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Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture X
The main goal of this lecture is to explain how, using the interaction representation, a systematic perturbation theory in the interaction strength can be developed for the Green’s function. We discussed connections of the one- and two-particle Green’s functions to experimental quantities in the previous lecture.
First we state the result, derived below: we can express the desired Heisenberg representation Green’s function in terms of the interaction representation operators: for t 1 > t 2 ,
Gαβ (t 1 , r 1 , t 2 , r 2 ) = −i〈Ψα(t 1 , r 1 )Ψ† β (t 2 , r 2 )〉 = −i〈S−^1 (t 1 , −∞)Ψ 0 α(t 1 , r 1 )S(t 1 , −∞)S−^1 (t 2 , −∞)Ψ† 0 β (t 2 , r 2 )S(t 2 , −∞)〉(1)
Here the unitary S operator is defined as (as before ¯h = 1)
S(t 1 , t 2 ) = T exp(−i
∫ (^) t 1
t 2
V 0 (t) dt), (2)
where V 0 (t) is the interaction representation of the interaction part V of the Hamiltonian H = H 0 + V. The purpose of S is to connect the interaction and Heisenberg representations: for a general operator Ψ in Heisenberg representation,
Ψ = S−^1 (t, −∞)Ψ 0 S(t, ∞). (3)
Let us quickly review where the above expressions come from. The fundamental definition of the interaction representation is that the operators evolve according to the unperturbed Hamiltonian: using the quantum field operator as an example,
Ψ 0 (t, r) = eiH^0 tψ(r)e−iH^0 t^ (4)
How do wavefunctions then transform? Using φ to denote the Schrodinger wavefunction, we know that φ evolves according to
i ∂φ ∂t
= (H 0 + V )φ. (5)
Since in the interaction representation, the H 0 part of the above time dependence was transferred to the operators, we might expect that in the interaction representation the wave function Φ 0 will evolve according only to V. Explicitly,
i
∂t = V 0 (t)Φ 0. (6)
Here V 0 is the interaction representation of V : V 0 (t) = eiH^0 tV e−iH^0 t. To see that this is correct, use the interaction representation expression Φ 0 (t) = eiH^0 tφ(t), as required for expectation values to be constant. Then
i
∂t
= i(iH 0 Φ 0 ) + ieiH^0 t^ ∂φ ∂t
= i(iH 0 Φ 0 ) + ieiH^0 t(−i(H 0 + V )φ) = −H 0 Φ 0 + eiH^0 t(H 0 + V )φ = eiH^0 tV φ = V 0 (t)eiH^0 tφ = V 0 (t)Φ 0 (t). (7)
The above can be written as
Φ 0 (t + dt) = (1 − i dt V 0 (t))Φ 0 (t) ≈ e−iV^0 (t)^ dt. (8)
So, applying this relation many times over a finite interval t 2 − t 1 divided into many small dt intervals, we obtain Φ 0 (t 2 ) = S(t 2 , t 1 )Φ 0 (t 1 ) (9)
with
S(t 2 , t 1 ) =
t∏=t 2
t=t 1
e−iV^0 (t)^ dt. (10)
Here the proper definition, as seen before in the Feynman path integral, is that we divide the interval from t 1 to t 2 into N subintervals and then take the limit N → ∞ We must be somewhat careful about combining these factors because H 0 and V may not commute. The solution is to introduce the time-ordered exponential,
S(t 2 , t 1 ) = T exp
( −i
∫ (^) t 2
t 1
V 0 (t) dt
) , (11)
where the definition of the time-ordering operator T is, as before, that the operators appearing when the Hamiltonian is expanded appear with earliest times to the right. Note that H and V 0 should be bosonic so that no fermionic exchanges, with associated minus signs, are needed in making this rearrangement.
Properties of S: clearly it is unitary so S−^1 = S†, and has the property
S(t 3 , t 1 ) = S(t 3 , t 2 )S(t 2 , t 1 ). (12)
Now equation (1) follows if we assume that at t = −∞ the Schrodinger and interaction representa- tions coincide, which is justified if we imagine that the perturbation Hamiltonian is adiabatically “turned on” at some time after −∞. So, again for t 1 > t 2 ,
Gαβ (t 1 , r 1 , t 2 , r 2 ) = −i〈S−^1 (t 1 , −∞)Ψ 0 α(t 1 , r 1 )S(t 1 , −∞)S−^1 (t 2 , −∞)Ψ† 0 β (t 2 , r 2 )S(t 2 , −∞)〉 (13)
We can combine the two interior S factors and rewrite the first to get
Gαβ (t 1 , r 1 , t 2 , r 2 ) = −i〈S−^1 (∞, −∞)S(∞, t 1 )Ψ 0 α(t 1 , r 1 )S(t 1 , t 2 )Ψ† 0 β (t 2 , r 2 )S(t 2 , −∞)〉 (14)
The advantage of this expression is that now the factors (except for the first term) are in proper chronological order. Writing S for S(∞, −∞), we have
Gαβ (t 1 , r 1 , t 2 , r 2 ) = −i〈S−^1 T {Ψ 0 α(t 1 , r 1 )Ψ† 0 β (t 2 , r 2 )S}〉. (15)
The above expression also holds for t 1 < t 2 , if we recall the sign convention in T. Under certain assumptions the factor S−^1 will just contribute an overall phase: if the ground state is nondegenerate, then adiabatic switching on and off of the perturbation Hamiltonian will leave the system in its ground state, and the factor S−^1 just becomes the exponential of the phase shift resulting from the energy change of the ground state. (Many of the minor complications that occur when the Green’s function formalism we are developing is generalized to either finite temperature or nonequilibrium involve this factor.) With the above assumptions,
Gαβ (t 1 , r 1 , t 2 , r 2 ) = −i
〈T {Ψ 0 α(t 1 , r 1 )Ψ† 0 β (t 2 , r 2 )S}〉 〈S〉
These four terms can be simplified into two pairs by interchanging the names of variables 3 and 4 in the integration. So we are left with
iG^112 =
∫ d^3 r 3 dt 3 d^3 r 4 dt 4 U (r 3 − r 4 )
[ inG^014 G^042 − G^013 G^034 G^042
]
. (23)
These two terms correspond to “Hartree” and “exchange” terms (exercise). They are tradition- ally diagramatically represented using a dotted line for U and solid lines for the G^0. These lines come together at “vertices”: for the above perturbation theory, each vertex joins a dotted line and two straight lines, one incoming and one outgoing.
Normally one works in momentum space for actual calculations. Also, as mentioned before the above formalism can be extended with a bit of work to calculate averages in other states than the ground state | 0 〉. Both finite-temperature calculations and even general nonequilibrium calculations (beyond linear response) are possible, but often the technical complexity in fully nonequilibrium problems is overwhelming. The number of diagrams increases rapidly with the desired order of accuracy: there are 10 diagrams at second order in the above perturbation theory. Many diagram- matic approximations are based on selecting out a particular subset of diagrams and finding some resummation trick.
In addition to doing perturbation theory in interaction strength, diagrammatic techniques are also very important for noninteracting or interacting particles in a random potential. We will use other techniques when we discuss such random problems later, but you should be aware that dia- grammatic perturbation theory has probably been the most important method for such problems. In particular, there is a famous “supersymmetry” technique for such problems developed by Efetov and others in the 1980s (cf. textbook of Efetov).
