• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,064 questions , 2,215 unanswered
5,347 answers , 22,734 comments
1,470 users with positive rep
818 active unimported users
More ...

  Where and how exactly does string theory and Q.E.D. use zeta function regularization?

+ 5 like - 0 dislike

In the video they mention it being used in many fields of physics inclusing String and QED theory.

But I remember reading somewhere that 1+2+3..=-1/12 is obviously a "mathematical trick" (something about stupidly equating incompatible sets), and if so, how does this turn out to be true for things that are real (like QED)?

This post imported from StackExchange Physics at 2014-03-26 12:48 (UCT), posted by SE-user laggingreflex
asked Mar 25, 2014 in Theoretical Physics by laggingreflex (50 points) [ no revision ]
retagged Mar 26, 2014
Watch the follow-up youtube.com/… and also youtube.com/watch?v=d6c6uIyieoo&feature=youtu.be and youtube.com/…

This post imported from StackExchange Physics at 2014-03-26 12:48 (UCT), posted by SE-user nephente

2 Answers

+ 2 like - 0 dislike

Zeta function regularization is used in other fields, and even in pure mathematics to obtain finite answers from otherwise divergent integrals. In bosonic string theory, the mass of states in lightcone gauge is,

$$M^2 = \frac{4}{\alpha'} \left[ \sum_{n>0} \alpha^{i}_{-n}\alpha^{i}_n + \frac{D-2}{2}\left( \sum_{n>0} n\right) \right]$$

where $\alpha'$ is the universal Regge slope, $D$ is the spacetime dimension, and $\alpha^{i}_n$ may be interpreted as Fourier coefficients of the expanded form of the embedding functions $X^{\mu}(\sigma)$ in the Polyakov action which provide a map from the worldsheet to the target space. We use the fact that

$$\sum_{n>0} n = 1+2+3+...=\zeta(-1)=-\frac{1}{12}$$

to write the expression for the mass of states as,

$$M^2 = \frac{4}{\alpha'} \left(N - \frac{D-2}{24} \right)$$

If we look at the ground state, corresponding to $N=0$, we see

$$M^2 = -\frac{1}{\alpha'}\frac{D-2}{6}$$

which corresponds to a particle with an imaginary mass, known as a tachyon. The demand that we preserve $SO(1,D-1)$ Lorentz symmetry forces us to choose that the first excited state $(N=1)$ be massless, and so we must choose the spacetime to be $D=26$. In other string theories, the critical dimension of the string may be lower, e.g. $10$ or $11$. For further details, I recommend Prof. Tong's lectures notes on string theory available at: http://www.damtp.cam.ac.uk/user/tong/string.html.

This post imported from StackExchange Physics at 2014-03-26 12:48 (UCT), posted by SE-user JamalS
answered Mar 25, 2014 by JamalS (895 points) [ no revision ]

And what happens with harmonic series ? we have $$ \sum_{n=1}^{\infty}n^{-1}= \gamma $ euler's constant but what would happen with $ \sum_{n=0}^{\infty}(n+a)^{-1} $

+ 1 like - 0 dislike

In addition to what JamalS said, there is also the Casimir energy in string theory.


answered May 11, 2014 by dimension10 (1,985 points) [ revision history ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights