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It appears in some quantum field theory calculations, in particular for Casimir Effect. In simple words Casimir Effect consists of a force that emerges between
by inlineint 10y ago
It appears in some quantum field theory calculations, in particular for Casimir Effect.
In simple words Casimir Effect consists of a force that emerges between two conductor planes that are parallel to each other. The force is proportional to sum of energies of all possible standing electromagnetic waves between the planes. In calculations for this force a divergent series of sum of all natural numbers (or their powers) appears and physicists use 1 + 2 + ... = -1/12 to calculate it (or continuation of zeta function in other points if appropriate).
https://en.wikipedia.org/wiki/Casimir_effect#Derivation_of_Casimir_effect_assuming_zeta-regularization https://en.wikipedia.org/wiki/Casimir_effect#Derivation_of_C...
and
https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_ef...
- cohomologo 10y agoI've never seen a physics book that treats this using the zeta function, except popular articles that try to present this calculation as mysterious. In practice, in my QFT class we needed to compute the sum lim \epsilon -> 0+ ( \sum_{n=1}^\infty n e^{- \epsilon n} + ...), that is the series was multiplied by a decaying exponential function with a rate of decay that goes to zero. This sum can easily be evaluated for small epsilon takes the form sum = 1/epsilon - 1/12 + O(epsilon). The 1/epsilon term (which goes to infinity) drops out of the final physical result when you do the calculation properly.