5.3 Matrix Distribution from a QFT Density Matrix
93
are the number operators. The total Hamiltonian is then H = H asy. + H int. , while the
total momentum, number and charge operators are given by
P :=
s
d 3 p
(2π) 3
p N
p,s ,
N :=
s
d 3 p
(2π) 3 N
p,s
Q :=
s
d 3 p
(2π) 3 q s N
p,s ,
(5.3.13)
respectively.
Let us now briefly discuss one of the “standard subtleties” of QFT about the
particle content and the operators given above (see [18, 19] for clarifications and more
details on this). To that end, we denote collectively by “λ” the coupling constants
in the QFT action, such that for λ = 0 the theory is a collection of free harmonic
oscillators. The | |
p, s states are the 1-particle states of the interacting theory, i.e.
eigenstates of H and
P that are irreducible representations of the Poincaré group.
Because of this, the | |
p, s set contains fundamental stable particles, i.e. particles
corresponding to some field operator in H (e.g. the electron), but also bound states
(e.g. the hydrogen atom). On the other hand, it does not contain 1-particle states of
the λ = 0 theory which, once λ = 0, become either strongly coupled at the energies
of interest (e.g. quarks) or unstable (e.g. the Higgs or excited bound states). In the
former case, such particles would only appear through bound states (e.g. pions) but,
precisely because the coupling is strong, a perturbative approach is only possible
through some effective action where the bound state is treated as a fundamental
particle. In the case of unstable particles, if they are long-lived ( L coll ) then it
makes sense to consider them as part of the spectrum instead of as a resonance, i.e.
as quasi-eigenstates of H .
Thus, the ladder operators introduced above are generally non-trivial functions of
the ones corresponding to the field operators. These combinations take into account
the virtual particles that “dress” the fundamental particles and sustain the bound
states. In particular, in the case of bound states these are effective ladder operators
obeying approximate canonical (anti-)commutation relations only at scales that do
not resolve the internal structure of the bound system. The “asymptotic” Hamiltonian
H asy. is therefore not the same as the Hamiltonian of the λ = 0 theory, i.e. the “free”
Hamiltonian H 0 , because it needs λ = 0 to have bound particle eigenstates and it also
contains the renormalized masses and couplings. This operator therefore matches H
on 1-particle states H | |
p, s = H asy. | |
p, s = E p,s | |
p, s, but not on the multi-particle
states of Eq. (5.3.7). This only works if one takes superpositions of them which
localize and separate in space the individual particles well enough, so that they
evolve as approximately free, i.e. asymptotic states. Keeping in mind this subtlety,
one can then work with these Fock states in the asymptotic regions.
We now have everything we need to express the phase space distribution in terms
of mesoscopic QFT operators. The expectation value of the s-particle number density
in the asymptotic regions is
f
in,out
s
(
p) := V
−1
N
p,s ρ in,out ≡ V
−1 Tr
ρ in,out N
p,s
,
(5.3.14)
where
93
are the number operators. The total Hamiltonian is then H = H asy. + H int. , while the
total momentum, number and charge operators are given by
P :=
s
d 3 p
(2π) 3
p N
p,s ,
N :=
s
d 3 p
(2π) 3 N
p,s
Q :=
s
d 3 p
(2π) 3 q s N
p,s ,
(5.3.13)
respectively.
Let us now briefly discuss one of the “standard subtleties” of QFT about the
particle content and the operators given above (see [18, 19] for clarifications and more
details on this). To that end, we denote collectively by “λ” the coupling constants
in the QFT action, such that for λ = 0 the theory is a collection of free harmonic
oscillators. The | |
p, s states are the 1-particle states of the interacting theory, i.e.
eigenstates of H and
P that are irreducible representations of the Poincaré group.
Because of this, the | |
p, s set contains fundamental stable particles, i.e. particles
corresponding to some field operator in H (e.g. the electron), but also bound states
(e.g. the hydrogen atom). On the other hand, it does not contain 1-particle states of
the λ = 0 theory which, once λ = 0, become either strongly coupled at the energies
of interest (e.g. quarks) or unstable (e.g. the Higgs or excited bound states). In the
former case, such particles would only appear through bound states (e.g. pions) but,
precisely because the coupling is strong, a perturbative approach is only possible
through some effective action where the bound state is treated as a fundamental
particle. In the case of unstable particles, if they are long-lived ( L coll ) then it
makes sense to consider them as part of the spectrum instead of as a resonance, i.e.
as quasi-eigenstates of H .
Thus, the ladder operators introduced above are generally non-trivial functions of
the ones corresponding to the field operators. These combinations take into account
the virtual particles that “dress” the fundamental particles and sustain the bound
states. In particular, in the case of bound states these are effective ladder operators
obeying approximate canonical (anti-)commutation relations only at scales that do
not resolve the internal structure of the bound system. The “asymptotic” Hamiltonian
H asy. is therefore not the same as the Hamiltonian of the λ = 0 theory, i.e. the “free”
Hamiltonian H 0 , because it needs λ = 0 to have bound particle eigenstates and it also
contains the renormalized masses and couplings. This operator therefore matches H
on 1-particle states H | |
p, s = H asy. | |
p, s = E p,s | |
p, s, but not on the multi-particle
states of Eq. (5.3.7). This only works if one takes superpositions of them which
localize and separate in space the individual particles well enough, so that they
evolve as approximately free, i.e. asymptotic states. Keeping in mind this subtlety,
one can then work with these Fock states in the asymptotic regions.
We now have everything we need to express the phase space distribution in terms
of mesoscopic QFT operators. The expectation value of the s-particle number density
in the asymptotic regions is
f
in,out
s
(
p) := V
−1
N
p,s ρ in,out ≡ V
−1 Tr
ρ in,out N
p,s
,
(5.3.14)
where