Extended Relativistic Invariance, Quantization of the Kinetic Momentum

Abstract It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.

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Is Einsteinian no-signalling violated in Bell tests?

Open Physics ◽

10.1515/phys-2017-0087 ◽

2017 ◽

Vol 15 (1) ◽

pp. 739-753 ◽

Cited By ~ 6

Author(s):

Marian Kupczynski

Keyword(s):

Quantum Electrodynamics ◽

Probability Distributions ◽

Selection Procedure ◽

Quantum Correlations ◽

Spin Projection ◽

Marginal Probability ◽

Strong Correlations ◽

Relativistic Invariance ◽

Long Range Correlations ◽

Apparent Violation

AbstractRelativistic invariance is a physical law verified in several domains of physics. The impossibility of faster than light influences is not questioned by quantum theory. In quantum electrodynamics, in quantum field theory and in the standard model relativistic invariance is incorporated by construction. Quantum mechanics predicts strong long range correlations between outcomes of spin projection measurements performed in distant laboratories. In spite of these strong correlations marginal probability distributions should not depend on what was measured in the other laboratory what is called shortly: non-signalling. In several experiments, performed to test various Bell-type inequalities, some unexplained dependence of empirical marginal probability distributions on distant settings was observed. In this paper we demonstrate how a particular identification and selection procedure of paired distant outcomes is the most probable cause for this apparent violation of no-signalling principle. Thus this unexpected setting dependence does not prove the existence of superluminal influences and Einsteinian no-signalling principle has to be tested differently in dedicated experiments. We propose a detailed protocol telling how such experiments should be designed in order to be conclusive. We also explain how magical quantum correlations may be explained in a locally causal way.

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Time reversal in field theory

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1955.0189 ◽

1955 ◽

Vol 231 (1187) ◽

pp. 479-495 ◽

Cited By ~ 81

Keyword(s):

Field Theory ◽

Time Reversal ◽

First Principles ◽

Relativistic Invariance ◽

Quantized Fields ◽

Transition Amplitudes

The idea of reversibility in time as applied to quantized fields is expounded from first principles. It is shown that in the usual theories reversibility of a certain kind is a concomitant of relativistic invariance and symmetry in space. Finally the relations between transition amplitudes consequent on the reversibility are derived.

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Spectral properties of the equation (∇ + ieA) × u = ±mu

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050000127x ◽

2001 ◽

Vol 131 (5) ◽

pp. 1065-1089

Author(s):

Daniel M. Elton

Keyword(s):

Spectral Properties ◽

Time Variable ◽

Two Dimensional ◽

Vector Product ◽

Relativistic Invariance ◽

Speed Of Light ◽

Large Parameter ◽

Real Vector ◽

Pauli Equation ◽

Minkowski Metric

We develop a spectral theory for the equation (∇ + ieA) × u = ±mu on Minkowski 3-space (one time variable and two space variables); here, A is a real vector potential and the vector product is defined with respect to the Minkowski metric. This equation was formulated by Elton and Vassiliev, who conjectured that it should have properties similar to those of the two-dimensional Dirac equation. Our equation contains a large parameter c (speed of light), and this motivates the study of the asymptotic behaviour of its spectrum as c → +∞. We show that the essential spectrum of our equation is the same as that of Dirac (theorem 3.1), whereas the discrete spectrum agrees with Dirac to a relative accuracy δλ/mc2 ~ O(c−4) (theorem 3.3). In other words, we show that our equation has the same accuracy as the two-dimensional Pauli equation, its advantage over Pauli being relativistic invariance.

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