Charge algebra for non-abelian large gauge symmetries at O(r)

Asymptotic symmetries of gauge theories are known to encode infrared properties of radiative fields. In the context of tree-level Yang-Mills theory, the leading soft behavior of gluons is captured by large gauge symmetries with parameters that are O(1) in the large r expansion towards null infinity. This relation can be extended to subleading order provided one allows for large gauge symmetries with O(r) gauge parameters. The latter, however, violate standard asymptotic field fall-offs and thus their interpretation has remained incomplete. We improve on this situation by presenting a relaxation of the standard asymptotic field behavior that is compatible with O(r) gauge symmetries at linearized level. We show the extended space admits a symplectic structure on which O(1) and O(r) charges are well defined and such that their Poisson brackets reproduce the corresponding symmetry algebra.

Eventual Quantities, Immediate Extensions, and Special Cuts

This chapter deals with eventual quantities, immediate extensions, and special cuts. It first considers the behavior of eventual quantities before discussing Newton weight, Newton degree, and Newton multiplicity as well as Newton weight of linear differential operators. It then establishes the following result: Every asymptotically maximal H-asymptotic field with rational asymptotic integration is spherically complete. The chapter proceeds by describing special (definable) cuts in H-asymptotic fields K with asymptotic integration and introducing some key elementary properties of K, namely λ‎-freeness and ω‎-freeness, which indicate that these cuts are not realized in K. It shows that has these properties. Finally, it looks at certain special existentially definable subsets of Liouville closed H-fields K, along with the behavior of the functions ω‎ and λ‎ on these sets.

Newtonianity of Directed Unions

This chapter considers the newtonianity of directed unions and proves an analogue of Hensel's Lemma for ω‎-free differential-valued fields of H-type: Theorem 15.0.1. Here K is an H-asymptotic field with asymptotic couple (Γ‎, ψ‎), and γ‎ ranges over Γ‎. The chapter first describes finitely many exceptional values, integration and the extension K(x), and approximating zeros of differential polynomials before proving Theorem 15.0.1, which states: If K is d-valued with ∂K = K, and K is a directed union of spherically complete grounded d-valued subfields, then K is newtonian. In concrete cases the hypothesis K = ∂K in the theorem can often be verified by means of Corollary 15.2.4.

Newtonian Differential Fields

This chapter deals with Newtonian differential fields. Here K is an ungrounded H-asymptotic field with Γ‎ := v(Ksuperscript x ) not equal to {0}. So the subset ψ‎ of Γ‎ is nonempty and has no largest element, and thus K is pre-differential-valued by Corollary 10.1.3. An extension of K means an H-asymptotic field extension of K. The chapter first considers the relation of Newtonian differential fields to differential-henselianity before discussing weak forms of newtonianity and differential polynomials of low complexity. It then proves newtonian versions of d-henselian results in Chapter 7, leading to the following analogue of Theorem 7.0.1: If K is λ‎-free and asymptotically d-algebraically maximal, then K is ω‎-free and newtonian. Finally, it describes unravelers and newtonization.

Asymptotic Fields and Asymptotic Couples

This chapter deals with asymptotic differential fields and their asymptotic couples. Asymptotic fields include Rosenlicht's differential-valued fields and share many of their basic properties. A key feature of an asymptotic field is its asymptotic couple. The chapter first defines asymptotic fields and their asymptotic couples before discussing H-asymptotic couples. It then considers asymptotic couples independent of their connection to asymptotic fields, along with the behavior of differential polynomials as functions on asymptotic fields. It also describes asymptotic fields with small derivation and the operations of coarsening and specialization, algebraic and immediate extensions of asymptotic fields, and differential polynomials of order one. Finally, it proves some useful extension results about asymptotic couples and establishes a property of closed H-asymptotic couples.

Evaluation of the Higher Order Terms of the Wedge-Splitting Specimen Based on the SBFEM

The scaled boundary finite element method (abbr. SBFEM) is a semi-analytical method developed by Wolf and Song. The analytical advantage of the solution in the radial direction allows SBFEM converge to the Williams expansion. The coefficients of the Williams expansion, including the stress intensity factor, the T-stress, and higher order terms can be calculated directly without further processing. In the paper the coefficients of higher order terms of the crack tip asymptotic field of typical wedge splitting specimens with two different loading arrangements are evaluated using SBFEM. Numerical results show the method has high accuracy and effectiveness. The results have certain significance on determining crack stability of the wedge-splitting specimen.

REVIEW OF THE N-QUANTUM APPROACH TO BOUND STATES

We describe a method of solving quantum field theories using operator techniques based on the expansion of interacting fields in terms of asymptotic fields. For bound states, we introduce an asymptotic field for each (stable) bound state. We choose the nonrelativistic hydrogen atom as an example to illustrate the method. Future work will apply this N-quantum approach to relativistic theories that include bound states in motion.