Functional Integration on Topological Groups II: Functional Mellin Transforms
Conventional functional/path integrals used in physics most often are deﬁned as inﬁnite-dimensional analogs of Fourier transforms. The inﬁnite-dimensional analogs of Mellin transforms similarly deﬁne a class of functional integrals. The associated functional integrals are useful tools for probing non-commutative function spaces in general and C ∗ -algebras in particular. Functional Mellin transforms can be used to deﬁne functional resolvents, functional complex powers, functional traces, functional logarithms, and functional determinants. Several aspects of these objects are explored and applied to various constructs in mathematical physics. Two are of particular note: 1) Certain fractional power operators (perhaps coincidentally) reproduce the channels and crossing symmetry of tree-level open tachyon string scattering processes, and their algebraic product parallels the gauge-gravity relation of Kawai-Lewellen-Tye. 2) The functional complex power of the generating functional of n -point functions in QFT and its functional logarithm are employed to justify the replica trick.