1 + 2 + 3 + 4 + ⋯
which increases without bound as n goes to infinity.
Although the full series may seem at first sight not to have any meaningful value, it can be manipulated to yield a number of mathematically interesting results, some of which have applications in other fields such as complex analysis, quantum field theory and string theory.
has a pole at x = 1.
The series can be summed by zeta function regularization. When the real part of s is greater than 1, the Riemann zeta function of s equals the sum . This sum diverges when the real part of s is less than or equal to 1, but when s = −1 then the analytic continuation of ζ(s) gives ζ(−1) as −1/12.
- "Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"
In bosonic string theory, the attempt is to compute the possible energy levels of a string, in particularly the lowest energy level. Speaking informally, each harmonic of the string can be viewed as a collection of independent quantum harmonic oscillators, where is the dimension of spacetime. If the fundamental oscillation frequency is then the energy in an oscillator contributing to the th harmonic is . So using the divergent series, the sum over all harmonics is . Ultimately it is this fact, combined with the Goddard–Thorn theorem, which leads to bosonic string theory failing to be consistent in dimensions other than 26.
A similar calculation is involved in computing the Casimir force.
See also↑Jump back a section
- Lepowsky, James (1999). "Vertex operator algebras and the zeta function". Contemporary Mathematics 248: 327–340. arXiv:math/9909178.
- Zee, A. (2003). Quantum field theory in a nutshell. Princeton UP. ISBN 0-691-01019-6. See pp. 65–6 on the Casimir effect.
- Zwiebach, Barton (2004). A First Course in String Theory. Cambridge UP. ISBN 0-521-83143-1. See p. 293.
- This Week's Finds in Mathematical Physics (Week 124), (Week 126), (Week 147)