相关论文: A geometric Hamiltonian description of composite q…
We survey some of the main conceptual developments in the study of PT-symmetric and pseudo-Hermitian Hamiltonian operators that have taken place during the past ten years or so. We offer a precise mathematical description of a quantum…
This note, in a rather expository manner, serves as a conceptional introduction to the certain underlying mathematical structures encoding the geometric quantization formalism and the construction of Witten's quantum invariants, which is in…
PT-symmetric quantum mechanics is an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose and complex conjugate) is replaced by the physically transparent condition of space-time reflection…
Smooth composite bundles provide the adequate geometric description of classical mechanics with time-dependent parameters. We show that the Berry's phase phenomenon is described in terms of connections on composite Hilbert space bundles.
Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries of the tetrahedron in R^3, we use…
We consider the space of probabilities {P(x)}, where the x are coordinates of a configuration space. Under the action of the translation group there is a natural metric over the space of parameters of the group given by the Fisher-Rao…
In the first part of this review we introduce the basics theory behind geometric phases and emphasize their importance in quantum theory. The subject is presented in a general way so as to illustrate its wide applicability, but we also…
We address several problems concerning the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space, in particular the geometry of the space of density states and canonical group actions on it. For quantum…
In this paper, we argue that quantum coherence in a bipartite system can be contained either locally or in the correlations between the subsystems. The portion of quantum coherence contained within correlations can be viewed as a kind…
Quantum machine learning (QML) seeks to exploit the intrinsic properties of quantum mechanical systems, including superposition, coherence, and quantum entanglement for classical data processing. However, due to the exponential growth of…
This paper introduces a formalism that aims to describe the intricacies of quantum computation by establishing a connection with the mathematical foundations of tensor theory and multilinear maps. The focus is on providing a comprehensive…
We generalize the Hamiltonian picture of General Relativity coupled to classical matter, known as geometrodynamics, to the case where such matter is described by a Quantum Field Theory in Curved Spacetime, but gravity is still described by…
A quantum theory is constructed for the system of a relativistic particle with mass m moving freely on the SL(2,R) group manifold. Applied to the cotangent bundle of SL(2,R), the method of Hamiltonian reduction allows us to split the…
We present a geometrical description of the space of density states of a quantum system of finite dimension. After presenting a brief summary of the geometrical formulation of Quantum Mechanics, we proceed to describe the space of density…
The tensor product postulate of quantum mechanics states that the Hilbert space of a composite system is the tensor product of the components' Hilbert spaces. All current formalizations of quantum mechanics that do not contain this…
These notes review a description of quantum mechanics in terms of the topology of spaces, basing on the axioms of Topological Quantum Field Theory and path integral formalism. In this description quantum states and operators are encoded by…
Familiar formulations of classical and quantum mechanics are shown to follow from a general theory of mechanics based on pure states with an intrinsic probability structure. This theory is developed to the stage where theorems from quantum…
A formulation of quaternionic quantum mechanics ($\mathbb{H}$QM) is presented in terms of a real Hilbert space. Using a physically motivated scalar product, we prove the spectral theorem and obtain a novel quaternionic Fourier series. After…
Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt…
The classical mechanics of a finite number of degrees of freedom requires a symplectic structure on phase space C, but it is independent of any complex structure. On the contrary, the quantum theory is intimately linked with the choice of a…