Negative Energies in the Dirac Equation (original) (raw)
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Dirac and Higher-Spin Equations of Negative Energies
2011
It is easy to check that both algebraic equation Det (\hat p - m) =0 and Det (\hat p + m) =0 for 4-spinors u- and v- have solutions with p_0= \pm E_p =\pm \sqrt{{\bf p}^2 +m^2}. The same is true for higher-spin equations. Meanwhile, every book considers the p_0=E_p only for both u- and v- spinors of the (1/2,0)\oplus (0,1/2)) representation, thus applying the Dirac-Feynman-Stueckelberg procedure for elimination of negative-energy solutions. Recent works of Ziino (and, independently, of several others) show that the Fock space can be doubled. We re-consider this possibility on the quantum field level for both s=1/2 and higher spins particles.
The Dirac equation is a cornerstone of quantum mechanics that fully describes the behaviour of spin ½ particles. Recently, the energy momentum relationship has been reconsidered such that |E|^2 = |(m0c^ 2)| 2 + |(pc)| 2 has been modified to: |E| 2 = |(m0c^2)|^2-|(pc)|^2 where E is the kinetic energy, moc^2 is the rest mass energy and pc is the wave energy for the spin ½ particle. This has been termed the 'Hamiltonian approach' and with a new starting point, the original Dirac equation has been derived: and the modified covariant form found is where h/2π = c = 1. The behaviour of spin ½ particles is found to be the same as for the original Dirac equation. The Dirac equation will also be expanded by setting the rest energy as a complex number, |(m0c 2)| e^jωt
On a General Spin Dirac Equation
2009
In its bare and natural form, the Dirac Equation describes only spin-1/2 particles. The main purpose of this reading is to make a valid and justified mathematical modification to the Dirac Equation so that it describes any spin particle. We show that this mathematical modification is consistent with the Special Theory of Relativity (STR). We believe that the fact that this modification is consistent with the STR gives the present effort some physical justification that warrants further investigations. From the vantage point of unity, simplicity and beauty, it is natural to wonder why should there exist different equations to describe particles of different spins? For example, the Klein-Gordon equation describes spin-0 particles, while the Dirac Equation describes spin-1/2, and the Rarita-Schwinger Equation describes spin-3/2. Does it mean we have to look for another equation to describe spin-2 particles, and then spin-5/2 particles etc? This does not look beautiful, simple, or at the very least suggest a Unification of the Natural Laws. Beauty of a theory is not a physical principle but, one thing is clear to the searching mind-i.e., a theory that possesses beauty, appeals to the mind, and is (posteriori) bound to have something to do with physical reality if it naturally submits itself to the test of experience. The effort of the present reading is to make the attempt to find this equation.
Mysteries and Insights of Dirac Theory
The Dirac equation has a hidden geometric structure that is made manifest by reformulating it in terms of a real spacetime algebra. This reveals an essential connection between spin and complex numbers with profound implications for the interpretation of quantum mechanics. Among other things, it suggests that to achieve a complete interpretation of quantum mechanics, spin should be identifled with an intrinsic zitterbewegung.
On the derivation of the Dirac equation
We point out that the anticommutation properties of the Dirac matrices can be derived without squaring the Dirac hamiltonian, that is, without any explicit reference to the Klein-Gordon equation. We only require the Dirac equation to admit two linearly independent plane wave solutions with positive energy for all momenta. The necessity of negative energies as well as the trace and determinant properties of the Dirac matrices are also a direct consequence of this simple and minimal requirement.
On the correspondence between the solutions of Dirac equation and electromagnetic 4-potentials
In this paper the inverse problem of the correspondence between the so- lutions of the Dirac equation and the electromagnetic 4-potentials, is fully solved. The Dirac solutions are classified into two classes. The first one consists of degenerate Dirac solutions corresponding to an infinite num- ber of 4-potentials while the second one consists of non-degenerate Dirac solutions corresponding to one and only one electromagnetic 4-potential. Explicit expressions for the electromagnetic 4-potentials are provided in both cases. Further, in the case of the degenerate Dirac solutions, it is proven that at least two 4-potentials are gauge inequivalent, and con- sequently correspond to different electromagnetic fields. An example is provided to illustrate this case.
In its bare and natural form, the Dirac Equation describes only spin-1/2 particles. The main purpose of this reading is to make a valid and justified mathematical modification to the Dirac Equation so that it describes any spin particle. We show that this mathematical modification is consistent with the Special Theory of Relativity (STR). From the vantage point of unity, simplicity and beauty, it is natural to wonder why should there exist different equations to describe particles of different spins? For example, the Klein-Gordon equation describes spin-0 particles, while the Dirac Equation describes spin-1/2, and the Rarita-Schwinger Equation describes spin-3/2. Does it mean we have to look for another equation to describe spin-2 particles, and then spin-5/2 particles etc? This does not look beautiful, simple, or at the very least suggest a Unification of the Natural Laws.
General Spin Dirac Equation (II)
In an earlier reading [1], we did demonstrate that one can write down a general spin Dirac equation by modifying the usual Einstein energy-momentum equation via the insertion of the quantity " s " which is identified with the spin of the particle. That is to say, a Dirac equation that describes a particle of spin 1 2 s S σ where is the normalised Planck constant, are the Pauli matrices and
The Dirac algebra and its physical interpretation
2000
A version of the Dirac equation is derived from first principles using a combination of quaternions and multivariate 4-vectors. The nilpotent form of the operators used allows us to derive explicit expressions for the wavefunctions of free fermions, vector bosons, scalar bosons; Bose-Einstein condensates, and baryons;annihilation, creation and vacuum operators; the quantum field integrals; and C, P and T transformations;
Journal of Physics A: Mathematical and Theoretical, 2010
A single Dirac particle is bound in d dimensions by vector V (r) and scalar S(r) central potentials. The spin-symmetric S = V and pseudospin-symmetric S = −V cases are studied and it is shown that if two such potentials are ordered V (1) ≤ V (2) , then corresponding discrete eigenvalues are all similarly ordered E (1) κν ≤ E (2) κν. This comparison theorem allows us to use envelope theory to generate spectral approximations with the aid of known exact solutions, such as those for Coulombic, harmonic-oscillator, and Kratzer potentials. The example of the log potential V (r) = v ln(r) is presented. Since V (r) is a convex transformation of the soluble Coulomb potential, this leads to a compact analytical formula for lower-bounds to the discrete spectrum. The resulting ground-state lower-bound curve E L (v) is compared with an accurate graph found by direct numerical integration.