3.4 Free One-Particle Green’s Function
41
one pole, located in the lower or upper complex ω-plane depending on whether p
is an occupied (n p = 1) or unoccupied (n p = 0) orbital in the ground-state Slater
determinant. The ionization energies and electron affinities are simply given by − p ,
that is, the negative orbital energies.
For the diagonal components G
0
pp (t, t
), the EOM assumes the simple form
i
∂
∂t
G
0
pp (t, t
) − ε p G
0
pp (t, t
) = δ(t − t
)
(3.55)
As solutions of the latter inhomogeneous differential equation, the functions
G
0
pp (t, t
) are denoted as (mathematical) Green’s functions, here for the differential
operator
i
∂
∂t
− p
. It is this proximity to mathematical Green’s functions which has
led to the designation “many-body Green’s functions” in the present context. Note
that Eq. (3.55) has two distinct solutions, differing with respect to their “causal”
behavior. The comparison with the definition (3.52) shows that either the “retarded”
solution, ∼θ(t − t
), or the “advanced” solution, ∼θ(t
− t), is adopted depending
on whether n p = 1 or n p = 1, respectively.
Exercises
3.1 Spin symmetry of the electron propagator: Write the spin-orbital indices in their
expanded form, p → pγ, and show that G pα,qα = G pβ,qβ and G pα,qβ = 0, supposing a non-degenerate ground state | 0 .
3.2 Revisit the 2E-2O model of Exercise 2.4 and
(a) Determine the ionization potentials and electron affinities;
(b) Evaluate explicitly the spectral representation (3.17) for G gα,gα (ω).
References
1. Migdal AB (1967) Theory of finite Fermi systems. Wiley-Interscience, New York
2. Abrikosov AA, Gorkov LP, Dzyaloshinski IE (1963) Methods of quantum field theory in statistical physics. Prentice-Hall, Englewood Cliffs
3. Thouless DJ (1961) The quantum mechanics of many-body systems. Academic Press, New York
41
one pole, located in the lower or upper complex ω-plane depending on whether p
is an occupied (n p = 1) or unoccupied (n p = 0) orbital in the ground-state Slater
determinant. The ionization energies and electron affinities are simply given by − p ,
that is, the negative orbital energies.
For the diagonal components G
0
pp (t, t
), the EOM assumes the simple form
i
∂
∂t
G
0
pp (t, t
) − ε p G
0
pp (t, t
) = δ(t − t
)
(3.55)
As solutions of the latter inhomogeneous differential equation, the functions
G
0
pp (t, t
) are denoted as (mathematical) Green’s functions, here for the differential
operator
i
∂
∂t
− p
. It is this proximity to mathematical Green’s functions which has
led to the designation “many-body Green’s functions” in the present context. Note
that Eq. (3.55) has two distinct solutions, differing with respect to their “causal”
behavior. The comparison with the definition (3.52) shows that either the “retarded”
solution, ∼θ(t − t
), or the “advanced” solution, ∼θ(t
− t), is adopted depending
on whether n p = 1 or n p = 1, respectively.
Exercises
3.1 Spin symmetry of the electron propagator: Write the spin-orbital indices in their
expanded form, p → pγ, and show that G pα,qα = G pβ,qβ and G pα,qβ = 0, supposing a non-degenerate ground state | 0 .
3.2 Revisit the 2E-2O model of Exercise 2.4 and
(a) Determine the ionization potentials and electron affinities;
(b) Evaluate explicitly the spectral representation (3.17) for G gα,gα (ω).
References
1. Migdal AB (1967) Theory of finite Fermi systems. Wiley-Interscience, New York
2. Abrikosov AA, Gorkov LP, Dzyaloshinski IE (1963) Methods of quantum field theory in statistical physics. Prentice-Hall, Englewood Cliffs
3. Thouless DJ (1961) The quantum mechanics of many-body systems. Academic Press, New York