相关论文: Revisiting the Algebraic and Analytic Descriptions…
The properties which give quantum mechanics its unique character - unitarity, complementarity, non-commutativity, uncertainty, nonlocality - derive from the algebraic structure of Hermitian operators acting on the wavefunction in complex…
This paper addresses the question why quantum mechanics is formulated in a unitary Hilbert space, i.e. in a manifestly complex setting. Investigating the linear dynamics of real quantum theory in a finite-dimensional Euclidean Hilbert space…
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review…
A finite dimensional system with a quadratic Hamiltonian constraint is Dirac quantized in holomorphic, antiholomorphic and mixed representations. A unique inner product is found by imposing Hermitian conjugacy relations on an operator…
Position deformation of a Heisenberg algebra and Hilbert space representation of both maximal length and minimal momentum uncertainties may lead to loss of Hermiticity of some operators that generate this algebra. Consequently, the…
We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert…
In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the…
A non-Hermitian operator $H$ defined in a Hilbert space with inner product $\langle\cdot|\cdot\rangle$ may serve as the Hamiltonian for a unitary quantum system, if it is $\eta$-pseudo-Hermitian for a metric operator (positive-definite…
Usually in quantum mechanics the Heisenberg algebra is generated by operators of position and momentum. The algebra is then represented on an Hilbert space of square integrable functions. Alternatively one generates the Heisenberg algebra…
The classical Hilbert space formulation of the axioms of Quantum Mechanics appears to leave open the question whether the Hermitian operators which are associated with the observables of a finite non-relativistic quantum system are uniquely…
We propose a novel framework for the quantum geometry of expectation values over arbitrary sets of operators and establish a link between this geometry and the eigenstates of Hamiltonian families generated by these operators. We show that…
Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure generated by unbounded metric operators in a Hilbert space. To that effect, we consider the notions of similarity and quasi-similarity…
By using a coherent state quantization of paragrassmann variables, operators are constructed in finite Hilbert spaces. We thus obtain in a straightforward way a matrix representation of the paragrassmann algebra. This algebra of finite…
Quantum mechanics of unitary systems is considered in quasi-Hermitian representation. In this framework the concept of perturbation is found counterintuitive, for three reasons. The first one is that in this formalism we are allowed to…
We show that QM can be represented as a natural projection of a classical statistical model on the phase space $\Omega= H\times H,$ where $H$ is the real Hilbert space. Statistical states are given by Gaussian measures on $\Omega$ having…
The necessity of complex numbers in quantum mechanics has long been debated. This paper develops a real Kahler space formulation of quantum mechanics [19], asserting equivalence to the standard complex Hilbert space framework. By mapping…
Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket, and a quasidensity operator. These are analogues of the star product, the Moyal bracket, and…
We re-examine the appearance of semiheaps and (para-associative) ternary algebras in quantum mechanics. In particular, we review the construction of a semiheap on a Hilbert space and the set of bounded operators on a Hilbert space. The new…
In the conventional Schr\"{o}dinger's formulation of quantum mechanics the unitary evolution of a state $\psi$ is controlled, in Hilbert space ${\cal L}$, by a Hamiltonian $\mathfrak{h}$ which must be self-adjoint. In the recent,…
In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum…