Using the second quantization procedure, we have so far only treated energy eigen- states with a trivial time dependence e−iω t, instant processes at a single time, and systems where interactions are approximated by time-independent mean-field theory.
But how does one then treat the general case of time dependence in second quanti- zation? That question will be addressed in this chapter, where the formal theory of time evolution is discussed using three representations, or “pictures:” the Schr¨odinger picture, the Heisenberg picture, and the interaction picture.
We also discuss some applications towards the end of the chapter, namely the theory of scattering, which is needed in later chapters and, furthermore, we explain Fourier transformations of retarded correlation functions, which are used extensively in this book.
5.1 The Schr¨odinger picture
The Schr¨odinger picture is useful when dealing with a time-independent Hamiltonian H, i.e., ∂tH = 0. Any other operator Amay or may not depend on time. The state vectors|ψ(t)ido depend on time, and their time evolution is governed by Schr¨odinger’s equation. The time-independence ofH leads to a simple formal solution:
i~∂t|ψ(t)i=H|ψ(t)i ⇒ |ψ(t)i=e−~iHt|ψ0i. (5.1) In the following we will measure the energy in units of frequency, such that~ drops out of the time-evolution equations: ε/~ → ε and H/~ → H. At the end of the calculations one can easily convert frequencies back to energies. With this notation we can summarize the Schr¨odinger picture with its states |ψ(t)iand operatorsAas:
The Schr¨odinger picture
States: |ψ(t)i=e−iHt|ψ0i.
Operators:A, may or may not depend on time.
H, does not depend on time.
(5.2)
To interpret the operatore−iHt we recall that a functionf(B) of any operatorB is defined by the Taylor expansion off,
f(B) = X∞ n=0
f(n)(0)
n! Bn. (5.3)
While the Schr¨odinger picture is quite useful for time-independent operatorsA, it may sometimes be preferable to collect all time dependencies in the operators and work with time-independent state vectors. We can do that using the Heisenberg picture.
80
THE HEISENBERG PICTURE 81
5.2 The Heisenberg picture
The central idea behind the Heisenberg picture is to obtain a representation where all the time dependence is transferred to the operators,A(t), leaving the state vectors|ψ0i time independent. The HamiltonianH remains time independent in the Heisenberg picture. If the matrix elements of any operator between any two states are identical in the two representations, then the two representations are fully equivalent. By using Eq. (5.2) we obtain the identity
hψ0(t)|A|ψ(t)i=hψ00|eiHtAe−iHt|ψ0i ≡ hψ00|A(t)|ψ0i. (5.4) Thus we see that the correspondence between the Heisenberg picture with time- independent state vectors|ψ0i, but time-dependent operatorsA(t), and the Schr¨odinger picture is given by the unitary transformation operator exp(iHt),
The Heisenberg picture
States: |ψ0i ≡eiHt|ψ(t)i. Operators:A(t)≡eiHtA e−iHt.
H does not depend on time.
(5.5)
As before, the original operatorAmay be time dependent. The important equation of motion governing the time evolution ofA(t) is easily established. SinceH is time independent, the total time derivative ofAin the Heisenberg picture is denoted by a dot, as in ˙A, while the explicit time derivative of the original Schr¨odinger operator is denoted as∂tA:
A(t) =˙ eiHt
iHA−iAH+∂tA
e−iHt⇒A(t) =˙ ih
H, A(t)i
+ (∂tA)(t), (5.6) whereX(t) always meanseiHtXe−iHt for any symbolX, in particular forX =∂tA.
In this way, an explicit time dependence ofAis taken into account. Note how carefully the order of the operators is kept during the calculation.
Both the Schr¨odinger and the Heisenberg picture require a time-independent Hamiltonian. In the general case of time-dependent Hamiltonians, we have to switch to the interaction picture.
5.3 The interaction picture
The third and last representation, the interaction picture, is introduced to deal with the situation where a system described by a time-independent HamiltonianH0, with known energy eigenstates|n0i, is perturbed by some, possibly time-dependent, inter- actionV(t),
H =H0+V(t), withH0|n0i=εn0|n0i. (5.7) The key idea behind the interaction picture is to separate the trivial time evolution due toH0 from the intricate one due toV(t). This is obtained by using onlyH0, not the full H, in the unitary transformation Eq. (5.5). As a result, in the interaction
82 TIME DEPENDENCE IN QUANTUM THEORY
picture both the state vectors |ψ(t)ˆ i and the operators ˆA(t) depend on time. The defining equations for the interaction picture are
The interaction picture
States: |ψ(t)ˆ i ≡eiH0t|ψ(t)i. Operators: ˆA(t) ≡eiH0tA e−iH0t.
H0 does not depend on time.
(5.8)
The interaction picture and the Heisenberg picture coincide whenV = 0, i.e., in the non-perturbed case. IfV(t) is a weak perturbation, then one can think of Eq. (5.8) as a way to pull out the fast, but trivial, time dependence due toH0, leaving states that vary only slowly in time due toV(t).
The first hint of the usefulness of the interaction picture comes from calculating the time derivative of|ψ(t)ˆ iusing the definition Eq. (5.8):
i∂t|ψ(t)ˆ i=
i∂teiH0t
|ψ(t)i+eiH0t
i∂t|ψ(t)i
=eiH0t(−H0+H)|ψ(t)i, (5.9) which by Eq. (5.8) is reduced to
i∂t|ψ(t)ˆ i= ˆV(t)|ψ(t)ˆ i. (5.10) The resulting Schr¨odinger equation for|ψ(t)ˆ ithus contains explicit reference only to the interaction part ˆV(t) of the full HamiltonianH. This means that in the interaction picture the time evolution of a state|ψ(tˆ 0)ifrom timet0 totmust be given in terms of a unitary operator ˆU(t, t0) which also only depends on ˆV(t). ˆU(t, t0) is completely determined by
|ψ(t)ˆ i= ˆU(t, t0)|ψ(tˆ 0)i. (5.11) WhenV and thus H are time independent, an explicit form for ˆU(t, t0) is obtained by inserting|ψ(t)ˆ i=eiH0t|ψ(t)i=eiH0te−iHt|ψ0iand |ψ(tˆ 0)i=eiH0t0e−iHt0|ψ0i into Eq. (5.11),
eiH0te−iHt|ψ0i= ˆU(t, t0)eiH0t0e−iHt0|ψ0i ⇒ Uˆ(t, t0) =eiH0te−iH(t−t0)e−iH0t0. (5.12) From this we observe that ˆU−1= ˆU†, i.e., ˆU is indeed a unitary operator.
In the general case with a time-dependent ˆV(t), we must rely on the differen- tial equation appearing when Eq. (5.11) is inserted in Eq. (5.10). We remark that Eq. (5.11) naturally implies the boundary condition ˆU(t0, t0) = 1, and we obtain:
i∂tUˆ(t, t0) = ˆV(t) ˆU(t, t0), U(tˆ 0, t0) = 1. (5.13) By integration of this differential equation we obtain the integral equation
Uˆ(t, t0) = 1 +1 i
Z t t0
dt0Vˆ(t0) ˆU(t0, t0), (5.14) which we can solve iteratively for ˆU(t, t0) starting from ˆU(t0, t0) = 1. The solution is
THE INTERACTION PICTURE 83
U(t, tˆ 0) = 1 + 1 i
Z t t0
dt1Vˆ(t1) + 1 i2
Z t t0
dt1Vˆ(t1) Z t1
t0
dt2Vˆ(t2) +ã ã ã (5.15) Note that in the iteration the ordering of all operators is carefully kept. A more compact form is obtained by the following rewriting. Consider for example the second- order term, paying special attention to the dummy variablest1 andt2:
Z t t0
dt1Vˆ(t1) Z t1
t0
dt2Vˆ(t2)
= 1 2
Z t t0
dt1Vˆ(t1) Z t1
t0
dt2Vˆ(t2) +1 2
Z t t0
dt2Vˆ(t2) Z t2
t0
dt1Vˆ(t1)
= 1 2
Z t t0
dt1
Z t t0
dt2Vˆ(t1) ˆV(t2)θ(t1−t2) +1 2
Z t t0
dt2
Z t t0
dt1Vˆ(t2) ˆV(t1)θ(t2−t1)
= 1 2
Z t t0
dt1
Z t t0
dt2
hVˆ(t1) ˆV(t2)θ(t1−t2) + ˆV(t2) ˆV(t1)θ(t2−t1)i
≡ 1 2
Z t t0
dt1
Z t t0
dt2Tt[ ˆV(t1) ˆV(t2)], (5.16) where we have introduced the time ordering operator Tt. Time ordering is easily generalized to higher-order terms. Innth order, where nfactors ˆV(tj) appear, alln!
permutationsp∈Sn of thentimestj are involved, and we define18 Tt[ ˆV(t1) ˆV(t2)ã ã ãVˆ(tn)]≡ X
p∈Sn
Vˆ(tp(1)) ˆV(tp(2))ã ã ãVˆ(tp(n)) ì (5.17) θ(tp(1)−tp(2))θ(tp(2)−tp(3)). . . θ(tp(n−1)−tp(n)).
Using the time ordering operator, we obtain the final compact form (see also Exer- cise 5.2):
Uˆ(t, t0) = X∞ n=0
1 n!
1 i
nZ t t0
dt1ã ã ã Z t
t0
dtnTt
Vˆ(t1)ã ã ãVˆ(tn)
=Tt
e−i
Rt
t0dt0Vˆ(t0) . (5.18) Note the similarity with a usual time-evolution factor e−iεt. This expression for U(t, tˆ 0) is the starting point for infinite-order perturbation theory and for intro- ducing the concept of Feynman diagrams; it is therefore one of the central equations in quantum field theory. A graphical sketch of the contents of the formula is given in Fig. 5.1.
18Forn= 3 we haveTt[ ˆV(t1) ˆV(t2) ˆV(t3)] =
Vˆ(t1) ˆV(t2) ˆV(t3)θ(t1−t2)θ(t2−t3) + ˆV(t1) ˆV(t3) ˆV(t2)θ(t1−t3)θ(t3−t2) + Vˆ(t2) ˆV(t3) ˆV(t1)θ(t2−t3)θ(t3−t1) + ˆV(t2) ˆV(t1) ˆV(t3)θ(t2−t1)θ(t1−t3) + Vˆ(t3) ˆV(t1) ˆV(t2)θ(t3−t1)θ(t1−t2) + ˆV(t3) ˆV(t2) ˆV(t1)θ(t3−t2)θ(t2−t1).
84 TIME DEPENDENCE IN QUANTUM THEORY
Uˆ(t, t0) =
t0
t
+
t0
Vˆ(t1) t
+
t0
Vˆ(t1) Vˆ(t2) t
+
t0
Vˆ(t1) Vˆ(t2) Vˆ(t3) t
+ . . .
Fig. 5.1. The time-evolution operator ˆU(t, t0) in the form of Eq. (5.15) can be viewed as the sum of additional phase factors due to ˆV on top of the trivial phase factors arising fromH0. The sum contains contributions from processes with 0,1,2,3, . . . scattering events ˆV, which happen during the evolution from timet0to time t.
5.4 Time-evolution in linear response
In many applications the perturbation ˆV(t) is weak compared toH0. It can therefore be justified to approximate ˆU(t, t0) by the first-order approximation
U(t, tˆ 0)≈1 + 1 i
Z t t0
dt0Vˆ(t0). (5.19)
This simple time-evolution operator forms the basis for the Kubo formula in linear response theory, which, as we shall see in the following chapters, is applicable to a wide range of physical problems.
5.5 Time-dependent creation and annihilation operators
It is of fundamental interest to study how the basic creation and annihilation operators a†ν and aν evolve in time given some set of basis states {|νi} for a time-independent HamiltonianH. As in Section 1.3.4 these operators can be taken to be either bosonic or fermionic. Let us first apply the definition of the Heisenberg picture, Eq. (5.5):
a†ν(t)≡eiHta†νe−iHt, (5.20a) aν(t)≡eiHtaνe−iHt. (5.20b) In the case of a general time-independent Hamiltonian with complicated interaction terms, the commutators [H, a†ν] and [H, aν] are not simple, and consequently the fundamental (anti-)commutator [aν(t1), a†ν(t2)]F,B involving two different times t1
andt2 cannot be given in a simple closed form:
[aν
1(t1), a†ν
2(t2)]F,B= eiHt1aν
1e−iH(t1−t2)a†ν
2e−iHt2±eiHt2a†ν
2e−iH(t2−t1)aν
1e−iHt1 = ?? (5.21) No further reduction is possible in the general case. In fact, as we shall see in the following chapters, calculating (anti-)commutators like Eq. (5.21) isthe problem in many-particle physics.
TIME-DEPENDENT CREATION AND ANNIHILATION OPERATORS 85 But let us investigate some simple cases to get a grasp of the time-evolution pictures. Consider first a time-independent Hamiltonian H which is diagonal in the
|νi-basis,
H =X
ν
ενa†νaν. (5.22)
The equation of motion, Eq. (5.6), is straightforward:19
˙
aν(t) =i[H, aν(t)] = ieiHt[H, aν]e−iHt
=ieiHtX
ν0
εν0h
a†ν0aν0, aνi
e−iHt = ieiHtX
ν0
εν0
−δνν0
aν0e−iHt
=−iενeiHtaνe−iHt = −iενaν(t). (5.23) By integration we obtain
aν(t) =e−iενtaν, (5.24)
which by Hermitian conjugation leads to
a†ν(t) =e+iενta†ν. (5.25) In this very simple case the basic (anti-)commutator Eq. (5.21) can be evaluated directly:
[aν
1(t1), a†ν
2(t2)]F,B =e−iεν1(t1−t2)δν
1ν2. (5.26)
For the diagonal Hamiltonian the time evolution is thus seen to be given by trivial phase factorse±iεt.
We can also gain some insight into the interaction picture by a trivial extension of the simple model. Assume that
H =H0+γH0, γ1, (5.27)
whereH0 is diagonalized in the basis{|νi}with the eigenenergiesεν. Obviously, the full HamiltonianH is also diagonalized in the same basis, but with the eigenenergies (1 +γ)ε. Let us however try to treatγH0 as a perturbationV to H0, and then use the interaction picture of Section 5.3. From Eq. (5.8) we then obtain
|ν(t)ˆ i=eiενt|ν(t)i. (5.28) But we actually know the time evolution of the Schr¨odinger state on the right-hand side of the equation, so
|ν(t)ˆ i=eiενte−i(1+γ)ενt|νi=e−iγενt|νi. (5.29) Here we clearly see that the fast Schr¨odinger time dependence given by the phase factoreiενt, is replaced in the interaction picture by the slow phase factoreiγενt. The reader can try to obtain Eq. (5.29) directly from Eq. (5.18).
19We are using the identities [AB, C] =A[B, C] + [A, C]Band [AB, C] =A{B, C} − {A, C}B, which are valid for any set of operators. Note that the first identity is particularly useful for bosonic operators and the second for fermionic operators (see Exercise 5.4).
86 TIME DEPENDENCE IN QUANTUM THEORY
Finally, we briefly point to the complications that arise when the interaction is given by a time-independent operatorV not diagonal in the same basis asH0. Consider for example the Coulomb-like interaction written symbolically as
H=H0+V =X
ν0
εν0a†ν0aν0+1 2
X
ν1ν2q
Vqa†ν
1+qa†ν
2−qaν
2aν
1. (5.30) The equation of motion for fermionic operatorsaν(t) is (see also Exercise 5.5):
˙
aν(t) =i[H, aν(t)]
=−iενaν(t) + i 2
X
ν1ν2q
Vqh a†ν
1+q(t)a†ν
2−q(t), aν(t)i aν
2(t)aν
1(t)
=−iενaν(t) + i 2
X
ν1ν2
Vν
2−ν−Vν−ν
1
a†ν
1+ν2−ν(t)aν
2(t)aν
1(t). (5.31) The problem in this more general case is evident. The equation of motion for the single operatoraν(t) contains terms with both one and three operators, and we do not know the time evolution of the three-operator producta†ν
1+ν2−ν(t)aν
2(t)aν
1(t). If we write down the equation of motion for this three-operator product we discover that terms are generated involving five-operator products. This feature is then repeated over and over again generating a never-ending sequence of products containing seven, nine, eleven, etc. operators. In the following chapters we will learn various approximate methods to deal with this problem.
5.6 Fermi’s golden rule
Fermi’s golden rule is a very useful expression in lowest-order perturbation theory for the transition rate Γf iassociated with a transition from an initial state|iito a different (and orthogonal) final state|fi. The transition is due to some weak perturbationV(t) that couples |ii and |fi. The two states are eigenstates of the unperturbed system described by a HamiltonianH0, so H0|νi=Eν|νiwithν=iorν =f.
The total Hamiltonian is H = H0+V(t), and the perturbation is thought to be adiabatically turned on at the distant past, so V(t) =V eηt, where V is a time- independent operator andη is a small positive rate. Suppose the system in question is in state|iiat timet0. At some later timet > t0 this initial state has evolved into
|i(t)iaccording to Eqs. (5.8) and (5.11),
|i(t)i=e−iH0tU(t, tˆ 0)eiH0t0|ii. (5.32) It is straightforward to determine the overlap between this state and a final state|fi. To first order inV, after lettingt0→ −∞, we obtain (see Exercise 5.9)
hf|i(t)i=−hf|V|iie−iEi(t−t0)eηt
Ef−Ei−iη. (5.33)
The probabilityPf(t) to find the system in state|fiat time tis given by|hf|i(t)i|2. The time-derivative dPf(t)/dt is therefore the change in probability per unit time,
THET-MATRIX AND THE GENERALIZED FERMI’S GOLDEN RULE 87 which we interpret as the transition rate Γf i between initial and final states. As shown in Exercise 5.9, we arrive at Fermi’s golden rule
Γf i= 2πhf|V|ii2δ(Ef−Ei). (5.34) If, furthermore, there is a continuum of equivalent final states one should sum over these and thus the density of states appears in the expression because ρ(Ei) = P
fδ(Ef−Ei). This version of Fermi’s golden rule is often used in scattering theory.
Here we give the more general expression and we emphasize that Eq. (5.34) is not limited to describing transitions between single-particle states, but is also valid for transitions between many-body states. This is exploited further in Chapter 10.
5.7 The T-matrix and the generalized Fermi’s golden rule
In this section, Fermi’s golden rule is generalized to include also higher-order pro- cesses where the initial and final states are coupled by multiple scatterings on the perturbation. However, the formal theory of scattering theory is rather involved, so we restrict ourselves to deriving the so-calledT-matrix using a physically motivated formulation. It is our first encounter with higher-order perturbation theory in this book.
In order to derive a generalized Fermi’s golden rule, we wish to include all higher orders in the perturbationV(t). Thus, instead of using Eq. (5.19) in Eq. (5.32), we apply the expansion from Eq. (5.15). The overlap in Eq. (5.33) then becomes
hf|i(t)i=
f 1
i Z t
t0
dt1Vˆ(t1) + 1 i2
Z t t0
dt1Vˆ(t1) Z t1
t0
dt2Vˆ(t2) +. . . i
e−iEfteiEit0, (5.35) where we omitted the zeroth-order term in Eq. (5.15) because the initial and final states are considered to be orthogonal.
At this point, we have to be careful with the formulation of how the perturbation is turned on in the distant past. As above, we assume that the interaction is turned on slowly to avoid complication due to transients that are not relevant for the discussion.
However, because many scattering events can take place between the timet0at which the perturbation is turned on and the timet of the measurement, we have to make sure that the “turning on” timeη−1 is well separated from the durationt−t0 of the interaction, i.e.,t−t0 η−1 and we thus take the limitt0 → −∞before the limit η→0, while keepingtfinite. One possible way to incorporate these conditions is to use a functional form ofV(t) different from the exponential that we used in the derivation of the lowest-order Fermi’s golden rule. We can for example useV(t) =f(t), where f(t) ={1 + exp[−η(t−t0)]}−1. This Fermi-function turns on the perturbation around timet0 during a characteristic time intervalη−1.
We now incorporate the turning-on functionf(t) explicitly in the expansion of the overlap, Eq. (5.35). For a specific termV(n) in this expansion,
V(n)= 1 in
Z t
−∞
dt1Vˆ(t1)f(t1)ã ã ã Z ti−1
−∞
dtiVˆ(ti)f(ti)ã ã ã Z tn−1
−∞
dtnVˆ(tn)f(tn), (5.36) we note that if the last factorf(tn) is turned on, most likely so are all the other factors f(ti) because they have time arguments larger thantnand because in the appropriate
88 TIME DEPENDENCE IN QUANTUM THEORY
limits their range is between t0 and t, which is much larger than η−1. Using this observation, we can replace f(ti) by unity wheni 6= n. Furthermore, now that we have only a single turning-on time, we can for simplicity go back to the exponential function, and letf(t)'exp(ηt) andt0=−∞. With these steps Eq. (5.35) becomes
|hf|i(t)i|=
X∞ n=1
1 in
f
Z t
−∞
dt1
Z t1
−∞
dt2ã ã ã Z tn−1
−∞
dtnVˆ(t1) ˆV(t2)ã ã ãVˆ(tn)eηtn i.
(5.37) Inserting the definition of the time-dependence in the interaction picture ˆV(t) = eiH0tV e−iH0tand performing the integrations, we obtain
|hf|i(t)i|=
eηt
Ei−Ef+iηhf|T|ii
, (5.38)
where
T =V +V 1
Ei−H0+iηV +V 1
Ei−H0+iηV 1
Ei−H0+iηV +. . . , (5.39) is called theT-matrix . TheT-matrix can also be written as
T =V +V 1
Ei−H0+iηT, (5.40)
because by iteration this expression generates the same series as in Eq. (5.39).
The transition rate is now evaluated in the same fashion as in the previous section, i.e., we identify Γf i with time derivative of|hf|i(t)i|2 . One finds
Γf i= 2πhf|T|ii2δ(Ef−Ei). (5.41) This expression is the generalized Fermi’s golden rule; clearly, to lowest order inV we recover Eq. (5.34). TheT-matrix and Eq. (5.41) is used in Chapter 10, where we discuss higher-order tunneling effects.
5.8 Fourier transforms of advanced and retarded functions
In this book, we encounter different kinds of correlation functions and so-called Green’s functions; the first appearing in the Kubo formula in the following chapter.
Often we will perform Fourier transforms of these functions and therefore a discussion of some general aspects of this is useful.
We consider first a so-called retarded function. The name ”retarded” means that, in accordance with normal causality, it is a measure of a physical observable due to the action of some force or interaction at times prior to the measurement. The general form of such functions is
CABR (t, t0) =−iθ(t−t0)D
A(t), B(t0)E
. (5.42)
Here we have used an example, where the operators are bosonic operators, while for fermionic operators the retarded functions are defined with an anti-commutators
FOURIER TRANSFORMS OF ADVANCED AND RETARDED FUNCTIONS 89 instead of the usual commutators, see also Eq. (8.28). In Eq. (5.42),A(t) andB(t) are operators in the Heisenberg picture and the average hãi means the thermal average defined in Eq. (1.118). The Hamiltonian thus enters in two places, namely through the time dependence of the operators and through the thermal average. In most cases, we will consider situation where the Hamiltonian is time independent and therefore the functionCABR (t, t0) only depends on the time differencet−t0. This is easily proved by writing out the definition of the average including the Heisenberg time-dependence,
A(t)B(t0)
= 1 ZTrh
e−βHeiHtAe−iHteiHt0Be−iHt0i
= 1 ZTrh
e−βHeiH(t−t0)Ae−iH(t−t0)Bi
, (5.43)
where we have used the cyclic properties of the trace and that the exponentialse−βH ande−iHt0 of course commute.
SinceCABR (t, t0) thus depends on one time variable, we write it as CR(t−t0) and define the Fourier transform as
CABR (ω) = Z ∞
−∞
dt eiωtCABR (t). (5.44) In order for the Fourier transform to be well-defined, the integrand must decay for both plus and minus infinity. Of course, for the retarded functions which are zero for negative times, only plus infinity could pose a problem. In the limit of infinite time difference,t−t0→ ∞, we expect on physical grounds that there can be no correlation betweenA(t) andB(t0) and thus we expect that
CABR (t, t0)→ −iθ(t−t0)h A(t)
B(t0)
− B(t0)
A(t)i
= 0. (5.45) If this is true, the Fourier transform in Eq. (5.44) is well defined. Whether this asymp- totic comes out correctly, depends on whether the theory or model that one uses has built in some relaxation mechanisms that destroy long time correlations. If not, corre- lations can exist at infinite ranges and the function will not decay and typically it will have some oscillatory dependence likeCABR (t−t0)∝exp[iε(t−t0)]. Nevertheless, even if the function is not decaying one can perform a Fourier transform if the frequency is allowed to be complex. By replacing the real frequency ω in Eq. (5.44) with the complex frequencyω+iη, where η is apositive real number, the Fourier transform then reads
CABR (ω) = Z ∞
−∞
dt eiωte−ηtCABR (t). (5.46) If the function is bounded in the sense that|CABR (t)|< M for alltand a positiveM then the integral in Eq. (5.46) is convergent.20The inverse Fourier transform should, of course, be modified correspondingly. However, if we letηbe a positive infinitesimal
20The transform can be defined even for an exponentially bounded function|CR(t)|<exp(αt) as long asη > α.
90 TIME DEPENDENCE IN QUANTUM THEORY
and take the limitη→0+at the end of the calculation, we can use the usual inverse Fourier transformation, because
CABR (t) = Z ∞
−∞
dω
2πe−iωtCABR (ω) = Z ∞
−∞
dω 2πe−iωt
Z ∞
−∞
dt0eiωt0e−ηt0CABR (t0)
=e−ηtCABR (t) −→
η→0+ CABR (t). (5.47)
Let us recapitulate: for retarded functions, which are not decaying at large times, we should define the Fourier transform with a complex frequencyω+iη, whereηis to be understood as a positive infinitesimal. The inverse Fourier transform is the same as usual, but in the end we must take the limitη→0+.
One example of functions where the procedure is necessary is the unperturbed Green’s functions that later in book turns out to be important building blocks in the formulation of many-body theory. Exercise 5.8 shows a typical function that we will encounter.
Another class of functions is the so-called advanced functions, which are defined similarly to Eq. (5.42)
CABA (t, t0) =iθ(t0−t)D
A(t), B(t0)E
. (5.48)
In this case the Fourier transformation is carried out in the same way, except now the frequency should be have a small negative imaginary part, i.e.,ω→ω−iη.
5.9 Summary and outlook
In this chapter, we have introduced the fundamental representations used in the de- scription of time evolution in many-particle systems: the Schr¨odinger picture, Eq. (5.2), the Heisenberg picture, Eq. (5.5), and the interaction picture, Eq. (5.8). Which pic- ture to use depends on the problem at hand. The first two pictures rely on a time- independent HamiltonianH, while the interaction picture involves a time-dependent HamiltonianH of the formH =H0+V(t), where H0 is a time-independent Hamil- tonian. For the state vectors and the operators in the interaction picture we found that
i∂t|ψ(t)ˆ i= ˆV(t)|ψ(t)ˆ i, A(t)ˆ ≡eiH0tA e−iH0t.
The time-evolution operator ˆU(t, t0) describing the evolution of an interaction picture state|ψˆ(t0)i, at timet0, to|ψ(t)ˆ i, at timet, was introduced,
|ψ(t)ˆ i= ˆU(t, t0)|ψ(tˆ 0)i, i∂tUˆ(t, t0) = ˆV(t) ˆU(t, t0), with ˆU(t0, t0) = 1.
We shall see in the following chapters how the operator ˆU(t, t0), due to its form Uˆ(t, t0) =
X∞ n=0
1 n!
1 i
nZ t t0
dt1ã ã ã Z t
t0
dtnTt
Vˆ(t1)ã ã ãVˆ(tn)
=Tt
e−iRtt0dt0Vˆ(t0) . plays an important role in the formulation of infinite-order perturbation theory and the introduction of Feynman diagrams, and how its first-order approximation,