  We discuss the quantum statistical fluctuations of energy in subsystems of
hot relativistic gas for both spin-zero and spin half particles. We explicitly
show the system size dependence of the quantum statistical fluctuation of
energy. Our results show that with decreasing system size quantum statistical
fluctuations increase substantially. As the consistency of the framework, we
also argue that the quantum statistical fluctuations give rise to the known
result for statistical fluctuation of energy in the canonical ensemble if we
consider the size of the subsystem to be sufficiently large. For a spin-half
particle quantum fluctuations show some interesting novel features. We show
that within a small sub-system quantum statistical fluctuation of energy for
spin half particles depends on the various pseudo-gauge choices of the
energy-momentum tensor. Interestingly, for sufficiently large subsystems
quantum fluctuations obtained for different pseudo-gauge choices converge and
we recover the canonical-ensemble formula known for statistical fluctuations of
energy. Our calculation is very general and can be applied to any branch of
physics whenever one deals with a thermal system. As a practical application,
we argue that our results can be used to determine a coarse-graining scale to
introduce the concept of classical energy density or fluid element relevant for
the strongly interacting matter, in particular for small systems produced in
heavy-ion collisions.
