex numbers. Connection of spin to the oscillatory part of
, of the amplitude 2im in the complex plane and the role 2im plays in spin magnetic moment and in the vacuum spin current are discussed here.
Charge Conjugation
Is symmetry between + and - solutions on the pure imaginary axes of the complex plane where the
does not represent the ground state of the field,
Field Propagators
The propagator has more formal symmetry and sensible behavior, where im represents oscillation with amplitude ( 2im ) in the complex plane around the ground state and contributes a spin current,
+ im |--------------- p0
---------|-------------------------- | p + im |
- im |--------------- p0
Applications
To Follow.
References
D A Ahmed, Quantum Dynamics of the Space, Unpublished 2005.
D A Ahmed, Theory of the Functional Space, Unpublished 2005.
D A Ahmed, Quantum Topology, physics/9812037, LANL e-print 1998.
M F Atiyah, Collected Works, Volume 5 Gauge Theories, Oxford U P 1988.
P A M Dirac, The Principles of Quantum Mechanics, Oxford U P 1982.
P A M Dirac, Directions in Physics, Wiley Interscience 1978.
P A M Dirac, Lectures on Quantum Field Theory, Yeshiva University 1966.
J D Bjorken, Drell, Relativistic Quantum Fields, McGraw Hill 1965.
J D Bjorken, Drell, Relativistic Quantum Mechanics, McGraw Hill 1964.