A compact review of the topics and references devoted to the path
integral formulations of such non-Hermitian quantum field theories
which may be made Hermitian via an interaction-dependent metric. The
text explains the details and it also offers several illustrative
examples. In particular, the author emphasizes the advantages of
using the Heisenberg representation of the dynamics in which, in a
way revealed by Jakubsk\'{y} [13] (not Jacubsk\'{y} as printed), the
metrics disappears, typically, from the equilibrium thermodynamics.
The paper discusses the extension of such a simplification to the
Schwinger-Dyson approach to field theories, keeping in mind the
Mostafazadeh's critical comment [11] and restricting attention,
typically, to the constructions of the Hermitization transformations
which appear simpler in the path-integral language.

The text is very closely related to the preceding paper by H. F.
Jones (cf. also extended abstract MR2780948).



MR2780949 Rivers, R. J. Path integrals for pseudo-Hermitian
Hamiltonians. Internat. J. Theoret. Phys. 50 (2011), no. 4,
1081--1096. 81S40 (81Qxx)