# Where does the habit of calling the elements of a projective Hilbert space “rays” originate from?

When describing the projective Hilbert space as the state space in quantum mechanics, physicists habitually refer to its elements as "rays in Hilbert space", while the mathematical literature seems to favour calling the elements of projective spaces "lines" in the original space. Granted, I've not done a systematic literature research on this, but personal experience and corroborating evidence (like googling "projective space ray" and observing that most of the results are indeed about Hilbert spaces) seem to support this.

So, the question is - who first referred to the elements of projective (Hilbert) space as "rays"? Can all later usage be traced back to this instance or did it arise in several instances separately?

• Usually, a ray is an half-line. – Mauro ALLEGRANZA Jul 17 '17 at 11:57
• Related to "visual ray" from optics. – Mauro ALLEGRANZA Jul 17 '17 at 12:02
• @MauroALLEGRANZA I'm aware of that, but the point is that what we mean here is not a half-line, since we're in a complex vector space it's more like a punctured plane, and even in the real case it would be union of two half-lines. Additionally, projective spaces not in the quantum context are not commonly described this consistently in terms of rays, people there talk about planes and lines as often - if not more often - as they talk about rays to describe the elements of the space. I'm interested in the origin of physicsts' consistent use of ray specifically for the case of Hilbert spaces. – ACuriousMind Jul 17 '17 at 12:20
• The English translation of Hilbert's Grundlagen uses the term half-ray: "All of the points of [a line] $a$ which lie upon the same side of $O$, when taken together, are called the half-ray emanating from $O$." – Mauro ALLEGRANZA Jul 17 '17 at 12:26
• "Ray" may be slightly inaccurate, but it's much shorter and easier to write and say than "punctured line" and "punctured plane." "Line" just has too much baggage. – Ben Crowell Jul 17 '17 at 20:48

Bargmann (1954, p. 1) calls this “Weyl’s terminology, cf. (1931, p. 4 and p. 20). Ray, ray space and ray representation (p. 181) stand for Weyl’s original Strahl, Strahlenkörper, Strahldarstellung, see (1928, pp. 7, 19, 161) or (1927):

(p. 5): Zwei von 0 verschiedene Vektoren gehören demselben Strahl an, wenn der eine aus dem anderen dutch Multiplikation mit einer (komplexen, von 0 verschiedenen) Zahl hervorgeht. (...) Fasse ich eine unitäre Abbildung (4) auf nicht als Abbildung des Vektor-, sondern des Strahlenkörpers (homogener Standpunkt), so soll sie kurz eine Drehung heißen.

(p.18): Denn in der Quantenmechanik haben nicht die Vektoren eine Bedeutung, sondern lediglich die Strahlen; sie kennzeichnen die verschiedenen reinen Fälle. (...) So soll das Wort Darstellung in Zukunft verstanden werden: als getreue Darstellung durch Drehungen des Strahlenkörpers*.

* Tiefgehende Untersuchungen über das Darstellungsproblem in diesem Sinne hat I. Schur angestellt: Crelles Journ. 127, 20, 1904 und 132, 85, 1907.

The word (apparently not used by Schur) likely originates in line geometry (a.k.a. optics) where rays were affine lines (W. R. Hamilton, Theory of Systems of Rays, 1828).