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First quantized pair interactions

Aaron Bennett

Published Jun 8, 2014• 2 citations
OpenAlex

Abstract

Dirac's theory of the electron admits the possibility of the electron dropping into states of negative energy without lower bound. Interpreting the negative-energy states as different particles, namely, positive-energy antiparticles (Dirac, P. A. M., 1930, Proc. Roy. Soc., {\bf 126}, No. 801, 360--365) presents a seemingly insurmountable difficulty for the necessarily single-particle Dirac equation. The observed annihilation of electron-positron pairs and creation of muon-antimuon pairs emphasizes the difficulty. Quantum Field Theory was developed in order to overcome this difficulty. Alternatively, the parametrized Dirac wave equation (variously attributed to Feynman) extends to many particles and is shown here to provide a first-quantized representation of such pair interactions. Unlike Quantum Field Theory, the parametrized Dirac equation also provides an elementary and unrestricted representation of quantum entanglement. Invariance with respect to $\mathcal{TPC}$ conjugation is another elementary property of the parametrized formalism.

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Physical Review•Apr 1, 1950•2,183 citations
Dirac's theory of the electron admits the possibility of the electron
dropping into states of negative energy without lower bound. Interpreting the
negative-energy states as different particles, namely, positive-energy
antiparticles (Dirac, P. A. M., 1930, Proc. Roy. Soc., {\bf 126}, No. 801,
360--365) presents a seemingly insurmountable difficulty for the necessarily
single-particle Dirac equation. The observed annihilation of electron-positron
pairs and creation of muon-antimuon pairs emphasizes the difficulty. Quantum
Field Theory was developed in order to overcome this difficulty. Alternatively,
the parametrized Dirac wave equation (variously attributed to Feynman) extends
to many particles and is shown here to provide a first-quantized representation
of such pair interactions. Unlike Quantum Field Theory, the parametrized Dirac
equation also provides an elementary and unrestricted representation of quantum
entanglement. Invariance with respect to TPC\mathcal{TPC}TPC conjugation is another
elementary property of the parametrized formalism.
Quantum Electrodynamics may be formulated as a Quantum Field Theory , and also as relativistic quantum mechanics by introduction of the Feynman-Stueckelberg parameter. As stated by M. Srednicki ({\it Quantum Field Theory}, Cambridge University Press, Cambridge and New York, 2007), "\dots any relativistic quantum physics that can be treated in one formalism can also be treated in the other". Entanglement of electrons at macroscopic spacelike separation has been observed (B. Hensen {\it et al.}, Nature, 526\bf{526}526, 2015, doi:10.1038.nature/15759), yet according to Quantum Field Theory is restricted to separations of the order of the Compton wavelength ℏ/mec\hbar/m_e cℏ/me​c (see e.g., S. J. Summers and R. Werner, Commun. Math. Phys. 110\bf{110}110,247,1987). There is no restriction on entanglement in parametrized relativistic quantum mechanics (A. F. Bennett, Ann. Phys. 345\bf{345}345, 1, 2014). The parametrized formalism is extended here to pair annihilation and pair creation. The {\it ansatz} used in the formalism to develop the correct higher order corrections is justified here with plane wave summations. The conflict over entanglement is further discussed in terms of the Klein-Gordon propagator, and an invariant Bell's inequality is developed for spin -1/2.
Abstract The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.
Considers the Dirac equation with external electromagnetic fields and gives a careful formulation of the Stuckelberg-Feynman interpretation. This electron-positron theory may be applied whenever electron and positron states can be separated, which is possible for a wide class of time-independent potentials vanishing at infinity. One then may introduce a system of observables with a position operator X(t) which is free of Zitterbewegung and yields the usual relativistic connection between momentum and velocity. The Stuckelberg-Feynman interpretation is usually associated with a scattering theory (propagator formalism), which differs from the one of a one-particle interpretation (formalism imitating non-relativistic techniques). For the above-mentioned potentials there is a simple relationship between these physically different formalisms. There are, however, more general potentials for which the Stuckelberg-Feynman interpretation may still be applied asymptotically. With the help of Klein's paradox the authors show that the Dirac equation may then have unphysical solutions not conserving the total charge.
The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.
We underscore the need of relativistic quantum mechanics in dealing with particles. Various notations used in relativity are defined and explained. We derive the Klein–Gordon equation for free particle as well as for particle in electromagnetic field. We obtain a solution and interpret it. We derive the Dirac equation for free particle as well as for particle in electromagnetic field. Plane wave solution to the Dirac equation is obtained. We show how spin of electron, idea of antiparticle , phenomena of pair production and pair annihilation follow from the Dirac equation.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.