# Can we regularize and use without any problem zeta regularization?

+ 3 like - 0 dislike
173 views

is zeta regularization used in physics (both theoretical and mathematical)?

i mean the use of the regularization of the infinite series ·$$1+2^{s}+3^{s}+........= \zeta (-s)$$

and for the Harmonic series , what would be valid ? (regularization)

$$\sum_{n=0}^{\infty} \frac{1}{n+a}= -\Psi (a)$$ or

$$\sum_{n=0}^{\infty} \frac{1}{n+a}= -\Psi (a) +loga$$

so we find no UV divergences

edited Mar 9, 2016

+ 2 like - 0 dislike

Zeta function regularization is one of the established tools in mathematics and physics for summing certain asymptotic series. In cases where the regularized series still diverges (such as for the harmonic series), a single series has no result but linear combination of several regularized series in which the leading divergent terms cancel make sense. It is only the latter that have a physical meaning. Thus in applications to singular interactions in quantum mechanics or quantum field theory one first must sum all relevant contributions before taking the limit where the argument attains its physical value.

In your explicit example, the first formula is valid and diverges; the second formula can be used only in combination with another, similar sum where the logarithm appears with the opposite sign, so that the divergent part cancels.

For the use of zeta function regularization in quantum field theory see, e.g., the book Analytic Aspects of Quantum Field by Bytsenko et al. (2003).

For an application in infinite-dimensional geometry see, e.g., here.

answered Mar 21, 2016 by (14,557 points)
edited Mar 21, 2016

 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): Email me at this address if my answer is selected or commented on: 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:p$\hbar$ysi$\varnothing$sOverflowThen 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). To avoid this verification in future, please log in or register.