Zeta function regularization: Difference between revisions

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In the case when ''a''_''n'' = ''n'', the zeta function is the ordinary [[Riemann zeta function]], and this method was used by [[Euler]] to "sum" the series [[1 + 2 + 3 + 4 + ...]] to ζ(−1) = −1/12.

Other values of ''s'' can also be used to assign values for the divergent sums

: [[1 + 1 + 1 + 1 + ...]] → ζ(0) = −1/2, 1 + 4 + 9 + ... → ζ(−2) = 0

and in general

: <math>\sum_{n=1}^\infty n^s=1^s + 2^s + 3^s + \cdots \rightarrow \zeta(-s) = -\frac{B_{s+1}}{s+1},</math>

where ''B''_''k'' is the ''k''th [[Bernoulli number]].<ref>{{cite web|url=http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/|title=The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation|first=Terence|last=Tao|date=10 April 2010}}</ref>

{{harvtxt|Hawking|1977}} showed that in flat space, in which the eigenvalues of Laplacians are known, the [[zeta function (operator)|zeta function]] corresponding to the [[partition function (quantum field theory)|partition function]] can be computed explicitly. Consider a scalar field ''φ'' contained in a large box of volume ''V'' in flat spacetime at the temperature ''T''&nbsp;=&nbsp;''β''⁻¹. The partition function is defined by a [[path integral formulation|path integral]] over all fields ''φ'' on the Euclidean space obtained by putting ''τ''&nbsp;=&nbsp;''it'' which are zero on the walls of the box and which are periodic in ''τ'' with period ''β''. In this situation from the partition function he computes energy, entropy and pressure of the radiation of the field&nbsp;''φ''. In case of flat spaces the eigenvalues appearing in the physical quantities are generally known, while in case of curved space they are not known: in this case asymptotic methods are needed.