A granular permutation-based representation of complex numbers and quaternions: elements of a possible realistic quantum theory

2004 ◽  
Vol 460 (2044) ◽  
pp. 1039-1055 ◽  
Author(s):  
T. N. Palmer
Keyword(s):  
Quantum Theory ◽  
Complex Numbers

i

2020 ◽  
pp. 93-103
Author(s):  
Marcel Danesi
Keyword(s):  
Quantum Theory ◽  
Electromagnetic Radiation ◽  
Quadratic Equation ◽  
Complex Numbers ◽  
Human History ◽  
Real Numbers ◽  
Electric Circuits ◽  
René Descartes ◽  
New Ideas ◽  
Imaginary Number

What kind of number is √−1? In a way that parallels the unexpected discovery of √2 by the Pythagoreans, when this number surfaced as a solution to a quadratic equation, mathematicians asked themselves what it could possibly mean. Not knowing what to call it, René Descartes named it an imaginary number. Like the irrationals, the discovery of i led to new ideas and discoveries. One of these was complex numbers—numbers having the form (a + bi), where a and b are real numbers and i is √−1. Incredibly, complex numbers turn out to have many applications. They are used to describe electric circuits and electromagnetic radiation and they are fundamental to quantum theory in physics. This chapter deals with imaginary numbers, which constitute another of the great ideas of mathematics that have not only changed the course of mathematics but also of human history.

SUPER-QUANTUM NONLOCAL CORRELATIONS IN QUATERNIONIC QUANTUM THEORY

2011 ◽  
Vol 09 (06) ◽  
pp. 1355-1362
Author(s):  
MATTHEW MCKAGUE
Keyword(s):  
Quantum Theory ◽  
Communication Complexity ◽  
Complex Numbers ◽  
Nonlocal Correlations ◽  
Information Causality ◽  
Rule Out ◽  
Nonlocal Box

We consider the nonlocal properties of naive quaternionic quantum theory, in which the complex numbers are replaced by the quaternions as the underlying algebra. Specifically, we show that it is possible to construct a nonlocal box. This allows one to rule out quaternionic quantum theory using assumptions about communication complexity or information causality while also providing a model for a nonlocal box using familiar structures.

Quantum theory of a non-Hermitian time-dependent forced harmonic oscillator having 𝒫𝒯 symmetry

2019 ◽  
Vol 34 (30) ◽  
pp. 1950187 ◽  
Author(s):  
I. A. Pedrosa ◽  
B. F. Ramos ◽  
K. Bakke ◽  
Alberes Lopes de Lima
Keyword(s):  
Quantum Theory ◽  
Wave Packet ◽  
Harmonic Oscillator ◽  
Driving Force ◽  
Time Dependent ◽  
Complex Numbers ◽  
Linear Potential ◽  
Expectation Values ◽  
Uncertainty Product ◽  
Dependent Mass

We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.

scholarly journals Probabilistic theories and reconstructions of quantum theory

2021 ◽  
Author(s):  
Markus Müller
Keyword(s):  
Probability Theory ◽  
Quantum Theory ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Classical Probability ◽  
Quantum Bit ◽  
Classical Probability Theory ◽  
Generalized Probabilistic Theories ◽  
Usual Representation ◽  
Probabilistic Theories

These lecture notes provide a basic introduction to the framework of generalized probabilistic theories (GPTs) and a sketch of a reconstruction of quantum theory (QT) from simple operational principles. To build some intuition for how physics could be even more general than quantum, I present two conceivable phenomena beyond QT: superstrong nonlocality and higher-order interference. Then I introduce the framework of GPTs, generalizing both quantum and classical probability theory. Finally, I summarize a reconstruction of QT from the principles of Tomographic Locality, Continuous Reversibility, and the Subspace Axiom. In particular, I show why a quantum bit is described by a Bloch ball, why it is three-dimensional, and how one obtains the complex numbers and operators of the usual representation of QT.

scholarly journals Quantum theory based on real numbers can be experimentally falsified

Nature ◽  
2021 ◽  
Author(s):  
Marc-Olivier Renou ◽  
David Trillo ◽  
Mirjam Weilenmann ◽  
Thinh P. Le ◽  
Armin Tavakoli ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Quantum States ◽  
Complex Numbers ◽  
Real Numbers ◽  
Quantum Formalism ◽  
Physical Experiments ◽  
Independent States

AbstractAlthough complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces1,2. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural3. In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states4. Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.

scholarly journals The ABC of Physics

2021 ◽  
Vol 3 (1) ◽  
pp. 9
Author(s):  
John Skilling ◽  
Kevin Knuth
Keyword(s):  
Quantum Theory ◽  
Gaussian Distribution ◽  
Complex Numbers ◽  
Time And Space ◽  
Fundamental Object ◽  
Pauli Matrices ◽  
Hilbert Sphere

Why quantum? Why spacetime? We find that the key idea underlying both is uncertainty. In a world lacking probes of unlimited delicacy, our knowledge of quantities is necessarily accompanied by uncertainty. Consequently, physics requires a calculus of number pairs and not only scalars for quantity alone. Basic symmetries of shuffling and sequencing dictate that pairs obey ordinary component-wise addition, but they can have three different multiplication rules. We call those rules A, B and C. “A” shows that pairs behave as complex numbers, which is why quantum theory is complex. However, consistency with the ordinary scalar rules of probability shows that the fundamental object is not a particle on its Hilbert sphere but a stream represented by a Gaussian distribution. “B” is then applied to pairs of complex numbers (qubits) and produces the Pauli matrices for which its operation defines the space of four vectors. “C” then allows integration of what can then be recognised as energy-momentum into time and space. The picture is entirely consistent. Spacetime is a construct of quantum and not a container for it.

Time reversal and complex numbers in quantum theory

1959 ◽  
Vol 13 (2) ◽  
pp. 326-343 ◽  
Author(s):  
E. Fabri
Keyword(s):  
Quantum Theory ◽  
Time Reversal ◽  
Complex Numbers
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