An operator-valued quantum phase space formula is constructed. The phase space formula of Quantum Mechanics provides a natural link between first and second quantization, thus contributing to the understanding of quantization problem. By the combination of quantization and hamiltonization of dynamics, a quantization hierarchy is introduced, beyond the framework of first and second quantization and generalizing the standard quantum theory. We apply our quantization method to quantum many-body system and propose an eclectic model, in which the dimension of Hilbert space does not scale exponentially with the number of particles due to the locality of interaction, and the evolution is a constrained Hamiltonian dynamics.
An operator-valued quantum phase space formula is constructed. The phase space formula of Quantum Mechanics provides a natural link between first and second quantization, thus contributing to the understanding of quantization problem. By the combination of quantization and hamiltonization of dynamics, a quantization hierarchy is introduced, beyond the framework of first and second quantization and generalizing the standard quantum theory. We apply our quantization method to quantum many-body system and propose an eclectic model, in which the dimension of Hilbert space does not scale exponentially with the number of particles due to the locality of interaction, and the evolution is a constrained Hamiltonian dynamics.
An approach is formulated to the problem of obtaining approximate solutions to many-body quantum mechanics. The starting point is the representation of quantum mechanics as Hamiltonian mechanics on a symplectic manifold (phase space). It is shown that Dirac's variation of an action integral provides a natural mechanism for constraining the dynamics to symplectic submanifolds and gives rise to a hierarchy of approximate many-body theories of which Hartree-Fock, random-phase approximation, time-dependent Hartree-Fock, and the double commutator equations of motion formalism are special cases.
We develop a general `classicalization' procedure that links Hilbert-space and phase-space operators, using Weyl's operator. Then we transform the time-dependent Schrödinger equation into a phase-space picture using free parameters. They include position Q and momentum P . We expand the phase-space Hamiltonian in an ℏ-Taylor series and fix parameters with the condition that coefficients of ℏ 0 , -iℏ 1 ∂/∂ Q and iℏ 1 ∂/∂ P vanish. This condition results in generalized Hamilton equations and a natural link between classical and quantum dynamics, while the quantum motion-equation remains exact . In this picture, the Schrödinger equation reduces in the classical limit to a generalized Liouville equation for the quantum-mechanical system state. We modify Glauber's coherent states with a suitable phase factor S ( Q , P , t ) and use them to obtain phase-space representations of quantum dynamics and quantum-mechanical quantities.
A central problem in modern mathematical physics is to study interacting systems beyond the reach of perturbation theory. This thesis studies interacting quantum systems using the mathematical theory of operator algebras. In the first part of this thesis, we consider how to construct quantum many-body systems from one-particle systems. This problem has a well-established solution - namely, the second quantization (or the Gaussian functor). However, it seems difficult for the second quantization to go beyond generalized free field constructions without the help of perturbation theory. The second quantization takes real Hilbert spaces as inputs. Using only information from one-particle Hilbert spaces, it is mathematically impossible for the second quantization to produce connected higher-point correlation functions. Moreover, from the perspectives of mathematical quantization and operator algebra theory, it is more natural to have a quantization procedure that takes an algebra along with a functional as the input. Such a pair of mathematical objects contains more information about the underlying one-particle system and includes the Hilbert space of states (via the GNS construction) as part of the data. Inspired by quantum probability theory, we introduce an alternative approach to construct quantum many-body systems. This approach is based on a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Mathematically, Poissonization is a functor from the category of von Neumann algebras with normal semi nite faithful weights to the category of von Neumann algebras with normal faithful states. It is a natural adaptation of the second quantization to the context of von Neumann algebras. This thesis discusses several properties of Poissonization, and uses Poissonization as a tool to construct various toy models of algebraic quantum field theories relevant to high energy physics. This collection of examples aims at demonstrating the versatility of Poissonization. In the second part of this thesis, we study the noisy interactions between an open quantum system and its environment. One of the challenges in quantum information science is to control open quantum systems with a large number of qubits. An important aspect of many-body systems is the emergence of collective phenomena. One collective noise model is an open atomic system in an electromagnetic environment. This model was considered by Dicke in the 50's. In this thesis, we study the entropic decay in Dicke's model and other related collective noise models. Specifically, we develop a general framework to estimate the spectral gap and modified logarithmic Sobolev constant of these collective noise models. In addition, we study the necessary mixing conditions a general Dicke's model must satisfy in order to have a unique equilibrium state.
The present paper is devoted to the study of the few–body quantum mechanical problem with singular finite rank cluster interactions. The corresponding Hamiltonians play an important role in mathematical physics, since few body Hamiltonians with more regular interactions present considerable difficulties which make impossible a detailed analytic study [ 13 ]. Also a numerical study of such Hamiltonians using Faddeev equations present very hard problems, since the interaction between the particles does not vanish at large distances. On the other hand few-body problems with singular cluster interaction are useful and intensively studied in statistical physics, since the corresponding Hamiltonians can be analyzed in detail even if the number of particles is very large. Models describing one dimensional particles are of particular interest, since the eigenfunctions of the many body Hamiltonians can often be calculated using Bethe Ansatz [ 14 ]. In fact the well-known Yang-Baxter equation was first written in connection with the study of system of several one-dimensional particles with pairwise delta interactions. Similar methods were used in atomic physics to study collisions of several particles [ 12 ] (in three dimensions). Few-body systems in applications are usually investigated by combining the classical description of the dynamics of heavy atoms with the quantum description of the dynamics of electrons and other light particles. The first attempt to construct a three body Hamiltonian describing three quantum particles in R 3 interacting through pairwise delta functional potentials is due to G.V. Skorniakov and K.A.Ter-Martirosian [ 29 ]. R.A.Minlos and L.D.Faddeev proved that the corresponding Hamiltonian is not bounded from below and therefore cannot be used in 278 the originally intended physical applications [ 24 , 25 ]. The operator was defined using the method of self–adjoint perturbations used by F.A.Berezin and L.D.Faddeev for investigating Schödinger operators with delta potential in R 3 [ 10 ]. Almost three decades later semibounded three-body operators in dimension three with generalized two-body interactions were constructed by extending the standard Hilbert space of square integrable functions in R 9 [ 17 , 18 , 19 , 27 , 30 ]. The most interesting model considered used the theory of generalized extensions suggested by B.Pavlov [ 7 , 26 ]. Different aspects of these models were analyzed recently [ 8 , 23 , 20 ]. A general approach to these operators is described in [ 7 , 22 ]. Let us mention that a realization of a many-body lower bounded Hamiltonian with point interactions for particles in R 3 has been obtained in the original Hilbert space by using the theory of Dirichlet forms [ 2 ]. The few-body systems of two dimensional particles with two-body interactions was studied in [ 11 ], where it was proven that the corresponding Hamiltonian is semibounded.
Efficiently simulating many-body quantum systems with Coulomb interactions is a fundamental question in quantum physics, quantum chemistry, and quantum computing, yet it presents unique challenges: the Hamiltonian is an unbounded operator; its Hilbert space dimension grows exponentially with particle number; and the Coulomb potential is singular, long-ranged, and non-smooth, violating the regularity assumptions of many prior state-of-the-art many-body simulation analyses. In this work, we establish rigorous error bounds for Trotter formulas applied to these systems. Our first main result shows that for general initial conditions in the domain of the Hamiltonian, second-order Trotter achieves a 1/4 convergence rate with explicit polynomial dependence of the error prefactor on the particle number. The polynomial dependence on system size suggests that the algorithm remains quantumly efficient, even without introducing any regularization of the Coulomb singularity. Such a worst-case rate has been observed in prior work for the hydrogen ground state, demonstrating its physical relevance. We further establish one-step local error lower bounds of order 5/4 for both formulas, showing that the local exponent is sharp. Our second main result identifies a set of physically meaningful conditions on the initial state under which the convergence rates improve, with the full first-order and second-order rates recovered at sufficiently high angular momentum. Our theoretical findings are consistent with prior numerical observations. Our third main result establishes improved convergence rates for many-body fermionic Coulomb systems with initial data of different Sobolev regularity. In particular, even without imposing any additional regularity beyond the natural domain of the Hamiltonian, both the first- and second-order Trotter formulas converge with rate 1/2, rather than the general 1/4 rate.
Allowing for virtual paths in phase space permits an extension of Hamilton’s principle of least action, of lagrangians and of hamiltonians to phase space. A subsequent canonical quantization, then, provides a framework for quantum statistical mechanics. The classical statistical mechanics and the conventional quantum mechanics emerge as special case of this formalism. Von Neumann’s density matrix may be inferred from it. Wigner’s functions and their evolution equation may also be obtained by a unitary transformation.