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S-Matrix Theory and Entanglement

Kazuo Fujikawa

Published Jun 29, 2015• 0 citations
Publisher

Abstract

But no good and simple criterion of entanglement for more complicated cases is known. This situation is rather strange since everybody talks about entanglement. How to produce and how to measure the entanglement are also interesting issues. When one measures the spin of the electron, for example, there are three major operators one can think of [1] a) Wigner spin, b) Dirac spin, c) Magnetic moment. The notion of chirality is important to distinguish these operators. Incidentally, many people talk about the Stern-Gerlach apparatus to measure the spin polarization of the electron. This is also strange since N. Bohr and N. F. Mott and other prominent people have shown many years ago that one cannot measure the electron spin polarization by a Stern-Gerlach apparatus, since the inhomogeneous magnetic field disturbs the motion of the electron so much and it is impossible to detect the spin orientation. The Stern-Gerlach particles are electrically neutral. One may ask if novel features appear in high energy physics? Chiral symmetry (including the use of weak interaction) can become important such as in a neutrino-antineutrino pair in Z decay

DOI
10.7566/jpscp.7.010013

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But no good and simple criterion of entanglement for more complicated cases is known. This situation is rather strange since everybody talks about entanglement. How to produce and how to measure the entanglement are also interesting issues. When one measures the spin of the electron, for example, there are three major operators one can think of [1] a) Wigner spin, b) Dirac spin, c) Magnetic moment. The notion of chirality is important to distinguish these operators. Incidentally, many people talk about the Stern-Gerlach apparatus to measure the spin polarization of the electron. This is also strange since N. Bohr and N. F. Mott and other prominent people have shown many years ago that one cannot measure the electron spin polarization by a Stern-Gerlach apparatus, since the inhomogeneous magnetic field disturbs the motion of the electron so much and it is impossible to detect the spin orientation. The Stern-Gerlach particles are electrically neutral. One may ask if novel features appear in high energy physics? Chiral symmetry (including the use of weak interaction) can become important such as in a neutrino-antineutrino pair in Z decay
Entanglement is studied in the framework of Dyson's S-matrix theory in relativistic quantum field theory, which leads to a natural definition of entangled states of a particle-antiparticle pair and the spin operator from a Noether current. As an explicit example, the decay of a massive pseudo-scalar particle into a pair of electron and positron is analyzed. Two spin operators are extracted from the Noether current. The Wigner spin operator characterizes spin states at the rest frame of each fermion and, although not measurable in the laboratory, gives rise to a straightforward generalization of low-energy analysis of entanglement to the ultrarelativistic domain. In contrast, if one adopts a (modified) Dirac spin operator, the entanglement measured by spin correlation becomes maximal near the threshold of the decay, while the entanglement is replaced by the classical correlation for the ultrarelativistic electron-positron pair by analogy to the case of neutrinos, for which a hidden-variables type of description is possible. Chiral symmetry differentiates the spin angular momentum and the magnetic moment. The use of weak interaction that can measure helicity is suggested in the analysis of entanglement at high energies instead of a Stern-Gerlach apparatus, which is useless for the electron. A difference between the electron spin at high energies and the photon linear polarization is also noted. The Standard Model can describe all of the observable properties of leptons.
Entanglement is the superposition of two or more particles in multiple states. An example is given for the double-well potential; another involves particles with spin 1/2. Measurement of spin is carried out by a Stern–Gerlach apparatus, using the deflection of a spin by a magnet. Entangled spin-1/2 states, namely triplet states having total spin 1, and the singlet state, with spin 0, are also examined; the latter state appears in same form when expressed in spin up/down along any Cartesian coordinate. The determination of one of the spins in an entangled singlet state immediately determines the other, resulting in a “spooky action at a distance.” Einstein, Podolsky, and Rosen (EPR) claimed to show, on the basis of local realism, that quantum mechanics must be incomplete because of this feature. Their logic suggests that one might explain the instantaneous results by invoking “hidden variables” not contained in a quantum treatment.
A thorough motion investigation and analysis was conducted again for the original Stern-Gerlach Experiment(SGE). The analysis is still based on the assumption that the atom passing the SG apparatus is a micromagnet due to the spin of its electron, the atoms in the source beam are with an orientation distribution of its angular momentum vectors, while the spin micromagnet is in the precession induced by an inhomogeneous magnetic field. It is the precession that keeps the silver beam with narrower orientation distribution of its angular momentum vectors after passing through the inhomogeneous magnetic field. The motion relativity was introduced in the SGE between the SG apparatus and the silver source beam, which interpreted all the detection phenomena in the multi-stage SGE. The research provided a new and convincing interpretation to the SGE with full satisfaction based on the classical physics without any mysterious or counter intuitive concepts introduced compared to quantum physics.
Abstract We propose a phenomenological theory of spin behavior in a magnetic field, which explains from the point of view of classical physics the two-valued result of the Stern-Gerlach experiment. The behavior of the spin and intrinsic magnetic moment of an electron wave of an atom in an external magnetic field is considered. We show that in a weak magnetic field, the intrinsic magnetic moment of an electron wave is always oriented parallel to the magnetic field strength vector, while in a strong magnetic field, depending on the initial orientation of the intrinsic magnetic moment, two orientations are realized: either parallel or antiparallel to the magnetic field strength vector. Within the framework of classical electrodynamics, the calculation of the motion of an atomic beam in an inhomogeneous magnetic field is carried out, which reproduces the results of the Stern-Gerlach experiment.
Spin is an important quantum degree of freedom in relativistic quantum information theory. This paper provides a first-principles derivation of the observable corresponding to a Stern-Gerlach measurement with relativistic particle velocity. The specific mathematical form of the Stern-Gerlach operator is established using the transformation properties of the electromagnetic field. To confirm that this is indeed the correct operator we provide a detailed analysis of the Stern-Gerlach measurement process. We do this by applying a WKB approximation to the minimally coupled Dirac equation describing an interaction between a massive fermion and an electromagnetic field. Making use of the superposition principle we show that the +1 and -1 spin eigenstates of the proposed spin operator are split into separate packets due to the inhomogeneity of the Stern-Gerlach magnetic field. The operator we obtain is dependent on the momentum between particle and Stern-Gerlach apparatus, and is mathematically distinct from two other commonly used operators. The consequences for quantum tomography are considered.