Nuclear reactions are complex, with a large number of possible channels. Understanding how different channels contribute to a given reaction is investigated by perturbing the continuous spectrum. Tools are developed to investigate reaction mechanisms by identifying the contributions from each reaction channel. Cluster decomposition methods, along with the spectral theory of proper subsystem problems, is used to identify the part of the nuclear Hamiltonian responsible for scattering into each channel. The result is an expression of the nuclear Hamiltonian as a sum over all scattering channels of channel Hamiltonians. Each channel Hamiltonian is constructed from solutions of proper subsystem problems. Retaining any subset of channel Hamiltonians results in a truncated Hamiltonian where the scattering wave functions for the retained channels differ from the wave functions of the full Hamiltonian by N-body correlations. The scattering operator for the truncated Hamiltonian satisfies an optical theorem in the retained channels. Because different channel Hamiltonians do not commute, how they interact determines their contribution to the full dynamics.

Light-front formulations of quantum field theories have many advantages for computing electroweak matrix elements of strongly interacting systems and other quantities that are used to study hadronic structure. The theory can be formulated in Hamiltonian form so non-perturbative calculations of the strongly interacting initial and final states are in principle reduced to linear algebra. These states are needed for calculating parton distribution functions and other types of distribution amplitudes that are used to understand the structure of hadrons. Light-front boosts are kinematic transformations so the strongly interacting states can be computed in any frame. This is useful for computing current matrix elements involving electroweak probes where the initial and final hadronic states are in different frames related by the momentum transferred by the probe. Finally in many calculations the vacuum is trivial so the calculations can be formulated in Fock space. The advantages of light front-field theory would not be interesting if the light-front formulation was not equivalent to the covariant or canonical formulations of quantum field theory. Many of the distinguishing properties of light-front quantum field theory are difficult to reconcile with canonical or covariant formulations of quantum field theory. This paper discusses the resolution of some of the apparent inconsistencies in canonical, covariant and light-front formulations of quantum field theory. The puzzles that will be discussed are (1) the problem of inequivalent representations (2) the problem of the trivial vacuum (3) the problem of ill-posed initial value problems (4) the problem of rotational covariance (5) the problem of zero modes and (6) the problem of spontaneously broken symmetries.

A discrete formulation of the real-time path integral as the expectation value of a functional of paths with respect to a complex probability on a sample space of discrete valued paths is explored. The formulation in terms of complex probabilities is motivated by a recent reinterpretation of the real-time path integral as the expectation value of a potential functional with respect to a complex probability distribution on cylinder sets of paths. The discrete formulation in this work is based on a discrete version of Weyl algebra that can be applied to any observable with a finite number of outcomes. The origin of the complex probability in this work is the completeness relation. In the discrete formulation the complex probability exactly factors into products of conditional probabilities and exact unitarity is maintained at each level of approximation. The approximation of infinite dimensional quantum systems by discrete systems is discussed. Applications to scattering theory and quantum field theory are illustrated.

We build a relativistic model to perform calculations of exclusive, semi-exclusive and inclusive unpolarized cross sections and various polarization observables in electron and neutrino scattering experiments with deuteron targets. The strong interaction dynamics is defined by an explicit dynamical unitary representation of the Poincaré group, where representations of space translations and rotations in the interacting and non-interacting representations are the same. The Argonne V18 potential is used to construct a relativistic nucleon-nucleon interaction reproducing the experimental deuteron binding energy and nucleon-nucleon scattering observables. Our formalism does not include the pion production channel and neglects two-body contributions in the electromagnetic as well as in the weak nuclear current operator. We show that it is applicable to processes at kinematics, where the internal two-nucleon energy remains below the pion production threshold but the magnitude of the three-momentum transfer extends at least to several GeV.

Phenomenological Poincaré invariant quantum mechanical models can provide an efficient description of the dynamics of strongly interacting particles that is frame independent and consistent with spectral and scattering observables. These models are representation dependent and in order to apply them to reactions with electromagnetic probes it is necessary to use a consistent electromagnetic current operator. The purpose of this work is to use local gauge invariance to construct consistent strong current operators. Current operators are constructed from a model Hamiltonian by replacing momentum operators in the Weyl representation by gauge covariant derivatives. The construction provides a systematic method to construct expressions for current operators that are consistent with relativistic models of strong interaction dynamics.

Background: The vacuum in the light-front representation of quantum field theory is trivial while vacuum in the equivalent canonical representation of the same theory is non-trivial. Purpose: Understand the relation between the vacuum in light-front and canonical representations of quantum field theory and the role of zero-modes in this relation. Method: Vacuua are defined as linear functionals on an algebra of field operators. The role of the algebra in the definition of the vacuum is exploited to understand this relation. Results: The vacuum functional can be extended from the light-front Fock algebra to an algebra of local observables. The extension to the algebra of local observables is responsible for the inequivalence. The extension defines a unitary mapping between the physical representation of the local algebra and a sub-algebra of the light-front Fock algebra. Conclusion: There is a unitary mapping from the physical representation of the algebra of local observables to a sub-algebra of the light-front Fock algebra with the free light-front Fock vacuum. The dynamics appears in the mapping and the structure of the sub-algebra. This correspondence provides a formulation of locality and Poincaré invariance on the light-front Fock space.

The scattering equivalence of quantum field theories formulated with light-front and instant-form kinematic subgroups is established using nonperturbative methods. The difficulty with field theoretic formulations of Dirac's forms of dynamics is that the free and interacting unitary representations of the Poincare group are defined on inequivalent representations of the Hilbert space, which means that the concept of kinematic transformations must be modified on the Hilbert space of the field theory. This work addresses this problem by assuming the existence of a field theory with the expected properties and constructs equivalent representations with instant and front-form kinematic subgroups. The underlying field theory is not initially associated with an instant form or light-front form of the dynamics. In this construction the existence of a vacuum and one-particle mass eigenstates is assumed and both the light-front and instant-form representations are constructed to share the same vacuum and one-particle states. If there is spontaneous symmetry breaking there will be a 0 mass particle in the mass spectrum (assuming no Higgs mechanism). The free field Fock space plays no role. There is no "quantization" of a classical theory. The property that survives from the perturbative approach is the notion of a kinematic subgroup, which means kinematic Poincar ' e transformations can be trivially implemented by acting on suitable basis vectors. This nonperturbative approach avoids dealing with issues that arise in perturbative treatments where is it necessary to have a consistent treatment of renormalization, rotational covariance, and the structure of the light-front vacuum. While addressing these issues in a computational framework is the most important unanswered question for applications, this work may provide some insight into the nature of the expected resolution and identifies the origin of some of differences between the perturbative and nonperturbative approaches.

Sharp-momentum transition matrix elements for scattering from a short-range Gaussian potential are computed using a real-time path integral. The computation is based on a numerical implementation of a new interpretation of the path integral as the expectation of a potential functional with respect to a complex probability distribution on cylinder sets of paths. The method is closely related to a unitary transfer matrix computation.

Contribution to Light Cone 2019: This contribution discusses some of the advantages and unique properties of relativistic quantum theories with kinematic light-front symmetries.

A class of phenomenological relativistic models of hadronic systems motivated by quantum chromodynamics that have dual representations as models of mesons and nucleons or quarks and gluons is investigated. These models are designed to provided qualitative insight into the role of sea quarks in hadronic structure and reactions. The model assumption is that the Hamiltonian can be divided into two parts: one that involves degrees of freedom in the same connected local and global color singlet and the remaining interactions that allow the connected local and global color singlets to interact. The first class of interactions results in infinite towers of bare ``particles'' with hadronic quantum numbers. All but a finite number of these remain stable when the second class of interactions is included. The model interactions are expressed in terms of subhadronic degrees of freedom, which determine the bare hadronic spectrum and the interactions involving the bare hadrons in terms of a small number of subhadronic model parameters. As a first test, this paper considers the simplest case of mesons that interact via a string-breaking interaction. One virtue of this model is that all of the bare meson masses and eigenfunctions can be computed analytically. In addition, the string-breaking interaction leads to production vertices that can also be computed analytically. The relativistic wave functions have a light-front kinematic symmetry. The goal is to find a simple relativistic quantum mechanical model based on subhadronic degrees of freedom that can provide an efficient, qualitatively consistent description of hadronic masses, lifetimes, cross sections, sea quark effects, and electromagnetic properties. The simplicity of the model makes it a potentially useful tool to study the impact of sea quarks on hadronic structure and reactions.

A formally exact discrete multi-resolution representation of quantum field theory on a light front is presented. The formulation uses an orthonormal basis of compactly supported wavelets to expand the fields restricted to a light front. The representation has a number of useful properties. First, light front preserving Poincar\'e transformations can be computed by transforming the arguments of the basis functions. The discrete field operators, which are defined by integrating the product of the field and the basis functions over the light front, represent localized degrees of freedom on the light-front hyperplane. These discrete fields are irreducible and the vacuum is formally trivial. The light-front Hamiltonian and all of the Poincar\'e generators are linear combinations of normal ordered products of the discrete field operators with analytically computable constant coefficients. The representation is discrete and has natural resolution and volume truncations like lattice formulations. Because it is formally exact it is possible to systematically compute corrections for eliminated degrees of freedom.

Relativistic invariance in Euclidean formulations of quantum mechanics is discussed. Relativistic treatments of quantum theory are needed to study hadronic systems at sub-hadronic distance scales. Euclidean formulations of relativistic quantum mechanics have some computational advantages. In the Euclidean representation the physical Hilbert space inner product is expressed in terms of Euclidean space-time variables with no need for any analytic continuation. The identification of the complex Euclidean group with the complex Poincar\'e group relates the infinitesimal generators of both groups. In this work explicit representations of the Poincar\'e generators in Euclidean space-time variables for all positive-mass positive-energy irreducible representations of the Poincar\'e group are derived. The commutation relations are checked, both hermiticity and self-adjointness are established, and reflection positivity of the kernels is verified.

Background: Relativistic treatments of quantum mechanical systems are important for understanding hadronic structure and dynamics at subnucleon scales. Relativistic invariance of a quantum system means that there is an underlying unitary representation of the Poincare group. This is equivalent to the requirement that the quantum observables (probabilities, expectation values, and ensemble averages) for equivalent measurements performed in different inertial reference frames are identical. Different representations are used in practice, including Poincare covariant forms of dynamics, representations based on Lorentz covariant wave functions, Euclidean covariant representations, and representations generated by Lorentz covariant fields. Purpose: The purpose of this work is to illustrate the relation between the different equivalent representations of states in relativistic quantum mechanics. Method: The starting point is a description of a particle of mass m and spin j using irreducible representations of the Poincare group. Since any unitary representation of the Poincare group can be decomposed into a direct integral of irreducible representations, these are the basic building blocks of any relativistically invariant quantum theory. The equivalence is established by constructing equivalent Lorentz covariant irreducible representations from Poincare covariant irreducible representations and constructing equivalent Euclidean covariant irreducible representations from Lorentz covariant irreducible representations. Results: Equivalent descriptions for positive mass representations of arbitrary spin are presented in each of these frameworks. Dynamical realizations of the different representations are briefly discussed. Conclusion: Poincare covariant, Lorentz covariant, and Euclidean covariant realizations of relativistic dynamics are shown to be equivalent by explicitly relating the positive-mass positive-energy irreducible representations of the Poincare group that appear in the direct integral.

Daubechies wavelets are used to make an exact multi-scale decomposition of quantum fields. For reactions that involve a finite energy that take place in a finite volume, the number of relevant quantum mechanical degrees of freedom is finite. The wavelet decomposition has natural resolution and volume truncations that can be used to isolate the relevant degrees of freedom. The application of flow equation methods to construct effective theories that decouple coarse and fine scale degrees of freedom is examined.

We investigate both theoretical and computational aspects of using wavelet bases to decouple physics on different scales in quantum field theory.

We discuss the formulation of the scattering asymptotic condition in a relativistic quantum theory formulated in terms of reflection positive Euclidean Green functions.

We discuss the relation between the trivial light-front vacuum and the non-trivial Heisenberg vacuum.

I discuss the role of relativistic quantum mechanics in few-body physics, various formulations of relativistic few-body quantum mechanics and how they are related.

Studying of the relativistic three-body bound state in a three-dimensional (3D) approach is a necessary first step in a process to eventually perform scattering calculations at GeV energies, where partial-wave expansions are not useful. To this aim we recently studied relativistic effects in the binding energy and for the first time, obtained the relativistic 3B wave function [1]. The relativistic Faddeev integral equations for the bound state are formulated in terms of momentum vectors, and relativistic invariance is incorporated within the framework of Poincaré invariant quantum mechanics.