Regularization of Infinite Sums: Part II « Diary of a Quantum Automata

June 28, 2009

Regularization of Infinite Sums: Part II

Filed under: math, research — Tags: divergent series, heat kernel regularisation, infinite series, infinite sum, laplace-beltrami, zeta regularisation — vivisheksudhir @ 7:57 pm

I continue from my previous post where I talked about the technique of analytical regularisation, one of the many fine weapons from mathematical analysis that helps the physicist in need. The previous method offers an elegant solution to the problem of “summing” classically divergent (the Cauchy sense) series. I wish to introduce a slightly different, and at times more powerful technique called zeta regularisation. The name comes from its relation to zeta functions of various kinds that crop up in the process of regularisation. The technique is so powerful in fact that it is used in the regularisation of infinite products (whose convergence is related to the convergence of an associated infinite sum) and more importantly in determining the trace, determinant and inverses of infinite dimensional operators residing in a Hilbert space (this is done using the spectral zeta function of the operator). I shall keep the exotica for another post and focus on sums alone here.

To regularise the sum $S= a_0 + a_1 + a_2 +... =\sum a_n$ , one constructs the zeta regularisation,

$S(z)={a_0}1^z + {a_1}2^z +{a_2}3^z +... =\sum_{n=0}^{\infty} a_n (n+1)^z$ ,

then the zeta regularised sum (ZRS) is given by $\lim_{z \rightarrow 0} S(z)$ . Note that $S(z)$ is a in general, the Dirichlet L-function $L(z,a)$ ; in most cases that occur in physics, these can be reduced to a Riemann or Selberg zeta function, the properties of which are extensively studied in the mathematics literature.

For example, let us take our old friend, the series $1+2+3+...$ and form its zeta regularisation, viz.,

$S(z)=1\cdot 1^z +2\cdot 2^z +3\cdot 3^z +...=\sum_n n^{z+1}$ .

Comparing it with the definition of the Riemann zeta function, $\zeta(z)=\sum_n n^{-z}$ , we immediately note that the zeta regularisation is simply $\zeta(-z-1)$ ; thus the ZRS is given by $\zeta(-1)$ . To calculate this value, we use the identity,

$\zeta (-n)=\frac{-B_{n+1}}{n+1}$ ; $n \in \mathbb{Z}_{-}$ ,

where $B_n$ is the $n^{th}$ Bernoulli number; we need, $\zeta (-1)=\frac{-B_2}{2}$ . Using the generating function of the Bernoulli numbers, viz.,

$\frac{x}{e^x -1}=\sum_{n=0}^{\infty} \frac{x^n}{n!}B_n$ ,

expanding the left side in Taylor series and comparing like powers of $x$ , we get $B_2 =1/6$ . This gives $\zeta(-1)=-1/12$ , giving the final ZRS $1+2+3+... =-1/12$ .

There are series which are still notoriously non-regularisable, a prominent example being the harmonic series, whose zeta regularisation, $\zeta(1)\rightarrow \infty$ .

Another important class of sum regularisation methods is the so-called heat kernel regularisation, so named since it uses the eigenvalues of the Laplace-Beltrami operator (the kernel of the heat equation). This offers an even more poweful analytical tool since the the theory of Laplace-Beltrami operators in various differentiable manifolds is well established. As such, the method is well suited to problems of vacuum polarisation calculations in various metric spaces. The basic tenet of the method is to form, for the series $S= a_0 + a_1 + a_2 +... =\sum a_n$ , the heat kernel regularisation (HKR), given by,

$S_{\omega}(z)=\sum_{n=0}a_n e^{-z\vert \omega_n \vert}$ ,

where $\omega_n$ are the eigenvalues of the suitable Laplace-Beltrami operator. Then $\lim_{z \rightarrow 0} S_{\omega}(z)$ gives the required regularised sum. The power of this method comes from the fact that the evalution of the above sum for $S_{\omega}(z)$ is facilitated by certain corollaries of the Selberg Trace Formula.

NOTES:

The relationship between the non-trivial zeros of the Riemann zeta function (the content of the 2nd Riemann hypothesis) and the distribution of primes has motivated physicists to look at physical “proofs” of the Riemann hypothesis! The general idea is to construct a suitable, physically realisable Hamiltonian (such as $\hat{H}=\hat{x}\hat{p}+\hat{p}\hat{x}$ , the Berry Hamiltonian) whose spectral zeta function coincides with the Riemann zeta and to see how the physics of the system (which can be experimentally observed) is related to the distribution of its zeros (via the Hilbert-Polya conjecture).
Certain divergent integrals can be regularised using a protracted zeta regularisation. This is achieved by using the Abel-Plana equation, $\sum_{n=0}^{\infty}f(n) - \int_{0}^{\infty}f(x) dx = \frac{f(0)}{2}+2i \int \frac{Im[f(it)]}{e^{2\pi t}-1} dt$ . The sum is zeta regularised initially and the ‘t’ integral is suitably performed along a suitable contour.
It has come to my notice that the Laplace-Beltrami operator has recently been extended to fractal geometries! i would like to read on that someday, once I get some ample free time at my disposal…

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Diary of a Quantum Automata

June 28, 2009