I'm reading about zeta-function regularization in physics and I have some mathematical doubt. I understand that, since a sum of infinite terms is not well defined in a field, a series that is considered divergent in the usual meaning can have a ''value'', defined in some less conventional way.
The zeta-function regularization is one of such ways that, as an example, assign the value $-\frac{1}{12}$ to the infinite series $S=1+2+3+4+...$ using the fact that the zeta-function $\zeta(s)$ is the analytic continuation of the series $\sum_{n=1}^\infty n^{-s}$, and $\zeta(-1)=-\frac{1}{12}$.
But how we can be sure that there does not exists other possible ''regularizations'' that gives different values to the same series?
And, if more than one value does exist, there is some criterion to select between them? Or the zeta-function regularization is preferred only for physical motivations (because the experiments confirm its values)?
On the web I've found a lot of posts less or more reliable about this topic, someone knows a reference to a well defined axiomatic approach to the problem of the value of divergent series?
Answer
Our problem is: Given a sequence $(a_n)_{n=0}^\infty$ of numbers, assign a "sum" to it, i..e, give a meaning to the pixel pattern $\sum_{n=0}^\infty a_n$.
This is easy the sequence of partial sums converges as we can simply assign the limit of these. This does not define $\sum_n a_n$ for all given sequences $(a_n)_n$, but only to a certain subset $S$ of the set of all sequences. But we have a collection of nice rules:
- $S$ is a vector space and $\sum$ is linear, that is
- If $(a_n)_n,(b_n)_n\in S$, then $(a_n+b_n)_n\in S$ and $\sum(a_n+b_n)=\sum a_n+\sum b_n$
- If $(a_n)\in S$ and $c\in\Bbb C$, then $(ca_n)_n\in S$ and $\sum ca_n=c\sum a_n$.
- $S$ is closed under adding/dropping/modifiying finitely many terms, which boils down to
- $(a_n)_n\in S$ if and only if $(a_{n+1})_n\in S$. In this case $\sum a_{n}=a_0+\sum a_{n+1}$
Regularization is the attempt to enlarge $S$ in a useful way. There are many ways to do so, and in particular if one ignores the above nice rules, one can do it almost arbitrarily (using the Axiom of Choice, perhaps).
On the other hand, if one wants to have permanence of the above rules (or perhaps others), then quite often the extension is uniquely determined.
For example, if we want to enlarge $S$ in such a way that it contains $(2^n)_n=(1,2,4,8,\ldots)$ and we assign (by any computational method at all) a value $\sum 2^n=c$, then $S$ must also contain $(2\cdot 2^n)_n=(2,4,8,16,\ldots)$ and $\sum 2^{n+1}=2c$, and $S$ must contiain this prepended with $1$ - which is again the original sequence - and so we obtain the equality $2c+1=c$. Therefore, if we want to assign a value to $\sum 2^n$ and want permanence of the above rule, we must agree to set $\sum 2^n=-1$. Similarly, it follows that $(1)_n$ cannot be $\in S$ because $c+1=c$ has no solution.
But can we extend $S$ to contain $(n)_n$? If so, then also $(n+1)_n\in S$ and their difference $(1)_n\in S$ - which we have just seen is impossible.
Thus any attempt to consistently assign a value to $1+2+3+4+\ldots$ must drop one of the very reasonable rules listed above.
But with these rules as goalposts removed, one would first have to agree what constitutes a "valid" extension of summation before one can make statements about whether a certain value (if it is assigned at all) is necessarily correct (as we did above for $\sum 2^n$).
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