Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 2

Invertible field transformations with derivatives: necessary and sufficient conditions

Pages: 309 – 325



Eugeny Babichev (Laboratoire de Physique Théorique, CNRS Université Paris-Sud, Université Paris-Saclay, Orsay, France)

Keisuke Izumi (Kobayashi–Maskawa Institute, and Department of Mathematics, Nagoya University, Nagoya, Japan)

Norihiro Tanahashi (Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan)

Masahide Yamaguchi (Department of Physics, Tokyo Institute of Technology, Meguro-ku, Tokyo, Japan)


We formulate explicitly the necessary and sufficient conditions for the local invertibility of a field transformation involving derivative terms. Our approach is to apply the method of characteristics of differential equations, by treating such a transformation as differential equations that give new variables in terms of original ones. The obtained results generalise the well-known and widely used inverse function theorem. Taking into account that field transformations are ubiquitous in modern physics and mathematics, our criteria for invertibility will find many useful applications.

E.B. acknowledges support from the research programs “Projet 80|PRIME CNRS” and PRC CNRS/RFBR (2018–2020) no. 1985.

This work is partially supported by the JSPS KAKENHI Grant Numbers JP17K1428 (K.I.), JP17H0109 (K.I.), JP18K03623 (N.T.), JP18K18764 (M.Y.), by the MEXT Grant-in-Aid for Scientific Research on Innovative Areas Nos. 15H05888, 18H04579 (M.Y.), and by the Mitsubishi Foundation (M.Y.)

K.I. is supported by Japan-Korea Bilateral Joint Research Projects (JSPS-NRF collaboration) String Axion Cosmology.

M.Y. is supported by JSPS and NRF under the Japan-Korea Basic Scientific Cooperation Program and would like to thank the participants attending the JSPS and NRF conference for useful comments.

Published 17 February 2022