Analytic Continuation and Dispersion Relations

A holomorphic function is so rigid that its values on a tiny arc determine it everywhere it can be extended. This analytic continuation lets one define a function by a formula in one region and follow it into another — the mathematics behind Wick rotation, dimensional regularization, and the dispersion relations that tie a physical amplitude's real and imaginary parts together via causality. This page builds on holomorphic.md and laurent-residues.md.

The identity theorem

Identity theorem. If two functions holomorphic on a connected domain $U$ agree on a set with a limit point in $U$ (e.g. a small arc, or a sequence converging inside $U$), they agree on all of $U$.

Holomorphic functions are thus determined by astonishingly little data. A consequence: there is at most one way to extend a holomorphic function from a region to a larger connected one. This uniqueness is what makes "the analytic continuation" a well-defined object, and it is why an identity proven for real arguments (e.g. a functional equation) automatically holds for complex ones by continuation.

Analytic continuation

If $f$ is holomorphic on $U$ and $g$ on $V$ with $U\cap V\ne \varnothing$ connected and $f=g$ there, then $g$ continues $f$ to $U\cup V$. Chaining continuations along a path extends a function as far as singularities permit. Two illustrations:

• The geometric series $\sum z^n$ converges only for $|z|<1$, but equals $\frac{1}{1-z}$ there — and $\frac{1}{1-z}$ is holomorphic on all of $\mathbb{C}\setminus \{1\}$. The rational function is the analytic continuation; the series was just one local representative.
• The factorial $n!$ continues to the Gamma function $\Gamma (z)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt$ (holomorphic for $\mathrm{Re}z>0$), then to all of $\mathbb{C}$ except simple poles at $z=0,-1,-2,\dots$ via $\Gamma (z)=\Gamma (z+1)/z$. This continuation is the backbone of dimensional regularization in QFT (continuing loop integrals in the spacetime dimension $d$).

Monodromy

Continuation along different paths can give different results when the paths enclose a branch point: the value depends on the homotopy class of the path. This monodromy is the multivaluedness of $\log z$ or $\sqrt{z}$ met in holomorphic.md: continuing $\log z$ once around $0$ adds $2\pi i$. The clean way to tame monodromy is to pass to a Riemann surface on which the function is single-valued (riemann-surfaces.md).

Germs, function elements, and natural boundaries

Continuation is made precise by the language of germs. A function element $(f,D)$ is a holomorphic function on a disc $D$; two elements define the same germ at a point if they agree near it. Continuation along a path is a chain of overlapping elements, and the collection of all germs reachable from a starting one — the complete analytic function — is a connected covering space over its domain, the abstract form of a Riemann surface. Locally this is the sheaf of holomorphic functions $\mathcal{O}$; monodromy is the failure of a global section to exist.

Monodromy theorem. If a germ can be continued along every path in a simply connected domain, the continuations agree — the result is a single-valued holomorphic function. Multivaluedness therefore requires a non-trivial loop (a branch point inside it).

Continuation can also simply stop: a natural boundary is a curve past which no extension exists. The lacunary series ${\sum }_n{z}^{2^n}$ is holomorphic on the unit disc but has the entire unit circle as a natural boundary — a dense wall of singularities — showing that "analytic continuation exists" is a genuine hypothesis, not automatic.

The Schwarz reflection principle

If $f$ is holomorphic in the upper half-plane, continuous up to a segment of the real axis, and real there, it continues across by $$f(\bar{z})=\overline{f(z)}.$$ This reflection principle is the analytic statement of a reality condition. In physics it gives the Schwarz reflection of amplitudes, $\mathcal{M}(s^{\ast })=\mathcal{M}(s{)}^{\ast }$, relating an amplitude above and below its branch cut — the input to dispersion relations below.

Dispersion relations

Causality forces a response function to be holomorphic in the upper half-plane (no response before the stimulus). Applying Cauchy's formula with a contour closed in the UHP relates the real and imaginary parts of $\chi (\omega )$ on the real axis — the Kramers–Kronig relations $$\mathrm{Re}\chi (\omega )=\frac{1}{\pi }\mathrm{P}{\int }_{-\infty }^{\infty }\frac{\mathrm{Im}\chi ({\omega }^{\prime })}{{\omega }^{\prime }-\omega }d{\omega }^{\prime },$$ and conversely. The physical content: the dispersive (real) part is fixed by an integral over the absorptive (imaginary) part at all frequencies. In QFT the same analyticity-in-energy of the S-matrix gives dispersion relations for amplitudes, tying the real part to the total cross section via the optical theorem — a rigorous, model-independent consequence of causality, used in observables/README.md.

Wick rotation

The sharpest physics use of continuation is Wick rotation: rotating the time contour $t\to -i\tau$ (equivalently the energy $p^0\to i{p}_E^0$) continues a Minkowski (Lorentzian) amplitude to a Euclidean one. The oscillatory $e^{iS}$ becomes a convergent $e^{-S_E}$, making the functional integral well-defined and identical in form to a statistical partition function. Physical results are recovered by continuing back. The rotation is legitimate precisely because the integrand is holomorphic in the relevant quadrant (the poles, displaced by $i\epsilon$, do not obstruct the contour rotation) — analytic continuation is what guarantees the Euclidean and Lorentzian theories compute the same thing.

References

• Stein & Shakarchi, Complex Analysis, Ch. 6 (Gamma function, continuation).
• Ahlfors, Complex Analysis, Ch. 6–7.
• Weinberg, The Quantum Theory of Fields, Vol. 1, §10.8 (dispersion relations).