G(t) = θ(t) e’γt sin(„¦t) = e’iωt . (5.85)

„¦2 ’ (ω + iγ)2

„¦ 2π

’∞

It perhaps surprising that this integral is identically zero if t < 0, and non-

zero if t > 0. This is one of the places where contour integral methods might

cast some light, but as long as we have con¬dence in the Fourier inversion

formula, we know that it must be correct.

We now observe that reversing the sign of γ on the right hand side of

(5.85) does more than just change e’γt ’ eγt on the left hand side. Instead

1 dω 1

∞

e’iωt = ’θ(’t) eγt sin(„¦t). (5.86)

„¦2 ’ (ω ’ iγ)2 2π „¦

’∞

This is obtained from (5.85) by noting that changing γ ’ ’γ in the denom-

inator integral is equivalent to complex conjugation followed by a change of

sign t ’ ’t. The result is an exponentially growing oscillation which is

suddenly silenced at t = 0.

132 CHAPTER 5. GREEN FUNCTIONS

t t

t=0 t=0

ιγ= +ιµ ιγ=’ιµ

The e¬ect on G(t), the Green function of an undamped oscillator, of changing

iγ from +iµ to ’iµ.

The e¬ect of taking the damping parameter γ from an in¬tesimally small pos-

tive value µ to an in¬nitesimally small negative value ’µ is therefore to turn

the causal Green function (no motion before the delta-function kick) of the

undamped oscillator into an anti-causal Green function (no motion after the

kick). Ultimately, this is because the the di¬erential operator corresponding

to a harmonic oscillator with initial -value data is not self-adjoint, and the

adjoint operator corresponds to a harmonic oscillator with ¬nal -value data.

This discontinuous dependence on an in¬nitesimal damping parameter is

the subject of the next few sections.

Physics Application: Caldeira-Leggett in Frequency Space

If we write the Caldeira-Leggett equations of motion (5.34) in Fourier fre-

quency space by setting

dω

∞

Q(ω)e’iωt ,

Q(t) = (5.87)

2π

’∞

and

dω

∞

qi (ω)e’iωt ,

qi (t) = (5.88)

2π

’∞

we have (after including an external force Fext to drive the system)

’ω 2 + („¦2 ’ ∆„¦2 ) Q(ω) ’ fi qi (ω) = Fext (ω),

i

(’ω 2 + ωi )qi (ω) + fi Q(ω) = 0.

2

(5.89)

5.5. ANALYTIC PROPERTIES OF GREEN FUNCTIONS 133

Eliminating the qi , we obtain

fi2

2 2 2

’ω + („¦ ’ ∆„¦ ) Q(ω) ’ Q(ω) = Fext (ω). (5.90)

2

ωi ’ ω 2

i

As before, sums over the index i are replaced by integrals over the spectral

function

fi2 2 ∞ ω J(ω )

’ dω , (5.91)

π 0 ω 2 ’ ω2

2

ωi ’ ω 2

i

and

fi2 2 J(ω )

∞

2

∆„¦ ≡ ’ dω . (5.92)

2

ωi π ω

0

i

Then

1

Q(ω) = Fext (ω), (5.93)

„¦2 ’ ω 2 + Π(ω)

where the self-energy Π(ω) is given by

2 J(ω ) ω J(ω ) 2 J(ω )

∞ ∞

dω = ’ω 2

’2

Π(ω) = dω .

ω (ω 2 ’ ω 2)

ω ’ ω2

π ω π

0 0

(5.94)

The expression

1

G(ω) ≡ (5.95)

„¦2 ’ ω 2 + Π(ω)

a typical response function. Analogous objects occur in all branches of

physics.

For viscous damping we know that J(ω) = ·ω. Let us evaluate the

integral occuring in Π(ω) for this case:

dω

∞

I(ω) = . (5.96)

ω 2 ’ ω2

0

We will assume that ω is positive. Now,

1 1 1 1

’

= , (5.97)

ω 2 ’ ω2 ω ’ω ω +ω

2ω

so ∞

1

ln(ω ’ ω) ’ ln(ω + ω)

I= . (5.98)

2ω ω =0

134 CHAPTER 5. GREEN FUNCTIONS

At the upper limit we have ln (∞ ’ ω)/(∞ + ω) = ln 1 = 0. The lower

limit contributes

1

’ ln(’ω) ’ ln(ω) . (5.99)