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Plan.tex
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Plan.tex
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\documentclass[12pt]{article}
\input{header.tex}
\newcommand{\sigdot}[1]{ \gv{\sigma} \hspace{-2pt} \cdot \hspace{-2pt} \v{#1} \,}
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\newcommand{\dotprod}[2]{ \v{#1} \hspace{-2pt} \cdot \hspace{-2pt} \v{#2} \,}
\newcommand{\beqa}{\begin{eqnarray*} }
\newcommand{\eeqa}{\end{eqnarray*} }
\newcommand{\pn}[2]{
#1
\hspace{1em} #2
}
\title{ Outline}
\author{}
\begin{document}
\maketitle
\[
H = \underbrace{\frac{p^2}{2m} + e\Phi }_{H_0} + V(x)
\]
\section{Introduction}
\subsection{ Description of the problem}
Describe the general problem we investigate:
Two charged particles in a loosely bound system, each particle having arbitrary spin
Calculate corrections to $g$-factor for particles in such a system
Talk about experimental motivation, measurements in O and C molecules.
Discuss how the problem is nonrelativistic, the precision we need, and the types of approximations we can then make.
\subsection{theoretical background}
Briefly discuss various theoretical approachs:
NRQED approach
Discuss prior work.
spin 1/2 approach
BMT equation
Khriplovich general spin formalism
Faustov ?
\section{Background for Nonrelativsitic Quantum Electrodynamics}
\subsection{Effective theories}
Some worked examples
\subsection{Nonrelativistic Quantum Electrodynamics}
General approach
Example with muonium
\section{The case of spin one-half}
General discussion of properties of the spin 1/2 case, in both relativistic and nonrelativistic cases.
Calculation with Foldy-Wouthyusen method, starting from equations of motions
Calculation with NRQED approach, calculations with diagrams.
\section{The case of spin one}
General discussion of properties of the spin 1 case, in both relativistic and nonrelativistic cases.
Contrast with spin 1/2.
Calculation with Foldy-Wouthyusen method, starting from equations of motions
Calculation with NRQED approach, calculations with diagrams.
\subsection{Spin-1 through diagrams}
The plan is to start from the exact spin-1 theory, and obtain the NRQED Lagrangian.
Start with relativistic Lagrangian.
Derive the electromagnetic vertices.
Discuss wave functions and find the connection between the relativistic and non-relativistic free theories.
Find one-photon terms by calculating scattering off an external field.
Find two-photon terms by calculating Compton scattering.
Write down NRQED Lagrangian.
\subsection{Spin-1 through equations of motion}
\subsubsection{Equations of motion}
Derive the Euler-Lagrange equations from the spin-1 relativistic lagrangian
Contrast the form of the wave functions here with in the previous approach.
Eliminate non-dynamic fields, and solve for the energy.
\subsubsection{Non relativistic wave functions}
Transform so that particle-antiparticle are uncoupled.
Find NR single-particle Hamiltonian.
Show that there are no corrections from a FW transformation that enter at our level of precision.
Finally find the nonrelativistic Hamiltonian.
\section{The case of general spin}
Describe the features of both the relativistic theory:
Definition of wave functions; spin degrees of freedom
Spin operators $S$ and $\Sigma$
Describe nonrelativistic theory along the same lines.
Describe the connection between the two free theories.
\subsection{NRQED Lagrangian for general spin}
Our goal is to calculate the NRQED Lagrangian.
Discuss what constraints we have: symmetries, hermiticity, etc.
Given assumptions about the strengths of the EM field and the momentum of the particles, we need up to $\frac{1}{m^3}$ terms.
Write down all allowed terms in the Lagrangian up to that order.
Note that only up to two-photon interactions appear, and they can be fixed by gauge invariance from the one-photon interactions.
\subsection{One-photon interaction in relativistic theory}
Now take the relativistic theory, and consider what the one-photon interaction will look like.
Constrained by Lorentz transformation properties and current conservation.
Show how only two bilinear terms are then allowed.
Their coefficients will be just charge and $g$-factor, with corrections at a higher order than we need.
Write down this general interaction.
\subsection{One-photon interaction in NRQED}
Express the current in terms of the nonrelativistic wave functions.
Thus, fix the NRQED coefficients.
Write down this general-spin NRQED Lagrangian.
\section{Corrections to $g$-factor in nonrecoil case}
Write the general NRQED Lagrangian.
Write as a Schroedinger like nonrelativistic Hamiltonian.
Calculate corrections to the $g$-factor for S-orbitals.
Show that no higher order terms in perturbation theory enter.
\section{Recoil case}
From the NRQED Lagrangian, we can calculate the effective Breit potential.
\subsection{NRQED calculation}
Calculate that potential from the one-photon interaction diagrams in NRQED.
\subsection{relativistic calculation}
We can calculate the same process in the relativistic theory, to make sure it agrees.
\subsection{CoM transformation}
Transform coordinates to the center of mass system.
When an external magnetic field is present, we need an additional unitary transformation.
\subsection{$g$-factor calculation}
Calculate the corrections to the g factor, now taking into account recoil effects.
\section{Conclusion}
\end{document}