nLab Mellin transform

Definition

For $\tau \mapsto f(\tau)$ a suitably well-behaved function of a single variable, its Mellin transform is the function given by the integral expression

$$s \mapsto \int_0^\infty \tau^{s-1} f(\tau) \; d\tau \,.$$

Examples

Of the exponential function

For the exponential function $\exp(-(-)A)$ the Mellin transform is proportional to the inverse power $A^{-(-)}$

$$\int_0^\infty \tau^{s-1} \exp(-\tau A) \,d\tau = (s-1)! \, A^{-s} \,.$$

For $s = 1$ this has a complex analougue given by the Fourier transform of the Heaviside distribution (this example).

In perturbative quantum field theory, especially in the computation of propagators such as the Feynman propagator, this formula is known as the Schwinger parameterization for $A^{-s}$, leading to the “worldline formalism”. See below at 1-Loop amplitudes.

Zeta functions

A zeta function/L-function is the analytic continuation of the Mellin transform of the corresponding theta function.

In particular it sends the Jacobi theta function to the (completed) Riemann zeta function:

$$\hat \zeta(s) = \int_0^\infty t^{s-1} \hat \theta(t) \, d t$$

More generally, the Mellin transform appears as a stage in the expression of zeta functions as adelic integrals in Iwasawa-Tate theory.

1-loop vacuum amplitudes

1-loop vacuum amplitudes in quantum field theory are analytically continued Mellin transforms of partition functions. Here the parameter $\tau$ is called the Schwinger parameter and the Mellin transform turns the worldline formalism-picture into the Feynman propagator-picture.

$$Tr H^{-s} = \int_0^\infty t^{s-1} Tr\, \exp(- t H) \, d t \,.$$

Examples

Zeta functions

• zeta function of an elliptic differential operator

• Riemann zeta function

Related concepts

• Laplace transform

• Fourier transform

• the generalization of Mellin transforms from automorphic forms to automorphic representation is the concept of automorphic L-function

References

• Wikipedia, Mellin transform

• Wikipedia, Schwinger parameterzation

• Joel Shapiro, Schwinger trick and Feynman parameter, 2007 (pdf)

For the adelic integral-version see at Iwasawa-Tate theory.

For function fields:

• David Goss, A formal Mellin transform in the arithmetic of function fields, Transactions of the AMS, volume 327, Number 2, October 1991 (pdf)

For the appearance in physics as integrals over Schwinger parameters producing Feynman propagators see

• Joel Shapiro, Schwinger trick and Feynman parameters (pdf)

• Stefan Weinzierl, section 4.2.1 of Mathematical aspects of particle physics, 2010 (pdf)