Narrow Results

Let H be a self-adjoint operator on a Hilbert space , T be a symmetric operator on and K(t)(t ∈ ℝ) be a bounded self-adjoint operator on . We say that (T, H, K) obeys the generalized weak Weyl relation (GWWR) if e^-itHD(T) ⊂ D(T) for all t ∈ ℝ and Te^-itHψ = e^-itH(T+K(t))ψ, ∀ψ ∈ D(T) (D(T) denotes the domain of T). In the context of quantum mechanics where H is the Hamiltonian of a quantum system, we call T a generalized time operator of H. We first investigate, in an abstract framework, mathematical structures and properties of triples (T, H, K) obeying the GWWR. These include the absolute continuity of the spectrum of H restricted to a closed subspace of , an uncertainty relation between H and T (a "time-energy uncertainty relation"), the decay property of transition probabilities |〈ψ,e^-itHϕ〉|² as |t| → ∞ for all vectors ψ and ϕ in a subspace of , where 〈·,·〉 denotes the inner product of . We describe methods to construct various examples of triples (T, H, K) obeying the GWWR. In particular, we show that there exist generalized time operators of second quantization operators on Fock spaces (full Fock spaces, boson Fock spaces and fermion Fock spaces) which may have applications to quantum field models with interactions.

A series of successive quantizations is considered, starting with the quantization of a non-relativistic or relativistic point particle: (1) quantization of a particle’s position, (2) quantization of wave function, (3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first-order equation in time derivative for complex wave function (or functional), the equivalent second-order equation for its real part was used. In such a way, at each level of quantization, the equation a quantum state satisfies is just like that of a harmonic oscillator, and wave function(al) is composed in terms of the pair of its canonically conjugated variables.

This paper describes an approach to quantum electrodynamics (QED) in curved space–time obtained by considering infinite-dimensional algebra bundles associated to a natural principal bundle $Q$ associated with any locally conformally flat space–time, with typical fibers including the Fock space and a space of fermionic multiparticle states which forms a Grassmann algebra. Both these algebras are direct sums of generalized Hilbert spaces. The requirement of $K$ covariance associated with the geometry of space–time, where $K$ is the structure group of $Q$, leads to the consideration of $K$ intertwining operators between various spaces. Scattering processes are associated with such operators and are encoded in an algebra of kernels. Intertwining kernels can be generated using $K$ covariant matrix-valued measures. Feynman propagators, fermion loops and the electron self-energy can be given well-defined interpretations as such measures. Divergence-free calculations in QED can be carried out by computing the spectra of these measures and kernels (a process called spectral regularization). As an example of the approach the precise Uehling potential function for the $H$ atom is calculated without requiring renormalization from which the Uehling contribution to the Lamb shift can be calculated exactly.

Quantum computers by their nature are many particle quantum systems. Both the many-particle arrangement and being quantum are necessary for the existence of the entangled states, which are responsible for the parallelism of the quantum computers. Second quantization is a very important approximate method of describing such systems. This lecture will present the general idea of the second quantization, and discuss shortly some of the most important formulations of second quantization.

The spectra of the second quantization and the symmetric second quantization of a strict Hilbert space contraction are computed explicitly and shown to coincide. As an application, we compute the spectrum of the nonsymmetric Ornstein–Uhlenbeck operator L associated with the infinite-dimensional Langevin equation

Semi-classical asymptotic expansions up to any finite orders of the partition function of an abstract Bose field model is considered.

We present a new method for treating the dynamics of molecular systems. The method has been named "quantum dressed" classical mechanics and is based on an expansion of the wave function in a time-dependent basis-set, the Gauss–Hermite basis-set. From here it is possible to proceed in two ways, one is in principle exact and the other approximate. In the exact approach one constructs a discrete variable representation (DVR) in which the grid points are defined by the Hermite part of the Gauss–Hermite basis set. In the approximate method a second order expansion of the potential around the classical trajectories is introduced and the quantum dymamics solved in a second quantization rather than a wave-function representation.

In this article, a stochastic SIR-type model for COVID-19 epidemic is built using the standard field theoretical language based on creation and annihilation operators. From the model, we derive the time evolution of the mean number of infectious (active cases) and deceased individuals. In order to capture the effects of lockdown and social distancing, we use a time-dependent infection rate. The results are in good agreement with the data for three different waves of epidemic activity in South Korea.

While collecting some personal memories about Julius Wess, I briefly describe some aspects of my recent work on many particle quantum mechanics and second quantization on noncommutative spaces obtained by twisting, and their connection to him.