We introduce a bi-oriented interlaced matrix product on 2 × 2 matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley-Dickson doubling. The opposite orientations ⋄ R and ⋄ L coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators Q, P on L 2 (R), a matrix D and its companion D satisfy, on the common Schwartz domain, We study the associated interlaced eigenvalue equations for operator-valued columns. The combinations Φ = A + iB and Ψ = A -iB decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space S(R 2 ) ⊂ L 2 (R 2 ) ⊂ S ′ (R 2 ); the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.
We introduce a bi-oriented interlaced matrix product on 2 × 2 matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley-Dickson doubling. The opposite orientations ⋄ R and ⋄ L coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators Q, P on L 2 (R), a matrix D and its companion D satisfy, on the common Schwartz domain, We study the associated interlaced eigenvalue equations for operator-valued columns. The combinations Φ = A + iB and Ψ = A -iB decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space S(R 2 ) ⊂ L 2 (R 2 ) ⊂ S ′ (R 2 ); the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.
We introduce a bi-oriented interlaced matrix product on 2×2 matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley--Dickson doubling. The opposite orientations XR and XL coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators Q,P on L2(R), a matrix D and its companion D satisfy, on the common Schwartz domain, DXRD=DXLD=2HId2, where H=(P2+Q2)/2. We study the associated interlaced eigenvalue equations for operator-valued columns. The combinations Φ=A+iB and Ψ=A−iB decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space S(R2)⊂L2(R2)⊂S′(R2); the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.
Abstract In this paper, we show that the position and the derivative operators, q^ q^ and D^ D^ , can be treated as ladder operators connecting various vectors of two biorthonormal families, Fφ Fφ and Fψ Fψ . In particular, the vectors in Fφ Fφ are essentially monomials inx, xk xk , while those in Fψ Fψ are weak derivatives of the Dirac delta distribution, δ(m)(x) δ(m)(x) , times some normalization factor. We also show how bi-coherent states can be constructed for these q^ q^ and D^ D^ , both as convergent series of elements of Fφ Fφ and Fψ Fψ , or using two different displacement-like operators acting on the two vacua of the framework. Our approach generalizes well- known results for ordinary coherent states.
Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to create generalized “Cartesian-like” positions, W, and momenta, Pw , with unit Poisson brackets. These are quantized by the usual replacement of the classical , x Px by quantum operators, leading to an infinite family of potential “operator observables”. However, all but one of the resulting operators are not Hermitian (formally self-adjoint) in the original position representation. Using either the chain rule or Dirac quantization, we show that the resulting operators are “quasi-Hermitian” relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, Pw , quantization yields a) continuous mutually unbiased bases (MUB), b) orthogonal bases (with Dirac delta normalization), c) biorthogonal bases (with Dirac delta normalization), d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states and Wigner distributions. The MUB lead to W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (±1,±i) , as well as “W-linear chirps”. As expected, W, Pw satisfy the uncertainty product relation: ΔWΔPw ≥1/2 , h=1.
In quantum mechanics, physical observables are represented by operators on a certain Hilbert space. The question of how such operators commute, has been a matter of discussion. In the standard perspective, the operators corresponding to the position and momentum of a system are assumed to satisfy the canonical commutation relations. It is known that these relations imply that the Hamilton and Heisenberg equations of motion are compatible as operator equations. However, Wigner showed that the inverse statement is not true. Therefore, it is a much weaker constraint to impose the compatibility of the equations of motion. For any physical system, this results in a set of compatibility conditions, which form the core of Wigner quantization. The key to finding operators that are subject to the compatibility conditions is provided by Lie superalgebras and their representations. Lie superalgebras can be defined as algebras generated by odd elements satisfying particular superbracket relations. Using these defining relations, Lie superalgebra generators can be found that obey the compatibility conditions. The Lie superalgebra elements act as operators on a vector space if we consider Lie superalgebra representations. Various physical systems are investigated in this thesis in the context of Wigner quantization. For each of these systems solutions are found in terms of Lie superalgebra generators, after which specific representations are examined. In such representations, the focus lies on determining particular physical properties of the system. Most of the studied systems are harmonic oscillator models. First, we examine a set-up of coupled harmonic oscillators, for which the interaction is represented by an interaction matrix. Then we focus on the angular momentum content of a 3N-dimensional harmonic oscillator. Finally, our attention goes to two one-dimensional systems, namely the free particle and the Berry-Keating-Connes Hamiltonian. The latter of these Hamiltonians is notorious for its possible connection with the Riemann hypothesis. All of the aforementioned Hamiltonians have been extensively investigated in the context of canonical quantization, so that our results can be compared to the well-known canonical case.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.