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Diagrammatic techniques in the Martin-Schwinger-Puff many-body theory
ANNALS
OF
PHYSICS:
75, 77-102 (1973)
Diagram’matic
Techniques in the Martin-Schwinger-Puff Many-Body Theory* JAMESH. CRICHTON
Seattle Pacific College Institute for Research, Seattle, Washington 98119
AND
SIMON Yu Department of Physics, University of Washington, Seattle, Washington 98105
ReceivedOctober22, 1971 In an attempt to understandthe connectionbetweenthe Martin-Schwinger-Puff Green’sfunctionapproachto the many-bodyproblemand the diagrammatic approach of BruecknerandGoldstone,the perturbationexpansionof the Green’sfunctiontheory is studied.This expansion leads naturally to a representation of the theory by two sets of time-dependent diagrams-the kinetic energy diagramsand the potential energy diagrams.Rulesare givenfor writing eachtime-dependent diagramasa sumof timeindependentdiagrams.Complications arisein the applicationof theserulesto bubble diagrams. A special simple treatment of these diagrams is then proposed. There exists a simple relation between families of kinetic diagrams and potential diagrams. This relation stems from a rather general thermodynamic identity. Both the relation and its
thermodynamicorigin are described.As a consequence of thesestudies,someconnections between the Brueckner-Goldstone made.
theory and the Green’s function theory are
1. INTRODUCTION In the Martin-Schwinger-Puff finite temperature Green’s function approach to the many-body problem [l], the physical quantities of interest-particle number and energy density-are very simply given in terms of G1 , the full single-particle propagator. To find G1, we have to solve an infinite hierarchy of Green’s function equations, which are essentially Schroedinger’s equation combined with particle statistics. The standard way of solving these equations, of course, is to terminate this infinite hierarchy by some approximation scheme[2]. Another way of solving these equations is to expand all interaction-dependent quantities in powers of the * Supported in part by the U. S. Atomic Energy Commission and the Research Corporation.
77 Copyright All rights
0 1973 by Academic Press, Inc. of reproduction in any form reserved.
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potential. This method of perturbation expansion is in principle equivalent to other perturbation theories. However, in practice, other perturbation theories, such as that of Brueckner and Goldstone [3], seem to be far simpler. Nevertheless, in the present paper, we wish to consider perturbation expansions in the Green’s function approach. The reason is the following: Much has been learned about many-body systems through studies in both the Green’s function approach as well as the Brueckner-Goldstone approach. However, the techniques developed within each of these two frameworks seem to be totally different. To understand how techniques in one formalism may be applied to the other, we wish to follow closely how an expression or a statement in one approach evolves into an expression within the framework of the other. To go from the Green’s function approach to the Brueckner-Goldstone approach, a perturbation expansion in the Green’s function approach seems to be the most natural first step. Carrying out an expansion of the Green’s function equations, the resulting perturbation theory is given in terms of an infinite set of diagrams. Two features characterize the diagrams in this theory. First, the diagrams represent integrals over momentum as well as imaginary time. This is natural since the Green’s function hierarchy, from which the diagrams stem, are integral equations over momentum and imaginary time. Second, it is most natural to write the energy in terms of two sets of diagrams, one set for the potential energy, and another set for the kinetic energy. In contrast to these two features, in the Brueckner-Goldstone theory, the energy is represented by only one set of diagrams which are integrals over momentum alone. The first step in our study is to carry out the integration over time. This is usually a tedious, even if straightforward, calculation. We have devised some diagrammatic rules to replace this calculation. The net result of these rules is to replace each time-dependent diagram by a set of time-dependent diagrams. Since there are two different kinds of time-dependent diagrams, corresponding to the potential energy and kinetic energy contributions, we have correspondingly two sets of time-independent diagrams. We have given explicit rules for these four different kinds of diagrams (which we call the vg diagram, tg diagram, v f diagram, and tf diagram, respectively). The relations between the g diagrams and the f diagrams are given. These rules in general simplify the calculations. However, the theory is hampered by one drawback: many of the f diagrams so defined are divergent due to zero denominators. These divergences appear in bubble diagrams, as well as in all kinetic diagrams. In principle, there is nothing wrong: all that one has to do is to combine several suchfdiagrams, and using either L’Hospital’s rule, or methods of regularization [4], one can arrive at finite results. However, this process destroys much of the simplicity of the rules.
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We have made a separate study of these bubble diagrams, and other bubble-like diagrams. The relevant feature characterizing these diagrams is what we call the n-chain. We have found that the bubble diagrams are related to the corresponding diagrams in which all the bubbles are “popped”. This relation allows us to bypass the problem of divergent integrals altogether. It is interesting that the same mathematics that is used to relate the bubble diagrams to the nonbubble diagrams leads to another useful relation. We find that within a certain well-defined set of Green’s function diagrams, a set which we call the family, a simple relation exists between the kinetic energy contributions and the potential energy contributions. If t and u are the kinetic and potential contributions, respectively, of a family of n-th order diagrams, the relation is given by nt=-
i
n-l-/I-f-
aP 1
V.
In the zero-temperature limit, the samerelation reducesto
nt = -(n - 1)~. Thus, as long as an entire family is included, we will have the total energy by studying the potential diagrams alone. Such a simple relation indicates a simple physical origin. Upon investigation, we found that indeed the relation stems from a rather general thermodynamic identity. In the zero-temperature limit, this thermodynamic identity reducesto the well-known Feynman identity [5]. Having devefoped these techniques, we make a very simple observation: except for a factor of l/2, the vf diagrams, i.e., the time-independent potential diagrams, give contributions exactly equal to those of the Brueckner-Goldstone diagrams. Gathering the results of these studies, we can identify a family of Green’s function diagrams with a set of Brueckner-Goldstone diagrams. We can then draw some connections between the two approaches. In Section 2 we describe perturbation expansions in the Green’s function approach. As the final product of this expansion, we can expressGin, the n-th order term in the expansion of the full single-particle propagator, as a sum of diagrams each of which is an integral over momentum and time, the integrand being a product of Gy’s, the single-particle propagator with no interaction, and V, the interaction, alone. In Section 3 we start by stating the precise rules for these integrals, which we will call “g” diagrams. Each g diagram will give a contribution to the potential energy, which we will call a “vg” diagram, as well as to the kinetic energy, which we will call the “fg” diagram. Carrying out the time integration, we show that each “vg” diagram gives rise to a set of “vf” diagrams, and each “tg” diagram gives 595/75/1-6
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CRICHTON AND YU
rise to a set of “tf” diagrams, where an f diagram is an integral over momentum alone. The rules for the f diagrams, as stated, give some divergent integrals. In Section 4 we will discuss how these diagrams can be treated. In Section 5, we show that within a family of g diagrams, a simple relation exists between the kinetic energy and the potential energy. We will indicate the thermodynamic origin of this relation. As a special, but important case of this relation, we consider the zero-temperature limit. In Section 6 we consider some consequences of these techniques in comparing Green’s function theories with Brueckner-Goldstone theories and also in Green’s function perturbation expansions. In Appendix 1 the proof of the thermodynamic relation of Section 5 is given. In Appendix 2 we prove the result of the zero-temperature limit of Section 5 by a different method.
2. THE GREEN'S FUNCTION PERTURBATION EXPANSION In the Green’s function formalism for the finite temperature many-fermion problem, we have a set of Green’s functions satisfying the following infinite hierarchy of coupled integral equations [l]: GA1 . . . n; 1’ . . . n’)
= G,O(l, n + 1) V(n + 1, n + 2) G,+,[23 ... n + 1 IZ + 2; (n + 2)+ 1’ *a. n’] +
5 (-l)j’+l j’zl’
GIO(lj’)
G&23
. . . n; 1’ -.. omitj’
..- n’).
Repeated indices imply integration over all space and the time interval 0 to T where T is related to the temperature T of the system by ir = l/kT = /I. If there are spin indices, they are also summed over. V(l2) is defined by V(12) = iv@, - rZ) s(t, - tz), and Go is defined to be 1. G,O is defined by the equation (i -&
+ g
+ p) G,O(l 1’) = S4(x, - x1’),
(2.2)
where TVis the chemical potential of the system. All of the Green’s functions satisfy the boundary condition GA1 . . . xi7 . . . n; 1’ . . . n’) = -G,(l
. . . @ . . . n; 1’ . . . +
(2.3)
For our purpose, the physical quantities of interest are the number and the energy. For reasons which will become obvious later, we prefer to consider the
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kinetic and potential contributions to the energy separately. The expectation value of all these quantities are related to G1 in very simple ways: lim G,(ll’), s d3x 1,7.1+
N = -i T = -i
s
d3x lim
(2.4)
( - g 1 G,(ll’),
1,-d+
V = - i s d3x lim (i -$- + g 1,-A+
c?.5) + p) G1(ll’).
1
(2.6)
In a Green’s function approximation, the infinite hierarchy Eq. (2.1) is modified to yield a finite and closed system of equations. We wish to make a perturbation expansion in terms of the interaction strength. We will do this for the exact theory. However, the following description can be easily modified in the cases of different approximations. Now the free system is characterized by TV> 0, and a bound system by p < 0. In order to have a valid expansion which, as one turns on the interaction, passes from the free system to the bound system, it is necessary to treat ~1as a function of the interaction. Conventionally, all quantities of physical interest can be expressed as a function of volume, temperature, the interaction strength (A), and either the number or the chemical potential. Since it is necessary to have the chemical potential dependent on the interaction strength it would be natural to have the number as the independent variable. Thus the chemical potential may be expressed as t.~ = ~(a, /3, A, N). Now the chemical potential for the free system would be p” = ~(v, /JO, N). This equation can be inverted to give N = N(v, /3, PO). Thus all physical quantities can be given as functions of v, /3, X and PO, which is our choice in this paper. Now, we assume that a power series expansion in terms of the interaction is valid, and write G, = CI=, G,” where G,” is the m-th order term in powers of the interaction. The m-th order term of a quantity A will in general be denoted by A”. We make a similar expansion for GIO, p, T and V, respectively. Substituting this power series expansion in Eq. (2.1) and equating all terms of the same order, we have m-1 Gn'V
. . . n;
1'
-*- n’) = C Gr(l n + 1) V(n + 1 n + 2) i=O
x G;;;-l(23 +
x
5
i
(Al
j'=l'
a.. n + 1, n + 2; (n + 2)+ 1’ ... n’) (-l)i'+l
@(I
j')
GF:i(23 *.* n; 1’ ... omit j’ =a*n’)
(2.7)
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Note that we have G,” in terms of (i) Gk,, where i < m - 1, (ii) Gk, where i < m, (iii) GO,i where i < m. Further iteration can be performed on terms of types (i) and (ii). By repeated use of Eq. (2.7) alone, we can express GIm as a sum of integrals over Gy, V, and Gy (i < m). Next, we wish to reduce Gy. We note that Eq. (2.2) gives rise to the following equations: (i G
+ p”) GT”( 11’) = s4(xI - x1’),
+ g
(2.8)
1
(i&
+ g
+ **. + $G;‘(ll’)
+ p”) Gy(11’) + ~lG~-‘(Il’)
= 0,
i > 1.
1 (2.9)
We could solve these equations trivially Gy(ll’)
= -#G;‘(12)
to get
Gf’(21’) - jk1G;‘(12)
G;l(21’) ... - $G;‘(12)
G,O”-l(21)‘. (2.10)
This gives e in terms of pi and lower order terms. Noting that N is independent of A, we have from Eq. (2.4) -i
I
d3x lim G!‘(ll’)
= N,
(2.11)
11-d+
s
d3x lim Gt(ll’)
= 0,
i 2 1.
(2.12)
l'+lf
To arrive at Eq. (2.1 l), we have used the trivial special case of Eq. (2.7) G;“” = Go, where G;“” is the zeroth order term of G1 . Combining Eq. (2.12) with Eq. (2.10), we have
fli =
J d’x, lim,,,, +(GIi(l 1’) - G,oi(ll’)) J d3xl lim,,+,+ Gy(12) Gp(21’) + terms involving
piA1 and G,oi-’ and lower order terms.
(2.13)
But from Eq. (2.7) we see that GIi(l 1’) - Gy(11’) can depend at most on e-l. Using Eqs. (2.10) and (2.13) concurrently, we can eventually reduce Gr and yi to terms depending on Gy and V alone. Finally, we can solve for G,O” by using Eq. (2.8) together with the boundary condition Eq. (2.3).
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The solution is given by the following G;‘(ll’)
= j $$
eiKdx~-x~‘)(l I’)~ ,
(11’)1 = ie- ie,(tl-f+B(tl
- &‘)fL’ + S(?,’ - &)&-I,
(2.14)
hi = l/(1 + eriTel), eE= (Kz2/2m) - PO. With this explicit solution for Gi”, we can in principle calculate Glm for each m. Having calculated Glm, the energy contributions of each order can be simply arrived at by applying Eqs. (2.11) and (2.12) to Eqs. (2.5) and (2.6). We have T” = -i p=
-;[j
I
d3x1 1,-11+ lim (- 2)
WV 1’1,
i -& + +$ ( 1
d3x, lim 1,+1+
+ PO)WV
1‘)] + ; p”N
m 3 1.
Since Gy(11’) is expressed as an integral over momentum, it is convenient at this point to fourier transform all quantities from coordinate spaceto momentum space.We then get tm = -i
p=-A
i
dK eG1”(K; 0-),
2 1 dK (i &
(2.15)
- e) Glm(K; O-) + i t~“p,
(2.16)
1
where t, v, and p are the kinetic energy density, potential energy density, and number density respectively, and Glm(k; t, - tl’) is the fourier transform of Glm(ll’). In this paper JdK will always be understood to mean Jd3K/(2r)3.
3. g DIAGRAMS,~DIAGRAMS
AND RELATIONS
BETWEEN
THEM
In the previous section, we saw that Gln(k; II’), tn and v” can be written as sums of momentum-time integrals over products of noninteracting single-particle propagators and the interaction. In this section, we begin by defining the g diagram which representsa term in Gln(k; 11’). We will then show that eachg diagram gives rise to 2 diagrams: the vg diagram which represents the contribution of the g diagram to the potential energy, and tg diagram, the contribution to the kinetic energy.
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A typical g diagram is an integral over momentum and time. If the time integration is performed, we have a sum of terms involving momentum alone. The latter terms will be calledfdiagrams. We give the rules for the fdiagrams and show the relation between the g diagrams and f diagrams. The g Diagram
An example of a g diagram is given in Fig. 1.
FIG.
1. An example of a g diagram.
Each diagram can be labeled as gz&(ll’), where n is the number of interaction vertices of the diagram, (y.is an index which characterizes the topology, ki is the momentum of the external line, and t1 and tl’ are the open ends of the incoming and outgoing line, respectively. The rules for each g diagram are as follows: (a) Assign a time variable to each vertex and a momentum variable to each internal line. (b) Corresponding to each line p between ti and tj , write (ij),, , where (ij)p is defined in Eq. (2.14). (c) If, in (ij), , ti = tj (i.e., a “bubble”) write (0-), . (d) Corresponding to each vertex, with p, p’ being the incoming lines and V, V’ the outgoing lines, write (pp’ I I/ 1 VV’) a(~ + CL’- v - v’), the fourier transform of the interaction. (e) Integrate over all internal time and momentum variables. (f) The overall sign is determined as in the usual Hugenholtz diagram [6]. In addition, put in overall factor i”. The energy contribution from each diagram is given by the following lemma. LEMMA 3.1. Each g diagram gives rise to two diagrams, the vg diagram, representing its potential energy contribution, and the tg diagram, representing its kinetic contribution.
The rules are as follows: The vg Diagram
(a’) Starting with a g diagram g$(l l’), draw the same diagram with the two original open ends joined together. The new diagram will be labeled as v( gz*).
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85
TECHNIQUES
Continuing with the example of Fig. 1, we have its contribution energy given by Fig. 2.
to the potential
e FIG.
2. An example of a ug diagram.
(b’) The contribution to the lines and vertices is given by rules (a)-(f) for g diagrams. Note that there are no external variables, and all momentum and time variables are to be integrated over. (c’) Put in an extra overall factor of -i/27. The tg Diagram (a”) Starting with g,“,(ll’), converging on a new “triangular to Fig. 3.
FIG.
draw the same diagram with the two open ends vertex”. Again, the example of Fig. 1 gives rise
3. An example of a rg diagram.
(b”) Assign to the triangular vertex a time variable. (c”) For the triangular vertex, write e&p - V) where p and v are the incoming and outgoing lines, respectively. (d”) The rules for the lines and old vertices are given by (a)-(f), noting’again that one should integrate over all momentum and time variables. (e’) Put in an extra overall factor of -i/T. Some Notations for the g Diagrams The diagrams can be represented in the following ways: gu”,(t,t,‘) = i”
I
d(I&
- KJ j di, Nan&,)
9:&J,
; Kit&,‘)
(3.1 (3.2) (3.3)
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where K,,, is the set of 2n momentum variables of integration, and in of y1time variables of integration. N,“(&J is the product of IZ interaction characteristic of the topology 01. SEi , VU”, and Tzi are products of all have made use of the fact that u(g,“,) is independent of i to write Van. rules given, it is manifest that
is the set vertices, (ij)p’s We From the
where
b&,
= j-’0 dt (th,
(tt&
.
(3.6)
We will now prove Lemma 3.1. By Eq. (2.16), the potential energy contribution of a g diagram is given by v(g,“J = - ; 1 dKi (i $
-
Q) gfi(toto+).
Now, using the form of gz given by Eqs. (3.1) and (3.4), we have at once fMi>
= - 7 x
(
Now fourier transforming
j diC,, j di, Nan(R2,) i$
- Q) Vm”(R2, , i,) 0
(totdq (4to+)Ki (Vkh& *
Eq. (2.8), we have
= qt, - t,rJ. (i & - ei)(tOtkhi 0
Performing
the tl, integration,
we have
Finally, noting that g,“I depends only on to - to+ = O- and therefore independent of to , we can put in l/~ s dt, to give 4g3
= - GjdiC,,
j di, N,IE(K2,) Van(&i,,).
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This expression agrees with Eq. (3.2), which is derived from the diagram rules. Similarly, from Eq. (2.15), the kinetic energy contribution of a g diagram is given by f( g,“J = -i
1 dKi eig:&,to+).
Again, invoking time translation invariance of g,“, , we insert l/~ s dt, . This gives at once t( g,“i) as in Eqs. (3.3) and (3.5). Before we proceed further, we will define two more kinds of diagrams. The vf Diagram Some examples of vf diagrams are given in Fig. 4.
FIG.
4.
Some examples of vf diagrams.
The rules with each diagram are as follows: (a) Assign to each line a momentum variable. Corresponding to line TV going from right to left, write -f, +. Corresponding to line v going from left to right, writef,-. (b)
For each vertex, write the same Q.L~’ 1 V 1vv’) 6(~ + p’ - v - v’).
(c) Each cut between two adjacent vertices i and j gives a factor of MC err - C e,) where p and v runs over all the lines intersecting the cut, and going towards the left and the right, respectively. (d)
Integrate over all momentum
variables.
(e)
The overall sign is determined as in Hugenholtz
(f)
Put in an overall factor of a.
diagrams.
Note that the ordering of vertices is crucial. We can label each vf diagram as f[v( gfi) P,J where P,, is a particular ordering of the IZ vertices. In the limit i7 + co each vf diagram is exactly 4 times a Brueckner-Goldstone diagram.
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The tf Diagram
Examples are given in Fig. 5.
FIG. 5. Some examples of tfdiagrams.
The rules are as follows: (a’) (b’)
The rules (a)-(e) for vfdiagrams apply. For the triangular vertex between K and K’, write e&K
- K’).
Each diagram can be labeled as f[t( g,“i) PntI], where P,+l is a permutation of the n + 1 vertices (including the triangular vertex). Some Notations for f Diagrams
We note that the f diagrams can be represented in the following ways (3.7)
(3.8) where qt”‘n equivalently defined as van(Pn(n), P,(n - I),..., P,(l)), to the product of “j” factors for the particular time ordering t,(n)
>
tP(n-1)
>
***
z=- tP,(l)
7
derived from VOn by replacing each (ij), by -f,+ if ti > tj and f,The definition of gtF+l(K2,J is similar except that we have variable associated with the triangular vertex. ~2~ (j) corresponds counting from the left. We define a Di factor assocyated with the t+ Di = e, + e,’ - e, - e,’
corresponds
if ti < tj . one more time to the j-th cut, vertex, given by (3.9)
where CL, CL’and v, v’ are the lines that go into and come out of the i-th vertex, respectively.
TECHNIQUES
89
zap,(j) = 2 DP”(i) .
(3.10)
DIAGRAMMATIC
We can easily verify that
i-1
Now, we are ready to state a theorem relating the g and f diagrams. THEOREM
3.2.
where P, runs over all possible permutations
of m vertices.
The theorem states that each g diagram is equal to all permutations of the corresponding vf diagram and similarly, each tg diagram is equal to all permutations of tf diagrams. We will first prove Theorem 3.2 for the potential energy part. Using time translation invariance, we can rewrite Eq. (3.2) as
v(g,“,)= -
7
j- dK2, j. 4L
- tn> JYc’YGd CY%n
, fn(tn = 0))
(3.11)
To evaluate the time integral, we will first divide the (n - 1)-dimensional time space into (n - I)! regions, within each of which, there is a well-defined time ordering. Then, using Eqs. (2.14) and (3.9)
s
d(f, - tn) ~~n(RZnfn(tn = 0))
x fxpn-lh
- 11, P,-,(n
- 2) )...) P,-,(l),
tn),
(3.12)
where
... stzdt, et,*,. [A,,An-,,...,AlIt= /’ dt,,et,*,stndt,-,etn-A-x 0
0
0
(3.13)
90
CRICHTON AND YU LEMMA
n-
3.3 [7].
Let A, = 0,
To prove the lemma, 1;then
use induction
on n. Suppose the statement
is true for
n-1 =
xc0
.,':(;"",)
2
-
z;
n:,k(A
3
-
4)
'
To evaluate the last sum, let h(d) = n;Zi (4 - d). Partial fractionating h with respect to &n , we have
This proves Lemma 3.3. We need one more lemma. LEMMA
3.4. exp i~(9~,(n = *(P(k)
1) - .9p,(k)) Ta;;“(P(n P(k -
1) *a* P(l), tJ
1) *a* P(1) t,P(n - 1) *m*P(k + 1)).
To prove the lemma, let ,u, CL’ and v, v’ be the incoming and outgoing lines of vertex i. If Tan(ijk ... r~) = Bjk...,,(-f,+)(-f=)(f,->(f,;), then Yan(jk so. rsi) = Bik...7,(fy-)(f;;>(-~+)(-f~).
Noting that f,+&- = eiTep, we have at once eiTD’ivan(ijk **ars) = Tan(jk *a*rsi) from which the statement of the lemma trivially follows. Using Lemmas 3.2 and 3.3, we can now rewrite Eq. (3.12) as
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Now note that the double sum together constitutes a complete set of permutations of n vertices, we have finally
=0))=cP, i”“7p(P,(n) sd(i,- in)Va”(R,,t,(t,
P&z - 1) **. P,(l)) n;:.
BP,(i)
’
where we have used the fact that =Qpn(n) = Cy=, Dpci) = 0. This proves the theorem for the ug diagrams. The proof for the tg diagrams is identical. There is a problem with zerodenominators, however, stemming from the fact that the “D” factor associated with the triangular vertex is zero. However, this wilI only be a special case of the issue we will now consider. 4. BUBBLE DIAGRAMS
If one applies directly the rules of Section 3 to every g diagram, one finds in many cases divergent f diagrams associated with zero-denominators. These divergent integrals can come either from bubble diagrams or the kinetic diagrams, or some more general structure. We give some examples of such diagrams in Fig. 6.
L) Bubble
b Kinetic
c “Generalized”
FIG.
6.
diagram
dmgram
bubble
d~agmm
Some examples of divergent fdiagrams.
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CRICHTON AND YU
Conventionally, one has to sum over all divergent diagrams related by cyclic permutation, and apply either the L’Hospital’s rule or methodsof regularization [4}. In this section, we will define a generalized structure which will include all the above cases, and state some theorems which will help us bypass altogether the problem of divergent integrals. As natural consequences of these theorems, one will learn how to sum over all bubble diagrams, and thereby have a perturbation theory in which the bubble diagrams do not appear. Using the same theorems, we will prove in the next section a relationship between the tg and ug diagrams. DEFINITION.
Let [ij]:
= Jd(t, - tl)(i2)K(23)K *-* (n - 1, n),(nj),
. We will call
[ij]: an n-chain. THEOREM 4.1.
[ij];t = [l/(n - 1) !](P-l/de”-‘)(ij)K Proof.
.
One can show the relation (d/de)(ij)K
= [ij]k .
(4.1)
Applying Eq. (4.1) directly to the n-chain, we have (d/de)[ij]k+
= (n - l)[ij]E.
Q.E.D.
Suppose we have a diagram gti . One can generate an (m + n)-th order diagram by inserting n bubbles into a propagator. This corresponds to replacing (ij), by [ij],“+‘(B(p))“, where B, = i
s
dK’ (K,K’
1 V 1K’K,,) (O-)K,.
(4.2)
We will label this diagram as gzTIPS,d . LEMMA 4.2.
We have a similar expression for the n-bubble g diagram and tg diagram. This follows trivially from Theorem 4.1. Recall that -iim+l/2T j d&,Nam(&,J s dtmV~~(&&,J is just u( g$). Thus Lemma 4.2 gives us the contribution to the n-bubble diagram in terms of the base diagram in which the bubbles are “popped”.
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TECHNIQUES
With this relation, one can write down a host of summation theorems. We give the simplest of such summation theorems, the summation over all bubble diagrams, in Lemma 4.3. LEMMA
4.3. Let ($), be (ij), with e,, replaced by e, + B, and “9~m(~z,J,,J be Yam
with each (ij), replaced by (t$), . Let i,J’$ be the sum of all diagrams generated by inserting bubblesinto a basediagram g,“,, . Then
We have a similar expression for g diagrams and tg diagrams. Proof.
Suppose we sum over all bubble insertions in one line. We have
The lemma trivially follows. Now, in the case of generalized bubble structures, such as that of Fig. 6C, Lemmas 4.2 and 4.3 will apply in exactly the same way. The only difference lies in replacing Bi , which corresponds to a bubble, by some other quantity, which corresponds to the substructure that replaces the bubble.
V. RELATIONS BETWEEN KINETIC DIAGRAMS
AND POTENTIAL
DIAGRAMS
In this section, we will prove some relations between the kinetic and potential diagrams. We will first write down a lemma which trivially follows from the work of the previous section. LEMMA
5.1. t(g3
= - G
j dK,,, j di, Nan@,,) 2ei -& 2 V~‘@J,,).
This follows from Eqs. (3.3), (3.5) and Theorem 4.1 applied to the case n = 2. Thus we can rewrite the energy density in terms of Ya,n(&,J,J alone, i.e.,
94
CFUCHTON
AND
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We will now consider a particular set of g diagrams. We will show that a simple relation between the kinetic energy and potential energy exists within this set. DEFINITION. Let gan = { g,: , where i runs over the 2n propagators We will call gU” a family of g diagrams. Thus, for example, Fig. 7 constitutes a family.
FIG.
THEOREM
A family
of g diagrams.
5.2.
nt&“) Proof.
7.
= - (n -
1 - /3 +,
Since v( g,“i) is independent of i v( &“)
Consider /3(8/a/3) v( gun). We can write
= 2nv(g3.
u&n).
in Van}.
DIAGRAMMATIC
95
TECHNIQUES
where
To evaluate II , we use the following identity 7 -$ (ij), = e,[ij]t + i(ti -
Writing
*Y-,“(~&~)
tj) e,(i&
.
(5.2)
as a product of 2n(ij)ds, Eq. (5.2) gives us at once
where n-1
I, = f d(i, -
t,) YE”,” c (- itjDj)*
0
j=l
Now I3 can be written as
x ?q(P(n Carrying out the differentiation we have at once 13 = (n -
- 1) P(?z - 2) ... P(1)
O)InS1.
(5.4)
with respect to h with the help of Lemma 3.3,
1) 1 WP(n-1) pn-1
x “p(P(n
t, =
2 iDP(n--8) ,...> iDP(
- 1) P(n - 2) .** P(1) t, = 0)
x TQyP(n -
1) P(n - 2) ... P(1) t,).
(5.5)
Using Lemma 3.3 again to evaluate Iz , we see that Iz =
c Pnml
nfl 7 exP iQP(12-1) - DP(k) ~(l’(tz K=l
n,“,-;
@‘PO)
-
DPM)
- 1) P(n - 2) *.- P(1) t,).
We see that the last term in Eq. (5.5) is exactly --I, .
(5.6)
96
CRICHTON
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Combining Eqs. (5.1), (5.3), (5.9, and (5.6), we have the desired result of Theorem 5.2. The Thermodynamic
Origin of Theorem 3.2
We now show that Theorem 5.1 stems from some general thermodynamic considerations together with the linear independence of different families, each considered as a functional of the interaction. Recall that the quantities of physical interest depend on /3, h and ,u”. We show in Appendix 1 that the following thermodynamic relation holds between the energy E and the potential energy V:
A(aE/aA)= v + #@v/a/3)- xp(ap/aA)(aN/ag).
(5.7)
To each order in h, we have nt” = -(n - 1) v* + /3(&P/a/I) - nj.P/3(+++9).
(5.8)
The only dependence of GIn(k; O-) on @’ comes from the term -~“[l l+]“, . One can show explicitly that Eq. (5.2) holds for the @ term. Defining Gln(k, 0-) = Gi(k, 0-) + p”[ll+]i one has from Eq. (5.2)
(5.9 Since Eq. (5.3) is true independent of the specific form of the interaction, we can consider it as an equation of functionals of the interaction. Using Eq. (3.1), we can write [8] (5.10) where LQ = -inei
- f (n - 1 - fl GF)(i
&
- ei).
Next, we collect all terms with the same topology. Then we can write Eq. (5.10) as
topological classes
o! the B&me toPologY
DIAGRAMMATIC
As functionals of the interaction, the topologically independent. Therefore, we have f
dKzn N/(&J
1 ,ll~,s
97
TECHNIQIJES
distinct classes must be linearly
j 9$Y~i/
= 0
for each cx.
(5.11)
olthesmne toPawY
Theorem 5.2 shows explicitly the same topology.
that the family constitutes the set of all terms of
The Zero-Temperature Limit
We will now derive a simple, but important COROLLARY
special case of Theorem 5.2.
5.3. At zero-temperature, nt( &“) = -(n - 1) V( goln).
We will prove lim,,, @?a( &“)/@I) = 0. Using Theorem 3.2, we can write V( gz) in such a way that the only p dependence comes in through&*. Now BWBf*
(5.12)
= 4W>f*.
Using the fact that j$
(5.13)
f* = Kte),
we have fqajag) f * = &es(e) + 0.
(5.14)
Corollary 5.3 stems from the well-known Feynman identity for a system in its ground state: x(&Y/a& = Y. This identity can be considered as a special case of the thermodynamic relation stated in the previous section.
6. SOME CONSEQUENCES, APPLICATIONS,
AND CONCLUDING
REMARKS
We will now state some direct consequences of the above techniques. 1. ComparisonbetweenGreen’sfunction diagramsand Brueckner-Goldstone Diagrams We will first state a theorem. THEOREM 6.1. The contribution to the ground state energy from a family of g diagrams is equal to the set of all permutations of Brueckner-Goldrtone diagrams of the sametopology.
98
CRICHTON
Proof.
AND
YU
From Corollary 5.3, we have
From Theorem 3.2, we have
Since each vf diagram is l/2 times a Brueckner-Goldstone is proved. As a second consequence of these techniques, we have
diagram, the theorem
LEMMA 6.2. A Green’s function approximation can be cast in the form of Brueckner-Goldstone diagrams tf and only if the approximation satisfies the thermodynamic relation Eq. (5.7). Furthermore, in these theories, the Brueckner-Goldstone diagrams of any topology are included only in the entire permuted set. This relation is satisfied in the Hartree-Fock and A,, theories, but not in A,, or A,, [2]. We jind empirically to fourth order in the interaction that the Hartree-Fock and A,, diagrams come in families, and can therefore be written down as classes of BruecknerGoldstone diagrams, while in Aoo and A,, , “broken” families are found, and “anomalous” terms (i.e., terms with “e’s” in the numerators of the integrands) which cannot be represented by Brueckner-Goldstone diagrams, were introduced. Conversely, to cast a statement in the Brueckner-Goldstone theory in the GreenS function .formalism, it is necessary that the statement be one which does not diflerentiate between d@erent time orderings. Thus, for example, the factorization theorem [6] cannot be cast in the Green’s function form. 2. Applications to Green’s Function Perturbative
Calculations.
The diagrammatic techniques of Theorem 3.2 should facilitate time integration in perturbative calculations. Theorem 5.2 and Corollary 5.3 implies that one can work with potential energy alone. Even in the case of approximations which do not satisfy Eq. (5.7), one can still use the result of Theorem 5.2 within complete families of diagrams. Some Concluding Remarks Our original motivation for the present study is to make a connection between the Green’s function approach and the Brueckner-Goldstone approach. These two
DIAGRAMMATIC
TECHNIQUES
99
approaches are in principle equivalent. By making connections then, we mean the facility of transforming a statement from one language to another. We have made a study of perturbation expansions in the Green’s function formalism. We feel that our study gives us sufficient understanding between Green’s function diagrams and Brueckner-Goldstone diagrams to make it relatively easy to go back and forth between the two perturbative approaches. To make direct connections between the Green’s function approach and the Brueckner-Goldstone approach, however, there is yet another link that should be studied in more detail: namely, the relation between Green’s function equations and Green’s function diagrams. In Section 2, we showed that in principle, we can derive Green’s function diagrams from Green’s function equations. In going through the process of expansion, we find that the same diagram may appear at various different stages of the expansion, and members of the same family may also fall out from various places, and yet after going through a complicated process of expanding and combining, we arrive eventually at very simple sets of Green’s function diagrams: they always appear in families with just the right proportionality constant for each member diagram. A simple final result suggests a simple method of arriving at it. We feel that a closer study can be made on the relation between Green’s function equations and Green’s function diagrams. In our program in connecting function theories to Brueckner-Goldstone theories we have to answer two questions: 1. What Green’s function diagrams are implied by the Green’s function equations ? 2. Which Green’s function diagrams relate to which Brueckner-Goldstone diagrams ? Our present study has been an attempt to answer question 2. We encourage studies for answers to question 1.
APPENDIX
1
In this appendix, we derive Eq. (5.7). We use the method of the grand canonical ensemble: the thermodynamic potential Q is defined by e+JQ
=
Tr(e-@‘)
where X = H,, + hV - $V. The expectation represented by an operator A is given by
>
value of any physical quantity
(Ah.rr,~ = Tr(Ae-@Ju”)/Tr(e-BJE4).
100
CRICHTON
AND
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The thermodynamic potential and all expectation values are thus functions of ,!I, p and h. Taking derivatives of Eq. (A. 1) with respect to /3, p and h, we have the following equations: aQ/aP = Wfo
- wo
- U/Pvk
(A-3) (A.4)
aqap = --(N), aqah = (i/x)(v).
(A.5)
Further differentiation and the assumption of interchangeability partial derivatives gives
h--wo aA
Apq
of the order of the
- (XV) = p + (AV),
a
W-9 (A.71
We are interested not in (H),
The first term on the right-hand side of Eq. (A.9) is seen to contain aN/ah, which vanishes identically. This is because our method for determining the power series expansion for p, using Eqs. (2.11) and (2.12), makes N independent of X. The expression in the parentheses in the second term is just aN/a/3. Thus
qaqah) = v + p(apj3) v - pA(ap/aA)(alv/ap).
DIAGRAMMATIC
TECHNIQUES
101
2
APPENDIX
To acquaint ourselves with manipulations off diagrams, and to shed further light on the t” - vn relation, we will now prove Corollary 5.3 for the family of Fig. 7 by working directly withfdiagrams. In this family there are eight g diagrams. Each g diagram gives rise to 24 vf diagrams, and 120 tf diagrams. We now divide the vf diagrams and tf diagrams each into 24 classes. Within each class of v f diagrams, we have eight identical diagrams corresponding to a particular permutation of the 4 interaction vertices. Corresponding to each class of diagrams, we have a class of tf diagrams composed of the 40 tf diagrams built out of the same ordering of the interaction vertices, but with the triangular vertex, inserted into all possible legs at all possible positions. We will prove that within the class tcn = -[(n
- 1)/n] v,“.
To prove the relation, consider the class of 40 tf diagrams. They can be classified into three groups according to the position of the triangular vertex and its two appending legs.
Subgroup
I
Subgroup
2
Subgroup
3
FIG. 8. A special grouping of tfdiagrams
for the proof of Appendix 2.
102
CRICHTON AND YU
Group 1. The triangular Group 2. The triangular
vertex is either to the extreme left or extreme right. vertex is not to the extreme left or right; appending
legs go in opposite directions. Group 3. The triangular legs go in the same direction.
vertex is not to the extreme left or right; appending
Group 2 gives no contribution since the two appending legs give a product e(e) t9(-e) = 0. Group 1 has two opposite appending legs together with a zero denominator. This gives in the limit a delta function. Together with the triangular vertex, we have es(e) = 0. Therefore, Group 1 also gives a zero contribution. Now divide Group 3 into three subgroups according to the position of the triangular vertex, e.g., we have the grouping of Fig. 8. Compare the contribution from each subgroup to a vf diagram. The sum of the four triangular vertices cancels exactly the denominators corresponding to the extra cut, except for an introduction of a minus sign. Since the vf diagram has an overall l/2 factor, each subgroup is therefore equivalent to minus two times the contribution from a vf diagram. The three subgroups together give minus six times that contribution. However, on the vf side we have eight diagrams in the class. Therefore, within the class, t,4 = -(6/8) UC4= -(3/4) v>. ACKNOWLEDGMENTS The authors wish to thank Professor Roger Anderson for his close collaboration in the initial phase of this work. They thank Professor David Yu for his assistance in identifying the Feynman identity, and Mr. P. Geren for his assistance in the proof of Theorem 3.1. They are indebted to Dr. B. Brandow, Professor R. Puff and Professor L. Wilets for fruitful discussions. This work was begun under a grant from the Research Corporation. For one of us (S.Y.), much of this work was done under the financial support of the Seattle Pacific College Institute of Research. He also acknowledges partial support of the Battelle Memorial Institute. REFERENCES 1. P. C. MARTIN AND J. SCHWINGER, P&r. Reu. 115 (1959), 1342; R. PUFF, Ann. P/ry& (N.Y.) 13 (1961), 317. 2. R. PUFF, A. S. REINER AND L. WILETS, Phys. Rev. 149 (1966), 778. 3. K. A. BRUECKNER,Phys. Rev. 100 (1955), 36; J. GOLDSTONE, Proc. Roy. Sot. A 239 (1947), 267. 4. C. BLOCH, General perturbation formalism for the many-body problem at nonzero temperatures, in “Lectures on the Many-Body Problem,” (E. R. Caianiello, Ed.). Academic Press, New York, 1962. 5. D. PINES, “The Many-Body Problem,” p. 43, Benjamin, New York, 1961. 6. N. M. HUGENHOLTZ, Phy.sica 23 (1957), 481. 7. For an alternate proof, see Ref. 141. 8. We have ignored the higher order CL*terms for simplicity.
OF
PHYSICS:
75, 77-102 (1973)
Diagram’matic
Techniques in the Martin-Schwinger-Puff Many-Body Theory* JAMESH. CRICHTON
Seattle Pacific College Institute for Research, Seattle, Washington 98119
AND
SIMON Yu Department of Physics, University of Washington, Seattle, Washington 98105
ReceivedOctober22, 1971 In an attempt to understandthe connectionbetweenthe Martin-Schwinger-Puff Green’sfunctionapproachto the many-bodyproblemand the diagrammatic approach of BruecknerandGoldstone,the perturbationexpansionof the Green’sfunctiontheory is studied.This expansion leads naturally to a representation of the theory by two sets of time-dependent diagrams-the kinetic energy diagramsand the potential energy diagrams.Rulesare givenfor writing eachtime-dependent diagramasa sumof timeindependentdiagrams.Complications arisein the applicationof theserulesto bubble diagrams. A special simple treatment of these diagrams is then proposed. There exists a simple relation between families of kinetic diagrams and potential diagrams. This relation stems from a rather general thermodynamic identity. Both the relation and its
thermodynamicorigin are described.As a consequence of thesestudies,someconnections between the Brueckner-Goldstone made.
theory and the Green’s function theory are
1. INTRODUCTION In the Martin-Schwinger-Puff finite temperature Green’s function approach to the many-body problem [l], the physical quantities of interest-particle number and energy density-are very simply given in terms of G1 , the full single-particle propagator. To find G1, we have to solve an infinite hierarchy of Green’s function equations, which are essentially Schroedinger’s equation combined with particle statistics. The standard way of solving these equations, of course, is to terminate this infinite hierarchy by some approximation scheme[2]. Another way of solving these equations is to expand all interaction-dependent quantities in powers of the * Supported in part by the U. S. Atomic Energy Commission and the Research Corporation.
77 Copyright All rights
0 1973 by Academic Press, Inc. of reproduction in any form reserved.
78
CRICHTON
AND
YU
potential. This method of perturbation expansion is in principle equivalent to other perturbation theories. However, in practice, other perturbation theories, such as that of Brueckner and Goldstone [3], seem to be far simpler. Nevertheless, in the present paper, we wish to consider perturbation expansions in the Green’s function approach. The reason is the following: Much has been learned about many-body systems through studies in both the Green’s function approach as well as the Brueckner-Goldstone approach. However, the techniques developed within each of these two frameworks seem to be totally different. To understand how techniques in one formalism may be applied to the other, we wish to follow closely how an expression or a statement in one approach evolves into an expression within the framework of the other. To go from the Green’s function approach to the Brueckner-Goldstone approach, a perturbation expansion in the Green’s function approach seems to be the most natural first step. Carrying out an expansion of the Green’s function equations, the resulting perturbation theory is given in terms of an infinite set of diagrams. Two features characterize the diagrams in this theory. First, the diagrams represent integrals over momentum as well as imaginary time. This is natural since the Green’s function hierarchy, from which the diagrams stem, are integral equations over momentum and imaginary time. Second, it is most natural to write the energy in terms of two sets of diagrams, one set for the potential energy, and another set for the kinetic energy. In contrast to these two features, in the Brueckner-Goldstone theory, the energy is represented by only one set of diagrams which are integrals over momentum alone. The first step in our study is to carry out the integration over time. This is usually a tedious, even if straightforward, calculation. We have devised some diagrammatic rules to replace this calculation. The net result of these rules is to replace each time-dependent diagram by a set of time-dependent diagrams. Since there are two different kinds of time-dependent diagrams, corresponding to the potential energy and kinetic energy contributions, we have correspondingly two sets of time-independent diagrams. We have given explicit rules for these four different kinds of diagrams (which we call the vg diagram, tg diagram, v f diagram, and tf diagram, respectively). The relations between the g diagrams and the f diagrams are given. These rules in general simplify the calculations. However, the theory is hampered by one drawback: many of the f diagrams so defined are divergent due to zero denominators. These divergences appear in bubble diagrams, as well as in all kinetic diagrams. In principle, there is nothing wrong: all that one has to do is to combine several suchfdiagrams, and using either L’Hospital’s rule, or methods of regularization [4], one can arrive at finite results. However, this process destroys much of the simplicity of the rules.
DIAGRAMMATIC
79
TECHNIQUES
We have made a separate study of these bubble diagrams, and other bubble-like diagrams. The relevant feature characterizing these diagrams is what we call the n-chain. We have found that the bubble diagrams are related to the corresponding diagrams in which all the bubbles are “popped”. This relation allows us to bypass the problem of divergent integrals altogether. It is interesting that the same mathematics that is used to relate the bubble diagrams to the nonbubble diagrams leads to another useful relation. We find that within a certain well-defined set of Green’s function diagrams, a set which we call the family, a simple relation exists between the kinetic energy contributions and the potential energy contributions. If t and u are the kinetic and potential contributions, respectively, of a family of n-th order diagrams, the relation is given by nt=-
i
n-l-/I-f-
aP 1
V.
In the zero-temperature limit, the samerelation reducesto
nt = -(n - 1)~. Thus, as long as an entire family is included, we will have the total energy by studying the potential diagrams alone. Such a simple relation indicates a simple physical origin. Upon investigation, we found that indeed the relation stems from a rather general thermodynamic identity. In the zero-temperature limit, this thermodynamic identity reducesto the well-known Feynman identity [5]. Having devefoped these techniques, we make a very simple observation: except for a factor of l/2, the vf diagrams, i.e., the time-independent potential diagrams, give contributions exactly equal to those of the Brueckner-Goldstone diagrams. Gathering the results of these studies, we can identify a family of Green’s function diagrams with a set of Brueckner-Goldstone diagrams. We can then draw some connections between the two approaches. In Section 2 we describe perturbation expansions in the Green’s function approach. As the final product of this expansion, we can expressGin, the n-th order term in the expansion of the full single-particle propagator, as a sum of diagrams each of which is an integral over momentum and time, the integrand being a product of Gy’s, the single-particle propagator with no interaction, and V, the interaction, alone. In Section 3 we start by stating the precise rules for these integrals, which we will call “g” diagrams. Each g diagram will give a contribution to the potential energy, which we will call a “vg” diagram, as well as to the kinetic energy, which we will call the “fg” diagram. Carrying out the time integration, we show that each “vg” diagram gives rise to a set of “vf” diagrams, and each “tg” diagram gives 595/75/1-6
80
CRICHTON AND YU
rise to a set of “tf” diagrams, where an f diagram is an integral over momentum alone. The rules for the f diagrams, as stated, give some divergent integrals. In Section 4 we will discuss how these diagrams can be treated. In Section 5, we show that within a family of g diagrams, a simple relation exists between the kinetic energy and the potential energy. We will indicate the thermodynamic origin of this relation. As a special, but important case of this relation, we consider the zero-temperature limit. In Section 6 we consider some consequences of these techniques in comparing Green’s function theories with Brueckner-Goldstone theories and also in Green’s function perturbation expansions. In Appendix 1 the proof of the thermodynamic relation of Section 5 is given. In Appendix 2 we prove the result of the zero-temperature limit of Section 5 by a different method.
2. THE GREEN'S FUNCTION PERTURBATION EXPANSION In the Green’s function formalism for the finite temperature many-fermion problem, we have a set of Green’s functions satisfying the following infinite hierarchy of coupled integral equations [l]: GA1 . . . n; 1’ . . . n’)
= G,O(l, n + 1) V(n + 1, n + 2) G,+,[23 ... n + 1 IZ + 2; (n + 2)+ 1’ *a. n’] +
5 (-l)j’+l j’zl’
GIO(lj’)
G&23
. . . n; 1’ -.. omitj’
..- n’).
Repeated indices imply integration over all space and the time interval 0 to T where T is related to the temperature T of the system by ir = l/kT = /I. If there are spin indices, they are also summed over. V(l2) is defined by V(12) = iv@, - rZ) s(t, - tz), and Go is defined to be 1. G,O is defined by the equation (i -&
+ g
+ p) G,O(l 1’) = S4(x, - x1’),
(2.2)
where TVis the chemical potential of the system. All of the Green’s functions satisfy the boundary condition GA1 . . . xi7 . . . n; 1’ . . . n’) = -G,(l
. . . @ . . . n; 1’ . . . +
(2.3)
For our purpose, the physical quantities of interest are the number and the energy. For reasons which will become obvious later, we prefer to consider the
DIAGRAMMATIC
81
TECHNIQUES
kinetic and potential contributions to the energy separately. The expectation value of all these quantities are related to G1 in very simple ways: lim G,(ll’), s d3x 1,7.1+
N = -i T = -i
s
d3x lim
(2.4)
( - g 1 G,(ll’),
1,-d+
V = - i s d3x lim (i -$- + g 1,-A+
c?.5) + p) G1(ll’).
1
(2.6)
In a Green’s function approximation, the infinite hierarchy Eq. (2.1) is modified to yield a finite and closed system of equations. We wish to make a perturbation expansion in terms of the interaction strength. We will do this for the exact theory. However, the following description can be easily modified in the cases of different approximations. Now the free system is characterized by TV> 0, and a bound system by p < 0. In order to have a valid expansion which, as one turns on the interaction, passes from the free system to the bound system, it is necessary to treat ~1as a function of the interaction. Conventionally, all quantities of physical interest can be expressed as a function of volume, temperature, the interaction strength (A), and either the number or the chemical potential. Since it is necessary to have the chemical potential dependent on the interaction strength it would be natural to have the number as the independent variable. Thus the chemical potential may be expressed as t.~ = ~(a, /3, A, N). Now the chemical potential for the free system would be p” = ~(v, /JO, N). This equation can be inverted to give N = N(v, /3, PO). Thus all physical quantities can be given as functions of v, /3, X and PO, which is our choice in this paper. Now, we assume that a power series expansion in terms of the interaction is valid, and write G, = CI=, G,” where G,” is the m-th order term in powers of the interaction. The m-th order term of a quantity A will in general be denoted by A”. We make a similar expansion for GIO, p, T and V, respectively. Substituting this power series expansion in Eq. (2.1) and equating all terms of the same order, we have m-1 Gn'V
. . . n;
1'
-*- n’) = C Gr(l n + 1) V(n + 1 n + 2) i=O
x G;;;-l(23 +
x
5
i
(Al
j'=l'
a.. n + 1, n + 2; (n + 2)+ 1’ ... n’) (-l)i'+l
@(I
j')
GF:i(23 *.* n; 1’ ... omit j’ =a*n’)
(2.7)
82
CRICHTON
AND
YU
Note that we have G,” in terms of (i) Gk,, where i < m - 1, (ii) Gk, where i < m, (iii) GO,i where i < m. Further iteration can be performed on terms of types (i) and (ii). By repeated use of Eq. (2.7) alone, we can express GIm as a sum of integrals over Gy, V, and Gy (i < m). Next, we wish to reduce Gy. We note that Eq. (2.2) gives rise to the following equations: (i G
+ p”) GT”( 11’) = s4(xI - x1’),
+ g
(2.8)
1
(i&
+ g
+ **. + $G;‘(ll’)
+ p”) Gy(11’) + ~lG~-‘(Il’)
= 0,
i > 1.
1 (2.9)
We could solve these equations trivially Gy(ll’)
= -#G;‘(12)
to get
Gf’(21’) - jk1G;‘(12)
G;l(21’) ... - $G;‘(12)
G,O”-l(21)‘. (2.10)
This gives e in terms of pi and lower order terms. Noting that N is independent of A, we have from Eq. (2.4) -i
I
d3x lim G!‘(ll’)
= N,
(2.11)
11-d+
s
d3x lim Gt(ll’)
= 0,
i 2 1.
(2.12)
l'+lf
To arrive at Eq. (2.1 l), we have used the trivial special case of Eq. (2.7) G;“” = Go, where G;“” is the zeroth order term of G1 . Combining Eq. (2.12) with Eq. (2.10), we have
fli =
J d’x, lim,,,, +(GIi(l 1’) - G,oi(ll’)) J d3xl lim,,+,+ Gy(12) Gp(21’) + terms involving
piA1 and G,oi-’ and lower order terms.
(2.13)
But from Eq. (2.7) we see that GIi(l 1’) - Gy(11’) can depend at most on e-l. Using Eqs. (2.10) and (2.13) concurrently, we can eventually reduce Gr and yi to terms depending on Gy and V alone. Finally, we can solve for G,O” by using Eq. (2.8) together with the boundary condition Eq. (2.3).
DIAGRAMMATIC
83
TECHNIQUES
The solution is given by the following G;‘(ll’)
= j $$
eiKdx~-x~‘)(l I’)~ ,
(11’)1 = ie- ie,(tl-f+B(tl
- &‘)fL’ + S(?,’ - &)&-I,
(2.14)
hi = l/(1 + eriTel), eE= (Kz2/2m) - PO. With this explicit solution for Gi”, we can in principle calculate Glm for each m. Having calculated Glm, the energy contributions of each order can be simply arrived at by applying Eqs. (2.11) and (2.12) to Eqs. (2.5) and (2.6). We have T” = -i p=
-;[j
I
d3x1 1,-11+ lim (- 2)
WV 1’1,
i -& + +$ ( 1
d3x, lim 1,+1+
+ PO)WV
1‘)] + ; p”N
m 3 1.
Since Gy(11’) is expressed as an integral over momentum, it is convenient at this point to fourier transform all quantities from coordinate spaceto momentum space.We then get tm = -i
p=-A
i
dK eG1”(K; 0-),
2 1 dK (i &
(2.15)
- e) Glm(K; O-) + i t~“p,
(2.16)
1
where t, v, and p are the kinetic energy density, potential energy density, and number density respectively, and Glm(k; t, - tl’) is the fourier transform of Glm(ll’). In this paper JdK will always be understood to mean Jd3K/(2r)3.
3. g DIAGRAMS,~DIAGRAMS
AND RELATIONS
BETWEEN
THEM
In the previous section, we saw that Gln(k; II’), tn and v” can be written as sums of momentum-time integrals over products of noninteracting single-particle propagators and the interaction. In this section, we begin by defining the g diagram which representsa term in Gln(k; 11’). We will then show that eachg diagram gives rise to 2 diagrams: the vg diagram which represents the contribution of the g diagram to the potential energy, and tg diagram, the contribution to the kinetic energy.
84
CRICHTON
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YU
A typical g diagram is an integral over momentum and time. If the time integration is performed, we have a sum of terms involving momentum alone. The latter terms will be calledfdiagrams. We give the rules for the fdiagrams and show the relation between the g diagrams and f diagrams. The g Diagram
An example of a g diagram is given in Fig. 1.
FIG.
1. An example of a g diagram.
Each diagram can be labeled as gz&(ll’), where n is the number of interaction vertices of the diagram, (y.is an index which characterizes the topology, ki is the momentum of the external line, and t1 and tl’ are the open ends of the incoming and outgoing line, respectively. The rules for each g diagram are as follows: (a) Assign a time variable to each vertex and a momentum variable to each internal line. (b) Corresponding to each line p between ti and tj , write (ij),, , where (ij)p is defined in Eq. (2.14). (c) If, in (ij), , ti = tj (i.e., a “bubble”) write (0-), . (d) Corresponding to each vertex, with p, p’ being the incoming lines and V, V’ the outgoing lines, write (pp’ I I/ 1 VV’) a(~ + CL’- v - v’), the fourier transform of the interaction. (e) Integrate over all internal time and momentum variables. (f) The overall sign is determined as in the usual Hugenholtz diagram [6]. In addition, put in overall factor i”. The energy contribution from each diagram is given by the following lemma. LEMMA 3.1. Each g diagram gives rise to two diagrams, the vg diagram, representing its potential energy contribution, and the tg diagram, representing its kinetic contribution.
The rules are as follows: The vg Diagram
(a’) Starting with a g diagram g$(l l’), draw the same diagram with the two original open ends joined together. The new diagram will be labeled as v( gz*).
DIAGRAMMATIC
85
TECHNIQUES
Continuing with the example of Fig. 1, we have its contribution energy given by Fig. 2.
to the potential
e FIG.
2. An example of a ug diagram.
(b’) The contribution to the lines and vertices is given by rules (a)-(f) for g diagrams. Note that there are no external variables, and all momentum and time variables are to be integrated over. (c’) Put in an extra overall factor of -i/27. The tg Diagram (a”) Starting with g,“,(ll’), converging on a new “triangular to Fig. 3.
FIG.
draw the same diagram with the two open ends vertex”. Again, the example of Fig. 1 gives rise
3. An example of a rg diagram.
(b”) Assign to the triangular vertex a time variable. (c”) For the triangular vertex, write e&p - V) where p and v are the incoming and outgoing lines, respectively. (d”) The rules for the lines and old vertices are given by (a)-(f), noting’again that one should integrate over all momentum and time variables. (e’) Put in an extra overall factor of -i/T. Some Notations for the g Diagrams The diagrams can be represented in the following ways: gu”,(t,t,‘) = i”
I
d(I&
- KJ j di, Nan&,)
9:&J,
; Kit&,‘)
(3.1 (3.2) (3.3)
86
CRICHTON
AND
YU
where K,,, is the set of 2n momentum variables of integration, and in of y1time variables of integration. N,“(&J is the product of IZ interaction characteristic of the topology 01. SEi , VU”, and Tzi are products of all have made use of the fact that u(g,“,) is independent of i to write Van. rules given, it is manifest that
is the set vertices, (ij)p’s We From the
where
b&,
= j-’0 dt (th,
(tt&
.
(3.6)
We will now prove Lemma 3.1. By Eq. (2.16), the potential energy contribution of a g diagram is given by v(g,“J = - ; 1 dKi (i $
-
Q) gfi(toto+).
Now, using the form of gz given by Eqs. (3.1) and (3.4), we have at once fMi>
= - 7 x
(
Now fourier transforming
j diC,, j di, Nan(R2,) i$
- Q) Vm”(R2, , i,) 0
(totdq (4to+)Ki (Vkh& *
Eq. (2.8), we have
= qt, - t,rJ. (i & - ei)(tOtkhi 0
Performing
the tl, integration,
we have
Finally, noting that g,“I depends only on to - to+ = O- and therefore independent of to , we can put in l/~ s dt, to give 4g3
= - GjdiC,,
j di, N,IE(K2,) Van(&i,,).
DIAGRAMMATIC
87
TECHNIQUES
This expression agrees with Eq. (3.2), which is derived from the diagram rules. Similarly, from Eq. (2.15), the kinetic energy contribution of a g diagram is given by f( g,“J = -i
1 dKi eig:&,to+).
Again, invoking time translation invariance of g,“, , we insert l/~ s dt, . This gives at once t( g,“i) as in Eqs. (3.3) and (3.5). Before we proceed further, we will define two more kinds of diagrams. The vf Diagram Some examples of vf diagrams are given in Fig. 4.
FIG.
4.
Some examples of vf diagrams.
The rules with each diagram are as follows: (a) Assign to each line a momentum variable. Corresponding to line TV going from right to left, write -f, +. Corresponding to line v going from left to right, writef,-. (b)
For each vertex, write the same Q.L~’ 1 V 1vv’) 6(~ + p’ - v - v’).
(c) Each cut between two adjacent vertices i and j gives a factor of MC err - C e,) where p and v runs over all the lines intersecting the cut, and going towards the left and the right, respectively. (d)
Integrate over all momentum
variables.
(e)
The overall sign is determined as in Hugenholtz
(f)
Put in an overall factor of a.
diagrams.
Note that the ordering of vertices is crucial. We can label each vf diagram as f[v( gfi) P,J where P,, is a particular ordering of the IZ vertices. In the limit i7 + co each vf diagram is exactly 4 times a Brueckner-Goldstone diagram.
88
CRICHTON AND YU
The tf Diagram
Examples are given in Fig. 5.
FIG. 5. Some examples of tfdiagrams.
The rules are as follows: (a’) (b’)
The rules (a)-(e) for vfdiagrams apply. For the triangular vertex between K and K’, write e&K
- K’).
Each diagram can be labeled as f[t( g,“i) PntI], where P,+l is a permutation of the n + 1 vertices (including the triangular vertex). Some Notations for f Diagrams
We note that the f diagrams can be represented in the following ways (3.7)
(3.8) where qt”‘n equivalently defined as van(Pn(n), P,(n - I),..., P,(l)), to the product of “j” factors for the particular time ordering t,(n)
>
tP(n-1)
>
***
z=- tP,(l)
7
derived from VOn by replacing each (ij), by -f,+ if ti > tj and f,The definition of gtF+l(K2,J is similar except that we have variable associated with the triangular vertex. ~2~ (j) corresponds counting from the left. We define a Di factor assocyated with the t+ Di = e, + e,’ - e, - e,’
corresponds
if ti < tj . one more time to the j-th cut, vertex, given by (3.9)
where CL, CL’and v, v’ are the lines that go into and come out of the i-th vertex, respectively.
TECHNIQUES
89
zap,(j) = 2 DP”(i) .
(3.10)
DIAGRAMMATIC
We can easily verify that
i-1
Now, we are ready to state a theorem relating the g and f diagrams. THEOREM
3.2.
where P, runs over all possible permutations
of m vertices.
The theorem states that each g diagram is equal to all permutations of the corresponding vf diagram and similarly, each tg diagram is equal to all permutations of tf diagrams. We will first prove Theorem 3.2 for the potential energy part. Using time translation invariance, we can rewrite Eq. (3.2) as
v(g,“,)= -
7
j- dK2, j. 4L
- tn> JYc’YGd CY%n
, fn(tn = 0))
(3.11)
To evaluate the time integral, we will first divide the (n - 1)-dimensional time space into (n - I)! regions, within each of which, there is a well-defined time ordering. Then, using Eqs. (2.14) and (3.9)
s
d(f, - tn) ~~n(RZnfn(tn = 0))
x fxpn-lh
- 11, P,-,(n
- 2) )...) P,-,(l),
tn),
(3.12)
where
... stzdt, et,*,. [A,,An-,,...,AlIt= /’ dt,,et,*,stndt,-,etn-A-x 0
0
0
(3.13)
90
CRICHTON AND YU LEMMA
n-
3.3 [7].
Let A, = 0,
To prove the lemma, 1;then
use induction
on n. Suppose the statement
is true for
n-1 =
xc0
.,':(;"",)
2
-
z;
n:,k(A
3
-
4)
'
To evaluate the last sum, let h(d) = n;Zi (4 - d). Partial fractionating h with respect to &n , we have
This proves Lemma 3.3. We need one more lemma. LEMMA
3.4. exp i~(9~,(n = *(P(k)
1) - .9p,(k)) Ta;;“(P(n P(k -
1) *a* P(l), tJ
1) *a* P(1) t,P(n - 1) *m*P(k + 1)).
To prove the lemma, let ,u, CL’ and v, v’ be the incoming and outgoing lines of vertex i. If Tan(ijk ... r~) = Bjk...,,(-f,+)(-f=)(f,->(f,;), then Yan(jk so. rsi) = Bik...7,(fy-)(f;;>(-~+)(-f~).
Noting that f,+&- = eiTep, we have at once eiTD’ivan(ijk **ars) = Tan(jk *a*rsi) from which the statement of the lemma trivially follows. Using Lemmas 3.2 and 3.3, we can now rewrite Eq. (3.12) as
DIAGRAMMATIC
91
TECHNIQUES
Now note that the double sum together constitutes a complete set of permutations of n vertices, we have finally
=0))=cP, i”“7p(P,(n) sd(i,- in)Va”(R,,t,(t,
P&z - 1) **. P,(l)) n;:.
BP,(i)
’
where we have used the fact that =Qpn(n) = Cy=, Dpci) = 0. This proves the theorem for the ug diagrams. The proof for the tg diagrams is identical. There is a problem with zerodenominators, however, stemming from the fact that the “D” factor associated with the triangular vertex is zero. However, this wilI only be a special case of the issue we will now consider. 4. BUBBLE DIAGRAMS
If one applies directly the rules of Section 3 to every g diagram, one finds in many cases divergent f diagrams associated with zero-denominators. These divergent integrals can come either from bubble diagrams or the kinetic diagrams, or some more general structure. We give some examples of such diagrams in Fig. 6.
L) Bubble
b Kinetic
c “Generalized”
FIG.
6.
diagram
dmgram
bubble
d~agmm
Some examples of divergent fdiagrams.
92
CRICHTON AND YU
Conventionally, one has to sum over all divergent diagrams related by cyclic permutation, and apply either the L’Hospital’s rule or methodsof regularization [4}. In this section, we will define a generalized structure which will include all the above cases, and state some theorems which will help us bypass altogether the problem of divergent integrals. As natural consequences of these theorems, one will learn how to sum over all bubble diagrams, and thereby have a perturbation theory in which the bubble diagrams do not appear. Using the same theorems, we will prove in the next section a relationship between the tg and ug diagrams. DEFINITION.
Let [ij]:
= Jd(t, - tl)(i2)K(23)K *-* (n - 1, n),(nj),
. We will call
[ij]: an n-chain. THEOREM 4.1.
[ij];t = [l/(n - 1) !](P-l/de”-‘)(ij)K Proof.
.
One can show the relation (d/de)(ij)K
= [ij]k .
(4.1)
Applying Eq. (4.1) directly to the n-chain, we have (d/de)[ij]k+
= (n - l)[ij]E.
Q.E.D.
Suppose we have a diagram gti . One can generate an (m + n)-th order diagram by inserting n bubbles into a propagator. This corresponds to replacing (ij), by [ij],“+‘(B(p))“, where B, = i
s
dK’ (K,K’
1 V 1K’K,,) (O-)K,.
(4.2)
We will label this diagram as gzTIPS,d . LEMMA 4.2.
We have a similar expression for the n-bubble g diagram and tg diagram. This follows trivially from Theorem 4.1. Recall that -iim+l/2T j d&,Nam(&,J s dtmV~~(&&,J is just u( g$). Thus Lemma 4.2 gives us the contribution to the n-bubble diagram in terms of the base diagram in which the bubbles are “popped”.
DIAGRAMMATIC
93
TECHNIQUES
With this relation, one can write down a host of summation theorems. We give the simplest of such summation theorems, the summation over all bubble diagrams, in Lemma 4.3. LEMMA
4.3. Let ($), be (ij), with e,, replaced by e, + B, and “9~m(~z,J,,J be Yam
with each (ij), replaced by (t$), . Let i,J’$ be the sum of all diagrams generated by inserting bubblesinto a basediagram g,“,, . Then
We have a similar expression for g diagrams and tg diagrams. Proof.
Suppose we sum over all bubble insertions in one line. We have
The lemma trivially follows. Now, in the case of generalized bubble structures, such as that of Fig. 6C, Lemmas 4.2 and 4.3 will apply in exactly the same way. The only difference lies in replacing Bi , which corresponds to a bubble, by some other quantity, which corresponds to the substructure that replaces the bubble.
V. RELATIONS BETWEEN KINETIC DIAGRAMS
AND POTENTIAL
DIAGRAMS
In this section, we will prove some relations between the kinetic and potential diagrams. We will first write down a lemma which trivially follows from the work of the previous section. LEMMA
5.1. t(g3
= - G
j dK,,, j di, Nan@,,) 2ei -& 2 V~‘@J,,).
This follows from Eqs. (3.3), (3.5) and Theorem 4.1 applied to the case n = 2. Thus we can rewrite the energy density in terms of Ya,n(&,J,J alone, i.e.,
94
CFUCHTON
AND
YU
We will now consider a particular set of g diagrams. We will show that a simple relation between the kinetic energy and potential energy exists within this set. DEFINITION. Let gan = { g,: , where i runs over the 2n propagators We will call gU” a family of g diagrams. Thus, for example, Fig. 7 constitutes a family.
FIG.
THEOREM
A family
of g diagrams.
5.2.
nt&“) Proof.
7.
= - (n -
1 - /3 +,
Since v( g,“i) is independent of i v( &“)
Consider /3(8/a/3) v( gun). We can write
= 2nv(g3.
u&n).
in Van}.
DIAGRAMMATIC
95
TECHNIQUES
where
To evaluate II , we use the following identity 7 -$ (ij), = e,[ij]t + i(ti -
Writing
*Y-,“(~&~)
tj) e,(i&
.
(5.2)
as a product of 2n(ij)ds, Eq. (5.2) gives us at once
where n-1
I, = f d(i, -
t,) YE”,” c (- itjDj)*
0
j=l
Now I3 can be written as
x ?q(P(n Carrying out the differentiation we have at once 13 = (n -
- 1) P(?z - 2) ... P(1)
O)InS1.
(5.4)
with respect to h with the help of Lemma 3.3,
1) 1 WP(n-1) pn-1
x “p(P(n
t, =
2 iDP(n--8) ,...> iDP(
- 1) P(n - 2) .** P(1) t, = 0)
x TQyP(n -
1) P(n - 2) ... P(1) t,).
(5.5)
Using Lemma 3.3 again to evaluate Iz , we see that Iz =
c Pnml
nfl 7 exP iQP(12-1) - DP(k) ~(l’(tz K=l
n,“,-;
@‘PO)
-
DPM)
- 1) P(n - 2) *.- P(1) t,).
We see that the last term in Eq. (5.5) is exactly --I, .
(5.6)
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CRICHTON
AND
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Combining Eqs. (5.1), (5.3), (5.9, and (5.6), we have the desired result of Theorem 5.2. The Thermodynamic
Origin of Theorem 3.2
We now show that Theorem 5.1 stems from some general thermodynamic considerations together with the linear independence of different families, each considered as a functional of the interaction. Recall that the quantities of physical interest depend on /3, h and ,u”. We show in Appendix 1 that the following thermodynamic relation holds between the energy E and the potential energy V:
A(aE/aA)= v + #@v/a/3)- xp(ap/aA)(aN/ag).
(5.7)
To each order in h, we have nt” = -(n - 1) v* + /3(&P/a/I) - nj.P/3(+++9).
(5.8)
The only dependence of GIn(k; O-) on @’ comes from the term -~“[l l+]“, . One can show explicitly that Eq. (5.2) holds for the @ term. Defining Gln(k, 0-) = Gi(k, 0-) + p”[ll+]i one has from Eq. (5.2)
(5.9 Since Eq. (5.3) is true independent of the specific form of the interaction, we can consider it as an equation of functionals of the interaction. Using Eq. (3.1), we can write [8] (5.10) where LQ = -inei
- f (n - 1 - fl GF)(i
&
- ei).
Next, we collect all terms with the same topology. Then we can write Eq. (5.10) as
topological classes
o! the B&me toPologY
DIAGRAMMATIC
As functionals of the interaction, the topologically independent. Therefore, we have f
dKzn N/(&J
1 ,ll~,s
97
TECHNIQIJES
distinct classes must be linearly
j 9$Y~i/
= 0
for each cx.
(5.11)
olthesmne toPawY
Theorem 5.2 shows explicitly the same topology.
that the family constitutes the set of all terms of
The Zero-Temperature Limit
We will now derive a simple, but important COROLLARY
special case of Theorem 5.2.
5.3. At zero-temperature, nt( &“) = -(n - 1) V( goln).
We will prove lim,,, @?a( &“)/@I) = 0. Using Theorem 3.2, we can write V( gz) in such a way that the only p dependence comes in through&*. Now BWBf*
(5.12)
= 4W>f*.
Using the fact that j$
(5.13)
f* = Kte),
we have fqajag) f * = &es(e) + 0.
(5.14)
Corollary 5.3 stems from the well-known Feynman identity for a system in its ground state: x(&Y/a& = Y. This identity can be considered as a special case of the thermodynamic relation stated in the previous section.
6. SOME CONSEQUENCES, APPLICATIONS,
AND CONCLUDING
REMARKS
We will now state some direct consequences of the above techniques. 1. ComparisonbetweenGreen’sfunction diagramsand Brueckner-Goldstone Diagrams We will first state a theorem. THEOREM 6.1. The contribution to the ground state energy from a family of g diagrams is equal to the set of all permutations of Brueckner-Goldrtone diagrams of the sametopology.
98
CRICHTON
Proof.
AND
YU
From Corollary 5.3, we have
From Theorem 3.2, we have
Since each vf diagram is l/2 times a Brueckner-Goldstone is proved. As a second consequence of these techniques, we have
diagram, the theorem
LEMMA 6.2. A Green’s function approximation can be cast in the form of Brueckner-Goldstone diagrams tf and only if the approximation satisfies the thermodynamic relation Eq. (5.7). Furthermore, in these theories, the Brueckner-Goldstone diagrams of any topology are included only in the entire permuted set. This relation is satisfied in the Hartree-Fock and A,, theories, but not in A,, or A,, [2]. We jind empirically to fourth order in the interaction that the Hartree-Fock and A,, diagrams come in families, and can therefore be written down as classes of BruecknerGoldstone diagrams, while in Aoo and A,, , “broken” families are found, and “anomalous” terms (i.e., terms with “e’s” in the numerators of the integrands) which cannot be represented by Brueckner-Goldstone diagrams, were introduced. Conversely, to cast a statement in the Brueckner-Goldstone theory in the GreenS function .formalism, it is necessary that the statement be one which does not diflerentiate between d@erent time orderings. Thus, for example, the factorization theorem [6] cannot be cast in the Green’s function form. 2. Applications to Green’s Function Perturbative
Calculations.
The diagrammatic techniques of Theorem 3.2 should facilitate time integration in perturbative calculations. Theorem 5.2 and Corollary 5.3 implies that one can work with potential energy alone. Even in the case of approximations which do not satisfy Eq. (5.7), one can still use the result of Theorem 5.2 within complete families of diagrams. Some Concluding Remarks Our original motivation for the present study is to make a connection between the Green’s function approach and the Brueckner-Goldstone approach. These two
DIAGRAMMATIC
TECHNIQUES
99
approaches are in principle equivalent. By making connections then, we mean the facility of transforming a statement from one language to another. We have made a study of perturbation expansions in the Green’s function formalism. We feel that our study gives us sufficient understanding between Green’s function diagrams and Brueckner-Goldstone diagrams to make it relatively easy to go back and forth between the two perturbative approaches. To make direct connections between the Green’s function approach and the Brueckner-Goldstone approach, however, there is yet another link that should be studied in more detail: namely, the relation between Green’s function equations and Green’s function diagrams. In Section 2, we showed that in principle, we can derive Green’s function diagrams from Green’s function equations. In going through the process of expansion, we find that the same diagram may appear at various different stages of the expansion, and members of the same family may also fall out from various places, and yet after going through a complicated process of expanding and combining, we arrive eventually at very simple sets of Green’s function diagrams: they always appear in families with just the right proportionality constant for each member diagram. A simple final result suggests a simple method of arriving at it. We feel that a closer study can be made on the relation between Green’s function equations and Green’s function diagrams. In our program in connecting function theories to Brueckner-Goldstone theories we have to answer two questions: 1. What Green’s function diagrams are implied by the Green’s function equations ? 2. Which Green’s function diagrams relate to which Brueckner-Goldstone diagrams ? Our present study has been an attempt to answer question 2. We encourage studies for answers to question 1.
APPENDIX
1
In this appendix, we derive Eq. (5.7). We use the method of the grand canonical ensemble: the thermodynamic potential Q is defined by e+JQ
=
Tr(e-@‘)
where X = H,, + hV - $V. The expectation represented by an operator A is given by
>
value of any physical quantity
(Ah.rr,~ = Tr(Ae-@Ju”)/Tr(e-BJE4).
100
CRICHTON
AND
YU
The thermodynamic potential and all expectation values are thus functions of ,!I, p and h. Taking derivatives of Eq. (A. 1) with respect to /3, p and h, we have the following equations: aQ/aP = Wfo
- wo
- U/Pvk
(A-3) (A.4)
aqap = --(N), aqah = (i/x)(v).
(A.5)
Further differentiation and the assumption of interchangeability partial derivatives gives
h--wo aA
Apq
of the order of the
- (XV) = p + (AV),
a
W-9 (A.71
We are interested not in (H),
The first term on the right-hand side of Eq. (A.9) is seen to contain aN/ah, which vanishes identically. This is because our method for determining the power series expansion for p, using Eqs. (2.11) and (2.12), makes N independent of X. The expression in the parentheses in the second term is just aN/a/3. Thus
qaqah) = v + p(apj3) v - pA(ap/aA)(alv/ap).
DIAGRAMMATIC
TECHNIQUES
101
2
APPENDIX
To acquaint ourselves with manipulations off diagrams, and to shed further light on the t” - vn relation, we will now prove Corollary 5.3 for the family of Fig. 7 by working directly withfdiagrams. In this family there are eight g diagrams. Each g diagram gives rise to 24 vf diagrams, and 120 tf diagrams. We now divide the vf diagrams and tf diagrams each into 24 classes. Within each class of v f diagrams, we have eight identical diagrams corresponding to a particular permutation of the 4 interaction vertices. Corresponding to each class of diagrams, we have a class of tf diagrams composed of the 40 tf diagrams built out of the same ordering of the interaction vertices, but with the triangular vertex, inserted into all possible legs at all possible positions. We will prove that within the class tcn = -[(n
- 1)/n] v,“.
To prove the relation, consider the class of 40 tf diagrams. They can be classified into three groups according to the position of the triangular vertex and its two appending legs.
Subgroup
I
Subgroup
2
Subgroup
3
FIG. 8. A special grouping of tfdiagrams
for the proof of Appendix 2.
102
CRICHTON AND YU
Group 1. The triangular Group 2. The triangular
vertex is either to the extreme left or extreme right. vertex is not to the extreme left or right; appending
legs go in opposite directions. Group 3. The triangular legs go in the same direction.
vertex is not to the extreme left or right; appending
Group 2 gives no contribution since the two appending legs give a product e(e) t9(-e) = 0. Group 1 has two opposite appending legs together with a zero denominator. This gives in the limit a delta function. Together with the triangular vertex, we have es(e) = 0. Therefore, Group 1 also gives a zero contribution. Now divide Group 3 into three subgroups according to the position of the triangular vertex, e.g., we have the grouping of Fig. 8. Compare the contribution from each subgroup to a vf diagram. The sum of the four triangular vertices cancels exactly the denominators corresponding to the extra cut, except for an introduction of a minus sign. Since the vf diagram has an overall l/2 factor, each subgroup is therefore equivalent to minus two times the contribution from a vf diagram. The three subgroups together give minus six times that contribution. However, on the vf side we have eight diagrams in the class. Therefore, within the class, t,4 = -(6/8) UC4= -(3/4) v>. ACKNOWLEDGMENTS The authors wish to thank Professor Roger Anderson for his close collaboration in the initial phase of this work. They thank Professor David Yu for his assistance in identifying the Feynman identity, and Mr. P. Geren for his assistance in the proof of Theorem 3.1. They are indebted to Dr. B. Brandow, Professor R. Puff and Professor L. Wilets for fruitful discussions. This work was begun under a grant from the Research Corporation. For one of us (S.Y.), much of this work was done under the financial support of the Seattle Pacific College Institute of Research. He also acknowledges partial support of the Battelle Memorial Institute. REFERENCES 1. P. C. MARTIN AND J. SCHWINGER, P&r. Reu. 115 (1959), 1342; R. PUFF, Ann. P/ry& (N.Y.) 13 (1961), 317. 2. R. PUFF, A. S. REINER AND L. WILETS, Phys. Rev. 149 (1966), 778. 3. K. A. BRUECKNER,Phys. Rev. 100 (1955), 36; J. GOLDSTONE, Proc. Roy. Sot. A 239 (1947), 267. 4. C. BLOCH, General perturbation formalism for the many-body problem at nonzero temperatures, in “Lectures on the Many-Body Problem,” (E. R. Caianiello, Ed.). Academic Press, New York, 1962. 5. D. PINES, “The Many-Body Problem,” p. 43, Benjamin, New York, 1961. 6. N. M. HUGENHOLTZ, Phy.sica 23 (1957), 481. 7. For an alternate proof, see Ref. 141. 8. We have ignored the higher order CL*terms for simplicity.