Phi

How to read it

Dictionary.com and the Collins English Dictionary give /faɪ/; Collins describes phi as a consonant, transliterated as ph or f. The small letter has two shapes, a looped φ and a stroked ϕ, and Unicode gives them separate code points; which shape a font draws for which code point varies.

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Capital and small forms

Unicode says U+03C6 is the ordinary Greek letter, “showing considerable glyph variation”, and that in mathematical contexts the loopy glyph is preferred for it, to contrast with the stroked ϕ at U+03D5, a technical symbol. NIST’s Guide for the Use of the SI lists both ϕ and φ as forms of the small phi. The Latin letter ɸ (U+0278) is a third look-alike.

Phi in maths and physics

In a wave or oscillation, φ is the phase shift, in radians: OpenStax’s University Physics writes a mass on a spring as x(t) = A cos(ωt + φ). Φ is flux, as in magnetic flux Φm in Faraday’s law. In number theory φ(n) is Euler’s totient function (NIST DLMF §27.2). NIST also uses φB for the volume fraction of a substance B.

In AI and machine learning

φ is a feature map: scikit-learn describes training vectors as implicitly mapped into a higher-dimensional space by the function φ. scikit-learn’s guide to support vector machines writes the kernel as K(xi, xj) = φ(xi)ᵀ φ(xj); for LinearSVC, φ is the identity function. In reinforcement learning, OpenAI’s Spinning Up writes the parameters of a policy as θ or φ.

In quantum physics

φ is a phase angle: multiplying a state |ψ⟩ by e^(iφ) changes its phase. MIT’s 8.06 notes point out that this leaves the density matrix unchanged. See Psi and Bra-ket notation (|ψ⟩).