Apparently this result is used in many areas physics including the extra dimensions of string theory, which is not the scope of the question. The result is apparently also used to understand the Casimir force and there are many different types of experimental validation for this force. But how does this result of adding all the natural numbers and getting -1/12 accomplish this?
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Actually the true equation is:
$$\int_0^\infty n dn - \sum_{n=1}^\infty n = \frac{1}{12}$$
Since the Casimir energy is the difference between the allowed wavelengths had the walls not been there and the allowed wavelengths with walls being there.
Actually the above sum is $\infty-\infty$ so you must regularise it using:
$$\lim_{\alpha\rightarrow 0}\left(\int_0^\infty ne^{-\alpha n} dn - \sum_{n=1}^\infty ne^{-\alpha n}\right) = \frac{1}{12}$$
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In QED the energy of electromagnetic waves are quantized $E=(n+1/2)\hbar\omega$. When calculating the total energy of vacuum you need to add up them altogether, and this leads to the infinite integer summation in 1-d QED, if consider 3-d case the regulation results in $\zeta(-3)$. For detail description, please refer to: https://en.wikipedia.org/wiki/Casimir_effect; https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension.
Kangle
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