How does one justify in String theory that 1+2+3+4+5+6+...=-1/12?
I'm not a string theorist, but I think the answer relates to a book I'm reading at the minute. I think it results from an analytic continuation of the Riemann Zeta function- take a function expressed as an infinite sum that converges for some arguments, then "bolt on" another function to areas where the original function was not defined in such a way that the result is holomorphic.
Hmm... I don't follow this. Can anyone explain in simple language how adding an infinite number of numbers -- a series that is clearly divergent -- become -1/12????
Very interesting... but now my head is spinning. Is this a math trick? We all know that 1+2+3+4+5+6+...= infinity. So how can this 1+2+3+4+5+6+...= -1/12 be justified?
The reference to Euler's 'proof' of this refers to a time before the notions we take for granted in analysis were formed and refined. one of the common ways for finding the 'sum' of one of these infinite series was to look for what we would call a formal power series in [tex] x [/tex] for which the terms of the original series were the coefficients. If such a power series could be found, and a closed form expression involving [tex] x [/tex] derived, the 'sum' of the series was taken to be the value the closed form formula gave when evaluated at [tex] x = 1 [/tex].
This obviously resulted in many problems (like this one), but also problems in which a single numerical series gave rise to two different power series and so, different closed forms, and so (again) different sums.
It is essentially a slightly fancier process than the old idea of grouping the terms of
1 - 1 + 1 - 1 + 1 - 1 + \bdots
in order to obtain different sums.
On the positive side, these discussions led to answering many of the questions about infinite series, convergence and divergence, and the subject of different summation methods. The classic tomes by Bromwich and Knopp (early 20th century, both) on issues related to infinite series and products, are excellent sources.
Thanks for the tip. I'm in the process of ordering the Knopp book. It sounds interesting.
In the meantime, I still have reservations about this infinite series that is used in String theory, and I'm wondering if this would disqualify it as a theory of reality.
I cannot address the physics application, if any, of this - my only background is in mathematics and statistics.
Thank you - I've just printed it and will look at it this weekend. I have no idea whether I'll make much of it, but it should be entertaining.
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