Although the Feynman's path-integral formulation of quantum
mechanics looks rather artificial, one of its key merits has been
soon recognized as lying in an immediate transfer and extension of
this approach to thermodynamics and, first of all, to
(relativistic) quantum field theory. In the latter context, this
formalism is virtually indispensable, e.g., for a practical and
feasible construction of the n-point functions of the theory via
the so called generating functionals Z[J]. Naturally, the
underlying computation methods are usually tested on the
quantum-mechanical or thermodynamical simpler versions of Z[J]. In
these tests, an interesting observation of an apparent
metric-independence of measurable quantities has recently been
reported by Jones and Rivers (cf. ref. [6]) and by Jakubsky (cf.
ref. [7]), respectively. Mostafazadeh makes a comment on these
results. In particular, he shows that for the specific class of
noninteracting models as considered by Jakubsky, the path-integral
formalism offers a very natural explanation of his observation of
the metric-independence of the thermodynamical quantities. Within
the former, more ambitious project of ref. [6] (cf. also [14]),
one has to be more careful. In particular, the functionals Z[J]
must be redefined in a self-consistent manner which leads, via the
corrected source term, to a return of the explicit
metric-dependence in Z[J]. In practice, this will imply a
reemergence of fairly serious practical computational difficulties
accompanying the use of the corrected formulae for the field's
n-point functions.


MR2358121 Mostafazadeh, A. Path-integral formulation of
pseudo-Hermitian quantum mechanics and the role of the metric
operator. Phys. Rev. D 76 (2007), no. 6, 067701, 4 pp. 81Q60
(81S40)