  In this article an introduction to the thermal field theory within imaginary
time has been discussed in details. The imaginary time formalism has been
introduced through both the operatorial and the functional integration method.
The prescription to perform frequency sum for boson and fermion has been
discussed. Green's function both in Minkowski time as well as in Euclidean time
has been derived. The tadpole diagram in $\phi^4$ theory and the self-energy in
$\phi^3$ theory have been computed. The basic features of general two point
functions for both fermions and bosons in presence of a heat bath have been
discussed. Then the free partition functions for scalar, fermion and gauge
field, and interacting scalar field have been obtained. The quantum
electrodynamics (QED) and gauge fixing have been discussed in details. The
one-loop self-energy for electron and photon in QED have been obtained in hard
thermal loop (HTL) approximation. The dispersion properties and collective
excitations of both electron and photon in a thermal medium have been
presented. The spectral representation of fermion and gauge boson propagators
have been obtained. In HTL approximation, the generalisation of QED results of
two point functions to quantum chromodynamics (QCD) have been outlined that
mostly involve group theoretical factors. As an effective field theory approach
the HTL resummation and the HTL perturbation theory have been introduced. The
leading order, next-to-leading order and next-to-next-leading order free energy
and pressure for deconfined QCD medium created in heavy-ion collisions have
been computed within HTLpt. The general features of the deconfined QCD medium
have also been outlined with non-perturbative effects like gluon condensate and
Gribov-Zwanziger action. The dilepton production rates from quark-gluon plasma
with these non-perturbative effects have been computed.
