  The definition of the classical transition state theory (TST) as a t= 0+
limit of the flux-side time correlation function relies on the assumption that
simultaneous measurement of population and flux is a well defined physical
process. However, the noncommutativity of the two measurements in quantum
mechanics makes the extension of such a concept to the quantum regime
impossible. For this reason, quantum TST (QTST) has been generally accepted as
any kind of quantum rate theory reproducing the TST in the classical limit, and
there has been a broad consensus that no unique QTST retaining all the
properties of TST can be defined. Hele and Althorpe (HA) [J. Chem. Phys. 138,
084108 (2013)] recently suggested that a true QTST can be defined as the exact
t=0+ limit of a certain kind of quantum flux-side time correlation function and
that it is equivalent to the ring polymer molecular dynamics (RPMD) TST. This
work seeks to question and clarify certain assumptions underlying these
suggestions and their implications. First, the time correlation function used
by HA as a starting expression is not related to the kinetic rate constant by
virtue of linear response theory, which is the first important step in relating
a t=0+ limit to a physically measurable rate. Second, a theoretical analysis
calls into question a key step in HA's proof which appears not to rely on an
exact quantum mechanical identity. The correction of this makes the true t=0+
limit of HA's QTST different from the RPMD-TST rate expression, but rather
equal to the well-known path integral quantum transition state theory rate
expression for the case of centroid dividing surface. An alternative path
integral quantum rate expression is then formulated based on the linear
response theory. It is shown that the t=0+ limit of the new rate expression
vanishes in the exact quantum limit.
