The Young Pedant's Guide to the Greek Alphabet and Acceptable Variable Nomenclature
Never use upsilon
You’re a young researcher, writing your very first real research paper. It involves some math, and you need a variable to represent time. What to pick? A? B? Upsilon? The word “time”?
There are precisely two acceptable choices for variables representing time or time intervals, and those are t and τ. Their upper case variants are also acceptable.
There are unwritten conventions for variable choice in physics which are usually learned by osmosis during an undergraduate degree. What follows is an guide to what (lowercase) Greek letters usually mean in a physics context. Disclaimer: My field(s) of expertise are Accelerator Physics and Optics, and so my variable choices mostly follow those conventions.
As you will discover by the tragic non-uniqueness of variable-to-quantity mappings, physics ran out of letters a long time ago. In my opinion, it’s time to start mining Egyptian hieroglyphs for new symbols. One is in common use already, the ☉ symbol in astronomy, used as a subscript for solar mass.
α - Alpha
The fine structure constant, which quantifies the strength of the electromagnetic force, roughly equal to 1/137
Angular acceleration, as in the rotational analogue of Newton’s second law
β - Beta
Normalized velocity, v/c where v is velocity and c is the speed of light
Spatial propagation constant for waves, β = 2 π / λ, where λ is the wavelength
γ - Gamma
The Lorentz factor of special relativity:
Also sometimes the propagation constant for waves. This use is discouraged by the Young Pedant’s Guide, as it is confusing in relativistic mechanics
δ - Delta
Almost always used to represent a small change or a perturbation in combination with another variable, e.g. δx is the “change in x,” δy is the “variation of y,” etc.
Every now and then it is used for the phase of a sine wave, when you run out of other Greek letters.
ε - Epsilon
A small quantity, perturbation, or error, not used in combination with other variables, e.g. x + ε is the quantity x with a small perturbation ε
An example of a crime against notation from my girlfriend’s robotics textbook:
e is Euler’s number! Use ε for error!! The Levi-Civita symbol, a totally antisymmetric tensor useful in many areas of math and physics
The electric permittivity of a material
Efficiency of a process
ζ - Zeta
Sometimes the “damping ratio” in the simple harmonic oscillator standard form, but usually overlooked because it is difficult to write by hand and not have it look like a squiggly line
η - Eta
Efficiency of a process
Impedance of free space, along with Z0
θ - Theta
Phase, as in the phase of sin(θ) is θ.
Similarly, θ is very often used for angle, which makes sense, as phase and angle are the same thing in a very fundamental sense.
The Heaviside Step Function, not to be confused with the Heaviside Layer, which confused the hell of out me when I saw the 2019 film Cats with no context. That was an experience I am unlikely to forget, and if you’re morbidly curious, I highly recommend watching both the film and Lindsay Ellis’ scathing review on Youtube.
ι - Iota
Too similar to i. Do not use.
Side note: i is the imaginary number, and don’t listen to any engineer that tells you j is the imaginary number — they are lying to you. It’s called the imaginary number, not the jmaginary number.
κ - Kappa
A coupling constant (e.g. coupled mode theory)
A spring constant
The curvature of a line in a curved coordinate system, rarely
Wavevector, or anything else usually called k, when you already have two or more k’s in your equation.
λ - Lambda
Wavelength. Always.
Okay sometimes it’s one-dimensional density, or just a placeholder variable, but mostly it’s wavelength.
μ - Mu
The magnetic permeability of a material
Tensor indices in general relativity, or just tensor indices in general
Reduced mass in classical mechanics
ν - Nu
Frequency of light (along with ω)
Tensor indices in general relativity, paired with μ
In general relativity, μ and ν represent indices ranging from 0 to 3, and i, j are indices ranging from 1 to 3.
ξ - Xi
I had a physics professor who told me this was the universal physicist’s symbol for “I don’t care what this is,” and I have never really found a contradictory example. Since ξ, like ζ, easily turns into a squiggly line when written, it’s often used as a placeholder for some long expression you don’t want to continually write out when doing algebra.
ο - Omicron
Nope. Identical to the roman letter o, and too close to 0. Not to be used as a variable on pain of permanent public shaming.
π - Pi
The constant π, obviously
Surprisingly, this also has other uses, commonly the canonical momentum operator in quantum mechanics and quantum field theory
The product operator, the multiplication equivalent of Σ for summations. (Technically, this one is the capital π, but they look identical)
ρ - Rho
Density (including mass and charge, often simultaneously)
Specifically, it is usually volume or three-dimensional density, distinct from σ and λ, which are usually two-dimensional and one-dimensional density, respectively.
Density matrices in quantum mechanics
Length coordinate in spherical or cylindrical coordinate systems
ρ is one of the most overused variables in physics. For example, consider a cylindrical coordinate system, with coordinates (ρ, θ, z). Within that coordinate system, you have copper wire with uniform mass density ρ. The wire is charged with radially varying charge density ρ(ρ), and has resistivity ρ. Oy.
The only other variable with which such confusion is possible is z. From my graduate course on electromagnetic waves, I present the wave impedance transformation, featuring five distinct z’s in a single equation:
σ - Sigma
Two-dimensional density, often surface charge density
Standard deviation (or standard error)
The Pauli matrices, the set of 2x2 matrices which generate the Lie algebra su(2) and form the basis for spin calculations in quantum mechanics.
τ - Tau
Time, usually an interval or time constant, or proper time in relativity
Specifically, the period of an oscillation is always T or τ, with almost no exceptions
Torque
υ - Upsilon
The upper case looks like Υ, the lower case looks like ν or u, and the letter pronunciation sounds nearly identical to epsilon. A uniquely terrible choice for a variable. Never use this.
ϕ - Phi
Like θ, this is nearly always the phase of a wave or an angle
Also common in spherical or cylindrical coordinates as one of the angular coordinates.
χ - Chi
Material susceptibility, a close relative of refractive index
A generic spinor in quantum mechanics
A common choice for a variable which is x, but also not x. A common source of frustration for TAs grading homework of those with poor handwriting.
ψ - Psi
The Wavefunction in quantum mechanics
ω - Omega
A variable so common in physics that I regularly mispronounce “w” as “ω.” It represents the angular frequency of an oscillation, always.
Choose wisely, my friends.