and the hermitian property, Eq. (C.14), leads to the relationship

GA = [GR — ]† .

ˆ ˆ (D.6)

E E

We introduce the Green™s operator

§R

ˆ

⎨ GE for mE > 0

ˆ

GE ≡ (D.7)

© ˆA

GE for mE < 0

for which we have the spectral representation

| » »|

1

ˆ

GE = = (D.8)

E’ »

ˆ

E’H »

in terms of the eigenstates, | , of the Hamiltonian

»

ˆ

H| |

= . (D.9)

» » »

The analytical properties of the retarded and advanced Green™s operators leads,

by an application of Cauchy™s theorem, to the spectral representations

∞ ˆ

dE AE

ˆ R(A) =

GE (D.10)

’∞ 2π E ’ E (’) i0

+

where we have introduced the spectral operator, the discontinuity of the Green™s

operator across the real axis

ˆ ˆE ˆE ˆ ˆ

≡ i(GR ’ GA ) = i(GE+i0 ’ GE’i0 )

AE

ˆ

= 2π δ(E ’ H) = 2π | » | δ(E ’ ») . (D.11)

»

»

Equivalently, we have the relationship between real and imaginary parts of, say,

position representation matrix elements

∞

m GR (x, x , E )

dE

e G (x, x , E) = P

R

(D.12)

E ’E

’∞ π

and

∞

e GR (x, x , E )

dE

m G (x, x , E) = ’P

R

. (D.13)

E ’E

’∞ π

The Kramers“Kronig relations due to the retarded propagator is analytic in the upper

half-plane.

The perturbation expansion of the propagator in a static potential is seen to be

equivalent to the operator expansion for the Green™s operator

1 1 1

ˆ

GE = = =

ˆ ˆ ˆ ˆ ˆ ˆ

(E ’ H0 )(1 ’ (E ’ H0 )’1 V )

E’H E ’ H0 + V

Appendix D. Analytic properties of Green™s functions 519

1 1

=

ˆ ˆ ˆ

1 ’ (E ’ H0 )’1 V E ’ H0

1

1 + (E ’ H0 )’1 V + (E ’ H0 )’1 V (E ’ H0 )’1 V + ...

ˆ ˆ ˆ ˆ ˆ ˆ

=

ˆ

E ’ H0

ˆ ˆ ˆˆ ˆ ˆˆ ˆˆ

= G0 (E) + G0 (E)V G0 (E) + G0 (E)V G0 (E)V G0 (E) + ... , (D.14)

where

1

ˆ

G0 (E) = (D.15)

ˆ

E ’ H0

is the free Green™s operator.

The momentum representation of the retarded (advanced) propagator or Green™s

function in the energy variable can be expressed as the matrix element

R(A)

ˆ |p

GR(A) (p, p , E) = p| GE (D.16)

of the retarded (advanced) Green™s operator

1

i0)’1

ˆ R(A) = ˆ

≡ (E ’ H +

GE (D.17)

(’)

ˆ (’) i0

E’H +

the analytical continuation from the various half-planes of the Green™s operator.

Other representations are obtained similarly, for example,

ˆ R(A)

x| GE |x .

GR(A) (x, x , E) = (D.18)

The hermitian property Eq. (D.6) gives the relationship

[GR (x, x , E)]— = GA (x , x, E — ) (D.19)

and similarly in other representations.

ˆ

Employing the resolution of the identity in terms of the eigenstates of H

ˆ | »|

I= (D.20)

»

»

we get the spectral representation in, for example, the position representation

—

ψ» (x)ψ» (x )

R(A)

G (x, x , E) = . (D.21)

E ’ » (’) i0

+

»

The Green™s functions thus have singularities at the energy eigenvalues (the energy

spectrum), constituting a branch cut for the continuum part of the spectrum, and

simple poles for the discrete part, the latter corresponding to states which are nor-

malizable (possible bound states of the system).

520 Appendix D. Analytic properties of Green™s functions

Along a branch cut the spectral function measures the discontinuity in the Green™s

operator

ˆ ˆ

≡ x|i(GE+i0 ’ GE’i0 )|x

A(x, x , E)

= i GR (x, x , E) ’ GA (x, x , E)

= ’2 mGR (x, x , E)

—

ψ» (x)ψ» (x ) δ(E ’

= 2π ») . (D.22)

»

ˆ

From the expression (P (x) = |x x|)

ˆ ˆ

A(x, x, E) = 2π T r(P (x)δ(E ’ H)) = 2π | x| |2 δ(E ’ ») (D.23)

»

»

we note that the diagonal elements of the spectral function, A(x, x, E), is the local

density of states per unit volume: the unnormalized probability per unit energy

for the event to ¬nd the particle at position x with energy E (or vice versa, the

probability density for the particle in energy state E to be found at position x).

Employing the resolution of the identity we have

δ(E ’ ≡ 2πN (E) ,

dx A(x, x, E) = 2π ») (D.24)

»

where N (E) is seen to be the number of energy levels per unit energy, and Eq. (D.24)

is thus the statement that the relative probability of ¬nding the particle somewhere

in space with energy E is proportional to the number of states available at that

energy.

We also note the completeness relation

dE

A(x, x , E) = δ(x ’ x ) (D.25)

2π

σ

where the integration (and summation over discrete part) is over the energy spectrum.

The position and momentum representation matrix elements of any operator are

related by Fourier transformation. For the spectral operator we have (assuming the

system enclosed in a box of volume V )

x|p A(p, p , E) p |x

A(x, x , E) =

pp

1 p·x’ i p ·x

i

= e A(p, p , E) , (D.26)

V

pp

and inversely we have

= N ’1 dx dx e’ p·x+ i p ·x

i

ˆ

p|AE |p

A(p, p , E) = A(x, x , E) , (D.27)

Appendix D. Analytic properties of Green™s functions 521

where the normalization depends on whether the particle is con¬ned or not, N =

V, (2π )d .

ˆ

For the diagonal momentum components of the spectral function we have (P (p) =

|p p|)

ˆ ˆ

A(p, p, E) = 2π T r(P (p) δ(E ’ H)) = 2π | p| |2 δ(E ’ ») (D.28)

»

»

describing the unnormalized probability for a particle with momentum p to have

energy E (or vice versa). Analogously to the position representation we obtain

A(p, p, E) = 2π N (E) . (D.29)

p

We have the momentum normalization condition

§

⎨ δ(p ’ p )

dE

A(p, p , E) = (D.30)

©

2π

δp,p

σ

depending on whether the particle is con¬ned or not.

Let us ¬nally discuss the analytical properties of the free propagator. Fourier

transforming the free retarded propagator, Eq. (C.22), we get (in three spatial di-

mensions for the pre-exponential factor to be correct), mE > 0,

√

’m e pE |x’x |

i

R

G0 (x, x , E) = , pE = 2mE (D.31)

2π 2 |x ’ x |

the solution of the spatial representation of the operator equation, Eq. (D.3),

2

E’ G0 (x, x , E) = δ(x ’ x ) , (D.32)

x

2m

which is analytic in the upper half plane.

√

The square root function, E, has a half line branch cut, which according to

the spectral representation, Eq. (D.21), must be chosen along the positive real axis,

the energy spectrum of a free particle, as we choose the lowest energy eigenvalue to

have the value zero. In order for the Green™s function to remain bounded for in¬nite

separation of its spatial arguments, |x ’ x | ’ ∞, we must make the following choice

of argument function

§√

⎨E for eE > 0

√

E≡ . (D.33)

©

|E| for

i eE < 0

rendering the free spectral function of the form

m sin( 1 pE |x ’ x |)

A0 (x, x , E) = θ(E) (D.34)

|x ’ x |

π2

522 Appendix D. Analytic properties of Green™s functions

and we can read o¬ the free particle density of states, the number of energy levels

per unit energy per unit volume,1

§ m

⎪ d=1

⎪ 2π 2 2 E

⎪

⎪

⎨

1 m

N0 (E) ≡ d=2 ,

A0 (x, x; E) = θ(E) (D.35)

2π 2

⎪

2π ⎪

⎪

⎪

© m√2mE

d=3

2π 2 3

where for completeness we have also listed the one- and two-dimensional cases.

The spectral function for a free particle in the momentum representation follows,

for example, from Eq. (D.28)

A0 (p, E) ≡ A0 (p, p, E) = 2π δ(E ’ p) , (D.36)

and describes the result that a free particle with momentum p with certainty has

energy E = p , or vice versa.

1 Thisresult is of course directly obtained by simple counting of the momentum states in a given

energy range, because for a free particle constrained to the volume Ld , there is one momentum

state per momentum volume (2π /L)d . However, the above argument makes no reference to a

¬nite volume.

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