# What is the story behind various uses of the word “spectrum”?

Here are five distinct uses of the word spectrum in physics and mathematics:

1. Spectrum (optics): The range of colors in the rainbow
2. Spectrum (particle physics): The range of electromagnetic frequencies emitted when an electron in an atom moves from a high energy state to a low energy state
3. Spectrum (functional analysis): the set of generalized eigenvalues of a linear map
4. Spectrum (algebra): the set of all prime ideals in a ring equipped with the Zariski topology
5. Spectrum (topology): a sequence $X_n$ of topological spaces equipped with inclusion maps from the suspension of $X_n$ into $X_{n+1}$
6. (Added later): Spectral Sequence (topology): a tool for calculating (co)homology, generalizing exact sequences

My assumption is that these uses of the word spectrum are all related, and I would like to tell a story about how and why the terminology was adapted to different contexts. Here is my speculation:

1 was introduced by Newton in the course of his study of optics. 2 was simply adapted from Newton's use of the word as physicists began to understand the structure of the atom. 3 arose in quantum physics (maybe due to Hilbert or von Neumann?) when it was realized that lines in atomic spectra correspond to eigenvalues of certain linear operators. 4 emerged from the theory of Banach algebras wherein it was realized that if $T$ is a linear operator on Hilbert space which commutes with its adjoint then the maximal ideal space of the commutative C*-algebra generated by $T$ coincides with the spectrum of $T$ in the sense of 3. I have no idea where 5 or 6 comes from (or if they are related).

Can anyone fill in the details of this historical narrative and correct it as necessary? In particular, does anybody know how the word spectrum came to be used in algebraic topology?

• See Spectrum on Earliest Uses for some history of the linear algebra meaning. – Jack M Nov 18 '14 at 12:21
• I wouldn't call #2 "particle physics", at least in the modern understanding of the term. It could be chemistry or atomic physics, but modern day particle physics isn't really concerned with atoms and hasn't been for many decades. As a particle physicist, the usage I encounter most is most closely related to #3 in your list. – Logan M Nov 19 '14 at 5:32

The connection between (2) and (3) is one of the surprising curiosities in the history of mathematics and physics.

Hilbert worked on integral equations mostly from 1903 to 1910. Here's what Constance Reid has to say about this in her biography:

The Courant-Hilbert book on mathematical methods of physics, which had appeared at the end of 1924, before both Heisenberg's and Schrödinger's work, instead of being outdated by the new discoveries, seemed to have been written expressly for the physicists who now had to deal with them....

To Hilbert himeself this was yet another example of that pre-established harmony which seemed to him almost the embodiment and realization of mathematical thought.

"I developed my theory of infinitely many variables from purely mathematical interests", he marvelled, "and even called it 'spectral analysis' without any presentiment that it would later find an application to the actual spectrum of physics!"

The source cited by Jack M in a comment suggests a link between (1) and (3):

There may be a link between Newton and Hilbert for, though the latter cited no previous writer for "Spektrum", J. Dieudonné History of Functional Analysis (1981, pp. 149-50) suggests he derived the term from W. Wirtinger "Beiträge zu Riemann’s Integrationsmethode für hyperbolische Differentialgleichungen, und deren Anwendungen auf Schwingungsprobleme", [Contributions to Riemann's integration methods for hyperbolic differential equations, and their application to oscillation problems] Mathematische Annalen, 48, (1897), 365-89. Wirtinger drew upon the similarity with the optical spectra of molecules when he used the term "Bandenspectrum" with reference to Hill’s (differential) equation.

• This is very interesting and surprising! – Paul Siegel Nov 18 '14 at 15:09
• 1 and 2 are related because as particles jump between energy states, they emit light of a given frequency concurrent with their energy. To wit, what we perceive as color, is essentially photons striking a surface and the energy that the surface does not absorb reflecting off. – Michael Brown Nov 18 '14 at 17:35

One crucial link in 1-5 is missing: spectrum of an atom. (Key words: spectral analysis in chemistry and physics, spectrometer, spectroscope, spectral lines etc.).

There is no doubt that all of them are closely related, and I can explain the relations of all but 5). The word "spectrum" was first used by Newton when he discovered that white light can be dissolved into colors. (I am not 100% sure, but I suppose that it was applied to the rainbow only AFTER that). Spectrum means ghost in Latin.

Then more than a century had to pass before Wollaston discovered the spectral lines in the emission spectra in 1802. This triggered a fascinating story of development of spectroscopy.

S. Sternberg, A history of 19th century spectroscopy, Appendix F in his book Group theory and physics.

(Probably THE most interesting paper I ever read on history of mathematics and physics. Strongly recommended to all who is interested in history of physics and mathematics!)

Gradually they started using this word in the sense "set of proper frequencies of an oscillator" and apply it to all oscillations. But originally this meant "set of frequencies of light emitted or absorbed by an atom".

Corresponding mathematical theory was developing in parallel starting with Lagrange and Fourier. This was known as "linear oscillations" in mathematics. Mathematically this is the same as the "set of eigenvalues of a linear operator". For example, the spectrum of a matrix is the set of its eigenvalues. From spectra of atoms we have spectra in quantum mechanics, and in the rest of modern physics which is based on quantum mechanics. This explains the connections 1-3.

Next development in mathematics was creation of the abstract theory of Banach Algebras (Gelfand, 1930-th). Original motivation of this theory was a) Fourier analysis and b) theory of linear operators. The revolutionary idea was that elements of any Banach algebra can be considered as functions of the "space of maximal ideals". This idea quickly penetrated the other area of mathematics, and this is how we obtain 4). The definition of the spectrum of a ring 4) is due to Grothendieck, and there is no doubt that he was motivated by Gelfand's theory of Banach algebras.

I am not an expert in 5) but this is due to J. Leray, and I suppose that it is based on the development of same idea in Topology.

• Thanks for filling in all the names and dates. Do you know if there is support in Grothendieck's work that his definition of the spectrum of a ring was based on an analogy with Banach algebras? The existence of this connection seems very likely, but I wonder how well documented it is. – Paul Siegel Nov 18 '14 at 15:12
• The notion terminology of a "spectrum" (number 5 in the list) was defined in the thesis of Elon Lima. The notion and terminology of a "spectral sequence" (which is different) was introduced by Leray. I don't know if there is any direct connection between the two choices of terminology. – Charles Rezk Nov 18 '14 at 16:52
• Spectrum also means image in Latin, which is more relevant here. – Michael Weiss Nov 18 '14 at 19:02
• @Paul Siegel: I definitely read it somewhere in some paper whose title was like "Evolution of the notion of space in math" or something similar, but I cannot recall exact reference. The idea that a ring of functions characterizes the space (as a set of maximal/prime ideals) is omnipresent in modern mathematics, and it originates in "Gelfand transform". – Alexandre Eremenko Nov 18 '14 at 20:41
• But of course "nothing is new below the Moon":-) it was Dedekind who introduces ideals... so te idea can be credited to him as well. – Alexandre Eremenko Nov 18 '14 at 20:44