  This is mainly a lecture note taken by myself following Weinberg's book, but
also contains some corrections to the abuse of mathematical treatment. This
article discusses projective unitary representations of Poincare group on the
single particle space, multi particle space also known as the Fock space,
creation and annilation operators, construction of free quantum fields and the
general relation between spin of state and spin of field. Both massive and
massless cases are considered. CPT is not considered. The first section briefly
reviews the basics of representation theory. This article further points out
some of the wrong treatment of mathematics in the book of Weinberg, and
reformulates them, including: Wigner's classification needs to be pass to the
universal cover via Bargmann's theorem, there is no projective representation
of Poincare group on Fock space in general, the Lorentz transformation of
fields need to be formulated with representations of the universal covers,
Dirac representation is not a linear representation of the Lorentz group. This
article also discusses the physical meaning of the state representation and its
relation with Schrodinger equation, compare its difference with state
representation, and the reason of equations of relativistic quantum mechanics
should be understood as a field equation rather than a wave function equation.
of equations of relativistic quantum mechanics should be understood as a field
equation rather than a wave function equation.
