The use of modified plane waves with the spherical incoming waves modification as final states in matrix elements used in calculating scattering cross sections is discussed from the standpoint of two techniques. The first method makes use of general properties of Green's functions for outgoing and incoming waves and their reciprocity relations. The second method is based on expanding Green's function in terms of modified plane waves with either incoming or outgoing spherical wave modification.

In this paper, we study the scattering theory for a 2×2 matrix Schrödinger operator P=−h 2 d 2 /dx 2 I 2 +V(x)+hR(x,hD x ) on L 2 (R)⌖L 2 (R), where V(x) is a real diagonal matrix, the eigenvalues of which are never equal. Under some assumptions of analyticity and decay at infinity of V, we describe the asymptotic behavior of the scattering matrix S=(s ij ) 1≤i,j≤4 associated with P when the semi-classical parameter h goes to zero. Moreover, we obtain the estimate ‖S 12 ‖+‖S 21 ‖=O(e −δ/h ), where S 12 and S 21 are the two off-diagonal elements of S and δ>0 is a constant which is explicitly related to the behavior of V(x) in the complex domain.

We study infrared dynamics in quantum electrodynamics to construct the well-defined S- matrix without infrared divergences. S-matrix is a fundamental quantity for the scattering theory of particles in quantum field theories. However, the conventional S-matrix for theories with massless particles is not well-defined due to the infrared divergences. This problem originates in the fact that the interactions mediated by low energy massless particles create infinitely long-range forces between charged particles. Therefore, the better understanding of the infrared dynamics is necessary for improving the S-matrix. In the first half of this thesis, we focus on the following subjects that capture the universal features of the infrared dynamics: asymptotic symmetry, soft theorem, and memory effect. We elucidate the fundamental properties of the charge conservation law associated with the asymptotic symmetry and also develop the new relations among the three subjects. In the last half, the proper asymptotic states for the infrared finite S-matrix is investigated. The Faddeev-Kulish(F-K) dressed state has been known as a candidate for such a state. However, there was an argument that the F-K dressed states are not gauge invariant. We resolve the problem by deriving a correct gauge invariant condition and showing that the F-K dressed state is a solution of the condition. We also discuss the relation between the asymptotic state and the asymptotic symmetry for QED.

Abstract The problem of extending quantum-mechanical formal scattering theory to a more general class of models that also includes quantum field theories is discussed, with the aim of clarifying certain aspects of the definition of scattering states. As the strong limit is not suitable for the definition of scattering states in quantum field theory, some other limiting procedure is needed. Two possibilities are considered, the abelian limit and adiabatic switching. Formulas for the scattering states based on both methods are discussed, and it is found that generally there are significant differences between the two approaches. As an illustration of the applications and the features of these formulas, S-matrix elements and energy corrections in two quantum field theoretical models are calculated using (generalized) old-fashioned perturbation theory. The two methods are found to give equivalent results.

In physics, the S-matrix or scattering matrix is a matrix that relates the initial state and the final state of a physical system undergoing a scattering In physics, the S-matrix or scattering matrix is a matrix that relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More formally, in the context of QFT, the S-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states In physics, the S-matrix or scattering matrix is a matrix that relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More formally, in the context of QFT, the S-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states and the out-states) in the Hilbert space of physical states: a multi-particle state is said to be free (or non-interacting) if it transforms under Lorentz transformations as a tensor product, or direct product in physics parlance, of one-particle states as prescribed by equation (1) below. Asymptotically free then means that the state has this appearance in either the distant past or the distant future. While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no event horizons, it has a simple form in the case of the Minkowski space. In this special case, the Hilbert space is a space of irreducible unitary representations of the inhomogeneous Lorentz group (the Poincaré group); the S-matrix is the evolution operator between t = − ∞ {\displaystyle t=-\infty } (the distant past), and t = + ∞ {\displaystyle t=+\infty } (the distant future). It is defined only in the limit of zero energy density (or infinite particle separation distance). It can be shown that if a quantum field theory in Minkowski space has a mass gap, the state in the asymptotic past and in the asymptotic future are both described by Fock spaces.

We provide practical simulation methods for scalar field theories on a quantum computer that yield improved asymptotics as well as concrete gate estimates for the simulation and physical qubit estimates using the surface code. We achieve these improvements through two optimizations. First, we consider a finite volume approach for estimating the elements of the S-matrix. This approach is appropriate in general for 1+1D and for certain low-energy elastic collisions in higher dimensions. Second, we implement our approach using a series of different fault-tolerant simulation algorithms for Hamiltonians formulated both in the field occupation basis and field amplitude basis. Our algorithms are based on either second-order Trotterization or qubitization. The cost of Trotterization in occupation basis scales as O ( λ N 7 | Ω | 3 / ( M 5 / 2 ϵ 3 / 2 ) ) where λ is the coupling strength, N is the occupation cutoff, | Ω | is the volume of the spatial lattice, M is the mass of the particles and ϵ is the uncertainty in the energy calculation used for the S -matrix determination. Qubitization in the field basis scales as O ( | Ω | 2 ( k 2 Λ + k M 2 ) / ϵ ) , where k is the cutoff in the field and Λ is a scaled coupling constant. We find in both cases that the bounds suggest physically meaningful simulations can be performed using on the order of 4 × 10 6 physical qubits and 10 12 T -gates which corresponds to roughly one day on a superconducting quantum computer with surface code and a cy