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, 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 (see e.g., S. J. Summers and R. Werner, Commun. Math. Phys.
110,247,1987). There is no restriction on entanglement in parametrized relativistic quantum mechanics (A. F. Bennett, Ann. Phys.
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.
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 conjugation is another
elementary property of the parametrized formalism.
There has been some recent interest in applying the techniques of Algebraic Quantum Field Theory (AQFT) to entanglement problems in perturbative QFT. In particular, the Hilbert space independence of this formulation makes it particularly interesting in the context of curved spacetimes and the emphasis on the algebra of observables makes the treatment of Bell inequalities in QFT resemble such treatment in non-relativistic Quantum Mechanics. In this work, we present the mathematical structures needed for formulating AQFT in terms of the Haag-Araki-Kastler (HAK) axioms and discuss their implications. Moreover, we discuss the algebraic approach to quantum entanglement in the form of Bell inequalities. We provide an extension of this formulation to general globally hyperbolic spacetimes using the so-called Locally Covariant approach to QFT, which extends the HAK axioms to general spacetimes by means of the Category Theory language.
In quantum time (QT) schemes, time is promoted to a degree of freedom, allowing Lorentz covariance to be made explicit for single particles. We ask whether this can be lifted to QFT, so that Lorentz covariance becomes manifest at the Hilbert-space level, rather than being hidden as in the standard canonical formulation. We address this question by proposing a second-quantized approach in which the elementary particle is the QT particle itself, leading naturally to the notion of spacetime field algebras and of quantum action. We show, however, that a naive many-body construction runs into inconsistencies. To pinpoint their origin we introduce a classical counterpart of the second-quantized formalism, spacetime classical mechanics (SCM), and prove a no-go theorem: Dirac quantization of SCM collapses back to standard QFT and therefore hides covariance. We circumvent this problem by presenting a quantum-action-based quantization that yields a spacetime version of quantum mechanics (SQM), making covariance manifest for (interacting) QFTs. Finally, we show that this resolution is tied to a genuine spacetime generalization of the notion of quantum state, required by causality and closely connected to recent ``states over time'' proposals and, in dS/CFT-motivated settings, to microscopic notions of timelike entanglement and emergent time.
Abstract Entanglement entropy of quantum fields in gravitational settings is a topic of growing importance. This entropy of entanglement is conventionally computed relative to Cauchy hypersurfaces where it is possible via a partial tracing to associate a reduced density matrix to the spacelike region of interest. In recent years Sorkin has proposed an alternative, manifestly covariant, formulation of entropy in terms of the spacetime two-point correlation function. This formulation, developed for a Gaussian scalar field theory, is explicitly spacetime in nature and evades some of the possible non-covariance issues faced by the conventional formulation. In this paper we take the first steps towards extending Sorkin’s entropy to non-Gaussian theories where Wick’s theorem no longer holds and one would expect higher correlators to contribute. We consider quartic perturbations away from the Gaussian case and find that to first order in perturbation theory, the entropy formula derived by Sorkin continues to hold but with the two-point correlators replaced by their perturbation-corrected counterparts. We then show that our results continue to hold for arbitrary perturbations (of both bosonic and fermionic theories). This is a non-trivial and, to our knowledge, novel result. Furthermore we also derive closed-form formulas of the entanglement entropy for arbitrary perturbations at first and second order. Our work also suggests avenues for further extensions to generic interacting theories.
The Wigner representation for quantum mechanics of particles is generalized to Bose fields. The standard Hilbert space quantization becomes, via the Weyl transform, a quantization method that consists of adding a Gaussian zeropoint field distribution to the vacuum. I comment on the possible advantages of the method in order to study quantum fields in curved spaces. I study a unified formulation of non-relativistic quantum electrodynamics in the Weyl–Wigner formalism, in terms of (classical-like) c-numbers.