2. Notation

https://doi.org/10.48550/arXiv.2403.13981

This section outlines some of the standard and non-standard notational conventions that are used in this work. More notation is introduced in later sections, as and when it is needed. Some of the appendices use notation that does not conform to the conventions introduced in this section.

2.1 Microscopic quantities vs macroscopic quantities

Boldface type distinguishes macroscopic quantities from microscopic quantities throughout this work. For example, the macroscopic analogues of the microscopic charge density $\rho$, the microscopic electric potential $\phi$, and the microscopic electric field $\me$, are denoted by ${\Rho}$, $\bphi$, and $\E$, respectively; and $\br$ and ${\rvec}$ denote points, positions, or displacements at the macroscale and the microscale, respectively.

In discussions of general features of the relationship between microscopic and macroscopic fields, $\nu$ will denote an arbitrary microscopic field and $\Nu$ will denote its macroscopic counterpart.

2.2 Mathematical sets and spaces

2.2.1 Sets of numbers

The symbols $\integer$, $\realone$, and $\complex$ denote the sets of integers, real numbers, and complex numbers, respectively.

The sets of positive, negative, non-positive, and non-negative integers or real numbers are defined and denoted as follows:

\begin{align*} \integerpos &\equiv\{x\in\integer:x>0\}, & \integernonneg &\equiv\{x\in\integer:x\geq0\}, \\ \integerneg &\equiv\{x\in\integer:x<0\}, & \integernonpos&\equiv\{x\in\integer:x\leq0\}, \\ \realpos &\equiv\{x\in\realone:x>0\}, & \realnonneg &\equiv\{x\in\realone:x\geq0\}, \\ \realneg &\equiv\{x\in\realone:x<0\}, & \realnonpos&\equiv\{x\in\realone:x\leq0\}. \end{align*}

2.2.2 Lattices

A one dimensional lattice with lattice spacing $a$ will be denoted by ${a\integer}$ if it includes the origin. If the origin is not one of the lattice points, it will be denoted by ${x_0+a\integer}$, for some ${x_0\in\realone}$. That is,

\begin{align*} x_0+a\integer\equiv \left\{x_0+ma: m\in\integer\right\}. \end{align*}

This notation generalizes to lattices in higher dimensions. For example, a two dimensional lattice with lattice spacings ${a_1}$ and ${a_2}$, and with one of the lattice points at a displacement ${\svec}$ from the origin, can be denoted by

\begin{align*} \svec+\left(a_1\integer\right)\times\left(a_2\integer\right); \end{align*}

and an $M$-dimensional lattice with lattice spacing $a$ in all directions can be denoted by ${\left(a\integer\right)^M}$.

2.2.3 Indexed sets

${\{A_\alpha\}}$ denotes the set containing ${A_\alpha}$ for every possible index $\alpha$; ${\{B_{\alpha\beta}\}}$ denotes the set containing ${B_{\alpha\beta}}$ for every possible pair of indices, ${\alpha\beta}$; and ${\{B_{\alpha\beta}\}_\beta}$ denotes the set containing ${B_{\alpha\beta}}$ for every possible index $\beta$ and a fixed value of index $\alpha$.

2.2.4 Intervals

If $u$ is any real-valued quantity with a continuous range of possible values (e.g., an $x$ coordinate or an average of field $\nu$), I use ${\interval(u,\Delta)}$, ${\interval[u,\Delta)}$ and ${\interval(u,\Delta]}$, and ${\interval[u,\Delta]}$ to denote interval subsets of this range, of width $\Delta$ and centered at $u$, which are open, half-open, and closed, respectively. I use the more conventional notation ${(u_1,u_2)}$, ${[u_1,u_2)}$ and ${(u_1,u_2]}$, and ${[u_1,u_2]}$, to specify intervals by their end points. For example, ${\interval(u,\Delta u]=(u-\Delta u/2,u+\Delta u/2]}$ is an interval that is open at its lower boundary and closed at its upper boundary.

2.2.5 Materials and regions in their interiors

An arbitrary material will be denoted by $\material$ and its surface will be denoted by ${\materialsurface}$. The volume of ${\material}$ will be denoted by ${\materialvolume}$. If $\material$ is a one dimensional material, or is being treated as one dimensional, ${\materialvolume}$ is its length.

A simply-connected region in the interior of a material will be denoted by ${V}$ and its measure will be denoted by ${\abs{V}}$. In a one dimensional material ${\abs{V}}$ is a length; and in a three dimensional material it is a volume.

A unit cell of a crystal will be denoted by $\unitcell$, and its length (1-d) or volume (3-d) will be denoted by $\volume$.

2.2.6 Tori

I use the term torus to mean a topological space. A 1-torus of ‘circumference’ ${C\in\realpos}$ is denoted by ${\onetorus(C)}$ or ${\onetorus}$ and defined as

\begin{align*} \onetorus=\onetorus(C)\equiv \realone/(C\integer). \end{align*}

Therefore if an arbitrary point in ${\onetorus(C)}$ is at position ${x\in\realone}$, it is also at all of the positions in the infinite set, ${\{x+qC: q\in\integer\}}$. In other words, each point in ${\onetorus(C)}$ is its own image under a displacement by an integer multiple of $C$. ${\onetorus(C)}$ is denoted by $\onetorus$ when the value of $C$ is clear from the context.

For any ${m\in\integerpos}$ and any set of $m$ circumferences, ${\{C_i\in\realpos:1\leq i \leq m\}}$,

\begin{align*} \onetorus^m(C_1\cdots C_m)\equiv \onetorus(C_1)\times\onetorus(C_2)\times\cdots\times\onetorus(C_m). \end{align*}

is an m-torus. Therefore, for any ${i\in\{1\cdots m\}}$ and any ${q\in\integer}$, the coordinates ${(x_1\cdots,x_i+qC_i,\cdots x_m)}$ and ${(x_1\cdots x_m)}$ identify the same point in ${\onetorus^m(C_1\cdots C_m)}$.

If $f$ is a function whose domain is an interval ${[a,a+C)\subset\realone}$, such as if it is the restriction to ${[a,a+C)}$ of a function whose domain is $\realone$, it can be ‘rolled up’ into a function ${g}$ with domain ${\onetorus}$ by identifying a point ${\tilde{a}\in\onetorus}$ as the counterpart in ${\onetorus}$ of ${a\in[a,a+C)}$; and defining

\begin{align*} g(\tilde{a}+(x-a))\equiv f(x),\forall x\in\closedopen{a,a+C}. \end{align*}

Conversely, any function ${f}$ whose domain is $\onetorus$ can be ‘unrolled’ into $\realone$ by defining a $C$-periodic function ${g}$, with domain $\realone$, as: ${g(x)\equiv f(x), \;\forall x\in\realone}$.

2.3 Function spaces

2.3.1 $p-$norms

If ${1\leq p \leq\infty}$, the p-norm of a function,

\begin{align*} f:\domain\to\image\in\{\realone,\complex\}, \end{align*}

from any measure space $\domain$ to either ${\realone}$ or ${\complex}$, is

\begin{align*} \norm{f}_p\equiv \left(\int_\domain \abs{f}^p \dd{\mu}\right)^{\frac{1}{p}}, \end{align*}

where $\mu$ is the measure of $\domain$. The integral ${\int_\domain\cdots\dd{\mu}}$ is a Lebesgue integral, but the difference between a Riemannn integral and a Lebesgue integral will rarely be relevant and will not be discussed.

2.3.2 Lebesgue spaces

A Lebesgue space or $\lebesguep$-space is a space of functions whose p-norm is finite. Therefore, for ${\image\in\{\realone,\complex\}}$, the set

\begin{align*} \lebesguep(\domain,\image)\equiv\left\{f:\domain\to\image \;s.t.\; \norm{f}_p<\infty\right\}. \end{align*}

is a ${\lebesguep}$-space.

2.3.3 Hilbert-Lebesgue spaces

A Hilbert-Lebesgue space is a $\lebesgue$-space. For example ${\lebesgue(\realone^m)}$ is the set of all functions ${f:\realone^m\to\complex}$ such that

\begin{align*} \norm{f}_2\equiv \left(\int_{\realone^m}\abs{f(\rvec)}^2 \dd[m]{r}\right)^{\frac{1}{2}}<\infty. \end{align*}

The norms of all of the Lebesgue spaces in this work are $2$-norms, ${\norm{\,\cdot\,}_2}$. They will be denoted as ${\lebesgue(\domain)}$ or ${\lebesgue(\domain,\realone)}$, where $\domain$ is usually ${\realone^m}$, ${\complex^m}$, or ${\onetorus^m}$ for some value of ${m\in\integerpos}$. Omission of the second argument of ${\lebesgue(\domain,\image)}$ implies that ${\image=\complex}$.

It follows from ${\realone\subset\complex}$ that ${\lebesgue(\domain,\realone)\subset\lebesgue(\domain)}$. Therefore a space may be denoted as ${\lebesgue(\domain)}$ despite the context suggesting that its elements are real-valued or can be chosen to be real-valued. A Hilbert-Lebesgue space will only be denoted as ${\lebesgue(\domain,\realone)}$ when it is necessary to exclude complex valued-elements (e.g., because they are unphysical).

2.3.4 Hilbert-Lebesgue spaces of functions with countable domains

The analogue of a $\lebesgue$-space for a set of functions whose domain ${\indexset}$ is a countable set, such as a lattice, is denoted by ${\seqlebesgue(\indexset,\image)}$ if ${\image\in\{\realone,\complex\}}$, or by ${\seqlebesgue(\indexset)}$ if ${\image=\complex}$. It is defined as

\begin{align*} \ell^2(\indexset,\image)\equiv\left\{ f:\indexset\to\image\;s.t.\; \norm{f}_2^2\equiv\sum_{i\in\indexset} \abs{f(i)}^2 < \infty\right\}. \end{align*}

Strictly speaking, all of the $\lebesgue$-spaces encountered in this work are really ${\seqlebesgue}$-spaces, because they are constructed as described in Appendix C.2. Therefore the only difference between the notation ${\lebesgue(\domain)}$ and the notation ${\seqlebesgue(\indexset)}$ is that ${f\in\lebesgue(\domain)}$ emphasizes that the domain of $f$ is continuous for all practical purposes, whereas ${f\in\seqlebesgue(\indexset)}$ is used to emphasize that its domain is countable.

2.3.5 Inner products

The inner product of space ${\lebesgue(\domain)}$ is

\begin{align*} \braketD{f}{g}\equiv \intdomain f^* g \dd{\mu}. \end{align*}

For example, the inner products of ${\lebesgue(\realone)}$, ${\lebesgue(\onetorus)}$, ${\lebesgue(\unitcell)}$ are, respectively,

\begin{align*} \braketR{f}{g}&\equiv\int_\realone f^*(x)g(x)\dd{x}, \\ \braketT{f}{g}&\equiv\int_\onetorus f^*(x)g(x)\dd{x}, \\ \braketcell{f}{g}&\equiv\int_\unitcell f^*(x)g(x)\dd{x}. \end{align*}

The inner products of all Hilbert-Lebesgue spaces may also be denoted with the ambiguous notation ${\braket{f}{g}}$.

2.3.6 Aside: The imaginary part of an `inner product'

If the reason to use ${\lebesgue(\domain,\complex)}$ rather than ${\lebesgue(\domain,\realone)}$ is to use $\complex$ as a proxy for ${\realone\times\realone}$ (e.g., with the real and imaginary parts of numbers representing Cartesian $x$- and $y$-components), then ${\braket{f}{g}}$ is not an inner product [Gallier, 2011; Heil, 2018; Garling, 2011; Scharlau, 2012; Renteln, 2013]. Its real part, ${\Re\{\braket{f}{g}\}}$, does meet the definition of an inner product, but its imaginary part does not have an obvious interpretation that is valid in all contexts.

If ${a,b\in\complex}$ represent vectors in ${\realone\times\realone}$, then ${\Re\{a^*b\}}$ is the vectors’ inner product and, if $i$ is regarded as a proxy for the unit pseudoscalar of the geometric algebra of ${\realone\times\realone}$, then ${i\Im\{a^*b\}}$ can be interpreted as their exterior product, which is a bivector [Lounesto, 2001; Vaz and da Rocha, 2016; Doran and Lasenby, 2003; Garling, 2011].