刘丹阳_QFT笔记之2(20170221)

Arbitrary \omega, what invariant
hamiltonian
canonical quantization
equal time commutation relations of canonical momentum and position in terms of operator a(x)
thus
is the total zero-point energy of all the oscillators per
unit volume, and the total volume is
when q=0
If we integrate in \epsilon over the whole range of k, the value of it is infinite
ultraviolet cutoff up to a \lamda, then
in the real world, the formalism of quantum field theory breaks down at some large
energy scale
choose
the ground state has energy eigenvalue zero
then the original assumption for hamiltonian is Lorentz invariant
some thing wrong with fermions
chapter4
free, spin-zero particles
let us consider adding terms to the hamiltonian that will result in
local, Lorentz invariant interactions
define a nonhermitian field
it and it’s conjugate are lorentz invariant
according the quantum
mechanic, the amplitude
between the initial and final
state is
write the hamiltonian in terms of lorentz invariant function
moreover
thus, the time dependent hamiltonian in terms of
Lorentz invariant, the time ordering must be frame independent, it is that their separation is timelike require
since the hamiltonian is the function of the wave function, so consider the commutation relations between the wave function
for
non-vanish
linearly combine them
for vanish the all commutator, must c hoose |λ| = 1, and commutate relations
the can build a suitable hamiltonian in terms of
start with the creation and annihilation operators a † (k) and a(k) as the fundamental objects has simply led back to the real, commuting, scalar field φ(x) as the fundamental object
there’s complex condition
the allowed choice is always commutators for fields of integer spin, and anticommutators for fields of half-integer spin.
Chapter5
construct initial and final states for scattering experiments
create a state of one particle
the Lorentz-invariant normalization
operator that (in the free theory) creates a particle localized in momentum space near k 1 , and localized in position space near the origin
it’s equivalent to the gauss average near the k1
create the initial and final state
then the amplitude is
and
then the amplitude is
for a+ create the single particle state, choose
to be zero
for a † 1 (±∞) to create a correctly normalized one-particle state in the interacting theory, choose。