Wikipedia:MOSPHYS

This proposal has become dormant through lack of discussion by the community.

It is inactive but retained for historical interest. If you want to revive discussion on this subject, try using the talk page or start a discussion at the village pump.

Rename this page to WP:Manual of Style/Physics when complete and universally agreed on by wikiproject physics.

This part of Wikipedia's Manual of Style contains guidelines for consistency of style in writing and editing articles on physics as well as physics-related parts of other articles. For consistency with the rest of Wikipedia, other manuals of style (in particular WP:MOSMATH for mathematics and WP:MOSCHEM for chemistry) apply when possible. Exceptions and additional conventions specific to physics topics are described here.

Typesetting of formulae

LaTeX versus HTML

There are a number of different techniques to produce mathematical formulae on Wikipedia, all with their advantages and disadvantages. Inevitably, this has led to a wide spectrum of preferences among editors and of styles used in articles. Even worse, there is a perpetual change of formatting by editors who don't like the current style of an article.

The main technical options are:

LaTeX (generated using <math>),
Wiki markup and HTML, together with Unicode characters,
the {{math}} and {{mvar}} templates.

The main principles to be considered when deciding about a particular formatting are consistency and consensus. Don't set one formula in a style very different from the rest of the article, and don't do mass changes from one style to another without prior discussion on the talk page. That said, the style recommended for physics articles is as a rule of thumb:

use wiki markup or the {{math}} template for inline formulae,
use <math>...</math> for displayed formulae.

This choice is aiming at a smooth integration of simple inline formulae in the surrounding text, while offering the extended possibilities of LaTeX formatting for more involved displayed formulae.

Roman versus italic

One of the most basic, though often ignored typesetting conventions for mathematical formulae concerns the use of upright (roman) versus italic typeface: as a general rule,

variables (including indices and physical quantities) should be set italic, while
names (including abbreviations, names of particles, chemical elements and units of measurement) should be set in roman type.

This rule explicitly applies also to subscripts and superscripts. Paying attention to this seemingly minor detail helps to immediately make clear the meaning of a particular notation, and should not be seen as nitpicking. The rule has a few notable exceptions, which will be explained below.

Greek letters

The Greek alphabet is extensively used throughout physics and mathematics. Due to technical limitations, the <math>...</math> environment always typesets lower case Greek letters in italic, and upper case ones in roman type. For consistency within an article, Greek letters produced using HTML or {{math}} should thus follow the same convention. This practice applies to both normal and bold font weight.

Lower case Greek letters that denote names of particles, and that don't appear in the same article within a <math> environment, should be set in roman type, following the general style for names.

Some Greek letters have a second, variant form, for example $\varepsilon$ vs. $\epsilon$ , $\varphi$ vs. $\phi$ , or $\vartheta$ vs. $\theta$ . If they are used, the same symbol should be used for the same quantity consistently throughout the article. (There is also a variant for the lowercase l in the Latin alphabet, $\ell$ , which can be useful for the distinction from an upper case I.)

Units and quantities

Quantities whose values are not given numerically, including both variables and physical constants, are denoted in a way similar to variables in mathematics. Examples are

the speed of light, $c$ = 299792458 m⋅s⁻¹;
the reduced Planck constant, $ħ$ = 1.054571817...×10⁻³⁴ J⋅s
the vacuum permittivity, $ε 0$ = 8.8541878188(14)×10⁻¹² F⋅m⁻¹
the elementary charge, $e$ = 1.602176634×10⁻¹⁹ C;
the electron rest mass, $m e$ = 9.1093837139(28)×10⁻³¹ kg.

One must carefully distinguish quantities, units, and dimensions, because different typographies apply to them. For example:

Voltage is a quantity and may be denoted as $V$ , an italic capital letter "V".
The volt is a unit and must be denoted as V, a roman capital letter. Brackets [] mean the units of, so [V] = V, and braces {} mean the numerical value of so V = {V}[V].
The dimension of voltage should be written in terms of the base dimensions in roman sans-serif type:^[1] ML²T⁻³I⁻¹.

When a physical constant serves as a unit in some systems, such as in with atomic units, it should be denoted as a constant (in italics). In particular, the elementary charge, even when treated as a unit, is denoted $e$ , notwithstanding that it was originally a unit (named the electron, with symbol $e$ ) (see Elementary charge § As a unit). These symbols are also never prefixed with SI prefixes. Related units that are denoted in roman type include

the unified atomic mass unit, symbol $u$ ;
the dalton, symbol $Da$ ;
the electronvolt, symbol $eV$ .

Generally, all symbols (abbreviations) of units are universally roman (upright). For dimensions, an exception occurs when denoting the dimension of an arbitrary quantity is necessary, as in "[quantity]"; see continuity equation for an example of this case. Representation of units and numerical quantities (where a numerical value and a unit are specified explicitly) should comply with WP:Manual of Style/Dates and numbers § Scientific notation, engineering notation, and uncertainty unless the present Manual specifies otherwise. See also: {{val}}.

The roman versus italic guideline applies to units. For example, write

P = P₀ + 9807 Pa/m × h,

but not

P = P₀ + 9807 Pa/m × h.

Common mathematical formulae

Examples:

Newton's second law:
$\mathbf {F} ={\frac {d\mathbf {p} }{dt}}$
Schrödinger equation: displayed (see below for angular brackets),
$i\hbar {\frac {d}{dt}}\left|\Psi \right\rangle ={\hat {H}}\left|\Psi \right\rangle \,.$

Representation of Common mathematical formulae (arithmetic, algebra (including vectors and tensors), summation, integration, differentiation, differential geometry, complex analysis ...) should comply with WP:Manual of Style/Mathematics § Typesetting of mathematical formulae unless the present Manual specifies otherwise. Notations that are popular in a certain context should be used; unusual or otherwise less common notation (for physics) should be avoided.

Functions

Exponential functions are very common in physics, representing growth and decay from solutions of differential equations, complex number representations, and group generators. As such they can have quite large/complicated variables.

For simple arguments on one line, the e notation is legitimate
$e^{i2\pi x/\lambda },e^{i\omega t}$

in which case fractions should use the forward slash /, not \frac{}{}.

For more detailed/complicated arguments, (see angular momentum and Schrödinger equation respectively), the "exp" notation is clearer to read:

R({\hat {n}},\phi )\equiv \exp \left(-{\frac {i}{\hbar }}\phi \,\mathbf {J} \cdot {\hat {\mathbf {n} }}\right)\,,\quad \exp \left[i(\mathbf {p} \cdot \mathbf {x} -Et)/\hbar \right]

however using the forward slash for division as above is acceptable, although extra brackets are usually needed,

R({\hat {n}},\phi )\equiv e^{-i\phi \,\mathbf {J} \cdot {\hat {\mathbf {n} }}/\hbar }\,,\quad e^{i(\mathbf {p} \cdot \mathbf {x} -Et)/\hbar }

whereas fractions using the horizontal stroke are not:

R({\hat {n}},\phi )\equiv e^{-{\frac {i}{\hbar }}\phi \,\mathbf {J} \cdot {\hat {\mathbf {n} }}}\,,\quad e^{{\frac {i}{\hbar }}(\mathbf {p} \cdot \mathbf {x} -Et)}

Vectors and vector spaces

For Euclidean vectors, there are numerous conventions. The better notations for vectors use:
- bold: easy to typeset and print, and stands out, easily compatible with other diacritics, or
- ${\overrightarrow {\text{overarrows}}}$ or ${\underline {\text{underlining}}}$ : easy to hand-write and indicates clearly a quantity with direction, although it becomes messy with other diacritics. Underlining is easily possible in HTML using the <u> </u> tags, like this: A, x. There is the template {{vec}} for HTML left/right/doubleheaded arrows over/under one letter, like this: a→, Z↔ etc.
- ${\overline {\text{overlining}}}$ : is often used for position vectors for example the position vector from point A to B would be written ${\overline {\text{AB}}}$ :
- In practice, bold italic is also employed, but is less common than upright. For consistency throughout articles, please use upright bold. In particular, please do not execute mass changes from bold style to bold-italic.
  $A$ , $\mathbf {A}$ , not $A$ , ${\boldsymbol {A}}$ , ${\overrightarrow {A}},{\overrightarrow {AB}}$
For "normal" or unit vectors, they can be bold, or a hat replaces the arrows:
$ê$ , $\mathbf {\hat {e}} ,{\hat {e}}$ , not ${\overrightarrow {\hat {e}}}$
Quantum state vectors, which are elements of a Hilbert space, should use the bra-ket notation, the standard in quantum mechanics, rather than bold, arrows, or hats. For the bra-ket notation, the {{langle}} and {{rangle}} templates may be used to generate HTML/Unicode equivalents for the \langle and \rangle glyphs of <math> mode. There are also specialized templates {{bra}} {{ket}} for creating bra and ket vectors, and {{bra-ket}} for inner products.
|ψ⟩, ⟨ψ|, $| ψ ⟩$ , $⟨ ψ |$ , $\left|\psi \right\rangle ,\left\langle \psi \right|$ , not ${\overrightarrow {\psi }},{\boldsymbol {\psi }},{\hat {\psi }}$
Operators usually have a hat, but not always ( $Â$ vs. $A$ ). Either is acceptable as long as the meaning is unambiguous. For example, when describing an eigenvalue of an operator using the same letter as for the operator itself, the operator should have a hat to distinguish the two.
For tensors not in index notation, use one of sans serif, bold sans serif, bold italic T ... (to be decided). Do not use 𝕓𝕝𝕒𝕔𝕜𝕓𝕠𝕒𝕣𝕕 𝕓𝕠𝕝𝕕, which are reserved for sets (see below sets and spaces).
$A$ , ${\boldsymbol {A}},\,{\mathsf {A}},\,{\boldsymbol {\mathsf {A}}}$ , not $\mathbb {A}$ , $𝔸$
For contraction in a Hilbert space, one should use bra-ket notation, not the Hermitian form notation:
$\langle \phi |\psi \rangle$ , not $\langle \phi ,\psi \rangle$ .
The use of dot product and del operators for Minkowski space may lead to confusion, especially so in indefinite-metric spaces even between $(1, 0)$ - and $(0, 1)$ -tensors. Also, if the article has to operate with both vectors and higher tensors, then the use of different styles for tensor fields of different types would be confusing too, even in 3-dimensional space:
$v μ \partial μ f$ or $v μ \nabla μ f$ , not $(\nabla f)\cdot v$ .

Subscripts and superscripts

Subscripts and superscripts follow the general rule about roman versus italic typeface. For example:

F ext

,

2\rho _{\mathrm {Al} }<\rho _{\mathrm {Fe} }

,

\mathbf {B} _{x_{0}}^{\mathrm {ext} }(t)

,

rather than

F ext

,

2\rho _{Al}<\rho _{Fe}

,

\mathbf {B} _{\mathrm {x_{0}} }^{\mathrm {ext} }(t)

.

Note that in the last example, "ext" is a text label (an abbreviation for "external") and is thus set roman, while x₀ is a variable and is this set in italic. This type of notation is sometimes used instead of B^ext(x₀, t) to indicate that only the functional dependence of B on t is of interest, while x₀ is a parameter held fixed (though conceptually still a variable).

An exception can be made when the <math> tag has to be used, and an intended glyph is not available in its renderers, such as non-italicized Greek letters. For example, when typesetting the mass of a neutrino ν (nu) in html or {{math}}, the ν can remain non-italic:

m ν

,

while in <math>;

m_{\mathrm {\nu } }

(with \mathrm applied to \nu)

looks no different than

m_{\nu }

(without the \mathrm)

and the latter may be used.

Index notation for tensors and spinors

Two ways to denote vectors, tensors, and spinors (also vector fields, tensor fields, and spinor fields) are:

as the full entity with no reference to specific components (which uses the boldface notations above), or
in components via index notation (cf. Ricci calculus and Van der Waerden notation), with or without reference to a basis.

Generally, the first is used in simpler contexts, while index notation is used to make advanced manipulations simpler. The linear operator notation is depreciated (with possible exception of theoretical mechanics and other domains where index notation is traditionally avoided).^{[clarification needed]}

Letters that are indices for tensors or spinors (see for example Ricci calculus and Van der Waerden notation) should be italicized. A specific script used as a tensor index may give a hint about nature of the corresponding linear space – this distinction should be used whenever possible. For example, spacetime-based (four dimensional) objects should be indexed with a subset of lowercase Greek letters:
$A μ$ or $A^{\mu }.$

Ideally - not all Greek letters are used for indices. For example, ϕ and ψ are excluded due to its heavy usage as wave functions and fields, and η is used to denote the standard (Minkowskian) metric.

Objects constructed of weight-1⁄2 spin representations of the Lorentz group should be indexed with capital Latin letters:^[2]
$φ A$ or $\varphi _{A}.$
Most of other linear spaces, including 3-dimensional space, use lowercase Latin letters:
$A i$ or $A_{i},.$

(although the letters

i, j

may cause confusion where complex numbers are used; this should be clarified in the context),

A k

or

A^{k}.

Note that {{ell}} is equivalent to ℓ. Distinction of indices is especially helpful when one tensor quantity is built on several linear spaces of different natures.

Calculus

If "d" is used in differentials or derivatives, it should be italic or upright throughout; mixed styles look irregular.
In vector expressions for the gradient ∇, with respect to a given basis, the basis vectors should be in front of the derivative operators. In 3d Cartesian coordinates:
$∇ = êx.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠∂/∂x⁠ + êy⁠∂/∂y⁠ + êz⁠∂/∂z⁠$ or $\nabla ={\hat {\mathbf {e} }}_{x}{\frac {\partial }{\partial x}}+{\hat {\mathbf {e} }}_{y}{\frac {\partial }{\partial y}}+{\hat {\mathbf {e} }}_{z}{\frac {\partial }{\partial z}}$

and not

\nabla = ⁠ \partial / \partial x ⁠ ê x + ⁠ \partial / \partial y ⁠ ê y + ⁠ \partial / \partial z ⁠ ê z

or

\nabla ={\frac {\partial }{\partial x}}{\hat {\mathbf {e} }}_{x}+{\frac {\partial }{\partial y}}{\hat {\mathbf {e} }}_{y}+{\frac {\partial }{\partial z}}{\hat {\mathbf {e} }}_{z}

and similarly for any other coordinates. It is clear this way that the derivatives are operators not acting on the basis vectors (which can have spatial dependence in a local coordinate frame).

When using <math>, use either $\nabla$ (\nabla) or $\bigtriangledown$ (\bigtriangledown ) throughout - it looks irregular to mix the styles.
The capital delta symbol is universally reserved for changes in quantities, for example Δx for change in position coordinate x.
The same symbol Δ is also used in mathematics to denote the Laplacian operator. Where possible, please use the "nabla squared" symbol ∇², which appears to be more common in physics, and is more intuitive; ∇² = ∇ ⋅ ∇ is a nicer notation, and less ambiguous for capital delta.
The symbol ∂_α is used for the four gradient operator (indexed components), as usual. There are other notations, including
- $\Box$ , which should be reserved for the D'Alembertian operator (see next point).
- $D$ , which should be reserved for some covariant derivative (see next point).
The symbol $\Box$ $\Box$ is used for the D'Alembertian operator. There are other notations, including
- $\Box ^{2}$ , to parallel the Laplacian ∇². The notation without the square is already known to denote the D'Alembertian. Either can be used (?)
- $D$ or $D 2$ (very rare), which should be reserved for some covariant derivative (?)

Setting	Not recommended	Recommended
Laplacian operator	$Δ$	$\nabla 2$
Four gradient	$D$ , $\Box$ etc.	$\partial α$
D'Alembertian operator	$D, D 2$ $\Box ^{2}$ etc.	$\Box ,\partial _{\alpha }\partial ^{\alpha }$

For integrals, either of the notations are employed in the literature:
$\int f (x) dx, \int dx f (x)$ or $\int f(x)\,dx,\,\int \,dxf(x)$

and either is acceptable in WP physics articles.

For {{math}}, integral symbols can be produced using the same syntax as for LaTeX using the {{intmath}} template.
For integrations over the boundary of a (hyper)volume V, the partial symbol $\partial$ denoting the boundary of a closed volume is encouraged (including a brief explanation such as "where $\partial V$ is the boundary of the volume V"); it is a powerful and compact notation:
$\int \partial V ψ(x) dV$ or $\int _{\partial V}\psi (\mathbf {x} )\,dV$

and removes the need to use another symbol for the boundary of the volume.

There is an abuse of notation where integrands are not enclosed in brackets, frequently the case when (say) Green's theorem in a 2d plane is applied:
$\oint \partial D p (x, y) dx + q (x, y) dy$ or $\oint _{\partial D}p(x,y)\,dx+q(x,y)\,dy$ ,

which really means:

\oint \partial D [p (x, y) dx + q (x, y) dy]

or

\oint _{\partial D}\left[p(x,y)\,dx+q(x,y)\,dy\right]

The brackets are often dropped by sources since it is known the integral symbol always includes the differentials (in the above example "

dx

" and "

dy

"). Nevertheless: for the sake of clarity an extra pair of brackets will not clutter, and should be included.

Ratios of differential (infinitesimal) quantities and derivatives

In physics, ratios of differential (infinitesimal) quantities frequently occur, and share the same notation with first order derivatives, which also frequently occur. For example, the local charge density of an electrically charged continuum is given by the ratio of the infinitesimal charge $dq$ in infinitesimal volume $dV$ :

\rho ={\dfrac {dq}{dV}}

However, this looks like a derivative of $q$ with respect to $V$ . In cases like this, notations can be misleading, as

the derivative is defined as a limiting difference quotient, but also an operator (cf. linear algebra),
calculating the differential of a function is a general way to determine infinitesimal changes of a quantity using a derivative.

So these are not exactly the same, differentials are more general. Following are the recommended applications of the notation.

Operations	Notation	Usage in the literature
Division of infinitesimal quantities $dy$ and $dx$	$⁠ dy / dx ⁠, dy ⁄ dx, dy / dx$	Ratios or derivatives
Derivative of $y = y (x)$ as a function of $x$	$⁠ d / dx ⁠ y$	Used only for derivatives, not ratios as above. Advantages include: makes clear that differentiation is an operator, clarifies notation for higher derivatives by repeated action of a derivative, $⁠ d n / dx n ⁠ y = ⁠ d / dx ⁠ ... ⁠ d / dx ⁠ ⁠ d / dx ⁠ y$

For partial derivatives this is not a problem, since differentials are never written as " $\partialx$ "; only the full symbol of a partial derivative, in any of the equivalent notations;

⁠ \partialF / \partialx ⁠, ⁠ \partial / \partialx ⁠ F, \partialF ⁄ \partialx, \partial ⁄ \partialx F, \partialF / \partialx,

has meaning.

Particles, substances and reactions

Symbols for subatomic particles and nuclei can be generated with {{Subatomic particle}} and Nuclide templates, otherwise they should be typographically similar.
Where chemical notation is used, it should comply with WP:MOSCHEM.
Formulae of (nuclear or particle) reactions should not be formatted with <math> or {{math}}. Use templates, wiki code formatting, and HTML tags to match the main text, both for inline and displayed formulae.
For all reactions, the arrow symbol → should be used, the equals sign = is usually wrong.

Examples:

Electron–positron annihilation:
e⁻
+
e⁺
→ 2
γ
.
β⁻ decay as nuclear reaction:
²⁴²
₉₅Am⁵⁺
→ ²⁴²
₉₆Cm⁶⁺
+
e⁻
+
ν
_e

and as particle reaction:

n⁰ → p⁺ +
e⁻
+
ν
_e

The roman versus italic guideline applies to substances:

\displaystyle P=P_{0}+9807{\frac {\mathrm {Pa} }{\mathrm {m} }}h\displaystyle 2\rho _{\mathrm {Al} }<\rho _{\mathrm {Fe} }

rather than

\displaystyle P=P_{0}+9807{\frac {Pa}{m}}h\,.

Sets and spaces

Number sets

Where needed, standard number sets should be set in blackboard bold, not just bold face:

inline (unicode): ℕ, ℚ, ℤ, ℝ, ℂ
displayed (LaTeX): $\mathbb {N,Q,Z,R,C}$
not recommended: N, Q, Z, R, C

Lie groups

Symbols for Lie groups shall be Roman, not italic:

inline (normal text): SU(n), SO(n), SL(2, ℂ), U(n), O(n)
displayed (\mathrm): $\mathrm {SU} (n),\mathrm {SO} (n),\mathrm {SL} (2,\mathbb {C} ),\mathrm {U} (n),\mathrm {O} (n)$
not recommended: $SU(n),SO(n),SL(2,\mathbb {C} ),U(n),O(n)$

Other

Hilbert spaces: H, ${\mathcal {H}}$ , or $ℋ$

Prerequisite knowledge

An article can usually not give a detailed explanation of all concepts and nomenclatures used to explain its topic. Some basic knowledge of physics and mathematics will generally be required of the reader, to an extent depending on the article's subject. However, where appropriate, links to more introductory articles and summary style descriptions of the essential concepts should be provided.

Physics

A large number of physical theories exist, and compatibility between them varies. Concepts as distance, time and mass appear to be universal, although their exact definitions can depend on the subject. Unless an article deals explicitly with these fundamental concepts, it should not explain the meaning of distance, time or mass.

Expected units

The following everyday units are presumed to be known and needn't be explained, although ideally linked at least once.

Physical quantity	SI unit	Decimal SI prefixes	Common non-SI unit
length	metre (m)	centimetre (cm)
time	second (s)		minute (min), hour (h),
mass	kilogram (kg)	gram (g)

Other well-known derived quantities include (and these should ideally be linked):

Physical quantity	SI unit	Common non-SI unit
energy	J	eV
velocity	m⋅s⁻¹
momentum	N⋅s or kg⋅m⋅s⁻¹
force	N or kg⋅m²⋅s⁻²
pressure	Pa or N⋅m⁻²
etc.

Old non-SI units (such as "erg", "dyne", "knots" etc.), and imperial units, ideally should only to be used for:

historical relevance,
within context relevant to those main articles,
if references (databooks, handbooks, appendices of books etc.) present measurements in terms of those units.

In general, SI units are usually assumed by most people. When using other unit systems which are extensively employed in practice although may be unfamiliar to laymen, including:

Gaussian units (especially wherever EM is applied), and
natural units (especially in fundamental physics; SR, GR, QM, QFT, particle physics etc.),

it should be explicitly stated they are going to be used before using them. There are several sets of natural units, each with their own applications, so which one must be clear. Ideally, natural units shouldn't be used, due to lack of familiarity and potential confusion by many readers. For example: giving masses of elementary particles in MeV when the reader thinks of kg (or decimal prefixes thereof) is confusing, a related point is that writing " $E = m$ ", without referring to natural units, is confusing when the reader expects the $c 2$ factor...

As a rule of thumb (for natural units):
They could be used in some calculations or explanations to illustrate how and why natural units are better than SI units in the process,

if the calculation/explanation can easily be done in SI units, use SI units, and not natural units for the sake of it.

Clarifying the relevant physics

Physics is such an enormous subject that an article should specify the relevant branch(es) of it, unless the concept is physically universal. Known academic paradigms, with usual abbreviations, are:

Mechanics (all types)
Classical physics:
- Galilean relativity, a.k.a. non-relativistic physics
- Newtonian mechanics (how well-known is Analytical mechanics?)
- Classical field theory (includes Newtonian gravitation and Maxwell's EM)
Theory of relativity (non-Galilean):
- Special relativity (SR)
- General relativity (GR)
Non-quantum physics: world lines and possibly classical fields, but no wave–particle duality or other intrinsic quantum phenomena like particle spin.
Quantum theory:
- Quantum mechanics (QM, relativistic or non-relativistic)
- Quantum field theory (QFT, implies QM, SR, background in classical field theory) – not necessary to mention if one of well-known derived theories is used directly:
  - Quantum electrodynamics (QED)
  - Quantum chromodynamics (QCD)
Particle physics (application of relativity and QFT):
- Variants of Yang–Mills theory (which namely?)
- Standard Model
- String theory (less known are the specific theories of "Heterotic strings", "D-branes", or "M theory", etc. so just link to string theory and take it from there?)

The lead of an article should specify the one of these, for example (see the wavefunction article) anything like:

In quantum mechanics, a branch of physics, a wave function is ...
In quantum mechanics, a wave function is ...

and not simply:

In physics, a wave function is ...

Mathematics

No abstract notations

Any abstract notation which requires additional irrelevant explanation (i.e. set builder notation, symbolic logic notations, especially quantifier notation, others?), are not to replace worded descriptions for the sake of compact mathematical statements - they simply create more work in explaining and linking the notations, while contributing no understanding to the physics at hand.

To exemplify:

Even if the context becomes abstract, statements like
"x is a number such that $x \in ℝ: x > 0$ , where $\in$ denotes "element of a set", the colon means "such that", and $ℝ$ denotes the real numbers"

are too long, cluttered, and confusing for non-expert mathematicians: simply write

"x is a real number greater than and not equal to zero"

since this way explains the context without introducing unfamiliar notations.

Often, integers or half-integers need to be listed (say, listing the number of normal modes for a standing wave, the orders of diffraction in physical optics and condensed matter physics, or quantum numbers). It's encouraged to simply list the numbers, and if needed followed by an ellipsis (three dots: ...), like so:
$j = 0, 1 ⁄ 2, 1, 3 ⁄ 2 ...$

which makes it very clear to anyone that

j

takes those values and the sequence of numbers is infinite. Even though the full set builder notation:

j \in {0, 1 ⁄ 2, 1, 3 ⁄ 2 ...},

or

j \in {n /2 : n \in ℕ 0},

etc.

is technically more correct, this is actually less clearer to a typical reader, since the symbols have to be explained by editors and/or read up by readers.

Writing compound statements, like the zeroth law of thermodynamics using logic notation:
"The zeroth law can be stated as $(T A = T C) \land (T B = T C) \Rightarrow (T A = T B)$ where $\land$ means "and", $\Rightarrow$ denotes "imples", and T_A, T_B, T_C are the temperatures of objects A, B, C."

may be compact, but is pointless since the notation has to be explained (constituting no thermodynamics). The law can be stated in words:

"Letting T_A, T_B, T_C denote the temperatures of objects A, B, C, if A is in thermal equilibruim with C (so

T A = T C

), and B is in thermal equilibruim with C (so

T B = T C

), then A and B are in thermal equilibruim with each other (

T A = T B

)."

Levels of difficulty

The tables below (mainly of numbers, operations, and functions required to build formulae) are roughly grouped into three levels of difficulty; school/(advanced) college level, then undergraduate level, finally graduate level (and beyond). Articles which fall into one (or between?) these sections should consider the reader to assume some familiarity within the scope of the section.

In all cases, the numbers, operations etc.:

should be linked to the main articles
not require any in-depth explanation (a few words or sentence at most).

Of course, other operations not tabulated can be used, provided their use is declared and linked. Most of the operations not tabulated below will be outside the scope of most of physics, such as some non-elementary functions like tetration, super-roots, and super-logarithms, and hence no prior knowledge of the reader is required.

Elementary level

The following are always presumed to be known:

Integers and real numbers, including common constants $π$ and $e$ ,
decimal notation for numbers, including scientific notation discussed above.

When complex numbers are expected, particular examples including;

waves,
electromagnetism (EM radiation, alternating current),
quantum mechanics,
solutions to differential equations,

etc., then $i$ (or $j$ ) is presumed to be known too, although the context should state it is the imaginary unit and not something else, like a summation dummy variable or tensor indices.

Operation	Notation	Notes
Arithmetic and elementary algebra
Addition Subtraction Additive inverse	$a + b$ $a - b$ $- a$	for scalars, vectors, operators, matrices
Multiplication (Scalar, of a vector)	$a b, a \cdot b, a \times b$ $λ v$	$a, b, λ$ are real numbers, "vector" $v$ means element of any linear space
Division	$a / b$ or $\displaystyle {\frac {a}{b}}$	$b$ may be a scalar only
Multiplicative inverse	$A -1$	for scalars, matrices, operators
Absolute value	$\| a \|$	including complex numbers
Plus-minus sign	$\pm a$	including complex numbers
Complex conjugate	$z$	complex scalars or 2-component spinors^[3]
Square roots (and nth roots?)	$\sqrt a, p 1/ q$	either $a \geq 0$ (then √a ≥ 0), or defined up to sign
Factorial	$n!$	Only for integers.
Summation	$\displaystyle \sum \limits _{k}a_{k}$	any linear space only finite sums?
Product	$\displaystyle \prod \limits _{k}a_{k}$	only finite products?
Elementary functions
Exponentiation	$a b$ (Due to conflict with tensor superscripts, make context clear this is exponentiation)	for scalar, matrix, operator $a$ for scalar $b$ only(?) either $a \geq 0$ or $b$ must be integer
Natural logarithm	$ln a$	multivalued unless $a > 0$
Trigonometric functions	All the primary ratios should be known; $sin x, cos x, tan x,$ The secondary are not essential (just reciprocals): $cosec x, sec x, cot x$
Hyperbolic functions	All the primary functions should be known; $sinh x, cosh x, tanh x,$ The secondary are not essential (just reciprocals): $cosech x, sech x, coth x$
Elementary calculus
Ordinary derivative (see notation above)	$⁠ d / dx ⁠ f (x)$
Integrals (indefinite, definite, and improper)	$\int f (x) dx, \int b a f (x) dx$
Basic linear algebra
Dot product	$a \cdot b$	3d Euclidean only, unless explained
Cross product	$a \times b$	3d Euclidean only, unless explained
Matrix multiplication	$A B$	Including operations with column- and row vectors

Undergraduate level

Operation	Notation	Notes
Multivariable calculus
Partial derivatives, nabla	$⁠ \partial 2 F / \partial y \partial x ⁠, \nabla$
Multiple integrals (including line, surface, and volume integrals)	$\int\int\int V ψ(x) dV$
Advanced linear algebra
Exterior product	$a \land b$	Euclidean only, unless explained
Tensor product	$a \otimes b, ab$	Euclidean only, unless explained
Tensor contraction	$A ... k ... ... B ... k ... ...$	possibly by multiple indices; see above
Bra-ket notation	$⟨ ϕ \|$ , $\| ψ ⟩$ , $⟨ ϕ \| ψ ⟩$ , $⟨ ϕ \| A \| ψ ⟩$	see above
Linear operator composition	$A B$	not always distinguished from matrix multiplication
Conjugate transpose Hermitian adjoint	$A †$	over complex numbers only, Usually denoted as $A *$ in pure mathematics.

Additionally:

Non-elementary functions or special functions, namely; the gamma function, error function, orthogonal polynomials, Bessel functions...
Integral transforms (at least in the simplist forms), including the Fourier and Laplace transforms.

Graduate level

Very advanced, abstract, and demanding topics, like;

Operation	Notation	Notes
Differential forms
Variational derivative, fractional derivative, covariant derivative
Functional integration, integration measures?
...

... exterior calculus on manifolds, analytic functions from operators, operator theory on Hilbert spaces, etc., will not be easily understood to the typical reader, and obviously never expected. Such topics need

to have enough explanation to make clear the relevance of the mathematics to physics (for instance, why a simpler formulation is not possible?),
to be thoroughly linked to the main subject articles containing the details.

Application of graphics and illustrations

Sidebar and navbox templates

Templates can be used for linking groups of articles, in the form of:

sidebars: boxes of links with collapsible sections, usually placed in the top right corner of an article, and
navboxes: boxes of links which are fully collapsible, usually placed at the very end of the article, even after the "see also", "references", and "external links" sections.

A decision at each article should be made for what kind of template of links to use. Sidebars are a matter of debate for their size and layout of links; some favour their use, others oppose or are neutral. Navboxes are usually less problematic due to their compactness.

Often, an article is well-served with a picture, diagram, or even an animation (see wavefunction for a good example), at the very top to provide an immediate visual depiction of what the article says.
- An image directly corresponding to the specific topic of an article should have priority for putting in the top right corner over a sidebar template.
- Placing an image in the center, while placing a template in the corner, creates large amounts of whitespace, and the table of contents only adds more whitespace. Both can be avoided by having the image floating in the top-right corner and the sidebar below, or replacing the sidebar with a navbox.
Some pages on abstract topics, where it's hard to find an image that's clearly associated with the article, (e.g. articles related to string theory), may benefit from an "icon" inside a sidebar. This can immediately inform the reader about the substance of the article, while including the links to related articles at the same, hence sidebars may be useful in cases like this.
Templates with a lot of links (say, 50 or more) are better as footers than as sidebars, since opening the sections of a sidebar makes it extend a long way down the page, and makes the box and article less readable this way.
It is legitimate to place navboxes in the "see also" or "references" sections if that's where the article ends. Placing navboxes in "see also" sections is beneficial as they display a large collection of related links in one place with minimum wikicode (only template syntax is needed).
If the contents of a template only has partial relevance to the article; they are likely to contribute to template creep, and hence should not be used.
If neither are suitable for any reason, the best resort is to just use the categories of the article.

Conventions in WP articles and sources

Aside from units, there are cases where specific choices of mathematical entities have to be made, notably the metric tensors in SR and GR, and the representations of the gamma matrices.

Sometimes, reliable and well-established sources use conventions which may lack a clear physical depiction, and WP editors should consider what the best conventions a physics article should have. Maxwell's equations in differential forms are an example (the 4-current is preferred as a 3-form, but sources (including Gravitation) use the 4-current as a 1-form, see the article Mathematical descriptions of the electromagnetic field for this).

Editors should:

decide what conventions are the easiest for a reader to follow and for internal consistency, and also state clearly (in a section or footnotes?) the prominent alternative conventions used by other authors,
decide if edit notices are necessary (see Template:Editnotices/Page/Mathematical descriptions of the electromagnetic field for an example) to reflect specific choices made and prevent other editors performing mass-changes to other conventions.

Footnotes

^ JCGM, International vocabulary of metrology – Basic and general concepts and associated terms (VIM)
^ See, for example, Penrose, Roger (1977), "The twistor programme", Reports on Mathematical Physics, 12 (1): 65–76, Bibcode:1977RpMP...12...65P, doi:10.1016/0034-4877(77)90047-7, MR 0465032
^ In pure mathematics, for a complex vector $v$ its conjugate $v$ is an element of the complex conjugate vector space. Although the complex conjugate of a vector is used in physics (for example, in $⟨ ψ | A | ψ ⟩$ ), the notation with overline is discouraged except for aforementioned two cases.