Feynman DiagramPart I - Interaction Picture

June 1, 2016

Contents

1 Note 1

2 Natural Units 1

3 Interaction Picture 23.1 Equation of Motion for Observables . . . . . . . . . . . . . . . . 23.2 Equation of Motion for wave function . . . . . . . . . . . . . . . 33.3 The evolution operator in Interaction picture . . . . . . . . . . . 33.4 Perturbative Solution for Û . . . . . . . . . . . . . . . . . . . . . 4

3.4.1 Time Ordering Operator T . . . . . . . . . . . . . . . . . 73.5 Adiabatic Switch on . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Todo 9

A Proof of Gell-Mann and Low Theorem 9

Abstract

In this article, we introduces the basic Interaction Picture in quan-tum mechanics, then introduce the idea of ”adiabatic turn on”, which issupported by Gell-Mann and Low theorem (proved in appendix).

1 Note

This note is intended for learning Feynman Diagram techniques in many-bodyphysics. However, due to the limited time and the growing size of this document,I have to split it into small parts. This first part provides one foundationalmaterial for the beginning of Feynman Diagrams, i.e. Interaction Picture ofquantum mechanics. Further updates will be found on my blog [1] within thissummer vocation.

The style of the article might be fastidious. Yet I have try to maintainsome physical viewpoints. Also, I will provide readers with reference to originalmaterials whenever possible.

1

2 Natural Units

For convenience, I will use the unit system s.t. ~ = 1. An example of such unitsystems is Plank units. 1 Usually, calculation down in such unit systems arecarried out mathematically first, then those normalized (being set to 1) units,such as ~ here, are inserted back to the equations by dimensional analysis. Forexample, in this article when one sees the imaginary unit i, one should replaceit with i~ or sometimes i~ to get back to S.I. units.

3 Interaction Picture

The Interaction Picture 2 is a very important way to view physical systems asinteracting systems growing from the underlying ground state systems. In Inter-action Picture, both the wave functions and the operators are time dependent.The Hamiltonian is separated into the ground state part and the interactionpart:

H = H0 + V

where

• H0 : is taken to be an exactly solvable Hamiltonian. Note that H0 shouldbe bilinear in C† and C for Wick’s theoren to be applicable. 3

The operators and states are related to Schrödinger’s Picture by

|ΨI〉 = eiH0t |Ψs〉 = eiH0te−iHt |ΨH〉 (3.0.1)Ô1 = e

iH0tÔSe−iH0t (3.0.2)

where subscripts denoting:

• I:Interaction Picture

• S: Schrödinger Picture

• H: Heisenberg Picture

Remark: we see that the ”inner product” value is unaltered by this:

〈ΨI |ÔS |ΨS〉 = 〈ΨS |e−iH0t · (eiH0tÔe−iH0t) · eiH0t|ΨS〉 = 〈ΨS |ÔS |ΨS〉

Physically, this leaves the probabilities (the only important value) unchanged.The evolution of states is governed by equation of motion. In Schrödinger

Picture, it is the Schrödinger’s equation. In Interaction Picture, we will mainlydeal the evolution operator.

1Readers could refer to [4] for details.2This part follows much of section 6 of [2]. Similar material could also be found on section

2.1 of [3].3See page 79, Sec 2.4 of [3]

2

3.1 Equation of Motion for Observables

Before we talk about evolution operator, we mention two useful results. Thefirst is a equation of motion for observables in Interaction Picture.

Since: ÔI(t) = eiĤ0tÔSe

−iĤ0t, we have:

i · ∂ÔI(t)∂t

= i[iĤ0e

iĤ0tÔSe−iĤ0t] + eiĤ0tÔS(−iĤ0)e−iĤ0t

]= −Ĥ0(eiĤ0tÔSe−iĤ0t) + eiĤ0tÔSe−iĤ0tĤ0= [ÔI(t), Ĥ0] (3.1.1)

This looks very similar to the Heisenberg’s equation, which is reasonable sincein both pictures the operators is time-dependent.

3.2 Equation of Motion for wave function

Another important result is the following equation:

i∂

∂t|ΨI(t)〉 =i

∂

∂t(eiĤ0t |ΨS(t)〉) (3.2.1)

=i(iĤ0) · |ΨI(t)〉+ eiĤ0t ·∂

∂t|ΨS(t)〉 (3.2.2)

=− Ĥ0 |ΨI(t)〉+ eiĤ0t(H0 + V ) |ΨS(t)〉 (3.2.3)

=[−Ĥ0 + eiH0t(Ĥ0 + V )e−iĤ0t] |ΨI(t)〉 (3.2.4)=V̂I |ΨI(t)〉 (3.2.5)

However, in Interaction Picture, there is one operator which is even moreimportant than the above equations. It is the evolution operator to be definedbelow.

3.3 The evolution operator in Interaction picture

Note: From now on, we work mostly in Interaction Picture. Hence the subscriptI will be neglected, when it is convenient, for operators or states in InteractionPicture.

The evolution operator Û(t, t0) is defined s.t.

|Ψ(t)〉 = Û(t, t0) |Ψ(t0)〉 (3.3.1)

We observe that Û behaves just like the time-evolution operator in SchrödingerPicture: it brings a state at time t0 to a later time t. For finite time, it can begiven a formal expression as follows:

3

By equation (3.0.1) on the preceding page we have

|ΨI(t)〉 = eiH0t |ΨS(t)〉 (3.3.2)= eiH0t · eiH(t−t0) |ΨS(t0)〉 (by Schrödinger equation)

(3.3.3)

= eiH0t · e−iH(t−t0) · eiH0t0 |ΨI(t0)〉 (by 3.0.1)(3.3.4)

so:

Û(t, t0) = eiH0te−iH(t−t0)eiH0t0 (3.3.5)

The evolution operator has following properties:

1. Û(t0, t0) = 1

Physically, this means that a state is not evolving.

2. Û(t2, t1)Û(t1, t0) = Û(t2, t0) :

Physically, an evolving state should obviously be continuous in time, hencethis property. It can also be demonstrated mathematically:

Û(t0, t1)Û(t1, t0) =eiH0t2e−iH(t2−t1)e−iH0t1 · eiH0t1e−iH(t1−t0)e−iH0t0

=eiH0t2e−iH(t2−t0)e−iH0t0

3. Û(t, t0) · Û(t0, t) = 1This is a result of 2.

4. Û(t, t0) is unitary:

Because U(t, t0) is a transformation of basis that brings all basis at timet0 to time t.

Also, one can easily see that mathematically: Û(t, t0) · Û†(t, t0) = 1

5. U−1(t, t0) = U†(t, t0) = U(t0, t)

This means that ”inverse” = ”dagger” for Û . This is result of 4.

3.4 Perturbative Solution for Û

The operator Û has another more useful form which is the basis for perturbativeanalysis of complex interacting systems. To obtain such form, we first derive an

4

integral equation for Û By equation (3.2.5) on the previous page:

i∂

∂t· Û(t, t0) |ΨI(t0)〉 = (V̂ )I Û(t, t0) |ΨI(t0)〉 (3.4.1)

→ i ∂∂tÛ(t, t0) = −iV̂IU(t, t0) (3.4.2)

→ Û(t, t0)− Û(t0, t0) = −i ·∫ tt0

V̂IU(t, t0)dt (3.4.3)

or Û(t, t0) = 1− i∫ tt0

V̂IU(t1, t0)dt1 (3.4.4)

The integral equation is solved by iterative methods. We first start with a (5.7.6) ofSakurai P356= (2.18) ofMakan P68

primitive guess:

Û (1)(t, t0) = 1−∫ tt0

V̂ (t′)dt′

Here, Û (1) is the first order solution.Then one improves on this with

Û (2)(t, t0) =1−∫ tt0

V̂ (t′)Û (1)(t′, t0)dt′ (3.4.5)

=1−∫ tt0

V̂ (t2)(1−∫ t2t0

V̂ (t1)dt1)dt2 (3.4.6)

This is called it the second order solution.And one improves again:

U (3) =1−∫ tt0

dt3 · V̂ (t3) · Û (2)(t3, t0) (3.4.7)

=1−∫ tt0

dt3V̂ (t3) · [1−∫ t3t0

dt2V̂ (t2) · (1−∫ t2t0

dt1V̂ (t1))dt1] (3.4.8)

Inductively we get the expression:

Û(t, t0) =

∞∑n=0

(−i)n∫ tt0

dt1

∫ t1t0

dt2 · · ·∫ tn−1t0

dtnV̂ (t1) · · · V̂ (tn) (3.4.9)

We remark that we have already obtain U in the form of

U = Σ1(−i)nan

If we can find another value bn = n!an, then

U =∑i

(−i)n

n!bn = (Formally writing as) e

−ib

5

This is related to other forms of Wick’s theorem, which we will not discussuntil the subsequent parts of this series.

Now we change the expression for Û a little bit. Let us take the case ofn = 2 as an example. The main part is the integral:

∫ tt0

dt2

∫ t2t0

dt1V̂I(t2)V̂I(t1) (3.4.10)

The integration is taken diagrammatically:

Figure 1: t2 > t1

If we relabel t1 to t2, t2 to t1, we get:∫ tt0

dt1

∫ t1t0

dt2 · V̂I(t1)V̂I(t2) (3.4.11)

the same? 3.4.10 and 3.4.11.Then the integration is changed into:

6

Figure 2: t1 > t2

We observe that doing the two integration is equivalent to doing a full inte-gral over the region marked by (t0 6 t1 6 t and t0 6 t2 6 t),

Figure 3: Whole region

while restricting the value of integration in the proper region and puttingthe operators in correct order.

For example:equation (3.4.10) on page 5 is:

∫ tt0dt2 ·

∫ t2t0dt1θ(t1 − t2)V̂I(t2)V̂I(t1)

equation (3.4.11) on the previous page is:∫ tt0dt1∫ t1t0dt2·θ(t1−t2)V̂I(t2)V̂I(t1)

7

where θ(t1 − t2) is restricting the to region t2 > t1. Also, I put the latesttime in the front.

3.4.1 Time Ordering Operator T

We introduce a time ordering operator T to signify the process we describedabove. When T acts on a set of operators, it automatically sorts them indescending time order.

For example, if t3 > t2 > t1, then

T (A(t1)B(t3)C(t2)) = B(t3)C(t2)A(t1)

Then

∫ tt0

dt1

∫ tt0

dt2 =T (V̂ (t1)V̂ (t2)) (3.4.12)

= (integration of V̂I(t1)V̂I(t2) when t1 > t2)

+(integration of V̂ (t2)V̂ (t1) when t2 > t1)

= (

∫ tt0

dt1

∫ t1t0

dt2V̂I(t1)V̂I(t2)

+(

∫ tt0

dt2

∫ t2t0

dt1V̂I(t2)V̂I(t1)) (3.4.13)

But equation (3.4.11) on page 6 is equal to equation (3.4.10) on page 5,

because∫ tt0dt2 ·

∫ t2t0dt1V̂I(t2)V̂I(t1) =

12 ·∫ tt0dt1∫ tt0dt2 · T [V̂2(t1)V̂2(t2)]

Similarly, for arbitrary n, we have∫ tt0

dtn

∫ tnt0

dt(n−1) · · ·∫ tt0

dt1·V̂I(tn) · · · V̂I(t1) =1

n!·∫ tt0

dtn · · ·∫ tt0

dt1·T (V̂I(tn) · · · V̂I(t1))

The factor n! is that we have in total n! different ways to rearrange t1, · · · , tn.And each arrangement ti1 , · · · , tin corresponds to a region in [t0, t0]

nwhere

t0 6 ti1 < · · · < tin 6 t.Thus we the most useful expression for operator Û :

Û(t, t0) =

∞∑n=0

(−i)n

n!

∫ tt0

dt1

∫ tt0

dt2 · · ·∫ tt0

dtnT [V̂ (t1) · · · V̂ (tn)] (3.4.14)

3.5 Adiabatic Switch on

The next idea for interacting many-body system is to use the notion of adiabaticswitch on. In the Interaction Picture, we assume our Hamiltonian is:

H = H0 + V

8

We wish to extract the eigenstate of H from that of H0. Hence we imaginethat we have

H ′ = H0 + e−�|t|V

So we have limt→∞H′ = limt→∞H

′ = H0.In this scenario, we have

|Ψ(t = ±∞)〉 = |Φ0〉where |Φ0〉 can be exactly solved.Then using the evolution operator U , we can bring up the states from −∞

to now t = 0.

|Ψ(t = 0)〉 =U�(0,−∞) |Ψ(t = −∞〉 (3.5.1)=U�(0,−∞) |Ψ(∞)〉 (3.5.2)

This is the solution to H = H0 + V , since e−�·0 = 0.

However, there is a problem with this. Because the real physical system isin the Hamiltonian H = H0 + V all the time. And our system dives in theHamiltonian H only at one instant of time. If the result of over calculation isto be meaningful, it must be independent on �.

In addition, we have a Gell-Mann and Low theorem, which requires a lengthproof provided in appendix A. It states that:

If the following quantity exists to all orders in perturbation theory, (denoting|Ψ0〉 :≡ U�(0,−∞) |Φ0〉. )

Û�(0,−∞) |Φ0〉〈Φ0| Û�(0,−∞) |Φ0〉

≡ |Ψ0〉〈Φ0|Ψ0〉

then |Ψ0〉 is an eigenstate of Ĥ in the following sense:

Ĥ|Φ0〉〈Φ0|Ψ0〉

= E|Ψ0〉〈Φ0|Ψ0〉

The equation (6.44) could be recast into a new form.

〈Φ0| Ĥ|Ψ0〉〈Φ0|Ψ0〉

= E · 〈Φ0|Ψ0〉〈Φ0|Ψ0〉

Since H = H0 + V and 〈Φ0|H0|Ψ0〉 = E0 · 〈Φ0|Ψ0〉, we have

〈Φ0|V |Ψ0〉〈Φ0|Ψ0〉

= E − E0 (3.5.3)

Thus, in the limit �→ 0 we have a way to calculate the energy of our physicalsystem’s E by 3.5.3. This is also intuitively satisfying, since when � is small, Ĥ ′

approximates Ĥ in a long enough time. If we imagine that we turn on H ′ fromt = −∞ very slowly, then we are confident that our calculation is giving us thereal wave function and its energy.

9

4 Todo

Here lists I want to add to this article.First I will define Green functions 4, which are related to a variety of physical

phenomenons. Then the above idea of adiabatic switch on will allow me to givea formula for Green functions. I will show that this formula, combined withperturbative solution to Û and the Wick’s thoerem 5, gives us a systematicalway of calculating Green functions. I will then explain Feynman Diagrams 6 ,which will give a graphical representation of each Green functions. I will try tounderstand the new path integral way of calculating Feynman Diagrams7.

A Proof of Gell-Mann and Low Theorem

The theorem states:

Theorem 1. If the following quantity exists to all orders in perturbation theory:

Û�(0,−∞) |Φ0〉〈Φ0| Û�(0,−∞) |Φ0〉

≡ |Ψ0〉〈Φ0|Ψ0〉

then |Ψ0〉 is an eigenstate of Ĥ in the following sense:

Ĥ|Φ0〉〈Φ0|Ψ0〉

= E|Ψ0〉〈Φ0|Ψ0〉

Proof:Here and afterwards, we will set |Ψ0(�)〉 ≡ Û�(0,−∞) |Φ0〉.Consider the following:

(Ĥ0 − E0) |Ψ0(�)〉 =(Ĥ0 − E0)Û�(0,−∞) |Ψ0〉 (A.0.4)

=(Ĥ0Û�(0,−∞)− Û�(0,−∞)E0

)|Ψ0〉 (A.0.5)

=(Ĥ0Û�(0,−∞)− Û�(0,−∞)Ĥ0

)|Ψ0〉 (A.0.6)

=[Ĥ0, Û�(0,−∞)] |Ψ0〉 (A.0.7)

The expression for Û�(0,−∞) is explicitly:

Û�(0,−∞) =∞∑n=0

(−i)n

n!

∫ 0−∞

dt1 · · ·∫ 0−∞

dtne�(t1+···+tn)T [V̂ (t1) · · · V̂ (tn)] (A.0.8)

4could be found on [2] or [3]5Proof of Wick’s theorem could be found on [2]6Both [2] and [3] explains Feynman Diagrams, though [2] explains in more detail.7Could be found on [5].

10

So we need [Ĥ0, V̂ (t1) · · · ˆV (tn)] to get [Ĥ0, Û�(0,−∞)].

Lemma 1. [H,ABC · · · ] = ([H,A]BC · · · ) + (A[H,B]C · · · ) + · · ·

Proof:Obviously we have:

[H,AB] = [H,A]B +A[H,B] (A.0.9)

Then for more terms we can, for example:

[H,ABC] =[H,AB]C +AB[H,C]

=[H,A]BC +A[H,B]C +AB[H,C]

Thus one may obviously obtain the lemma by using equation (A.0.9) recur-sively.

Using above lemma, one has:

[Ĥ0, V̂ (t1) · · · V̂ (tn)] = [Ĥ0, V̂ (t1)]V̂ (t2) · · ·+ · · ·+ V̂ (t1) · · · V̂ (tn−1)[Ĥ0, V̂ (tn)] (A.0.10)

Also, in Interaction Picture, one has:

i∂V̂ (t)

∂t= [V̂ (t), Ĥ0]

or:1

i

∂V̂ (t)

∂t= [Ĥ0, V̂ (t)]

Hence the above commutators are turned into partial derivatives:

[Ĥ0, V̂ (t1) · · · V̂ (tn)] =1

i

(∂

∂t1+ · · ·+ ∂

∂tn

)V̂ (t1) · · · V̂ (tn) (A.0.11)

We need an additional fact:

T [

(∂

∂t1+ · · ·+ ∂

∂tn

)V̂ (t1) · · · V̂ (tn)] =

(∂

∂t1+ · · ·+ ∂

∂tn

)T [V̂ (t1) · · · V̂ (tn)]

(A.0.12)

This can be seen on an example. Notice that T [V̂ (t1)V̂ (t2)] = θ(t1−t2)V̂ (t1)V̂ (t2)+θ(t2 − t1)V̂ (t2)V̂ (t1), where θ(t) is Heaviside step function. We also have:∂∂tθ(t) = δ(t). So when t1 6= t2, T

∂∂t1

= ∂∂t1T , i.e. it commutes with T .When t1 = t2, this is obviously also correct as well. Hence:

T [∂

∂t1V̂ (t1)V̂ (t2)] =

∂

∂t1T [V̂ (t1)V̂ (t2)]

11

Similar arguments will give us equation (A.0.12), since T is essentially a seriesof θ(t) functions.

Using the above fact and A.0.11, one turns A.0.7 into (Denote∫ 0−∞ dt1 · · ·

∫ 0−∞ dtn

as (Integrate)):

(Ĥ0 − E0) |Ψ0(�)〉 =[Ĥ0, 1 +

∞∑n=0

(−i)n

n!(Integrate) e�(t1+···+tn)T [V̂ (t1) · · · V̂ (tn)]

]=

∞∑n=1

(−i)n

i · n!(Integrate) e�(t1+···+tn)

(∂

∂t1+ · · ·+ ∂

∂tn

)T [V̂ (t1) · · · V̂ (tn)]

=

∞∑n=1

− (−i)n−1

n!(Integrate) e�(t1+···+tn)

(∂

∂t1+ · · ·+ ∂

∂tn

)T [V̂ (t1) · · · V̂ (tn)]

(A.0.13)

Integration by parts gives us, for example:

∫ 0−∞

dt1e�t1

∂

∂t1T [V̂ (t1)] = e

�t1T [V̂ (t1)]

∣∣∣∣0∞− �∫ 0∞dt1e

�t1T [V̂ (t1)]

Similarly:

(Integrate) e�(t1+···+tn)∂

∂t1T [V̂ (t1) · · · V̂ (tn)]

= V̂ (0) (Integrate without t1) e�(t2+···+tn)T [V̂ (t2) · · · V̂ (tn)]

− � (Integrate) e�(t1+···+tn)T [V̂ (t1) · · · V̂ (tn)] (A.0.14)

For other ” ∂∂ti ”, the procedure is essentially the same, and they all contributeto the same value, as can be seen by relabeling the dummy indices ”t1 · · · tn”.Hence A.0.13 is:

(Ĥ0 − E0) |Ψ0〉 =∞∑n=1

− (−i)n−1

n!· n×{

V̂ (0) (Integrate without t1) e�(t2+···+tn)T [V̂ (t2) · · · V̂ (tn)]

− � (Integrate) e�(t1+···+tn)T [V̂ (t1) · · · V̂ (tn)]}|Φ0〉

(n for n same contributions from

∫ 0−∞

dt1 · · ·∫ 0−∞

dtn.)

= −V̂ (0)Û�(0,−∞) |Φ0〉 (the first term)

+ �

∞∑n=1

(−i)n−

(n− 1)!(Integrate) e�(t1+···+tn)T [V̂ (t1) · · · V̂ (tn)] |Φ0〉

12

The second term is dealt with as follows:

(−i)n−1

(n− 1)!(Integrate) T [V̂ (t1) · · · V̂ (tn)](assuming V̂ = g · V̂ ′)

=(−i)n−1

(n− 1)!(Integrate) gnT [V̂ ′(t1) · · · V̂ ′(tn)]

= ig∂

∂g

(−i)n

n!gn (Integrate) T [V̂ ′(t1) · · · V̂ ′(tn)]

= ig∂

∂g

(−i)n

n!gn (Integrate) T [V̂ (t1) · · · V̂ (tn)]

Hence:

second term = �

∞∑n=1

ig∂

∂g

(−i)n

n!(Integrate) T [V̂ (t1) · · · V̂ (tn)] |Φ0〉

= i�g∂

∂g

∞∑n=0

(−i)n

n!(Integrate) T [V̂ (t1) · · · V̂ (tn)] |Φ0〉

(since∂

∂g(zeroth term) = 0)

= i�g∂

∂gÛ�(0,−∞) |Φ0〉

= i�g∂

∂g|Ψ0(�)〉

Together we have (noting that V̂ (0) is actually the interacting part of realHamiltonian V̂ ):

(Ĥ0 − E0) |Ψ0(�)〉 = −V̂ |Ψ0(�)〉+ i�∂

∂g|Ψ0(�)〉 (A.0.15)

Write it in another way:

(Ĥ0 + V̂ − E0) |Ψ0(�)〉 = (Ĥ − E0) |Ψ0(�)〉

= i�g∂

∂g|Ψ0(�)〉 (A.0.16)

Multiplying left with 〈Φ0|〈Φ0|Ψ0(�)〉 , we have:

RHS = i�g〈Φ0| ∂∂g |Ψ0(�)〉〈Φ0|Ψ0(�)〉

(since∂

∂g〈Φ0| = 0)

= i�g

∂∂g 〈Φ0|Ψ0(�)〉〈Φ0|Ψ0(�)〉

= i�g∂

∂gln 〈Φ0|Ψ0(�)〉 (A.0.17)

13

while:

LHS =〈Φ0|Ĥ|Ψ0(�)〉〈Φ0|Ψ0(�)〉

= E − E0 (A.0.18)

So we have:

i�g∂

∂gln 〈Φ0|Ψ0(�)〉 = E − E0 (A.0.19)

From above expression, we notice that lim�→0+ � ln 〈Φ0|Ψ0(�)〉 is finite, otherwisewe would have E = E0 when �→ 0+, which is absurd. Hence, ln 〈Φ0|Ψ0(�)〉 ∝ 1� ,or 〈Φ0|Ψ0(�)〉 ∝ e1/�.

Now we calculate

(H − E0 − i�g∂

∂g)|Ψ0(�)〉〈Φ0|Ψ0(t)〉

=(Ĥ − E0 − i�g ∂∂g ) |Ψ0(�)〉

〈Φ0|Ψ0(�)〉+ i�g

|Ψ0(�)〉 ∂∂g 〈Φ0|Ψ0(�)〉

〈Φ0|Ψ0(�)〉2

= (by A.0.16 the first term vanishes) 0 +|Ψ0(�)〉〈Φ0|Ψ0(�)〉

· i�g ∂∂g

ln 〈Φ0|Ψ0(�)〉

= (by A.0.19)|Ψ0(�)〉〈Φ0|Ψ0(�)〉

· (E − E0) (A.0.20)

That is: (Ĥ − E − i�g ∂

∂g

)|Ψ0(�)〉〈Φ0|Ψ0(�)〉

= 0 (A.0.21)

Now by assumption |Ψ0(�)〉〈Φ0|Ψ0(�)〉 is finite when �→ 0+, we have:

Ĥ|Φ0〉〈Φ0|Ψ0〉

= E|Ψ0〉〈Φ0|Ψ0〉

(A.0.22)

Q.E.D.

References

[1] http://we-taper.github.io/

[2] Alexander L. Fetter, John Dirk Walecka, Quantum Theory of Many-ParticleSystems

[3] Gerald D. Mahan, Many-Particle Physics (3rd)

[4] https://en.wikipedia.org/wiki/Planck_units (accessed June 1, 2016).

[5] Negele & Orland, Quantum Many-particle Systems.

14

http://we-taper.github.io/https://en.wikipedia.org/wiki/Planck_units

NoteNatural UnitsInteraction PictureEquation of Motion for ObservablesEquation of Motion for wave functionThe evolution operator in Interaction picturePerturbative Solution for Time Ordering Operator T

Adiabatic Switch on

TodoProof of Gell-Mann and Low Theorem