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arXiv:1406.2310

First quantized pair interactions, fermion loops and entanglement

Aaron Bennett

Published Jun 8, 2014• 0 citations
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Abstract

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, $\bf{526}$, 2015, doi:10.1038.nature/15759), yet according to Quantum Field Theory is restricted to separations of the order of the Compton wavelength $\hbar/m_e c$ (see e.g., S. J. Summers and R. Werner, Commun. Math. Phys. $\bf{110}$,247,1987). There is no restriction on entanglement in parametrized relativistic quantum mechanics (A. F. Bennett, Ann. Phys. $\bf{345}$, 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.

DOI
10.48550/arxiv.1406.2310

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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.
The formulation of quantum electrodynamics as parametrized relativistic quantum mechanics [Ann. Phys. {\bf 345} (2014) 1-16] is extended here to The Standard Model of the electroweak interaction. It is shown here that parametrization preserves the Standard interactions of the Higgs field with fermions and with the weak bosons. That is, the masses of the Higgs boson, the fermions and the gauge bosons are Standard. The principal advantage of the parametrized first quantized formulation is that quantum entanglement is unrestricted in spacetime [Found. Phys. {\bf 46} (2016) 1090-1108]. Thus, the formulation does not have the shortcoming of Quantum Field Theory in which entanglement is restricted to spacetime separations of the order of the Compton wavelength. A covariant Bell's inequality is derived here for fermions at arbitrary separation in spacetime.
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.
The quantum field theories (QFT) constructed in [1,2] include phenomenology of interest. The constructions approximate: scattering by 1/r1/r1/r and Yukawa potentials in non-relativistic approximations; and the first contributing order of the Feynman series for Compton scattering. To have a semi-norm, photon states are constrained to transverse polarizations and for Compton scattering, the constructed cross section deviates at large momentum exchanges from the cross section prediction of the Feynman rules. Discussion includes the incompatibility of canonical quantization with the constructed interacting fields, and the role of interpretations of quantum mechanics in realizing QFT.
Entanglement has long been the subject of discussion by philosophers of quantum theory, and has recently come to play an essential role for physicists in their development of quantum information theory. In this paper we show how the formalism of algebraic quantum field theory (AQFT) provides a rigorous framework within which to analyze entanglement in the context of a fully relativistic formulation of quantum theory. What emerges from the analysis are new practical and theoretical limitations on an experimenter’s ability to perform operations on a field in one spacetime region that can disentangle its state from the state of the field in other spacelike-separated regions. These limitations show just how deeply entrenched entanglement is in relativistic quantum field theory, and yield a fresh perspective on the ways in which the theory differs conceptually from both standard nonrelativistic quantum theory and classical relativistic field theory. “...despite its conservative way of dealing with physical principles, algebraic QFT leads to a radical change of paradigm. Instead of the Newtonian view of a space-time filled with a material content one enters the reality of Leibniz created by relation (in particular inclusions) between ‘monads ’ (∼ the hyperfinite type III1 local von Neumann factors A(O) which as single algebras are nearly void of physical meaning) ” Schroer (1998, p. 302).
In this thesis, entanglement under fully relativistic settings are discussed. The thesis starts with a brief review of the relativistic quantum mechanics. In order to describe the effects of Lorentz transformations on the entangled states, quantum mechanics and special relativity are merged by construction of the unitary irreducible representations of Poincar{é} group on the infinite dimensional Hilbert space of state vectors. In this framework, the issue of finding the unitary irreducible representations of Poincar{é} group is reduced to that of the little group. Wigner rotation for the massive particles plays a crucial role due to its effect on the spin polarization directions. Furthermore, the physical requirements for constructing the correct relativistic spin operator is also studied. Then, the entanglement and Bell type inequalities are reviewed. Special attention has been devoted to two historical papers, by EPR in 1935 and by J.S. Bell in 1964. The main part of the thesis is based on the Lorentz transformation of the Bell states and the Bell inequalities on these transformed states. It is shown that entanglement is a Lorentz invariant quantity. That is, no inertial observer can see the entangled state as a separable one. However, it was shown that the Bell inequality may be satisfied for the Wigner angle dependent transformed entangled states. Since the Wigner rotation changes the spin polarization direction with the increased velocity, initial dichotomous operators can satisfy the Bell inequality for those states. By choosing the dichotomous operators taking into consideration the Wigner angle, it is always possible to show that Bell type inequalities can be violated for the transformed entangled states.