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
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Vectors are set in bold, \(\vect{x}\); operators wear no hats unless ambiguity demands them.
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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).
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Einstein summation over repeated indices is understood unless stated otherwise.
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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).
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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.
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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 |
Single letters are reused in this treatise, as in every physics text: \(e\) is Euler's number and the elementary charge, \(\pi\) is the circle constant and the Cantor pairing map and a meson, \(k\) is the Boltzmann constant and a wavevector, \(\alpha\) is the fine-structure constant and a generic index. Reuse is disambiguated by context and by the declaration each statement makes; only \(\ee\) and \(\ii\) are protected by dedicated upright macros, and they are protected because they sit inside formulas alongside the italic variables they would otherwise be confused with.
\(\varphi\) is worth one note because it is the newest of these. Almost everywhere it is a bound variable — a homomorphism in Algebraic Structures, a formula of a formal language in Logic, Sets, and Maps, an azimuthal angle, a test function — and it means nothing beyond what the surrounding statement declares. Where it is a constant it is always the golden ratio of Definition 7.90, introduced by name at each use (Proposition 125.2, Proposition 32.55, and the listing in Physical Constants and SI Units); the two readings never occur in one statement. It carries no upright macro because the universal symbol for the golden ratio is the italic letter, and inventing another would cost the reader more than the overload does.
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.