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josephpalazzo
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How does one justify in String theory that 1+2+3+4+5+6+...=-1/12?
muppet said: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.
atyy said:
statdad said: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
[tex]
1 - 1 + 1 - 1 + 1 - 1 + \bdots
[/tex]
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.
statdad said:I cannot address the physics application, if any, of this - my only background is in mathematics and statistics.
String Theory is a theoretical framework in physics that seeks to unify the four fundamental forces of nature (gravity, electromagnetism, strong nuclear force, and weak nuclear force) into one consistent theory. The equation Sum = -1/4 is a key component of String Theory and represents the sum of all possible energy states in a system.
String Theory proposes that the universe is made up of tiny, vibrating strings rather than point-like particles. These strings vibrate at different frequencies, giving rise to different types of particles and forces. This theory also suggests that there may be multiple dimensions beyond the three we experience in our everyday lives.
Currently, there is no direct evidence that confirms the predictions of String Theory. However, some of its principles have been used to explain certain phenomena in physics, such as the properties of black holes and the behavior of subatomic particles. Additionally, String Theory has shown promise in resolving some long-standing paradoxes in physics, such as the black hole information paradox.
The equation Sum = -1/4 is a crucial component of String Theory as it represents the sum of all possible energy states in a system. This sum is calculated using a mathematical technique called zeta function regularization, which helps to eliminate certain mathematical inconsistencies in the theory. The equation also plays a role in defining the fundamental interactions between strings and their vibrations.
No, there are other theories that also aim to unify the fundamental forces, such as Loop Quantum Gravity and Grand Unified Theory. However, String Theory is currently the most widely studied and well-developed theory in this area. It is also the only theory that incorporates the principles of quantum mechanics and general relativity, making it a promising candidate for a theory of everything.