Effective interaction in unified perturbation theory

Annals of Physics 364 (2016) 200–247

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Effective interaction in unified perturbation theory Kazuo Takayanagi Department of Physics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102, Japan

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Article history: Received 29 April 2015 Accepted 17 November 2015 Available online 2 December 2015

abstract We present a unified description of the Bloch and Rayleigh– Schrödinger perturbation theories of the effective interaction in both algebraic and graphic representations. © 2015 Elsevier Inc. All rights reserved.

Keywords: Effective interaction Bloch perturbation theory Rayleigh–Schrödinger perturbation theory Folded diagram representation Bracketing representation

1. Introduction The effective interaction has been an important concept in various quantum many-body problems [1–17]. We first divide a large Hilbert space into a model space (P-space) of a tractable size and its complement (Q -space). Then we define the effective interaction v in the P-space to describe a set of selected eigenstates of the full Hamiltonian H. There are two different theories to derive the perturbation expansion of v , i.e., the Bloch [7–9] and the Rayleigh–Schrödinger (RS) [17] theories. Though both theories have their own algebraic (bracketing) and graphic (folded diagram) representations for v , their interrelation has not been clarified to date in either representation. In this uncomfortable situation, we present a unified description of the Bloch and RS perturbation theories in both algebraic and graphic representations generally in nondegenerate P-space. We present the above scenario in the following three steps. The first step describes the Bloch perturbation theory strictly in terms of the ‘‘effective transition potential’’ u between the P- and Q -spaces. At this step, the Bloch perturbation theory acquires its precise description which compares well with the RS counterpart. The second step introduces the ‘‘sequence representation’’ which is a translator between

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aop.2015.11.006 0003-4916/© 2015 Elsevier Inc. All rights reserved.

K. Takayanagi / Annals of Physics 364 (2016) 200–247

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the Bloch and RS theories. Here, we also derive the ‘‘main frame expansion’’ which presents v in a simple form that leads directly to both Bloch and RS expansions. The last step derives the Bloch and RS perturbation theories in parallel using the main frame expansion in the sequence representation. The above derivation proves the ‘‘unified representation’’ which exhibits both perturbation theories simultaneously, clarifying their interrelation. The present work is organized as follows. In Section 2, we describe the effective interaction v fixing the notation. In Section 3, we reconstruct the Bloch theory in terms of the effective transition potential u. In Sections 4 and 5, respectively, we explain the algebraic (bracketing) and graphic (folded diagram) representations for u, which are then summarized in Section 6. Then, by translating the above results for u into those for v , we establish the precise description of the Bloch perturbation theory of v in Section 7. Next, we set out to unify the perturbation theories. In Sections 8 and 9, we introduce the sequence representation and the main frame expansion, respectively. In Sections 10 and 11, respectively, we derive the Bloch and RS perturbation theories of v via the main frame expansion in the sequence representation. In Section 12, we introduce the unified representation which presents these two perturbation theories in a unified fashion in both algebraic and graphic ways, as summarized in Fig. 43. In Section 13, we compare the present approach based on the Bloch theory with the standard RS approach based on the expansion of propagators, to confirm great advantages of the present approach. Finally in Section 14, we summarize the present work. 2. Effective interaction In this section, we briefly review the effective interaction v using the notation of Refs. [10–12,17]. 2.1. Model space We describe a quantum system in a Hilbert space of dimension D with the following Hamiltonian: H = H0 + V ,

(1)

where H0 is the unperturbed Hamiltonian and V is the perturbation. By diagonalizing the full Hamiltonian H in the whole Hilbert space, we obtain D eigenstates of H. In many cases, however, the above diagonalization is beyond the current computer capacity, and we are not usually interested in all of the D eigenstates. Therefore, we divide the Hilbert space of dimension D into a model space (P-space) of tractable dimension d and its complement (Q -space); we are to describe the physics in the d-dimensional P-space. The projection operators onto these spaces are denoted as P and Q , respectively, which satisfy P 2 = P, Q 2 = Q , and PQ = QP = 0. Here we require that the P-space be spanned by a set of d eigenstates {|i⟩, i = 1, . . . , d} of H0 that satisfy H0 |i⟩ = ϵi |i⟩,

i = 1, . . . , d.

(2)

Then, the Q -space is spanned by the other eigenstates {|I ⟩, I = d + 1, . . . , D} satisfying H0 |I ⟩ = ϵI |I ⟩,

I = d + 1, . . . , D.

(3)

Throughout this work, the P- and Q -space basis states are denoted by lowercase and uppercase letters, respectively. Then the projection operators P and Q are given by

P =

d 

|i⟩⟨i|,

Q =

i=1

D 

|I ⟩⟨I |,

(4)

I =d+1

and satisfy the following relations.

[H0 , P ] = [H0 , Q ] = 0,

PHQ = PVQ ,

QHP = QVP .

(5)

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2.2. Model space wave function In accordance with the d-dimensional P-space, we choose d eigenstates of H to describe: H |Ψα ⟩ = Eα |Ψα ⟩,

α = 1, . . . , d.

(6)

Then, we split the eigenstate |Ψα ⟩ in Eq. (6) into the P- and Q -space components as

|Ψα ⟩ = P |Ψα ⟩ + Q |Ψα ⟩ = |φα ⟩ + |Φα ⟩,

α = 1 , . . . , d.

(7)

Correspondingly, we write Eq. (6) in the following block form:



PHP QHP

PHQ QHQ



   |φα ⟩ |φα ⟩ = Eα , |Φα ⟩ |Φα ⟩

α = 1, . . . , d.

(8)

We can easily solve Eq. (8) for |φα ⟩ and |Φα ⟩ to obtain H BH (Eα )|φα ⟩ = Eα |φα ⟩,

α = 1, . . . , d,

(9)

Q

α = 1, . . . , d.

(10)

|Φα ⟩ =

Eα − H

VP |φα ⟩,

Here, we have defined the energy-dependent Bloch–Horowitz Hamiltonian H BH (E ) as [4,5,18–21] H BH (E ) = PHP + PVQ

Q E−H

QVP ,

(11)

Q 1 where E − is an abbreviation for Q E −QHQ Q . In the Bloch–Horowitz theory, we first solve Eq. (9) for Eα H and |φα ⟩. Then, we obtain also |Φα ⟩ immediately via Eq. (10). We can thus write down |Ψα ⟩ of Eq. (7) as

|Ψα ⟩ = |φα ⟩ +

Q Eα − H

V |φα ⟩,

α = 1, . . . , d.

(12)

At the end, note that the above Bloch–Horowitz scenario is crucially dependent on the solution of the eigenvalue problem (9) which requires special numerical techniques [22] because of the energydependence of H BH (E ). 2.3. Effective interaction v For the d chosen eigenstates {|Ψα ⟩, α = 1, . . . , d} in Eq. (6), we define the energy-independent effective Hamiltonian H eff that describes their projections onto the P-space, |φα ⟩ = P |Ψα ⟩, by H eff |φα ⟩ = Eα |φα ⟩,

α = 1, . . . , d.

(13)

{|φα ⟩, α = 1, . . . , d} is not generally an orthogonal set, we define its biorthogonal set Because  | φα ⟩, α = 1, . . . , d that satisfies ⟨ φβ |φα ⟩ = δβ,α ,

P =

d 

|φα ⟩⟨ φα |.

(14)

α=1

Then, we can express H eff of Eq. (13) formally as H eff =

d 

Eα |φα ⟩⟨ φα |.

(15)

α=1

At the end, we define the effective interaction, v = P v P, via H eff = PH0 P + v. The purpose of this work is to establish a unified description of the above v .

(16)

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Fig. 1. Schematic representation of projection operator Ω = Ω P in Eq. (17). Ω and P are the ‘‘inverse’’ to each other.

3. Bloch theory In this section, we describe the Bloch theory for the effective interaction v in terms of the ‘‘effective transition potential’’ u. 3.1. Bloch equation for Ω In the Bloch approach [5,7–9,20,21], we first introduce the wave operator Ω = Ω P that generates the true eigenstate |Ψα ⟩ of Eq. (6) from its projection |φα ⟩ onto the P-space as

|Ψα ⟩ = Ω |φα ⟩,

α = 1 , . . . , d,

(17)

which in turn signifies P Ω = P Ω P = P. Note that Ω is an ‘‘inverse’’ of the projector P in the sense shown in Fig. 1. From Eqs. (6) and (17), it is easy to see PH Ω |φα ⟩ = Eα |φα ⟩, and therefore H

eff

α = 1 , . . . , d,

(18)

and v in Eq. (16) are given by

H eff = PH Ω = PH0 P + v,

(19)

v = PV Ω .

(20)

Eq. (20) clearly shows that the perturbation theory of v reduces to that of Ω . Using the fact that |Ψα ⟩ is an eigenstate of H, we now derive an equation for Ω . First, by combining Eqs. (6) and (17), we have H0 Ω |φα ⟩ + V Ω |φα ⟩ = Eα |Ψα ⟩,

α = 1 , . . . , d.

(21)

Second, by multiplying Eq. (6) by Ω P = Ω , we obtain

Ω H0 |φα ⟩ + Ω V Ω |φα ⟩ = Eα |Ψα ⟩,

α = 1, . . . , d.

(22)

Then, the difference between Eqs. (21) and (22) gives

[Ω , H0 ] = V Ω − Ω V Ω ,

(23) 1

which is referred to as the (generalized) Bloch equation [8,9].

1 In the field of quantum chemistry, the present approach was studied in detail by Löwdin in a series of papers starting from Ref. [20], and is known as the partitioning technique [21].

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3.2. Bloch equation for u Let us note here that the right hand side of Eq. (23) has nonvanishing matrix elements only between the Q - and P-spaces; matrix elements within the P-space identically vanish as P (V Ω − Ω V Ω )P = PV Ω − PV Ω = 0,

(24)

because our definition of Ω guarantees Ω P = Ω and P Ω = P. Accordingly, we define the operator u = QuP as the right hand side of Eq. (23), i.e., u = V Ω − ΩV Ω,

(25)

and rewrite Eq. (23) as

[ Ω , H 0 ] = u.

(26)

To express the solution to Eq. (26) in a simple manner, we introduce the bracketing notation [8,9]; for an arbitrary operator A, we define (A) = Q (A)P by I

(A)j = ⟨⟨I |(A)|j⟩ =

1

ϵj − ϵI

⟨⟨I |A|j⟩ = RjI I ⟨⟨A⟩j ,

(27)

where the single and double angle brackets, ‘‘⟨· · ·⟩’’ and ‘‘⟨⟨· · ·⟩⟩’’, represent the P- and Q -space matrix j 1 elements, respectively. Note that we adopt the convention to place RI = ϵ −ϵ to the left of I ⟨⟨A⟩j j

I

in Eq. (27). When necessary, we refer to I (A)j and (A) as the labeled and unlabeled bracketings, respectively.2 It is now easy to see that Eq. (26) determines ⟨⟨I |Ω |j⟩ as

⟨⟨I |Ω |j⟩ =

1

ϵj − ϵI

⟨⟨I |u|j⟩ = RjI I ⟨⟨u⟩j = ⟨⟨I |(u)|j⟩,

(28)

which proves the following operator equation. Q Ω P = (u).

(29)

Because P Ω P = P by definition, Ω = P Ω P + Q Ω P can be written as

3

Ω = P + (u),

(30)

which transforms Eq. (17) into

|Ψα ⟩ = |φα ⟩ + (u)|φα ⟩,

α = 1 , . . . , d.

(31)

Note here that u is quite similar to the (half-on-shell) T -matrix in the scattering theory. This can be seen most easily in degenerate P-space where ϵ1 = · · · = ϵd = ϵ . In this case, Eq. (31) reduces to

|Ψα ⟩ = |φα ⟩ +

Q

ϵ − H0

u|φα ⟩,

α = 1, . . . , d,

(32)

which looks very like the Lippmann–Schwinger equation [23–25]. By comparing Eqs. (32) and (12), we realize that u is an ‘‘effective transition potential’’ between the Q - and P-spaces; the difference between the full propagator E Q−H and the free propagator ϵ−QH is compensated by replacing V α 0 with u. At the end, we eliminate Ω in Eq. (25) using Eq. (30), to obtain u = QVP + QV (u) − (u)VP − (u)V (u),

(33)

2 In Appendix A, we present a detailed explanation on the bracketing notation. 3 In the usual notation in the effective interaction theory [10–12,15], we express Ω as Ω = P +ω. This in turn means ω = (u).

K. Takayanagi / Annals of Physics 364 (2016) 200–247

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or equivalently

(u) = (V ) + (V (u)) − ((u)V ) − ((u)V (u)),

(34)

which we refer to as the Bloch equation for u. In the following, we use Eq. (34) rather than Eq. (33) for the sake of symmetry; Eq. (34) contains u only as (u) on both hand sides while Eq. (33) does not, which explains that Eq. (34) suits iterative solutions better than Eq. (33). Once the effective transition potential u is obtained via Eq. (34), the effective interaction v of Eq. (20) can be expressed simply as

v = V + V (u).

(35)

In Eq. (35) and in what follows, we assume implicitly that each expression for v is multiplied by the projector P from the left and right. 3.3. u versus Ω In the above, we have derived the Bloch equation (23) for Ω and (34) for u. The standard description [8,9] of the Bloch theory is based on Eq. (23), and gives the perturbation expansion of Ω . In this work, however, we develop the perturbation theory using the Bloch equation (34) for u. Here we explain why we prefer u to Ω ; there are two reasons. First, unlike the dimensionless operator Ω , u has the dimension of energy E in the same way as V and v , and therefore has a simple perturbative description. In order to realize the above point, let us briefly review the usual perturbation expansion of Ω :

Ω = Ω (0) + Ω (1) + Ω (2) + · · · ,

P Ω P = Ω (0) = P ,

(36)

(n)

where Ω stands for the nth order term in powers of V . By substituting the expansion (36) into the Bloch equation (23) for Ω , we obtain n 

[Ω (n) , H0 ] = V Ω (n−1) −

Ω (m−1) V Ω (n−m) ,

n = 1, 2, . . . .

(37)

m=2

Using the bracketing notation in Section 3.2, we can solve Eq. (37) for Ω (n) as

Ω (n) = (V Ω (n−1) ) −

n 

(Ω (m−1) V Ω (n−m) ),

n = 1, 2, . . . .

(38)

m=2

Note here that Eq. (38) is an expansion of Ω (n) in terms of {Ω (n−1) , . . . , Ω (1) } and V , which have different dimensions. We can easily imagine that such an incoherent expansion of Ω would lead to a clumsy expression, which in fact is the present situation of the Bloch theory. In contrast, we shall soon see the coherence of the expansion of (u) in Section 4. Second, the expression (u) on the right hand side of Eq. (29) explicitly shows its structure as given by Eq. (27). On the other hand, the expression Q Ω P on the left hand side of Eq. (29) does not explicitly carry the above information. When we introduce the sequence representation in Section 8, we need to define the transformation (u) −→ Ru, which makes explicit use of the compound structure of (u). This means that, if we stick to working with Ω , we cannot go beyond Section 8. In order to unify the Bloch and RS perturbation theories in Section 12, it is essential to use (u) in place of Q Ω P of the standard approach. 4. Bracketing — algebraic perturbation Here we develop the perturbation theory of u via the Bloch equation (34).

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K. Takayanagi / Annals of Physics 364 (2016) 200–247

4.1. Bracketing expansion of u We expand the effective transition potential u in powers of V as u = u(1) + u(2) + · · · ,

(39)

(n)

where u stands for the nth order term in the perturbation. Then an iterative formula for u derived easily from Eq. (34) as

(u(n) ) = (V (u(n−1) )) − ((u(n−1) )V ) −

n −1 

((u(m−1) )V (u(n−m) )),

n = 2, 3, . . . .

(n)

can be

(40)

m=2

Note that Eq. (40) is an expansion of u(n) in terms of {u(n−1) , . . . , u(1) = V } all of which have the same dimension, making a sharp contrast to the expansion of Ω based on Eq. (38). Starting with (u(1) ) = (V ), Eq. (40) immediately gives the following solution [7].

(u(1) ) = (V ), (u(2) ) = (V (u(1) )) − ((u(1) )V ) = (V (V )) − ((V )V ), (3)

(2)

(2)

(41) (1)

(1)

(u ) = (V (u )) − ((u )V ) − ((u )V (u )) = (V (V (V ))) − (V ((V )V )) − ((V (V ))V ) + (((V )V )V ) − ((V )V (V )). By construction, each ‘‘)V ’’ carries a minus sign in Eq. (41). In this work, we call the expansion (41) in terms of the round brackets ‘‘(· · ·)’’ of Eq. (27) as the Bloch bracketing expansion, or simply as the bracketing expansion.4 4.2. Valid bracketing We call a bracketing to be Bloch-valid or simply valid, if it is generated iteratively via Eq. (40) as in Eq. (41). Other bracketings are invalid. Using the above definition of valid bracketing, we express the bracketing expansion of (u) as

(u) =

∞  n =1

(n)

(u ) =

n V ′s

∞ 



n=1

bracketing

   (VV · · · V V ),

(42)

where the summation is over all possible valid bracketings. Here a natural question arises; how can we distinguish valid bracketings from invalid ones? This is the question that we answer below. 1

2

n

Let us examine an nth order valid bracketing composed of {V , V , . . . , V }. Here, we number n V ’s for later convenience, and assume that they appear in ascending order in each bracketing. For example, 1

2

3

(V ((V )V )) in (u(3) ) of Eq. (41) is to be expressed as (V ((V ) V )). Eq. (40) shows that there is a oneto-one correspondence between the outmost pair of brackets and the single V enclosed directly by it, classifying each nth order valid bracketing as n V ′s

1 n m  ( · · · ) −resolution −−−−→ (i) ( V (· · ·)), (ii) ((· · ·) V ), (iii) ( (· · ·) V (· · ·)) , 1

1

n

n

m

(43)

m

which we refer to as the resolution of bracketing. Here, we have assigned the same number to the n V ′s

 outmost pair of brackets and the corresponding V . Now, note the following; if the bracketing ( · · · ) 4 Note that the term ‘‘bracketing’’ is usually reserved for the angle brackets ‘‘⟨· · ·⟩’’ in the RS perturbation theory [7,9,17]. To avoid possible confusion, therefore, we will refer to the angle brackets ‘‘⟨· · ·⟩’’ as the P-form in later sections.

K. Takayanagi / Annals of Physics 364 (2016) 200–247

207

on the left hand side is valid, so is each inner bracketing (· · ·) on the right hand side, and vice versa. Then, by using the resolution (43) recursively, we can express the condition for valid bracketing as follows. Proposition A (Condition for Valid Bracketing). A bracketing scheme of nth order is valid, if and only if the resolution can be performed properly for each pair of brackets, i.e., if and only if each pair of brackets encloses a single V directly in one of the three types of bracketing in (43), establishing the one-to-one correspondence between n V ’s and n pairs of brackets. Let us use condition A to distinguish valid bracketing schemes. Given a third order ‘‘bracketing’’

((V )V (V )) as an example, we perform the resolution as follows. 1

2

3

1

2

3

((V ) V (V )) −→ ((V ) V (V )) 2

2 1

2

3

−→ ( ( V ) V ( V )), 21

1

3

(44)

32

where the corresponding pair of brackets and V are assigned the same number at each step of the 1

2

3

2

resolution. In the given bracketing ((V ) V (V )) in the first line, we find that V is the only V that is enclosed directly by the outmost pair of brackets in type (iii) bracketing of (43). We can thus confirm 2

the one-to-one correspondence between V and the outmost pair of brackets which is now denoted as ( · · ·) . Now we move onto the inner bracketings. Because the resolution of the first order bracketings 2

2 1

3

1

3

(V ) and (V ) simply gives ( V ) and ( V ) , we arrive at the last expression in the second line, which 1

1

3

3

shows clearly that each pair of brackets directly encloses a single V properly as indicated by numbers. We can thus perform the entire resolution of the given bracketing ((V )V (V )) properly, to conclude that it is a valid bracketing via condition A. Next, let us consider a fifth order ‘‘bracketing’’ (((V )V (V ))V (V )). Then, we perform the resolution successively as follows. 1

2

3

4

5

1

2

3

4

1

2

3

4

5

(((V ) V (V )) V (V )) −→ (((V ) V (V )) V (V )) 4

4 5

−→ ( ((V ) V (V )) V ( V )) 42

2 1

2

5

3

54

4

5

−→ ( ( ( V ) V ( V )) V ( V )) . 421 1

2

3

4

1

3

32

5

5

(45)

54 4

In the bracketing (((V ) V (V )) V (V )) in the first line, V is the only V that is enclosed directly by the outmost pair of brackets in type (iii) bracketing of (43). We thus obtain the right hand side of the first 1

2

3

5

line. Now we move onto the inner bracketings, and perform the resolution of ((V ) V (V )) and (V ) in the same way as in Eq. (44) to arrive at the last expression. Because we have demonstrated the entire resolution of the given bracketing (((V )V (V ))V (V )) properly, we conclude that it is a valid bracketing via condition A. As another example of valid bracketing, let us consider ((V ((V )V (V ))V (V )). By using condition A, we can easily confirm that this is valid because we can perform the entire resolution as follows. 1

2

3

4

5

6

1

2

3

4

5

6

((V ((V ) V (V )) V (V ))) −→ ( ( V ( ( V ) V ( V ))) V ( V )) . 51

32

2

4

431

6

65

Let us turn to invalid bracketing schemes. We now take ((V )(V )V (V (V ))) as an example, and try to perform its resolution by looking for the V that is enclosed by the outmost pair of brackets only. Then we find 1

2

3

4

5

3

((V )(V ) V (V (V ))) - - > ((· · ·)(· · ·) V (· · ·)),

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K. Takayanagi / Annals of Physics 364 (2016) 200–247

Fig. 2. Left: labeled diagram representing I (u)j . Right: unlabeled diagram for (u). Thin vertical lines imply that the horizontal order of the corresponding elements is the same in the diagram and bracketing. See the text.

which shows that ((V )(V )V (V (V ))) is not classified into the three types in (43) and does not satisfy condition A. We conclude, therefore, that this is not a valid bracketing. As another example, let us take ((V )V (V )V ((V ))). Then, its resolution gives 1

2

3

4

5

2

4

((V ) V (V ) V ((V ))) - - > ((· · ·) V (· · ·) V (· · ·)), 2

4

which proves that this is inv alid because the outmost pair of brackets encloses directly V and V at the same time. 5. Folded diagram — graphic perturbation Here we visualize the bracketing expansion of u developed in Section 4. 5.1. Definition of folded diagram We introduce the folded diagram by the following rules (I)–(III). (I) We define the labeled folded diagram representing I (u)j = ⟨⟨I |(u)|j⟩ as shown in the left panel in Fig. 2. Here the Q -space state ‘‘I’’ is expressed by a solid line with a double arrow pointing upwards to the left (Q -line), and the P-space state ‘‘j’’ is denoted by a solid line with a single arrow pointing downwards to the left (P-line).5 The effective transition potential u is expressed j by an open circle, and the propagator RI is denoted by a horizontal dashed line that crosses the Q -line ‘‘I’’ at the ‘‘Q -cross’’ and the P-line ‘‘j’’ at the ‘‘P-cross’’. Then, we define the unlabeled folded diagram representing (u) by dropping all P- and Q -indices from the labeled diagram for I (u)j , as shown in the right panel in Fig. 2. As indicated by thin lines, we draw the elements in the diagram – u, Q - and P-crosses – right above their corresponding elements – u, ‘‘(’’, and ‘‘)’’ – in the bracketing (u). (II) We introduce the unlabeled diagram for (V (u)), ((u)V ), and ((u)V (u)) on the right hand side of Eq. (34) as shown in Fig. 3, where V is expressed by a solid circle. Note that type (i), (ii), and (iii) diagrams in Fig. 3 respectively stand for type (i), (ii), and (iii) bracketings in (43). Here we attribute the minus sign carried by ((u)V ) and ((u)V (u)) in Eq. (34) to the internal P-line in the diagram. In the same way as in Fig. 2, we draw each element of the diagram right above its corresponding element in the bracketing. Then, by construction, the horizontal order of elements in the diagram is the same as in the bracketing. Note that these unlabeled diagrams can be easily converted into labeled ones by allotting the P- and Q -indices appropriately, as will be demonstrated in Section 6.2. (III) In type (iii) diagram for ((u)V (u)), we draw the right block of (u) above the left block of (u) without any vertical overlap. We stress that this convention fixes the vertical order of all elements in the diagram. Note that this is not the case in the usual description of the Bloch theory [7–9] which does not require rule (III).

5 The definition here is not the only possibility. One could have chosen the so-called folded resolvent notation [9], for example. However, the simplest correspondence between the Bloch and RS theories can be obtained via the present convention.

K. Takayanagi / Annals of Physics 364 (2016) 200–247

209

Fig. 3. Unlabeled folded diagrams representing (V (u)), ((u)V ), and ((u)V (u)) in Eq. (34). Solid circles represent V . Other notation is the same as in Fig. 2. Three diagram types (i), (ii), and (iii) correspond, respectively, to the three bracketing types (i), (ii), and (iii) in Eq. (43).

Fig. 4. Graphic expression for Eq. (40). u(n) is denoted by an open circle with ‘‘n’’.

To summarize, we have introduced the folded diagram that visualizes each bracketing in the Bloch equation (34). Our diagram rule completely specifies both horizontal and vertical orders of each element in the diagram. 5.2. Folded diagram expansion of u Using the unlabeled folded diagram defined in Section 5.1, we can express Eq. (40) graphically as shown in Fig. 4. Starting from (u(1) ) = (V ), we can use Fig. 4 iteratively to obtain the diagrams in Fig. 5 representing (u(2) ) and (u(3) ) in Eq. (41). It is clear that we can continue the above process to generate (u(n) ) for an arbitrary n, establishing the folded diagram expansion of (u) corresponding to Eq. (42). By virtue of our folded diagram rule in Section 5.1, each term in Eq. (40) fixes both horizontal and vertical orders of all graphic elements, u, V , Q - and P-crosses, in its folded diagram in Fig. 4. Because each step of the iteration with Eq. (40) and Fig. 4 conserves the above property, each valid bracketing in the perturbation expansion – which is the final stage of the iteration – completely fixes the order of V , Q - and P-crosses both horizontally and vertically in its diagrammatic expression. As an immediate consequence of the above observation, note that V ’s and propagators appear alternately in the vertical direction, as can be confirmed in Fig. 5. 5.3. Valid folded diagram We call a folded diagram to be Bloch-valid or simply valid, if it is constructed by using Fig. 4 iteratively as demonstrated in Fig. 5. Other diagrams are called invalid. In this section, we examine the valid folded diagram in connection with the valid bracketing in Section 4.2.

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Fig. 5. Graphic expression for (u(1) ), (u(2) ), and (u(3) ) in Eq. (41).

Clearly, the three bracketing types in (43) are expressed by the three diagram types in Fig. 3. Then, the algebraic one-to-one correspondence between the outmost pair of brackets and the explicit V in (43) is translated into the graphic one-to-one correspondence between the uppermost propagator 1

n

line and the uppermost V . Suppose we are given a valid diagram which is composed of {V , . . . , V } connected by a zigzag and n propagator lines. Then, we simply take the uppermost V and the corresponding uppermost propagator line in the diagram, which classifies the given valid diagram into the three types in Fig. 3. We refer to the above process as the graphic resolution of folded diagram which corresponds to the algebraic resolution of bracketing in Section 4.2. It is then easy to understand the following proposition by translating condition A for valid bracketing in Section 4.2 into the graphic expression. Proposition B (Condition for Valid Folded Diagram). A folded diagram is valid, if and only if the resolution can be performed properly for each block representing (u), i.e., if and only if each block of (u) can be resolved into one of the three types in Fig. 3. In order to see how to use condition B, let us look into Fig. 6 which represents the graphic resolution corresponding to the algebraic resolution in Eq. (45). We start with the leftmost diagram in Fig. 6, 4

where the fifth order block in the dotted square is to be resolved. Noting that V is the uppermost V 4

4

in the diagram, we number the uppermost propagator right above V as ‘‘4’’. Then, V separates the diagram into the left and right blocks indicated by dotted squares in the second diagram, and the right block is placed above the left block in accordance with type (iii) diagram in Fig. 3. We have thus resolved properly the first diagram into the second in Fig. 6. It is now easy to proceed in the same way as in Eq. (45) to arrive at the last diagram. We can thus complete the graphic resolution of the leftmost diagram properly, showing that it is a valid folded diagram constructed using Fig. 4 iteratively. As another example, let us consider the diagram in Fig. 7. Then, at the first step of the resolution, 4

we find that the left and right blocks separated by the uppermost interaction V overlap vertically. This means that the leftmost diagram cannot be resolved into a type (iii) diagram in Fig. 3, and is invalid. At the end, let us stress the following. If we had not required rule (III) in Section 5.1, both diagrams in Figs. 6 and 7 would be valid, which in fact is the standard convention. We shall see, however, that rule (III) is indispensable in unifying the Bloch and RS perturbation theories.

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Fig. 6. Graphic resolution corresponding to the algebraic resolution (45) shown below the diagram. The graphic resolution here shows that the leftmost diagram is valid. Blocks to be resolved at each step are indicated by dotted squares. See the text.

Fig. 7. A fifth order invalid folded diagram. The graphic resolution cannot be performed properly in contrast to Fig. 6. See the text.

6. Bloch perturbation theory of effective transition potential u In the above, we have established the Bloch perturbation expansion of u in both algebraic and graphic representations. In this section, we first summarize the correspondence between these two representations, and then explain how to evaluate the matrix elements of u in practice. 6.1. Correspondence between bracketing and folded diagram Using the one-to-one correspondence between the algebraic and graphic resolutions in Sections 4.2 and 5.3, we now explain the following one-to-one correspondence:

v alid folded diagram ←→ v alid bracketing.

(46)

Let us start with the left direction ←− in (46). Given a valid bracketing, the algebraic resolution identifies the given valid bracketing as one of the three types in (43), which in turn specifies the graphic resolution of the target diagram as the corresponding diagram in Fig. 3. Then, we move onto the inner bracketings, and repeat the same process. We can thus draw the target diagram representing the given bracketing. The above correspondence can be confirmed clearly by the examples in Fig. 8, where we have shown two valid folded diagrams and their corresponding valid bracketing schemes. For example, given the bracketing (((V )V (V ))V (V )), its algebraic resolution (45) in Fig. 6 explains how to draw the corresponding diagram in the right panel in Fig. 8.

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K. Takayanagi / Annals of Physics 364 (2016) 200–247

Fig. 8. Correspondences (46), (70), and (75) among folded diagram, bracketing, and sequence representations. The left and right diagrams correspond to the resolutions (44) and (45), and also to the identities (72) and (74), respectively. See the text.

Next, let us turn the right direction −→ in (46). Because the graphic resolution of a given valid folded diagram determines its algebraic resolution of the target bracketing, we can explain the right direction −→ in (46) in the same way as in the above, as can be confirmed easily by the examples in Fig. 8.6 6.2. Matrix elements j

Here we study the matrix element I (u)j = RI ⟨⟨I |u|j⟩ for each term of the perturbation expansion in Eq. (41). Let us start with the algebraic representation of (u(2) ). Using the definition (27) repeatedly as explained in Appendix A, we obtain the following expressions for the two terms that compose (2) )j . I (u 1

I

2

(V (V ))j =



j

1

2

j

RI I ⟨⟨V ⟩⟩J RI J ⟨⟨V ⟩j ,

J 1

I

2

((V ) V )j =



j

2

1

RI RkI I ⟨⟨V ⟩k k ⟨V ⟩j .

(47)

k

Next, we examine the graphic representation. We attach external indices ‘‘I’’ and ‘‘j’’, and internal dummy indices to the unlabeled diagrams for (u(2) ) in Fig. 5, to end up with the labeled diagrams in j Fig. 9. Because our convention places R to the left of ⟨⟨· · ·⟩ as prescribed in Eq. (27), we assign RI to the Q -cross (not to the P-cross) in the diagram to maintain the horizontal correspondence of elements. 1

2

j

1

j

2

Then, in the labeled diagram of I (V (V ))j in Fig. 9, for example, we find RI , I ⟨⟨V ⟩⟩J , RI , and J ⟨⟨V ⟩j from left to right, in the same order as in the algebraic expression in Eq. (47). We can thus write down the right hand side of Eq. (47) in both algebraic and graphic ways.

6 In practice, the easiest way to obtain the bracketing for a given folded diagram is the projection. Because the horizontal order of each element in the diagram is the same as in the bracketing by definition, we simply project the diagram onto the horizontal line, and replace each solid circle with V , and each of Q - and P-crosses with ‘‘(’’ and ‘‘)’’, respectively, to arrive at the target bracketing. The above correspondence is clearly seen in Fig. 8,

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213

Fig. 9. Folded diagram representation of I (u(2) )j of Eq. (47) and I (u(3) )j of Eq. (48). Each R is assigned to the Q -cross on the Q -line.

Now we turn to (u(3) ) in Eq. (41). It is straightforward to show the following expressions for the five terms that compose (u(3) ) and the corresponding diagrams in Fig. 9. 1

I

2

3

(V (V (V )))j =



j

1

j

j

1

j

2

3

j

RI I ⟨⟨V ⟩⟩J RJ J ⟨⟨V ⟩⟩K RK K ⟨⟨V ⟩j ,

JK 1

I

2

3

(V ((V ) V ))j =



2

3

2

3

RI I ⟨⟨V ⟩⟩J RJ RkJ J ⟨⟨V ⟩k k ⟨V ⟩j ,

Jk 1

I

2

3



3



((V (V )) V )j =

1

j

RI RkI I ⟨⟨V ⟩⟩J RkJ J ⟨⟨V ⟩k k ⟨V ⟩j ,

Jk 1

I

2

(((V ) V ) V )j =

1

j

2

3

RI RlI RkI I ⟨⟨V ⟩k k ⟨V ⟩l l ⟨V ⟩j ,

kl 1

I

2

3

((V ) V (V ))j =



1

j

2

j

3

RI RkI I ⟨⟨V ⟩k k ⟨V ⟩⟩J RJ J ⟨⟨V ⟩j .

(48)

kJ

At the end, we show the following fifth order term in Fig. 10. It is now easy to confirm the correspondence between the algebraic and graphic expressions. 1

I

2

3

4

5

(((V ) V (V )) V (V ))j =



j

1

2

3

4

j

5

RI RlI RkI I ⟨⟨V ⟩k k ⟨V ⟩⟩J RlJ J ⟨⟨V ⟩l l ⟨V ⟩⟩K RK K ⟨⟨V ⟩j .

(49)

JKkl

7. Bloch perturbation theory of effective interaction v Having established the perturbation expansion of u, we can easily obtain the perturbation expansion of v in Eq. (35). 7.1. Bracketing expansion of v We first expand v of Eq. (35) in powers of V as

v = v (1) + v (2) + v (3) + · · · .

(50)

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K. Takayanagi / Annals of Physics 364 (2016) 200–247

Fig. 10. Folded diagram representation of Eq. (49) that contributes to I (u(5) )j .

Fig. 11. Diagrammatic expression for Eq. (51). v (n) is expressed by an open square denoted by ‘‘n’’.

Then, the expansion (39) of u gives

v (1) = PVP ,

v (n) = PV (u(n−1) ),

n = 2, 3, . . . ,

(51)

which is diagrammatically shown in Fig. 11. Combining Eqs. (51) and (42), it is easy to write down the following bracketing expansion of v .

v=

∞ 

v (n) =

n=1

n−1 V ′ s

∞ 



n=1

bracketing

   V (VV · · · V V ).

(52)

Here, the first few terms are obtained explicitly from Eq. (41) as

v (1) = V , v (2) = V (V ), v (3) = V (V (V )) − V ((V )V ), v (4) = V (V (V (V ))) − V (V ((V )V )) − V ((V (V ))V ) + V (((V )V )V ) − V ((V )V (V )).

(53)

Let us evaluate the following matrix element of v in Eq. (52):

i

⟨v⟩j =

∞  i

(n)

⟨v ⟩j =

n =1

n −1 V ′ s

∞ 



n=1

bracketing

   i ⟨V (VV · · · V V )⟩j .

(54)

For example, i ⟨v (1) ⟩j and i ⟨v (2) ⟩j are obviously given by i

⟨v (1) ⟩j = i ⟨V ⟩j ,

i

⟨v (2) ⟩j = i ⟨V (V )⟩j =

0

1

0

 i I

1

⟨V ⟩⟩I RjI I (V )j .

(55)

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215

Fig. 12. Graphic expression for i ⟨v (1) ⟩j , i ⟨v (2) ⟩j , and i ⟨v (3) ⟩j in Eqs. (55) and (56).

It is straightforward to go to higher order terms; to write down the matrix element i ⟨v (n) ⟩j , we simply

multiply I (u(n−1) )j in Section 6.2 by i ⟨V ⟩⟩I and sum over the dummy index ‘‘I’’. For example, the third order term i ⟨v (3) ⟩j can be calculated as follows. 0

i

1

2

0

1

2

⟨v (3) ⟩j = i ⟨V (u(2) )⟩j = i ⟨V (V (V ))⟩j − i ⟨V ((V ) V )⟩j  0  0 1 2 1 2 = i ⟨V ⟩⟩I I (V (V ))j − i ⟨V ⟩⟩I I ((V ) V )j I

I 0



=

i

1

2

⟨V ⟩⟩I RjI I ⟨⟨V ⟩⟩J RjJ J ⟨⟨V ⟩j −

0



IJ

i

1

2

⟨V ⟩⟩I RjI RkI I ⟨⟨V ⟩k k ⟨V ⟩j ,

(56)

Ik

where we have used Eq. (47) in going to the last line. Note that the leftmost V in Eq. (52), which transforms (u) into V (u), is assigned the number ‘‘0’’. 7.2. Folded diagram expansion of v Let us turn to the graphic representation. In order to draw the diagram of i ⟨v (n) ⟩j , we simply need

to attach i ⟨V ⟩⟩I on top of the Q -line ‘‘I’’ of the diagram for I (u(n−1) )j , as is clear from Fig. 11. We show the folded diagrams which represent i ⟨v (1) ⟩j , i ⟨v (2) ⟩j , and i ⟨v (3) ⟩j in Fig. 12. At the end, we give two more examples; the first example is the following fourth order term: 0

i

1

2

3

⟨V ((V ) V (V ))⟩j =

0

 i

1

2

3

⟨V ⟩⟩I I ((V ) V (V ))j

I

=

i k

2

1

0



3

⟨V ⟩⟩I RjI RkI I ⟨⟨V ⟩k k ⟨V ⟩⟩J RjJ J ⟨⟨V ⟩j ,

(57)

IJ

which is expressed by the left diagram in Fig. 13. We can easily obtain the above results using Eq. (48) 1

2

2

3

3

and Fig. 9 for I ((V ) V (V ))j . The second example is the following sixth order term: 0

i

1

4

5

⟨V (((V ) V (V )) V (V ))⟩j =

0

 i

1

2

3

4

5

⟨V ⟩⟩I I (((V ) V (V )) V (V ))j

I

=

0

 i kl

⟨V ⟩⟩

j I RI

1

2

3

4

j

5

RlI RkI I ⟨⟨V ⟩k k ⟨V ⟩⟩J RlJ J ⟨⟨V ⟩l l ⟨V ⟩⟩K RK K ⟨⟨V ⟩j ,

(58)

IJK

which is expressed by the right diagram in Fig. 13. It is now straightforward to obtain the above result 1

2

3

4

5

using Eq. (49) and Fig. 10 for I (((V ) V (V )) V (V ))j .

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K. Takayanagi / Annals of Physics 364 (2016) 200–247

Fig. 13. Left: graphic expression for Eq. (57). Right: the same for Eq. (58).

7.3. Summary of bloch perturbation theory To summarize Sections 3–7, we have derived the Bloch perturbation theory in a strict manner, which now compares well with the RS counterpart [17] at precise description. We now set out, therefore, to unify the Bloch and RS perturbation theories. 8. Sequence representation In this section, we introduce the ‘‘sequence representation’’ in the Bloch theory, which is mathematically equivalent to the bracketing representation we have described in Section 4, and at the same time serves as a translator between the Bloch and RS perturbation theories. 8.1. Motivation for sequence representation Before introducing the sequence representation, we explain in Section 8.1.1 why it is necessary, and in Section 8.1.2 why it is nontrivial. 8.1.1. Why is it necessary? — P-form Here, we compare the Bloch and RS perturbation theories from the viewpoint of the matrix structure in P-space, and explain why we need the sequence representation in addition to the bracketing representation in Section 4. Let us take Eq. (57) as an example in the Bloch theory. The second line is the final expression for actual calculation that specifies all P- and Q -indices, which we refer to as the PQ -form. Now, by introducing the propagator Rj = ϵ −QH , we rewrite Eq. (57) as j

i

⟨V ((V )V (V ))⟩j =

 i

0

⟨V R R V ⟩k k ⟨V Rj V ⟩j . j

k

(59)

k

We refer to the right hand side of Eq. (59) as the (labeled) Bloch P-form, for it specifies the P-indices only.7 Here, note that the transformation of the bracketing into its P-form in Eq. (59) is not direct; it is via the PQ -form in Eq. (57). In the RS theory in Section 11, on the other hand, we can write down the corresponding RS P-form

 i

⟨V Rj k ⟨V Rj V ⟩j Rk V ⟩k ,

k

7 In this work, we use ‘‘labeled’’ and ‘‘unlabeled’’ P-forms in the Bloch and RS theories. For convenience, they are tabulated in Table B.1 in Appendix B.

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217

directly at the beginning, as we will see in Eq. (108). In the following, we develop our theory in terms of the P-form which reveals its matrix structure in P-space; the PQ -form is used in actual calculations only. Then, the above observation indicates the following. To establish a robust correspondence between the Bloch and RS theories, the Bloch theory needs a new representation that leads to its P-form directly, i.e., an alternative to V ((V )V (V )) which turns immediately into the right hand side of Eq. (59). This is the sequence representation that we are going to introduce. 8.1.2. Why is it nontrivial? — Entanglement and irreducibility If one is familiar with the effective interaction theory in degenerate P-space [3–7], one might think that one could easily define the desired representation that leads directly to its P-form. In nondegenerate P-space, however, this is not true because of the irreducibility of the P-form as we explain below. Suppose we work in degenerate P-space where ϵ1 = · · · = ϵd = ϵ in Eq. (2). Then, we can remove the P-indices of propagators in Eq. (59) as Rj =

Q

ϵj − H0

−→ R =

Q

ϵ − H0

,

(60)

to obtain i

⟨V ((V )V (V ))⟩j =

 i

⟨V R R V ⟩k k ⟨V R V ⟩j

k

= i ⟨V R R V V R V ⟩j ,

(61)

in place of Eq. (59), where VV in the second line means VPV . Let us compare the P-forms in Eqs. (59) and (61). The P-form in Eq. (59) is irreducible in the sense that it cannot be reduced to a product of independent P-space matrices [17]; the first matrix j k j i ⟨V R R V ⟩k is dependent on the index ‘‘j’’ of the second matrix k ⟨V R V ⟩j . The origin of the irreducibility can be traced back to the P-indices of R’s, and therefore to the nondegenerate P-space. On the other hand, the P-form in Eq. (61) is reducible because R’s do not carry the P-indices, and can be transformed easily into the second line to yield the following operator equation in degenerate P-space. V ((V )V (V )) = VRRVVRV .

(62)

The right hand side of Eq. (62) is a usual product of operators which transforms immediately into its P-form in Eq. (61). Then, one might tend to conclude that VRRVVRV in Eq. (62) is the desired representation which is to replace V ((V )V (V )). The above discussion holds, however, in degenerate P-space only. Let us go back to the irreducible P-form (59) in nondegenerate P-space. Though the P-indices of R’s do not allow the above simple argument leading to Eq. (62), we can prove that Eq. (62) is completely legitimate also in nondegenerate P-space. The right hand side of Eq. (62), which we will refer to as the sequence representation, is not any more a usual product of operators, but is an entangled product; the sequence VRRVVRV as a whole is a single operator which gives the irreducible P-form in Eq. (59). In the following, we prove the above nontrivial statement. 8.2. Valid sequence We define the sequence representation of (u) as follows. In each valid bracketing in Eq. (42), we replace each left bracket ‘‘(’’ with R, and eliminate each right bracket ‘‘)’’. We refer to the resultant expression as the sequence representation of (u), for each term of the expansion is a ‘‘sequence’’ of V ’s and R’s. We call a sequence is valid if it is obtained from a valid bracketing via the above process, as shown by the following examples. RRVVRV ←− ((V )V (V )), RRRVVRVVRV ←− (((V )V (V ))V (V )).

(63)

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K. Takayanagi / Annals of Physics 364 (2016) 200–247

By construction, a sequence does not stand for a normal product of V ’s and R’s, but represents an entangled product in nondegenerate P-space; a sequence as a whole represents a single operator as its corresponding bracketing. Similarly, we can transform the Bloch equation (34) into the sequence representation as Ru = RV + RVRu − RRuVRu.

(64)

It is then clear that the bracketing expansion (42) of (u) = Ru transforms into the following sequence expansion.

Ru =

∞ 

Ru

(n)

=

n =1

n V ′ s and n R′ s

∞ 



n =1

sequence

  

R ··· V ,

(65)

where the summation is over all possible valid sequences. The first few terms in Eq. (65) are given by Ru(1) = RV , Ru(2) = RVRV − RRVV , Ru

(3)

(66)

= RVRVRV − RVRRVV − RRVRVV + RRRVVV − RRVVRV ,

which corresponds to Eq. (41) in the bracketing representation. Note here that each combination ‘‘VV ’’, which necessarily derives from ‘‘)V ’’ in the bracketing representation, carries a minus sign on the right hand side. Now, we are at an important stage; in order to assert that the sequence is really a ‘‘representation’’ in the same way as the bracketing, we have to answer the following questions properly. First, given a sequence of n V ’s and n R’s, is it possible to tell whether it is a valid sequence or not? The answer is ‘‘Yes’’ as shown in Section 8.3. Second, is there a one-to-one correspondence between a valid bracketing and a valid sequence? The answer is ‘‘Yes’’ as shown in Section 8.4, and therefore it is always possible to translate a valid sequence into its corresponding valid bracketing, and vice versa. 8.3. Condition for valid sequence Here, we prove the following condition with which we can distinguish valid sequences from invalid ones. Proposition C (Condition for Valid Sequence). A sequence ‘‘R · · · V ’’ composed of n R’s and n V ’s represents a valid sequence, if and only if the following inequality holds at each V . NR ≥ NV ,

(67)

where NV is the order of the chosen V in the sequence, and NR is the number of R’s placed to the left of the chosen V . Before giving the proof, let us look into the following examples. 1 2

3

2 2

3

NV → NR →

RR V V R V ,

NV → NR →

R V V RR V ,

1 2

3

1 1

3

1 2

3 4

5

3 3

4 4

5

RRR V V R V V R V , 1 2 3 4

5

3 3 3 3

5

RRR V V V V RR V ,

— valid — invalid

where we have shown NV and NR explicitly for each V . Two examples in the first line represent valid sequences given in Eq. (63); we can confirm that they satisfy inequality (67) at each V . In the 2

second line, on the other hand, the first and second examples do not satisfy inequality (67) at V and 4

V , respectively. Then, Proposition C tells that they are invalid sequences. Now we give the proof of the proposition. In a valid sequence, because of the one-to-one correspondence between n V ’s and n pairs of brackets explained in Proposition A of Section 4.2, each

K. Takayanagi / Annals of Physics 364 (2016) 200–247

219

V has its corresponding R to its left. This means that inequality (67) is a necessary condition for valid sequence. Next, we prove by induction that inequality (67) is also a sufficient condition. First, we can easily observe in Eq. (66) that the proposition is true for n = 1, 2. Second, assuming that the proposition holds up to the (n − 1)th order, we examine an nth order sequence ‘‘R · · · V ’’ that satisfies inequality (67). Reading the sequence ‘‘R · · · V ’’ from left to right, we single out the V that first satisfies NV = NR , which is denoted as V . At the same time, we replace the leftmost R with a pair of brackets that encloses the whole sequence. For example, the above process transforms the sequences in Eq. (63) as 1 2

3

2 2

3

2

NV → NR →

RR V V R V −→ ( RV V RV ),

NV → NR →

RRR V V R V V R V −→ ( RRVVRV V RV ),

2

1 2

3 4

5

3 3

4 4

5

2 4

4

(68)

4

where V and the brackets are assigned the same number. In the first and second examples, we single 2

4

out V and V , respectively, to arrive at the right hand sides. Generally, the above step of choosing V classifies the given nth order valid sequence into the following three forms. n V ′ s and n R′ s n

1

  

m

−resolution −−−−→ (i) ( V · · ·), (ii) ( · · · V ), (iii) ( · · · V · · ·) ,

R ··· V

1

1

n

n

m

(69)

m

which we refer to as the resolution of sequence. A little consideration shows the following; with the above choice of V , each ‘‘· · · ’’ on the right hand side of (69) satisfies inequality (67) in itself, and is therefore a valid sequence by assumption.8 This means that the three forms of valid sequence in (69) are identified as the three types of valid bracketing in (43), showing that the given nth order sequence is valid. We can now conclude, by induction, that inequality (67) is a sufficient condition for valid sequence, which completes the proof of the proposition.  8.4. Correspondence to other representations In Section 8.3, we have proven the one-to-one correspondence between the resolution (69) of the sequence and the resolution (43) of the bracketing. Then, using the same discussion as in Section 6.1, we can easily show the one-to-one correspondence of each sequence to the bracketing and folded diagram, which in turn justifies the sequence expansion (65) of (u) = Ru. 8.4.1. Correspondence between sequence and bracketing Here we confirm the following one-to-one correspondence in practice.

v alid sequence ←→ v alid bracketing.

(70)

The left direction ←− is trivial by definition of the valid sequence. Therefore, we now explain the right direction −→ using examples.

8 We can easily confirm the following for the second example in Eq. (68). First, if we singled out 2 that does not satisfy V 2

NV = NR , we would have (RRV V RVVRV ) which looks like form (iii) in (69). However, this form does not correspond to the valid bracketing type (iii) in (43); both sequences RRV and RVVRV are invalid because they are not composed of the same 5

number of V ’s and R’s. Second, if we singled out V , which is not the first but the second V that satisfies NV = NR , we would 5

arrive at (RRVVRVVR V ) which looks like form (ii) in (69). However, this form does not correspond to the valid bracketing type (ii) in (43), because the sequence RRVVRVVR does not satisfy inequality (67) and is therefore invalid.

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Let us look into the valid sequences in Eq. (68). The first sequence, RRVVRV , can be transformed into its corresponding valid bracketing using the resolution (69) successively as follows. 1 2

3

2 2

3



2



RRVVRV −→ RR V V R V −→ ( RV V RV ) 1

2

2

2

3

−→ ((V ) V ( V )) . 21

1

3

(71)

32

In the first line, we proceed in the same way as in Eq. (68). Then, we find two inner sequences ‘‘RV ’’ 2

separated by V , which are indicated by braces for clarity. Then, because the resolution of RV gives (V ), we arrive at the valid bracketing in the second line. We can thus write down the following identity: 1

2 3

1

2

3

R R V R V V = ( ( V ) V ( V )), 2 1

3

21

1

3

(72)

32

where the left and right hand sides express the same operator in different representations. For clarity, R’s on the left hand side are numbered in accordance with the bracketing on the right hand side. Let us turn to the second sequence, RRRVVRVVRV , which can be treated in the same way to give 1 2

3 4

5

  

4



RRRVVRVVRV −→ RRR V V R V V R V −→ ( RRVVRV V RV ) 3 3 1

4 4

2

5

3

4

4

4

5

−→ ( ( ( V ) V ( V )) V ( V )) . 421

1

3

32

5

(73)

54

4

In the first line, we single out V as in Eq. (68) to make two inner valid sequences RRVVRV and RV 4

separated by V . Then, by using Eq. (71) for RRVVRV and replacing RV with (V ), we arrive at the last expression to prove 1 2

3 4

5

1

2

3

4

5

R R R V V R V V R V = ( ( ( V ) V ( V )) V ( V )) . 4 2 1

3

5

421

1

3

32

5

(74)

54

The above identities (72) and (74) replace each left arrow ←− in Eq. (63) by a double pointed arrow ←→, confirming the one-to-one correspondence (70). 8.4.2. Correspondence between sequence and folded diagram Because the one-to-one correspondences (46) and (70) hold, a valid sequence is uniquely expressed by a valid folded diagram, and vice versa, showing the following one-to-one correspondence:

v alid sequence ←→ v alid folded diagram.

(75)

We can confirm the above relation (75) in Fig. 8, which shows the identities (72) and (74) with their corresponding diagrams. 8.5. Sequence representation of v Now, we present v of Eq. (35) in the sequence representation. 8.5.1. Sequence expansion of v For (u), we have proven that the identity (72), RRVVRV = ((V )V (V )), holds in nondegenerate Pspace. For v , this means that the identity (62), VRRVVRV = V ((V )V (V )), is completely legitimate not only in degenerate P-space but also in nondegenerate P-space.

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221

The above observation can be easily generalized; using the expansion (65) of (u) = Ru, we can express v of Eq. (35) by the following sequence expansion.

v=

n−1 V ′ s and n−1 R′ s

∞ 

  



V

R ··· V

,

(76)

sequence

n =1

which reads explicitly as

v = V + VRV + VRVRV − VRRVV + VRVRVRV − VRVRRVV − VRRVRVV + VRRRVVV − VRRVVRV + · · · .

(77)

8.5.2. P-form of v The matrix element of Eq. (76) can be expressed as i

⟨v⟩j =

 i

⟨VR · · · V ⟩j .

(78)

P-form

Here, we show how to transform each sequence in Eq. (76) into its P-form in Eq. (78) in nondegenerate P-space, which is composed of two steps (i) and (ii). Let us take the sequence VRRVVRV in Eq. (62) as an example, which we now transform into its P-form in Eq. (59) as VRRVVRV

= ⟨VRRV ⟩⟨VRV ⟩  j k j −→ i ⟨V R R V ⟩k k ⟨V R V ⟩j .

(79)

k

In the first step (i) of the transformation, we put angle brackets in all possible places, showing where to take the P-space matrix elements. For obvious reasons, we refer to the expression ⟨VRRV ⟩⟨VRV ⟩ as the unlabeled P-form.9 Then, we draw the corresponding unlabeled P-form diagram on the left hand side of Fig. 14 representing VRRVVRV = ⟨VRRV ⟩⟨VRV ⟩; we first draw the diagram of RRVVRV via the resolution as explained in Section 8.4, and then put PVQ on top of the leftmost Q -line to arrive at the desired diagram. In the second step (ii), we label all P-lines with the P-indices properly, to fix the P-indices of all propagators alloted to the Q -crosses. We thus arrive at the labeled P-form diagram on the right hand side of Fig. 14, which in turn gives the labeled P-form in the second line of Eq. (79) immediately. At the end, we make several points. First, Eq. (79) explicitly shows that the sequence representation VRRVVRV , which is trivially equivalent to the unlabeled P-form ⟨VRRV ⟩⟨VRV ⟩, is much ‘‘closer’’ to its labeled P-form than the bracketing counterpart V ((V )V (V )) in Eq. (59). To transform a sequence to its labeled P-form, we simply need to attach the P-indices to the corresponding unlabeled P-form. The above observation shows that the sequence representation meets the motivation explained in Section 8.1.1. Second, we have studied the labeled P-form of i ⟨v⟩j in degenerate and nondegenerate P-spaces in Sections 8.1.2 and 8.5.2, respectively. Here we contrast these results using Eqs. (61) and (79) in Table 1. We can see that the sequence VRRVVRV is a normal product of operators in degenerate Pspace giving the reducible P-form i ⟨VRRV ⟩k k ⟨VRV ⟩j , while it is an entangled product in nondegenerate P-space yielding the irreducible P-form i ⟨VRj Rk V ⟩k k ⟨VRj V ⟩j .

9 The unlabeled P-form is a q-number (matrix), while the labeled P-form is a c-number (matrix element), as summarized in Appendix B.

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Fig. 14. Diagrammatic expression for the transformation (79) from the unlabeled to the labeled P-form. V ’s and R’s are numbered in the same way in both graphic and algebraic representations, which can be explained via Eq. (72).

Table 1 Sequence representation of v and its labeled P-form i ⟨v⟩j in degenerate and nondegenerate P-spaces, respectively, in the Bloch perturbation theory. The table shows the examples in Eqs. (61) and (79). See the text. Degenerate P-space

Nondegenerate P-space

Sequence

Normal product VRRVVRV

Entangled product VRRVVRV

Labeled P-form

Reducible i ⟨VRRV ⟩k k ⟨VRV ⟩j

i

v i

⟨v⟩j

Irreducible ⟨VRj Rk V ⟩k k ⟨VRj V ⟩j

9. Main frame expansion Here we introduce the ‘‘main frame expansion’’ from which we can derive directly both Bloch and RS perturbation expansions. We first introduce the main frame expansion in the bracketing representation for clarity, and then translate it into the sequence representation. The main frame expansion shall turn out to be an intermediate step on the way from the Bloch equation to the perturbation expansion of u and v .

9.1. Main frame expansion in bracketing representation

9.1.1. Main frame equation We have derived the Bloch perturbation theory of v in Section 7 starting from Eqs. (34) and (35). Here we transform these two equations into the following system of equations for u and v .

(u) = (V ) + (V (u)) − ((u)v), v = V + V (u),

(80) (81)

which are shown diagrammatically in Fig. 15. In the following, we refer to Eq. (80) as the main frame equation. Note the following point in Fig. 15; our diagram rule (III) in Section 5.1 guarantees that v is placed above (u) in the diagram of ((u)v) in Eq. (80), i.e., both lines entering and leaving v are P-lines in Fig. 15. We will soon see that the graphic iteration of Eq. (80) is crucially dependent on this property.

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223

Fig. 15. Diagrammatic expression for (u) in Eq. (80) and v in Eq. (81).

Fig. 16. Graphic expression for processes (q) and (p) of (83) in the bracketing representation and (90) in the sequence representation.

9.1.2. Main frame expansion The main frame equation (80) is a linear equation of u for a fixed v , and its iterative solution can be obtained immediately as

(u) = (V ) + (V (V )) − ((V )v) + (V (V (V ))) − (V ((V )v)) − ((V (V ))v) + (((V )v)v) + (V (V (V (V )))) − (V (V ((V )v))) − (V ((V (V ))v)) + (((V (V ))v)v) + ((V ((V )v))v) − ((((V )v)v)v) + · · · .

(82)

Note that the expansion (82) is obtained by repeated applications of the following two processes to (V ) in an arbitrary order.

(q) (· · ·) −→ (V (· · ·)),

(p) (· · ·) −→ −((· · ·)v),

(83)

which correspond to the second and third terms on the right hand side of the main frame equation (80), and are graphically shown in Fig. 16. Our diagram rule guarantees that processes (q) and (p) add a V on the Q -line and a v on the P-line, respectively, in Fig. 16. Then, by using these processes (q) and (p) repeatedly, we can easily draw the diagrams in Fig. 17 which displays several terms in the expansion (82). Now we present Eq. (82) in the following general expression:

(u) =

∞  ∞  

q V ′s

p v′ s

      (−1)p (V · · · (V · · · (V ) · · · v) · · · v),

(84)

q=0 p=0 order

which is shown diagrammatically in Fig. 18, and is referred to as the main frame expansion of (u) for its graphic structure. In Fig. 18, a general term is composed of the pivot V = QVP that appears as (V ) in Eq. (84), and q V ’s on the (main) Q -line, and p v ’s on the (main) P-line. Such a term is constructed by using process (q) of (83) q times and process (p) p times in all possible ‘‘orders’’ as indicated by

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Fig. 17. Graphic expression for main frame. Four diagrams represent ((V (V ))v), (V ((V )v)), ((V ((V )v))v), and (V (V ((V )v))) in Eq. (82) in the bracketing representation, and the corresponding sequences in Eq. (91) in the sequence representation.

Fig. 18. Graphic expression for the main frame expansion in Eqs. (84) and (92). In a general term, there are q V ’s and p v ’s on the main Q - and P-lines, respectively, which are arranged in all possible vertical orders. order in Eq. (84). For example, both ((V (V ))v) and (V ((V )v)) in Eq. (82) correspond to p = q = 1 in Eq. (84). These two terms are distinguished by the order of applications of processes (q) and (p), as can be confirmed in Fig. 17. A general term on the right hand side of the main frame expansion (84), which we refer to as the main frame, is given by the following steps.



p+q V ′ s

   • Take a bracketing (V (V (V · · · (V ) · · · ))), which is composed of p + q V ’s and the pivot V . • Choose p V ’s among the p + q V ’s. Then, eliminate each of the chosen V ’s as ‘‘(V ( −→ ((’’, and insert −v between the corresponding right brackets as ‘‘)) −→ −)v)’’. For example, the main frame ((V ((V )v))v) in Eq. (82) is obtained as follows: 1

2

3

4

2

4

3

1

(V (V (V ( V ) ) ) ) −→ (−1)2 ( = V (V ( = V ( V ) v) ) v), 1

3

(85) 1

3

where we have eliminated V and V between left brackets, and inserted − v and − v between the corresponding right brackets. Correspondingly, in the graphic representation of the main frame expansion in Fig. 18, a general term may be given as follows.

• Take a main frame diagram where the pivot V = QVP connects the main Q -line with p + q V ’s and the empty main P-line.

• Choose p V ’s among the p + q V ’s. Then, eliminate each of the chosen V ’s on the Q -line, and add −v on the P-line at the same vertical level.

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225

Fig. 19. Graphic expression for the process (85) in the bracketing representation, and (93) in the sequence representation.

1

3

In Fig. 19, we show the graphic expression for the process (85), which removes V and V from the Q 1

3

line, and puts −v and −v on the P-line at the corresponding vertical levels, leading to the right hand side of Fig. 19 that represents ((V ((V )v))v).

9.1.3. Main frame versus perturbation expansions By comparing the main frame and perturbation expansions, we notice the following two points. First, the structure of a general term is much simpler in the main frame expansion (84) than in the perturbation expansion (42). In the bracketing representation, the main frame is made up only with nestings; there are no multiple brackets like (· · · (· · ·) · · · (· · ·) · · · ). Accordingly, a general diagram in Fig. 18 is featured by the simple main frame structure that is composed of a single (main) Q line and a single (main) P-line connected at the pivot V . On the other hand, a general term in the perturbation expansion (42) has both nestings and multiple brackets, and its folded diagram is given by a complicated zigzag. Second, the correspondence between the algebraic and graphic representations is much simpler in the main frame expansion than in the perturbation expansion. Given a bracketing in the main frame expansion, we can draw the corresponding diagram immediately as follows. Let us note that two successive left brackets, which correspond to two successive Q -crosses in the diagram, appear in the bracketing in one of the following two forms:

(q) (V (,

(p) ((,

(86)

which are generated by processes (q) and (p) in (83), respectively. Then, we can easily see that form (q) indicates a V on the main Q -line between the two Q -crosses, and form (p) implies a v on the P-line at the corresponding vertical level. The above observation is sufficient to draw the diagram for a given main frame. We simply need to look into the left half of the main frame composed of the left brackets and V ’s; we do not need to perform the resolution of the whole bracketing, which is indispensable for the perturbation expansion (42) as explained in Section 6.1. Let us confirm the above two points using the example (V (((V ((V )v))v)v)) in Fig. 20. First, we note that the diagram shows the simple main frame structure without making a complicated zigzag, 4

which reflects the absence of multiple brackets. Second, we can see that ‘‘( V (’’ in the bracketing 4 4

5 5

indicates V on the main Q -line between the propagators 4 and 5, and ‘‘( (’’ implies v on the main P56

line between the propagators 5 and 6, and so on. It is clear that the left half (V (((V (( of the bracketing solely specifies the diagram completely in a much simpler manner than in the perturbation expansion in Section 6.1.

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Fig. 20. Simple correspondence between the main frame (V (((V ((V )v))v)v)) = RVRRRVRRV vvv and its graphic expression. Note that V ’s and v ’s are numbered as in Fig. 19. See the text.

9.2. Main frame expansion in sequence representation Here we present the main frame expansion in the sequence representation using the results in the bracketing representation in Section 9.1. The one-to-one correspondence (70) between the bracketing and sequence representations holds for the main frame expansion in the same way as for the perturbation expansion. It is then easy to see the following correspondences. RRVRRV vv ←→ ((V ((V )v))v), RVRVRRv ←→ (V (V ((V )v))),

(87)

which are shown in Fig. 17. Similarly, it is straightforward to translate all the results in Section 9.1 into the sequence representation, as we show below. 9.2.1. Main frame equation It is clear that Eqs. (80) and (81) take on the following forms in the sequence representation. Ru = RV + RVRu − RRuv,

(88)

v = V + VRu,

(89)

where Eq. (88) is the main frame equation. 9.2.2. Main frame expansion For a fixed v , we can solve Eq. (88) iteratively for u in the same way as in Section 9.1. We first rewrite processes (q) and (p) in (83) into the sequence representation as follows. (··· )

(··· )

      (q) R · · · −→ RV R · · · ,

(··· )

(··· )

      (p) R · · · −→ −R R · · · v,

(90)

where each brace indicates the sequence corresponding to each bracketing (· · ·) in (83). Then, by applying processes (q) and (p) in an arbitrary order to (V ) = RV , we obtain the following iterative solution for (u) = Ru: Ru = RV + RVRV − RRV v + RVRVRV − RVRRV v − RRVRV v + RRRV vv

+ RVRVRVRV − RVRVRRV v − RVRRVRV v + RRRVRV vv + RRVRRV vv − RRRRV vvv + · · · ,

(91)

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227

which is obviously the sequence representation of Eq. (82). We now express the expansion (91) in the following general way:

Ru =

q V ′ s and p+q+1 R′ s

∞  ∞  

(−1)

p





p v′ s



   R · · · RVR · · · R V v · · · v,

(92)

q=0 p=0 order

which is the main frame expansion in the sequence representation corresponding to Eq. (84), and is shown in Fig. 18. A general term of the expansion (92) is given by the following steps. p+q (RV )′ s

  • Take a sequence RVRV · · · RV RV . • Eliminate p V ’s in the sequence except for the rightmost (pivot) V , and multiply the obtained expression by p (−v)’s from the right. 

For example, the sequence RRVRRV vv in Eq. (87) is obtained as 1

2

3

4

R V R V R V R V

2

4

3

1

−→ (−1)2 R = VR V R= V R V v v,

(93)

which is the sequence counterpart to the process (85), and is shown in Fig. 19. 9.2.3. Main frame versus perturbation expansions The comparison of the main frame and the perturbation expansions can be summarized by the following two points. First, a general term has a much simpler form in the main frame expansion (92) than in the perturbation expansion (65); it reads from left to right as the Q -space operator R · · · RVR · · · R, the transition potential QVP between Q - and P-spaces, and the P-space operator v · · · v . The above plain structure (main frame structure) of the sequence reflects the absence of multiple brackets in the corresponding bracketing scheme. Second, the correspondence between the sequence and folded diagram representations is much simpler in the main frame expansion than in the perturbation expansion; for a given main frame in the sequence representation, we can immediately draw its folded diagram without the resolution of sequence. By translating the two forms in (86) into

(q) RVR,

(p) RR,

(94)

we realize that the diagram for the given sequence can be written down simply by looking into the Q -space operator R · · · RVR · · · R only. Let us confirm the above points using the example RVRRRVRRV vvv in Fig. 20. In the same way as in 4

4

the bracketing representation, we notice that ‘‘R V R ’’ in the sequence indicates V on the main Q -line 4

5

5

between the propagators 4 and 5, and ‘‘R R’’ implies the presence of v on the main P-line between 5 6

the propagators 5 and 6, and so on. Given the sequence RVRRRVRRV vvv , we can thus write down its diagram in Fig. 20 immediately by looking into its Q -space operator part RVRRRVRR only. This is in marked contrast to the situation of the perturbation expansion discussed in Section 8.4. 10. Bloch perturbation theory via main frame expansion Here, we derive the Bloch perturbation theory of v in Section 7 from a different viewpoint using the main frame expansion in the sequence representation, which gives a new insight into the Bloch theory.

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10.1. Bloch main frame expansion By substituting Ru = (u) of Eq. (92) into Eq. (89), we obtain the following main frame expansion of v .

v =V+

∞  ∞  

q V ′s





p v′ s



   (−1) V R · · · RVR · · · R V v · · · v . p

(95)

q=0 p=0 order

In the following, we transform Eq. (95) into its labeled P-form using steps (i) and (ii) explained in Section 8.5.2. As an example, let us take VRRVRRV vv in the expansion (95). In step (i), we draw the diagram representing VRRVRRV vv = ⟨VRRVRRV ⟩⟨v⟩⟨v⟩; we simply attach PVQ on top of the diagram for RRVRRV vv in Fig. 19. In step (ii), we fix all P-indices to obtain the diagram in Fig. 21, which yields the labeled P-form as VRRVRRV vv

= ⟨VRRVRRV ⟩⟨v⟩⟨v⟩ −→ i ⟨VRj Rl VRl Rk V ⟩k k ⟨v⟩l l ⟨v⟩j ,

(96)

where we have implicitly assumed the summation over repeated P-indices. The above example shows the following. First, step (i) here is much simpler than the corresponding step (i) in Section 8.5.2; we can draw the diagram for each main frame straightforwardly without the resolution of sequence, as we stressed in Section 9.2.3. Consequently, we can obtain the labeled P-form of the main frame expansion much more easily than that of the perturbation expansion. Second, R’s are visible only in the first factor j l l k i ⟨VR R VR R V ⟩k which stands for the main Q -line; the first factor represents the whole irreducibility of the P-form, which reflects the simple main frame structure stressed in Section 9.2.3. Therefore, the other factors k ⟨v⟩l and l ⟨v⟩j can be calculated as independent matrices, which is to simplify the iterative expansion in the next section in a drastic way. By generalizing the above discussion on the example (96), we can transform the main frame expansion (95) of v immediately into the following labeled P-form: q V ′s i

⟨v⟩j = i ⟨V ⟩j +

∞  ∞  





p ⟨v⟩′ s



   (−1) i ⟨V R · · · RVR · · · R V ⟩k k ⟨v⟩l · · · ⟨v⟩j p

j

k

q=0 p=0 order

=

 i

⟨VRj · · · RVR · · · Rk V ⟩k k ⟨v⟩l · · · ⟨v⟩j ,

(97)

P-form

where the second line is an abbreviation of the first line. We can easily see that Fig. 22 gives the graphic expression for Eq. (97).

10.2. Bloch perturbation expansion Here we derive the P-form of the perturbation expansion of i ⟨v⟩j by performing the iteration with the labeled P-form (97) of the main frame expansion.10 It is easy to see that the first few terms of the

10 Note the following. By substituting Eq. (84) into Eq. (81), we obtain the bracketing representation of Eq. (95). However, its iterative solution is the bracketing expansion (52) of v , which is not convenient to discuss the P-form of v as explained in Section 8.1.1. Therefore, we do not proceed in this direction.

K. Takayanagi / Annals of Physics 364 (2016) 200–247

Fig. 21. Diagrammatic expression for the labeled Bloch P-form i ⟨VRj Rl VRl Rk V ⟩k Eq. (96).

k

229

⟨v⟩l l ⟨v⟩j of the main frame VRRVRRV vv in

Fig. 22. Graphic expression for the labeled P-form (97) of the main frame expansion of i ⟨v⟩j . This is obtained simply by putting PVQ on top of the Q -line in Fig. 18.

resultant expansion are given by i

⟨v (1) ⟩j = i ⟨V ⟩j ,

i

⟨v (2) ⟩j = i ⟨VRj V ⟩j ,

i

i

⟨v (3) ⟩j = i ⟨VRj VRj V ⟩j −i ⟨VRj Rk V ⟩k k ⟨v (1) ⟩j = i ⟨VRj VRj V ⟩k −i ⟨VRj Rk V ⟩k k ⟨V ⟩j , ⟨v (4) ⟩j = i ⟨VRj VRj VRj V ⟩j −i ⟨VRj VRj Rk V ⟩k k ⟨v (1) ⟩j −i ⟨VRj Rk VRk V ⟩k k ⟨v (1) ⟩j + i ⟨VRj Rl Rk V ⟩k k ⟨v (1) ⟩l l ⟨v (1) ⟩j −i ⟨VRj Rk V ⟩k k ⟨v (2) ⟩j = i ⟨VRj VRj VRj V ⟩j −i ⟨VRj VRj Rk V ⟩k k ⟨V ⟩j −i ⟨VRj Rk VRk V ⟩k k ⟨V ⟩j + i ⟨VRj Rl Rk V ⟩k k ⟨V ⟩l l ⟨V ⟩j −i ⟨VRj Rk V ⟩k k ⟨VRj V ⟩j .

(98)

Correspondingly, by performing the iteration with the labeled P-form diagram in Fig. 22, we can easily obtain Fig. 23 which represents Eq. (98).

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Fig. 23. Labeled Bloch P-form diagrams for i ⟨v (2) ⟩j , i ⟨v (3) ⟩j , and i ⟨v (4) ⟩j in Eq. (98).

Carrying out the expansion in Eq. (98) up to infinite order, we obtain the labeled P-form of the Bloch perturbation expansion of i ⟨v⟩j as i

⟨v⟩j =

 i

⟨VRj · · · Rk V ⟩k k ⟨VR · · · Rl V ⟩l · · · ⟨VRj · · · Rj V ⟩j ,

(99)

P −form

which is shown graphically in Fig. 24. The greatest merit of this approach lies in the fact that the assignment of the P-index to each R is automatic; at each step of the iteration, we expand the labeled P-form (97) in which each visible R has been assigned the P-index already. This means that the present derivation of the P-form (99) is much more efficient than the direct derivation in Section 8.5. To realize the above statement, we now examine several examples of the iterative expansion of Eq. (97) to obtain Eq. (99). As a first example, let us look into the final term in i ⟨v (4) ⟩j in Eq. (98), which is the P-form of VRRVVRV obtained via the following expansion. i

⟨VRj Rk V ⟩k k ⟨v (2) ⟩j −→ i ⟨VRj Rk V ⟩k k ⟨VRj V ⟩j ,

(100)

which is expressed diagrammatically in Fig. 25. Here we start with the labeled P-form j k (2) ⟩j of the main frame on the left hand side of Eq. (100). Then, by expanding k ⟨v (2) ⟩j i ⟨VR R V ⟩k k ⟨v

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Fig. 24. Graphic expression for the Bloch perturbation expansion (99) of i ⟨v⟩j .

Fig. 25. Graphic expression for the expansion in Eq. (100).

as k ⟨VRj V ⟩j , we arrive at the labeled P-form of the perturbation expansion on the right hand side. The above example confirms the following advantages of the present method. First, given the main frame VRRV v , we can write down the left hand side of Eq. (100) immediately at the very beginning; we can assign the P-indices to R’s straightforwardly by drawing the main frame diagram on the left hand side of Fig. 25 without the resolution as explained in Section 10.1. Second, we can use the second order result k ⟨v (2) ⟩j = k ⟨VRj V ⟩j , which is obtained independently in advance, in going to the right hand side of Eq. (100). Here, it is particularly worth noting that Fig. 25 illustrates an efficient method to draw the folded diagram of a given P-form of the perturbation expansion. Suppose we are given the P-form j k j i ⟨VR R V ⟩k k ⟨VR V ⟩j on the right hand side of Eq. (100). Then, it is easy to go back to the main frame i

⟨VRj Rk V ⟩k k ⟨v (2) ⟩j on the left hand side, which in turn gives the main frame diagram on the left hand

side of Fig. 25 immediately. Then, by expanding k ⟨v (2) ⟩j diagrammatically as shown in the figure, we arrive at the desired P-form diagram on the right hand side of Fig. 25. In this method, we first reduce each term of the perturbation expansion to the main frame from which it derives, which we refer to as the reduction. We emphasize that the present method via the reduction is much more feasible than the method via the resolution in Section 8.5.2. As a second example, let us consider the following expansion which gives the Bloch P-form of VRRVRRVVRVVRV . i

⟨VRj Rl VRl Rk V ⟩k k ⟨v (2) ⟩l l ⟨v (2) ⟩j −→ i ⟨VRj Rl VRl Rk V ⟩k k ⟨VRl V ⟩l l ⟨VRj V ⟩j .

(101)

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Fig. 26. Graphic expression for the labeled Bloch P-form on the right hand side of Eq. (101).

Fig. 27. Graphic expression for the labeled Bloch P-form in the last line of Eq. (102).

Here we start with the labeled P-form i ⟨VRj Rl VRl Rk V ⟩k k ⟨v (2) ⟩l l ⟨v (2) ⟩j of the main frame on the left

hand side which is shown in Fig. 21. Then, by substituting k ⟨v (2) ⟩l = k ⟨VRl V ⟩l and l ⟨v (2) ⟩j = l ⟨VRj V ⟩j , we arrive at the labeled P-form of the perturbation expansion on the right hand side which is displayed in Fig. 26. Here again, we realize the advantage of starting from the main frame VRRVRRV vv ; if we started with VRRVRRVVRVVRV in the perturbation expansion, we would have to make all the way through the resolution of step (i) in Section 8.5.2 to arrive at Fig. 26 and the right hand side of Eq. (101). Note also the following; given the right hand side of Eq. (101), we can draw the diagram in Fig. 26 easily via the reduction, i.e., by reducing the right hand side of Eq. (101) to its main frame on the left hand side which is shown in Fig. 21. As a third example, we show the following two-step expansion which yields the Bloch P-form of VRRVVRRVVRV . i

⟨VRj Rk V ⟩k k ⟨v (4) ⟩l −→ i ⟨VRj Rk V ⟩k k ⟨VRj Rl V ⟩l l ⟨v (2) ⟩j −→ i ⟨VRj Rk V ⟩k k ⟨VRj Rl V ⟩l l ⟨VRj V ⟩j . j k

Here we start with the labeled P-form i ⟨VR R V ⟩k k ⟨v

(4)

(102) (4)

⟩l . Then, we expand k ⟨v ⟩l as k ⟨VR R V ⟩l l ⟨v (2) ⟩j j l

at the first iteration, and express l ⟨v (2) ⟩j as l ⟨VRj V ⟩j at the second iteration. We thus arrive at the last expression which is displayed in Fig. 27. Also in this example, we can easily confirm the advantage of the present method as in the previous examples. To summarize, we have derived the Bloch perturbation expansion (99) via the main frame expansion, which is much more feasible in practice than the direct derivation in Section 8.5. We have

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Fig. 28. Transformation of the main frame VRRVRRV vv = ⟨VRRVRRV ⟩⟨v⟩⟨v⟩ into the labeled RS P-form. See the text.

also obtained an efficient method via the reduction to draw the diagram for a given P-form of the perturbation theory. 11. RS perturbation theory via main frame expansion Here, we derive the RS perturbation theory which presents the Bloch P-form (99) in a different fashion. The present derivation via the main frame expansion (95) parallels the Bloch counterpart in Section 10, and is much simpler than the usual RS approach [17] based on the expansion of propagators especially in nondegenerate P-space. 11.1. RS main frame expansion In this section, by rewriting the labeled Bloch P-form (97) of the main frame expansion, we derive its RS counterpart. Let us describe the transformation of the main frame expansion (95) into the labeled P-form (97) in two steps, i.e., we start in degenerate P-space, and then set out for nondegenerate P-space. In other words, we assign the P-indices firstly to angle brackets, and secondly to propagators. For example, we present the transformation (96) in two steps as follows.

⟨VRRVRRV ⟩⟨v⟩⟨v⟩ −→ i ⟨VRRVRRV ⟩k k ⟨v⟩l l ⟨v⟩j −→ i ⟨VRj Rl VRl Rk V ⟩k k ⟨v⟩l l ⟨v⟩j .

(103)

First, in degenerate P-space, the main frame VRRVRRV vv = ⟨VRRVRRV ⟩⟨v⟩⟨v⟩ is simply a normal product of operators, and can be transformed immediately into the labeled P-form as shown in the first line. Second, we move onto nondegenerate P-space, i.e., we assign the P-indices to R’s by drawing Fig. 21, to end up with the desired P-form in the second line in Eq. (103). Here, we change the above second step from the first line to the second in Eq. (103). Instead of drawing the diagram in Fig. 21, we proceed as shown in Fig. 28. We start with the Bloch P-form ② in degenerate P-space, and move ⟨v⟩’s into the spaces between R’s in the P-space matrix i ⟨VRRVRRV ⟩k in the reverse order as shown by arrows, to arrive at the expression ③. Note that propagators (R’s) and interaction vertices (V ’s and ⟨v⟩’s) appear alternately in ③. Because the main frame is constructed via the processes (q) and (p) in Fig. 16, the above transformation of ② into ③ rearranges V ’s, ⟨v⟩’s, and R’s from right to left in the order in which they appear in Fig. 21 from bottom to top. This means that the expression ③ gives the P-indices of R’s in the same way as the folded diagram in Fig. 21; we simply look into the ‘‘range’’ of each P-line shown below ③, and attach the corresponding P-index to each R in that ‘‘range’’. For example, we find two R’s in the ‘‘range’’ of the P-line ‘‘l’’ denoted by l ⟨· · · · · ·⟩l , and

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Fig. 29. Diagrammatic expression for the labeled RS P-form of the main frame in Eq. (104). By projecting the diagram onto the horizontal line, we obtain the RS P-form as shown below the diagram.

assign the index ‘‘l’’ to these two R’s.11 We can thus arrive at the last expression ④ in nondegenerate P-space, establishing the following transformation of ① into ④.

⟨VRRVRRV ⟩⟨v⟩⟨v⟩ −→ i ⟨VRj l ⟨v⟩j Rl VRl k ⟨v⟩l Rk V ⟩k .

(104)

We refer to the right hand side of Eq. (104) as the labeled RS P-form, which is a rewrite of the labeled Bloch P-form in the second line of Eq. (103). Now, we make several points on Fig. 28. First, the P-space matrix multiplications in ② and ③ are performed in opposite directions. In the Bloch P-form ②, ⟨v⟩’s are arranged in the order of P-space matrix multiplications; the repeated P-indices appear as ‘‘· ·⟩k k ⟨··’’. In ③, on the other hand, ⟨v⟩’s are arranged in the opposite order; the repeated P-indices appear as ‘‘k ⟨· · · ·⟩k ’’, which we refer to as the ‘‘P-form product rule’’ in the RS theory [17].12 The above observation shows that, by preparing the P-form ③ in degenerate P-space via the P-form product rule, we can determine the P-indices of explicit R’s to get the P-form ④ in nondegenerate P-space. In the RS main frame, the P-form product rule not only determines the P-indices of angle brackets, but also fixes the P-indices of R’s in a simple manner. Second, we emphasize that the above derivation of the P-form in Eq. (104) is crucially dependent on the simple structure of the main frame; the fact that all explicit R’s are contained only in the first factor ⟨VRRVRRV ⟩ in ① has allowed the transformation in Fig. 28. Third, we define the RS diagram as follows. Let us recall that the order of V ’s, ⟨v⟩’s, and R’s in the RS P-form ④ is the same as the vertical order of the corresponding elements in the Bloch P-form diagram in Fig. 21. We define, therefore, the diagram for the RS P-form ④ by Fig. 29, which is obtained by rotating the Bloch diagram in Fig. 21 by π /2.13 Then, the horizontal order of each element in the RS diagram is the same as in the algebraic expression ④, as can be easily confirmed in Fig. 29; we can trace its origin back to our diagram rule (III) in Section 5.1 which fixes the vertical order of graphic elements in the Bloch diagram.

11 The internal P-indices ‘‘k’’ and ‘‘l’’ are assigned to R’s using the ‘‘diagonal range’’ as explained here. On the other hand, the external P-index ‘‘j’’ is assigned by looking into the ‘‘off-diagonal range’’ i ⟨· · · · · ·⟩j . 12 In Ref. [17], we call the P-form product rule as the ‘‘bracket product rule’’. In order to avoid possible confusion, we refrain from using ‘‘bracket’’ here because ‘‘bracket’’ means ‘‘round bracket’’ if not otherwise specified in this work. 13 Though we have drawn the Q -line horizontally in Ref. [17], here we draw the Q -line to point downwards to the left. The present convention is essential in unifying the Bloch and RS perturbation theories in Section 12.3.

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Fig. 30. Diagrammatic expression for the labeled RS P-form (105) of the main frame expansion of i ⟨v⟩j .

By generalizing the above discussion with Fig. 28, we can transform the main frame expansion (95) straightforwardly into the following labeled RS P-form: q V ′ s and p ⟨v⟩′ s i ⟨v⟩j = i ⟨V ⟩j +

∞  ∞  



  (−1)p i ⟨V Rj · · · ⟨v⟩j · · · RVR · · · k ⟨v⟩l · · · Rk V ⟩k

q=0 p=0 order

=

 i

⟨VRj · · · ⟨v⟩j · · · RVR · · · k ⟨v⟩l · · · Rk V ⟩k ,

(105)

P-form

which is shown graphically in Fig. 30.14 At the end, we stress the one-to-one correspondence between Eqs. (105) and (97) via the P-form product rule, and between Figs. 30 and 22 via the rotation of π /2. 11.2. RS perturbation expansion Having obtained the RS P-form (105) of the main frame expansion, we can derive the RS P-form of the perturbation expansion immediately.15 Let us solve Eq. (105) iteratively for i ⟨v⟩j . In this approach, each explicit R is automatically assigned the correct P-index because the labeled P-form is expanded at each step of the iteration as in the Bloch theory in Section 10.2. Then, it is easy to obtain i

⟨v (1) ⟩j = i ⟨V ⟩j ,

i

⟨v (2) ⟩j = i ⟨VRj V ⟩j ,

i

i

⟨v (3) ⟩j = i ⟨VRj VRj V ⟩j −i ⟨VRj k ⟨v (1) ⟩j Rk V ⟩k = i ⟨VRj VRj V ⟩k −i ⟨VRj k ⟨V ⟩j Rk V ⟩k , ⟨v (4) ⟩j = i ⟨VRj VRj VRj V ⟩j −i ⟨VRj VRj k ⟨v (1) ⟩j Rk V ⟩k −i ⟨VRj k ⟨v (1) ⟩j Rk VRk V ⟩k + i ⟨VRj l ⟨v (1) ⟩j Rk k ⟨v (1) ⟩l Rl V ⟩k −i ⟨VRj k ⟨v (2) ⟩j Rk V ⟩k = i ⟨VRj VRj VRj V ⟩j −i ⟨VRj VRj k ⟨V ⟩j Rk V ⟩k −i ⟨VRj k ⟨V ⟩j Rk VRk V ⟩k + i ⟨VRj l ⟨V ⟩j Rl k ⟨V ⟩l Rk V ⟩k −i ⟨VRj k ⟨VRj V ⟩j Rk V ⟩k .

(106)

In the graphic expression, the perturbation expansion of i ⟨v⟩j can be obtained by using Fig. 30 iteratively, as presented in Fig. 31. We thus obtain the following labeled P-form of the RS perturbation expansion. i

⟨v⟩j =

 i

⟨VRj · · · ⟨VRj · · · Rj V ⟩j · · · k ⟨VR · · · Rl V ⟩l · · · Rk V ⟩k ,

(107)

P-form

14 Fig. 30 corresponds to Fig.10 in Ref. [17]. 15 In the standard description of the RS perturbation theory, the RS P-form here is referred to as the RS ‘‘bracketing’’ [7,9,17].

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Fig. 31. Labeled RS P-form diagrams for i ⟨v (2) ⟩j , i ⟨v (3) ⟩j , and i ⟨v (4) ⟩j in Eq. (106). V ’s are numbered in accordance with the Bloch diagrams in Fig. 23.

which is the RS counterpart to Eq. (99), and is presented in Fig. 32. Let us recall here that the algebraic expression (105) of the main frame expansion is simply the projection of its graphic expression in Fig. 30. Then, because the iteration conserves the above property at each step, the RS perturbation expansion (107) is obtained by projecting Fig. 32 onto the horizontal line, which can be confirmed by the examples in Fig. 31. To explain the iterative expansion of Eq. (105) to obtain Eq. (107), we now look into the following examples. As a first example, let us consider the final term of ⟨v (4) ⟩ in Eq. (106). It is obtained as i

⟨V Rj k ⟨v (2) ⟩j Rk V ⟩j −→ i ⟨VRj k ⟨VRj V ⟩j Rk V ⟩k ,

(108)

which represents the RS counterpart to Eq. (100) of the Bloch theory, and is displayed in Fig. 33. Here we realize the following advantage of the present approach. First, as we explained in detail in Section 11.1, we can easily assign the P-index correctly to each explicit R on the left hand side of Eq. (108). Second, we can use the second order result, k ⟨v (2) ⟩j = k ⟨VRj V ⟩j which is calculated independently in advance, in going to the right hand side. Note here the following; in the same way as in the Bloch perturbation expansion of Section 10.2, the present example illustrates an efficient method to draw the folded diagram of a given RS Pform. Given the right hand side of Eq. (108), we can immediately go back to the RS main frame j (2) ⟩j Rk V ⟩j on the left hand side, which we call the process of reduction. Then, we can easily i ⟨V R k ⟨v

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Fig. 32. Graphic expression for the RS perturbation expansion (107) of i ⟨v⟩j .

Fig. 33. Graphic expression for the expansion in Eq. (108).

draw the corresponding main frame diagram on the left hand side of Fig. 33. At the end, by expanding (2) ⟩j diagrammatically, we arrive at the desired P-form diagram on the right hand side of Fig. 33. k ⟨v As a second example, let us consider the following seventh order term. i

⟨VRj l ⟨v (2) ⟩j Rl VRl k ⟨v (2) ⟩l Rk V ⟩k −→ i ⟨VRj l ⟨VRj V ⟩j Rl VRl k ⟨VRl V ⟩l Rk V ⟩k ,

(109)

which is shown in Fig. 34. Here we start with the main frame on the left hand side, which is displayed in Fig. 29. Then, by using k ⟨v (2) ⟩l = k ⟨VRl V ⟩l and l ⟨v (2) ⟩j = l ⟨VRj V ⟩j which can be calculated independently of other factors, we arrive at the RS P-form of the perturbation expansion on the right hand side. Fig. 34 shows that the RS P-form of Eq. (109) is simply the projection of the corresponding diagram onto the horizontal line. Note that Eq. (109) and Fig. 34 correspond to Eq. (101) and Fig. 26 in the Bloch theory. As a third example, we show the following sixth order term. i

⟨VRj k ⟨v (4) ⟩l Rk V ⟩k −→ i ⟨VRj k ⟨VRj l ⟨v (2) ⟩j Rl V ⟩l Rk V ⟩k −→ i ⟨VRj k ⟨VRj l ⟨VRj V ⟩j Rl V ⟩l Rk V ⟩k ,

(110)

which is shown in Fig. 35. We can easily understand that Eq. (110) and Fig. 35 are the RS counterpart to Eq. (102) and Fig. 27 of the Bloch theory. To summarize, we have presented the derivation of the RS perturbation expansion (107) via the main frame expansion, which parallels the Bloch counterpart in Section 10.

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Fig. 34. Graphic expression for the labeled RS P-form in Eq. (109). V ’s are numbered in accordance with the corresponding Bloch diagram in Fig. 26.

Fig. 35. Graphic expression for the labeled RS P-form in Eq. (110). V ’s are numbered in accordance with the corresponding Bloch diagram in Fig. 27.

12. Unified description of perturbation theories We have derived both Bloch and RS perturbation theories in Sections 10 and 11, respectively, via the main frame expansion. Here we establish the ‘‘unified representation’’ which clearly shows the relation between the Bloch and RS perturbation theories in both algebraic and graphic representations. 12.1. Main frame expansion in unified representation In this section, we present the Bloch and RS main frame expansions of v in a unified way on the basis of the results in Sections 10.1 and 11.1. In the algebraic representation, we have shown that the Bloch and RS P-forms (97) and (105) for i ⟨v⟩j , which we show below again for convenience, transform into each other via the P-form product rule in Section 11.1. i

⟨v⟩j =

 i

⟨VRj · · · Rk V ⟩k k ⟨v⟩l · · · ⟨v⟩j ,

i

⟨VRj · · · ⟨v⟩j · · · k ⟨v⟩l · · · Rk V ⟩k .

(97)

P −form

=

 P −form

(105)

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239

Fig. 36. Unified representation of Eq. (96) and Fig. 21 in the Bloch theory and Eq. (104) and Fig. 29 in the RS theory. By projecting the diagram onto the abscissa and ordinate, we obtain the Bloch and RS P-forms, respectively.

Fig. 37. Unified representation of Eq. (97) and Fig. 22 in the Bloch theory and Eq. (105) and Fig. 30 in the RS theory.

In the graphic representation, the Bloch and RS P-forms in Figs. 22 and 30 transform into each other by a rotation of π /2 by definition. Moreover, the above algebraic expression in each theory is the projection of its corresponding graphic expression onto the horizontal line. The above correspondence can be confirmed by comparing Eq. (96) and Fig. 21 in the Bloch theory to Eq. (104) and Fig. 29 in the RS theory, all of which represent the same labeled P-form in different theories and representations. Now, let us look into Fig. 36, where Fig. 29 is superimposed on Fig. 21 after a rotation of π /2. The notable point is that Fig. 36 presents the above four expressions in a unified fashion; the Bloch and RS P-forms (96) and (104) are obtained simply by projecting the single folded diagram onto the abscissa (Bloch axis) and the ordinate (RS axis), respectively. We refer to the representation in Fig. 36 as the unified representation. At the end, we generalize the above discussion straightforwardly to obtain Fig. 37, which shows Eq. (97) and Fig. 22 of the Bloch theory and Eq. (105) and Fig. 30 of the RS theory in the unified representation. 12.2. Perturbation expansion in unified representation In this section, we present the Bloch and RS perturbation expansions of v in a unified fashion using the results in Sections 10.2 and 11.2. The Bloch iteration with Eq. (97) and Fig. 22 can be performed in parallel with the RS iteration with Eq. (105) and Fig. 30 using the unified representation in Fig. 37. Then, we obtain naturally the unified

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Fig. 38. Unified representation of Eq. (99) and Fig. 24 in the Bloch theory and Eq. (107) and Fig. 32 in the RS theory.

Fig. 39. Unified representation for Eq. (111). The Bloch and RS P-forms are projections of a single folded diagram onto the abscissa (Bloch axis) and the ordinate (RS axis), respectively.

representation of the perturbation expansion in Fig. 38, which displays simultaneously Eq. (99) and Fig. 24 in the Bloch theory and Eq. (107) and Fig. 32 in the RS theory clarifying their interrelations. Here, the Bloch and RS P-forms (99) and (107), which we show below again, are simply the projections of the single diagram onto the abscissa (Bloch axis) and the ordinate (RS axis), respectively. i

⟨v⟩j =

 i

⟨VRj · · · Rk V ⟩k k ⟨VR · · · Rl V ⟩l · · · ⟨VRj · · · Rj V ⟩j ,

i

⟨VRj · · · ⟨VRj · · · Rj V ⟩j · · · k ⟨VR · · · Rl V ⟩l · · · Rk V ⟩k .

(99)

P −form

=



(107)

P −form

Moreover, the unified representation shows clearly how to move from one representation to another. For example, the transformation of a given Bloch P-form into the RS P-form can be done as follows; we first draw the folded diagram of the given Bloch P-form, and then project the diagram onto the ordinate (RS axis) to obtain the desired RS P-form. The above transformation makes full use of the unified representation, and is much more feasible than using the P-form product rule repeatedly.16 To get familiarized with the unified representation, we show several examples in the following.

16 Because the Bloch and RS P-forms of the main frame expansion transform into each other via the P-form product rule, the Bloch and RS P-forms of the perturbation expansion turn into each other by using the P-form product rule repeatedly. This is not, however, very feasible for higher order terms.

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Fig. 40. Unified representation for Eq. (112). For simplicity, we have omitted R’s in the diagram and P-indices in the algebraic expressions. Other notation is the same as in Fig. 39.

The first example is the following fourth order term in Fig. 39 which presents the P-forms in Figs. 25 and 33 in a unified fashion. 0

i

1

2

3

⟨V Rj Rk V ⟩k k ⟨V Rj V ⟩j

0

=

i

1

2

3

⟨V ((V ) V (V ))⟩j 0

2

3

1

←→ i ⟨V Rj k ⟨V Rj V ⟩j Rk V ⟩k ,

(111)

where the left and right hand sides are the Bloch and RS P-forms in Eqs. (100) and (108), respectively. On the Bloch side, we show also the bracketing representation for comparison; we realize that the sequence (not the bracketing) representation of the Bloch theory clarifies its relation to the RS perturbation theory. The second example is the seventh order P-form in Fig. 40 which present Eq. (101) and Fig. 26 in the Bloch theory and Eq. (109) and Fig. 34 in the RS theory in the unified representation. 0

i

1

2

3

5

4

6

⟨V Rj Rl V Rl Rk V ⟩k k ⟨V Rl V ⟩l l ⟨V Rj V ⟩j

=

0

i

1

2

3

4

5

6

⟨V (V ((V ) V (V )) V (V ))⟩j 0

5

6

1

3

4

2

←→ i ⟨V Rj l ⟨V Rj V ⟩j Rl V Rl k ⟨V Rl V ⟩l Rk V ⟩k . (112) The third example is the following sixth order P-form shown in Fig. 41. 0

i

1

2

3

4

5

⟨V Rj Rk V ⟩k k ⟨V Rj Rl V ⟩l l ⟨V Rj V ⟩j

=

0

i

1

2

3

4

5

⟨V ((V ) V ((V ) V (V )))⟩j 0

2

4

5

3

1

←→ i ⟨V Rj k ⟨V Rj l ⟨V Rj V ⟩j Rl V ⟩l Rk V ⟩k .

(113)

It is now easy to see that Fig. 41 represents Eq. (102) and Fig. 27 in the Bloch theory and Eq. (110) and Fig. 35 in the RS theory in a unified way. 12.3. Summary of unified description We can summarize the results in Sections 8–12 by the following two points; first, we have derived the Bloch and RS perturbation theories in a unified fashion. Second, we have clarified their interrelations.

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Fig. 41. Unified representation for Eq. (113). Other notation is the same as in Fig. 40.

Fig. 42. Unified derivation of the Bloch and RS perturbation theories via the main frame expansion. See the text.

The first point is shown schematically in Fig. 42. We have derived the perturbation expansion by developing the main frame expansion iteratively in the sequence representation, which can be carried out in parallel for the Bloch and RS theories. The second point is summarized in Fig. 43. We have established the unified representation shown in the middle of the figure, which gives all four representations of v simultaneously (graphic and algebraic representations in Bloch and RS theories), clarifying their relations explained in Sections 10– 12.17

17 In Fig. 43, the correspondence between the graphic and algebraic representations of the Bloch theory is denoted by the projection and reduction as explained in Section 10, though it was first proven via the resolution in Section 8.4. This is because the correspondence via the projection and reduction is simpler and more feasible than via the resolution in practice.

K. Takayanagi / Annals of Physics 364 (2016) 200–247

243

Fig. 43. Unified description of effective interaction v . The unified representation in the middle presents all four expressions of

v with their interrelations. See the text.

13. Discussion In the present approach, we have used the main frame expansion to derive the Bloch and RS perturbation theories of v . In the conventional RS approach, on the other hand, we use the expansion of various propagators around the free propagator.18 Here we examine our results from the conventional RS viewpoint in the hierarchy of perturbation theories classified by the definition of P-space, i.e., onedimensional P-space, degenerate multi-dimensional P-space, and the most general nondegenerate multi-dimensional P-space.

13.1. One-dimensional P-space In one-dimensional P-space (d = 1), Eq. (6) becomes H |Ψ ⟩ = E |Ψ ⟩,

(114)

and Eq. (12) reads as

|Ψ ⟩ = |φ⟩ +

Q E−H

V |φ⟩.

(115)

In this case, the effective interaction v is given by the following c-number:

v = ∆E = E − ϵ = ⟨φ|V |Ψ ⟩,

(116)

where the unperturbed energy ϵ is assigned to the P-space state |φ⟩ which is normalized as ⟨φ|φ⟩ = 1.

18 Note the distinction between the ‘‘RS perturbation theory’’ and the ‘‘RS approach’’. The ‘‘RS perturbation theory’’ is the theory developed in Section 11. The ‘‘RS approach’’ means that we expand various propagators around the free one.

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Now, we derive the main frame expansion in the RS approach. By substituting Eq. (115) into Eq. (116), we obtain



Q



v = ⟨V ⟩ + V V E−H         −v −v −v = V + VRV + VR RV + VR R RV + · · · V

V

V

 = ⟨VRV · · · R(−v)R · · · R VR · · · R(−v)R · · · RV ⟩,

(117)

term

where the angle brackets ⟨· · ·⟩ mean ⟨φ| · · · |φ⟩. In the second line, we have expanded the full propagator around the free propagator R = ϵ−QH as 0

Q E−H where

  −v V

=

Q

ϵ + v − H0 − V

=R+R

  −v V

R+R

    −v −v V

R

V

R + ··· ,

(118)

means that we choose either −v or V in writing down a specific term in the expansion,

and we sum up over all possibilities. Then, we recognize that the resultant expression, the third line of Eq. (117), is the RS main frame expansion (105) for v in one-dimensional P-space. We thus obtain the Bloch and RS main frame expansions as follows.

v =



⟨VR · · · RV ⟩ v · · · v,

(119)

⟨VR · · · v · · · v · · · RV ⟩,

(120)

P −form

=

 P −form

which correspond to Eqs. (97) and (105), respectively, and are trivially identical because v is a c-number. At the end, let us stress that Eq. (117) clarifies the meaning of the main frame expansion Q in one-dimensional P-space; it is simply the double series expansion of E − = ϵ+v−QH0 −V in powers H of V and v . Let us turn to the perturbation expansion. By carrying out the iteration with Eqs. (119) and (120), we obtain the Bloch and RS P-forms as

v =



⟨VR · · · RV ⟩ ⟨VR · · · RV ⟩ · · · ⟨VR · · · RV ⟩,

(121)

⟨VR · · · ⟨VR · · · RV ⟩ · · · ⟨VR · · · RV ⟩ · · · RV ⟩,

(122)

P −form

=

 P −form

which represent Eqs. (99) and (107), respectively, with all P-indices removed. Here, the second line is the well-known RS ‘‘bracketing’’ expansion for v = ∆E [9,17], and the first line is its Bloch counterpart. In one-dimensional P-space, the RS approach thus yields both Bloch and RS perturbation expansions straightforwardly. 13.2. Degenerate multi-dimensional P-space In degenerate P-space where ϵ1 = · · · = ϵd = ϵ in Eq. (2), we have the free propagator R = ϵ−QH , 0 and therefore we can expand the full propagator around R. It is then straightforward to generalize the RS approach in one-dimensional P-space of Section 13.1 to degenerate multi-dimensional Pspace [17]; we can embed the P-space matrix structure in Eqs. (119) and (120), to obtain the Bloch and RS main frame expansions as i

⟨v⟩j =

 i

⟨VR · · · RV ⟩k k ⟨v⟩l · · · ⟨v⟩j ,

(123)

i

⟨VR · · · ⟨v⟩j · · · k ⟨v⟩l · · · RV ⟩k .

(124)

P −form

=

 P −form

K. Takayanagi / Annals of Physics 364 (2016) 200–247

245

Then, by expanding Eqs. (123) and (124) iteratively, we arrive immediately at the following P-forms of the perturbation expansion via the RS approach. i

⟨v⟩j =

 i

⟨VR · · · RV ⟩k k ⟨VR · · · RV ⟩l · · · ⟨VR · · · RV ⟩j ,

(125)

i

⟨VR · · · ⟨VR · · · RV ⟩j · · · k ⟨VR · · · RV ⟩l · · · RV ⟩k ,

(126)

P −form

=

 P −form

which represent Eqs. (99) and (107), respectively, with all propagators being stripped of the P-indices. 13.3. Nondegenerate multi-dimensional P-space In nondegenerate P-space, we are to realize the difference between the present and the RS approaches. In the present approach, we have derived both Bloch and RS perturbation expansions (99) and (107) in a straightforward manner directly in nondegenerate P-space. In the RS approach, on the other hand, we find that the situation changes largely from Section 13.2. Because the P-space states do not have the common reference energy ϵ , the system does not define the free propagator R = ϵ−QH . This means that we cannot use the RS approach in Section 13.2 as it stands. 0 Instead, we have to make the following detour [17]. First, we introduce an artificially degenerate Pspace at ϵ˜ to define the free propagator R˜ = ϵ˜ −QH . Second, we proceed as in Section 13.2, to arrive at the 0

˜ Third, we introduce the perturbation perturbation expansion (126) with each R being replaced with R. P (H0 − ϵ˜ )P to infinite order, to prove the RS expression (107) in nondegenerate P-space. 13.4. Present versus RS approaches The above discussion has clarified the following. In degenerate P-space, the RS approach gives the main frame and the perturbation expansions as immediately as the present approach. In nondegenerate P-space, on the other hand, the present approach has considerable advantages over the RS approach, which we summarize below. First, the present approach is straightforward, while the RS approach, though completely legitimate, makes a long detour by introducing an artificially degenerate P-space to arrive at the final results in nondegenerate P-space. Second, the present approach yields the Bloch and RS perturbation expansions in parallel as the iterative solutions of the main frame expansion as summarized in Fig. 42. The RS approach, on the other hand, does not give such a unified description. Consequently, the present approach alone clarifies the interrelation between the Bloch and RS perturbation theories in Fig. 43. 14. Summary The perturbation expansion of the effective interaction v has been formulated in the Bloch and Rayleigh–Schrödinger (RS) theories. However, the interrelation between these two theories has been left unclear to date. In this situation, we have presented the following results in this work. First, we have reconstructed the Bloch theory using the ‘‘effective transition potential’’, in such a way that establishes a robust one-to-one correspondence between the algebraic (Bloch bracketing) and graphic (Bloch folded diagram) representations; we have made the description of the Bloch perturbation theory as precise as the RS counterpart. Second, we have introduced the ‘‘sequence representation’’, an alternative to the Bloch bracketing representation, that plays the role of a translator between the Bloch and RS theories. Then, we have derived the ‘‘main frame expansion’’ of v , which in turn has given the Bloch and RS perturbation expansions in the sequence representation in parallel as shown in Fig. 42.

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Third, we have established the ‘‘unified representation’’ of v which presents both Bloch and RS perturbation theories simultaneously, clarifying their interrelation in the algebraic and graphic representations as shown in Fig. 43. At the end, we have shown that our approach, being applicable directly in general nondegenerate P-space, is simpler and more coherent than the standard RS approach based on the expansion of propagators. Having presented the unified description of the perturbation theories of the effective interaction in nondegenerate P-space, we have increased the universality of the concept of the effective interaction more than ever. We believe that the present approach is to take the place of the conventional RS approach in every perturbative investigation of quantum systems. Appendix A. Matrix element of bracketing We have defined the bracketing notation in Eq. (27). Here, we explain several matrix elements which appear in the text in the bracketing notation. First, let us consider the following matrix element. I

(AB)j = ⟨⟨I |(AB)|j⟩ =

1

ϵj − ϵI

I

⟨⟨AB⟩j = RjI I ⟨⟨AB⟩j ,

(A.1)

where A and B are arbitrary operators. Eq. (A.1) shows that a bracketed product of operators is not a usual product of operators any more. Second, let us consider a nested bracketing scheme (A(B)). In evaluating its matrix element, we first lift the outmost brackets via Eq. (27), and then move onto the inner brackets as follows. I

(A(B))j = ⟨⟨I |(A(B))|j⟩ = RjI ⟨⟨I |A(B)|j⟩ =



j

RI ⟨⟨I |A|J ⟩⟩ ⟨⟨J |(B)|j⟩

J

=



j RI I

⟨⟨A⟩⟩

j J RJ J

⟨⟨B⟩j .

(A.2)

J

In the same way, we can show I

((A)B)j = ⟨⟨I |((A)B)|j⟩ = RjI ⟨⟨I |(A)B|j⟩ =



j

RI ⟨⟨I |(A)|k⟩ ⟨k|B|j⟩

k

=



j

RI RkI I ⟨⟨A⟩k k ⟨B⟩j .

(A.3)

k

Third, we consider a multiple bracketing scheme ((A)V (B)). Its matrix element can be evaluated as I

((A)V (B))j = ⟨⟨I |((A)V (B))|j⟩ = RjI ⟨⟨I |(A)V (B)|j⟩  j = RI ⟨⟨I |(A)|k⟩⟨k|V |J ⟩⟩⟨⟨J |(B)|j⟩ kJ

=



j

j

RI RkI I ⟨⟨A⟩k k ⟨V ⟩⟩J RJ J ⟨⟨B⟩j .

(A.4)

kJ

The above examples are sufficient to explain all matrix elements in the text. Appendix B. Labeled and unlabeled P-forms Here, we tabulate the labeled and unlabeled P-forms in the Bloch and RS theories in Table B.1. An unlabeled P-form represents a matrix and is an operator in P-space, and a labeled P-form is its matrix element.

K. Takayanagi / Annals of Physics 364 (2016) 200–247 Table B.1 Examples of unlabeled and labeled P-forms in the Bloch and RS theories.

Bloch P-form RS P-form

Unlabeled (matrix)

Labeled (matrix element)

⟨VRRV ⟩⟨VRV ⟩ ⟨VR⟨VRV ⟩RV ⟩

i

i

⟨VRj Rk V ⟩k k ⟨VRj V ⟩j ⟨VRj k ⟨VRj V ⟩j Rk V ⟩k

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