Related papers: Tagged vector space, Part I: Dirac notation as ori…
Drawing inspiration from Dirac's work on functions of non commuting observables, we develop a fresh approach to phase space descriptions of operators and the Wigner distribution in quantum mechanics. The construction presented here is…
The definition of quantum states in terms of tagged vector spaces is generalized to incorporate the spatiotemporal and spin degrees of freedom. Considering a tagged vector space where the index space is a function space, representing the…
The phase-space formulation of quantum mechanics has recently seen increased use in testing quantum technologies, including metho ds of tomography for state verification and device validation. Here, an overview of quantum mechanics in phase…
In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that…
The formalism of classical and quantum mechanics on phase space leads to symplectic and Heisenberg group representations, respectively. The Wigner functions give a representation of the quantum system using classical variables. The…
Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations…
An extension of the Weyl-Wigner-Moyal formulation of quantum mechanics suitable for a Dirac quantized constrained system is proposed. In this formulation, quantum observables are described by equivalent classes of Weyl symbols. The Weyl…
The Dirac wave function in a curved spacetime is usually defined as a quadruplet of scalar fields. It can alternatively be defined as a four-vector field. We describe these two representations in a common geometrical framework and we prove…
We develop a theory of ordered *-vector spaces with an order unit. We prove fundamental results concerning positive linear functionals and states, and we show that the order (semi)norm on the space of self-adjoint elements admits multiple…
We describe how self-adjoint ordered operator spaces, also called non-unital operator systems in the literature, can be understood as $*$-vector spaces equipped with a matrix gauge structure. We explain how this perspective has several…
Geometric properties of operators of quantum Dirac constraints and physical observables are studied in semiclassical theory of generic constrained systems. The invariance transformations of the classical theory -- contact canonical…
We prove that Wigner functions contain a symplectic connection. The latter covariantises the symplectic exterior derivative on phase space. We analyse the role played by this connection and introduce the notion of local symplectic…
The thesis is devoted to the phase space representation of relativistic quantum mechanics. For a class of observables with matrix-valued Weyl symbols proportional to the identity matrix, the Weyl-Wigner-Moyal formalism is proposed. The…
Expressing the Wigner distribution function in Dirac notation reveals its resemblance to a classical trajectory in phase space.
The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group $G$ is developed in detail. Several New features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion…
Operators in quantum mechanics - either observables, density or evolution operators, unitary or not - can be represented by c-numbers in operator bases. The position and momentum bases are in one to one correspondence with lagrangian planes…
We introduce a definition of symmetry generating vector fields on manifolds which are equipped with a first-order reductive Cartan geometry. We apply this definition to a number of physically motivated examples and show that our newly…
We show that the definition of an algebraic basis for a vector space allows the construction of an isomorphism with the one here called Algebraic Vector Space. Although the concept does not bring anything new, we mention some of the…
In the canonical approach to general relativity it is customary to parametrize the phase space by initial data on spacelike hypersurfaces. However, if one seeks a theory dealing with observations that can be made by a single localized…
A vector space over a field $\mathbb{F}$ is a set $V$ together with two binary operations, called vector addition and scalar multiplication. It is standard practice to think of a Euclidean space $\mathbb{R}^n$ as an $n$-dimensional real…