Related papers: Functionals and the Quantum Master Equation
A definition of quantum mechanics on a manifold $ M $ is proposed and a method to realize the definition is presented. This scheme is applicable to a homogeneous space $ M = G / H $. The realization is a unitary representation of the…
Noncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this…
In this paper a generalization of Weyl quantization which maps a dynamical operator in a function space to a dynamical superoperator in an operator space is suggested. Quantization of dynamical operator, which cannot be represented as…
Effective equations are often useful to extract physical information from quantum theories without having to face all technical and conceptual difficulties. One can then describe aspects of the quantum system by equations of classical type,…
We introduce a suitable notion of integral operators (comprising the fractional Laplacian as a particular case) acting on functions with minimal requirements at infinity. For these functions, the classical definition would lead to divergent…
In many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and…
The authors previous derivation of a variational principle from the total work functional, as a generalization of the first variation of an action functional, is extended by deriving a corresponding generalization of the Hamiltonian…
A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into…
We define an infinite class of unitary transformations between position and momentum fractional spaces, thus generalizing the Fourier transform to a special class of fractal geometries. Each transform diagonalizes a unique Laplacian…
The von Neumann trace form of quantum statistical mechanics is transformed to an integral over classical phase space. Formally exact expressions for the resultant position-momentum commutation function are given. A loop expansion for wave…
A new derivation is given of Branson's factorization formula for the conformally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the…
Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of…
Let $Top_c$ be the category of compact spaces and continuous maps and $Top_f\subset Top_c$ be the full subcategory of finite spaces. Consider the covariant functor $Mor:Top_f^{op}\times Top_c\to Top_c$ that associates any pair $(X,Y)$ with…
Structures of quantum Fokker-Planck equations are characterized with respect to the properties of complete positivity, covariance under symmetry transformations and satisfaction of equipartition, referring to recent mathematical work on…
We give a review of concepts related to connection of classical and quantum theories, from the phase space perspective. Quantum theory is described by non-commutative operators of coordinates and momenta, results in values having a certain…
Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed and their Taylor towers are computed. We also show that these functors factor through…
We introduce a master constraint operator $\hat{\mathbf{M}}$ densely defined in the diffeomorphism invariant Hilbert space in loop quantum gravity, which corresponds classically to the master constraint in the programme. It is shown that…
The notion of quantum matrix pairs is defined. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of…
Stochastic Master equations or quantum filtering equations for mixed states are well known objects in quantum physics. Building a mathematically rigorous theory of these equations in infinite-dimensional spaces is a long standing open…
This is a survey on a notion of invariant operators, or Fourier multipliers on Hilbert spaces. This concept is defined with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. In particular this…