Louis Marchildon

Finite-Dimensional Bicomplex Hilbert Spaces

This paper is a detailed study of finite-dimensional modules defined on bicomplex numbers. A number of results are proved on bicomplex square matrices, linear operators, orthogonal bases, self-adjoint operators and Hilbert spaces, including the spectral decomposition theorem. Applications to concepts relevant to quantum mechanics, like the evolution operator, are pointed out.

Why Should We Interpret Quantum Mechanics

The development of quantum information theory has renewed interest in the idea that the state vector does not represent the state of a quantum system, but rather the knowledge or information that we may have on the system. I argue that this epistemic view of states appears to solve foundational problems of quantum mechanics only at the price of being essentially incomplete.

Scattering by a dielectric obstacle in a rectangular waveguide

A method for the computation of S-parameters associated with a rectangular waveguide with a rectangular or cylindrical obstacle of arbitrary complex scalar permittivity is presented. The method uses modal analysis and integral relationships to connect appropriate components of the field. In this way, convergence is achieved faster than by point-matching techniques. Our method is well adapted to resonance problems, as illustrated by comparison with both theoretical and experimental results. >

Boundary elements and analytic expansions applied to H-plane waveguide junctions

We propose a method to calculate field distribution and S-parameters in a planar n-port junction with rectangular waveguides. We use boundary elements on metallic walls, combined with modal expansion in waveguides and analytic representations for the field in dielectric samples or ferrites. Our approach uses fewer nodal points than either the finite-element or the boundary-element method. It is applicable to an H-plane junction with quite arbitrary geometry. The junction may contain several homogeneous or piecewise homogeneous circular cylindrical dielectric samples or ferrites, or samples with more general shape if a simple analyticity condition is met. >

Causal Loops and Collapse in the Transactional Interpretation of Quantum Mechanics

Cramers transactional interpretation of quantum mechanics is reviewed, and a number of issues related to advanced interactions and state vector collapse are analyzed. Where some have suggested that Cramers predictions may not be correct or definite, I argue that they are, but I point out that the classical-quantum distinction problem in the Copenhagen interpretation has its parallel in the transactional interpretation.

Two-particle interference in standard and Bohmian quantum mechanics

The compatibility of standard and Bohmian quantum mechanics has recently been challenged in the context of two-particle interference, both from a theoretical and an experimental point of view. We analyse different setups proposed and derive corresponding exact forms for Bohmian equations of motion. The equations are then solved numerically, and shown to reproduce standard quantum-mechanical results.

New dielectric method for the measurement of physical adsorption of gases at high pressure

A new method for the measurement of gas adsorption at high pressure is described in detail. The method is based on dielectric virial coefficients and it takes advantage of the dielectric technique for the accurate measurement of the compressibility factor of gases at high pressure. The method is simple, self‐sufficient, easy to use, and permits precise measurement of the density of nonideal gases. Adsorption measurements of methane on BPL‐activated carbon as a function of pressure at 25 °C are reported.

The graded Lie groups SU(2,2/1) and OSp(1/4)

We study the graded Lie groups corresponding to the graded Lie algebras SU(2,2/1) and OSp(1/4). General finite group transformations are parametrized, and nonlinear representations are obtained on coset spaces. Jordan and traceless algebras are constructed which admit these groups as automorphism groups.

Why I am not a QBist

Quantum Bayesianism, or QBism, is a recent development of the epistemic view of quantum states, according to which the state vector represents knowledge about a quantum system, rather than the true state of the system. QBism explicitly adopts the subjective view of probability, wherein probability assignments express an agent’s personal degrees of belief about an event. QBists claim that most if not all conceptual problems of quantum mechanics vanish if we simply take a proper epistemic and probabilistic perspective. Although this judgement is largely subjective and logically consistent, I explain why I do not share it.

Hilbert Space of the Bicomplex Quantum Harmonic Oscillator

Bicomplex numbers are pairs of complex numbers with a multiplication law that makes them a commutative ring. The problem of the quantum harmonic oscillator is investigated in the framework of bicomplex numbers. Starting with the commutator of the bicomplex position and momentum operators, we find eigenvalues and eigenkets of the bicomplex harmonic oscillator Hamiltonian. Coordinate‐basis eigenfunctions of the Hamiltonian are then obtained in terms of hyperbolic Hermite polynomials, and some of them are graphically illustrated. These eigenfunctions form a basis of an infinite‐dimensional module over bicomplex numbers, and this module can be given the structure of a bicomplex Hilbert space.