Jones, H. F.; Mateo, J.; Rivers, R. J.

========================================

Path-integral derivation of the anomaly for the Hermitian
equivalent of
  the complex $PT$-symmetric quartic Hamiltonian.

========================================

 Phys. Rev. D (3) 74 (2006), no. 12, 125022, 6 pp. 81Q05 (81S40)

========================================

Exciting reading for many fresh fans of new developments in the so
called PT-symmetric Quantum Theory where the coordinate
(typically, $x$ in Schroedinger equation $(p^2+V(x))\psi(x) =
E\psi(x)$) is allowed to become complex. Exciting for several
reasons:

(1) The authors pay attention to the really exceptional model of
this class, with the apparently repulsive $V(x) \sim -x^4$ which
ceases to be repulsive inside unusual, ``physical" complex wedges
of $x$ (cf. the paper [6] by Buslaev and Grecchi, with several
later rediscoveries).

(2) In a way extending the work of some of their predecessors, the
authors employ the Feynman's path integral approach where they are
able to get an agreement with expectations.

(3) On this occasion, an amazing number of underlying technical
difficulties is given a compact explanation.

(4) A hope is revitalized that one could sucessfully continue and
try to aply the method in more dimensions, up to the case of field
theory where a similar project is particularly ambitious (cf. a
comment on alternatives to the Higgs mechanism in [7]).

(5) Although the quantity of interest is here the partition
function rather than Green functions, the gap has already
partially been filled (cf. H. F. Jones, R. J. Rivers, The
disappearing $Q$ operator, Phys.Rev. D 75 (2007) 025023).