Notation
This appendix is the treatise's single, canonical reference for notation: general typographic conventions (General conventions), the macro-to-meaning dictionary (Macro-to-meaning dictionary), and a pointer to the dimension-and-index ladder used throughout the geometric chapters (Index conventions for general dimension). The editorial rules that govern how the treatise is written — the SI axiom, the proofs-always rule, the experiment-detail rule — are stated once, in the front matter (Editorial Conventions); this appendix does not repeat them, only the symbols.
General conventions
-
Vectors are set in bold, \(\vect{x}\); operators wear no hats unless ambiguity demands them.
-
The upright \(\dd\) denotes the differential, \(\ii\) the imaginary unit, \(\ee\) Euler's number — never the italic \(d\), \(i\), \(e\) used for ordinary variables (Epistemology and the Scientific Method macro policy).
-
Einstein summation over repeated indices is understood unless stated otherwise.
-
The metric signature in relativistic chapters is \((-,+,+,+)\). Geometric chapters of Part II are written for general dimension \(D = p + q\) with signature \((p,q)\) (Index conventions for general dimension).
-
A tensor's type is written \((r,s)\): \(r\) contravariant (upper) and \(s\) covariant (lower) slots. The pair \((p,q)\) is reserved for metric signature, never for tensor type.
-
Label prefixes identify what a cross-reference points to:
ch:chapter,sec:section,eq:equation,fig:figure,tab:table,exp:experiment,phen:phenomenon,app:appendix section,part:part,thm:/def:/prop:/cor:/lem:the corresponding theorem-family environment.
Index conventions for general dimension
Chapters that develop geometry in general dimension \(D = p+q\) (Part II, Chapters 13 and 14) fix a specific ladder of index letters, embedding spaces, and symmetry groups — \(\mu,\nu,\ldots\) versus \(a,b,\ldots\) for diffeomorphism- and local-Lorentz-covariant objects, extended to \(m,n,\ldots\)/\(A,B,\ldots\) and \(M,N,\ldots\)/\(I,J,\ldots\) for the \((p{+}q{+}1)\)- and \((p{+}q{+}2)\)-dimensional embeddings of the (anti-)de Sitter and conformal geometries. Rather than duplicate that material, this appendix points to it: the full statement, with the symmetry-ladder table (Poincaré, Lorentz, translations, anti-de Sitter, de Sitter, conformal), is Section 13.1 of Differentiable Manifolds, Tensors, and Curvature.
Macro-to-meaning dictionary
Every macro used in the mathematics of this treatise is defined once, in
preamble/macros.tex, and used identically in every chapter; none
is redefined locally. Table E.1 is the complete dictionary.
| Macro | Renders as | Meaning |
|---|---|---|
\dd | $\dd$ | differential (upright) |
\ii | $\ii$ | imaginary unit (upright) |
\ee | $\ee$ | Euler's number (upright) |
\pp | $\pp$ | partial derivative symbol |
\dv{f}{x} | $\dv{f}{x}$ | total derivative |
\pdv{f}{x} | $\pdv{f}{x}$ | partial derivative |
\abs{x} | $\abs{x}$ | absolute value / modulus |
\norm{x} | $\norm{x}$ | norm |
\set{x} | $\set{x}$ | set-builder braces |
\gen{x} | $\gen{x}$ | generated subgroup/subspace |
\avg{x} | $\avg{x}$ | ensemble or expectation average |
\ket{\psi} | $\ket{\psi}$ | quantum ket |
\bra{\psi} | $\bra{\psi}$ | quantum bra |
\braket{\phi}{\psi} | $\braket{\phi}{\psi}$ | inner product |
\ketbra{\phi}{\psi} | $\ketbra{\phi}{\psi}$ | outer product |
\ev{A}{\psi} | $\ev{A}{\psi}$ | expectation value |
\comm{A}{B} | $\comm{A}{B}$ | commutator |
\acomm{A}{B} | $\acomm{A}{B}$ | anticommutator |
\pb{A}{B} | $\pb{A}{B}$ | Poisson bracket |
\tr | $\tr$ | trace |
\transpose | {(superscript)} | matrix transpose |
\sgn | $\sgn$ | sign function |
\diag | $\diag$ | diagonal matrix |
\im | $\im$ | image of a map |
\id | $\id$ | identity map |
\grad, \curl, \Div | $\grad,\curl,\Div$ | gradient, curl, divergence |
\SO, \SU, \ISO | $\SO,\SU,\ISO$ | special orthogonal, unitary, inhomogeneous |
\U, \Ogrp, \GL, \SL | $\U,\Ogrp,\GL,\SL$ | unitary, orthogonal, general/special linear |
\Sp, \Spin | $\Sp,\Spin$ | symplectic, spin group |
\R, \C, \N, \Z, \Q | $\R,\C,\N,\Z,\Q$ | reals, complexes, naturals, integers, rationals |
\identity | $\identity$ | identity operator |
\Lag | $\Lag$ | Lagrangian (never \L) |
\Ham | $\Ham$ | Hamiltonian |
\vect{x} | $\vect{x}$ | vector (bold) |
\ec | $\ec$ | equation-final comma |
\ep | $\ep$ | equation-final period |
\nn | {(no output)} | suppress an equation number |
The kernel commands \d, \i, \c, \L are never
redefined in this treatise (CLAUDE.md macro policy): old ported sources
that redefined them as differential/imaginary-unit/speed-of-light/
Lagrangian have been translated on port to \dd, \ii, plain
\(c\), and \Lag respectively. The physics package is banned
outright, since its \qty collides with siunitx v3's.