Resummation

In mathematics and theoretical physics, resummation is a procedure to obtain a finite result from a divergent sum (series) of functions. Resummation involves a definition of another (convergent) function in which the individual terms defining the original function are re-scaled, and an integral transformation of this new function to obtain the original function. Borel resummation is probably the most well-known example. The simplest method is an extension of a variational approach to higher order based on a paper by R.P. Feynman and H. Kleinert.^[1] In quantum mechanics it was extended to any order here,^[2] and in quantum field theory here.^[3] See also Chapters 16–20 in the textbook cited below.

References

1. Feynman R.P., Kleinert H. (1986). "Effective classical partition functions" (PDF). Physical Review A. 34 (6): 5080–5084. Bibcode:1986PhRvA..34.5080F. doi:10.1103/PhysRevA.34.5080. PMID 9897894.
2. Janke W., Kleinert H. (1995). "Convergent Strong-Coupling Expansions from Divergent Weak-Coupling Perturbation Theory" (PDF). Physical Review Letters. 75 (6): 287. arXiv:quant-ph/9502019. Bibcode:1995PhRvL..75.2787J. doi:10.1103/physrevlett.75.2787.
3. Kleinert, H., "Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions". Physical Review D 60, 085001 (1999)

Books

• Hagen Kleinert, Critical Properties of φ⁴-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4658-7 (also available online) (together with V. Schulte-Frohlinde).