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Analytic Continuation

Analytic Continuation is a technique in complex analysis that extends the domain of a given holomorphic function to a larger region. It is fundamental in number-theory and quantum-field-theory for assigning finite values to divergent series and integrals.

Key Concepts

• Uniqueness: If two analytic functions agree on a set with an accumulation point within their domain, they are identical on the connected component containing that set.
• Riemann Zeta Function: The classic example where $\zeta (s)={\sum }_{n=1}^{\infty }{n}^{-s}$ converges only for $\text{Re}(s)>1$. Analytic continuation allows evaluation at $s=-1$, yielding $\zeta (-1)=-1/12$.
• Regularization: A process used in physics to handle infinities, often relying on analytic continuation to extract meaningful physical quantities from divergent expressions.

Applications in Quantum Physics

• Casimir Effect: The calculation of vacuum energy between plates relies on summing zero-point energies, which diverge. Analytic continuation provides a finite, measurable result.
• String Theory: Critical dimensions and partition functions often require analytic continuation of the zeta function to ensure consistency.
• Zeta Function Regularization: A method to define the determinant of operators with infinite eigenvalues by using ${\zeta }^{\prime }(0)$.

Notable Derivations

• The result $1+2+3+\text{[}[concepts/dots-agent|dots]]=-1/12$ is not a standard sum but the value of the analytically continued zeta function at $s=-1$.
• This value appears in the Casimir effect and bosonic string theory calculations.
• For a detailed breakdown of this derivation in the context of quantum physics, see: 12 for Quantum Physics Applications

References

• Analytic Continuation: Deriving -1/12 for Quantum Physics Applications