Fortschritte der Physik / Progress of Physics: Band 25, Heft 10 1977 [Reprint 2021 ed.] 9783112519547, 9783112519530

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H E F T 10

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B A N D 25

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B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der D D R an eine Buchhandlung oder an den Akademie -Verlag, D D R -10,8 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der B R D und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle K U N S T UND W I S S E N , Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 701 Leipzig, Postfach 160, oder • an den Akademie -Verlag, D D R - 108 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortachritte der Physik" Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Lösche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 223 6221 und 22362 29; Telex-Nr. 114420; Postscheckkonto: Berlin 35021; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 74 Altenburg, Carl-von-Ossietzky-Straßc 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je Ilctt 15,— M (Preis für die D D R : 1 0 , - 11). Bestellnummer dieses Heftes: 1027/25/10. (C) 1977 by Akademie-Verlag Berlin. Printed in the German Demooratic Republic. A N (EDV) 57 618

Fortschritte der Physik 25, 5 7 9 - 6 4 8 (1977)

The S-Matrix Formulation of the Cluster Expansion in Statistical Mechanics1) P. FEE2)

Center for Interdisciplinary

Research,

University

of Bielefeld,

F. R.

Germany

Contents 1. Introduction

579

First Part — The N on-Relativistic Case 2. General Definitions of Thermodynamical Functions for Quantum Systems 581 3. The Cluster Expansion in Quantum Statistical-Mechanics 581 a) Mathematical Preliminaries 584 b) Cluster-Expansion for the Ideal Gas 589 c) Cluster-Expansion for the Interacting Gas 596 4. Decoupling of Thermodynamics from Dynamics: The iS-Matrix Structure of the LevelDensity 603 5. (S-Matrix Formulation of the Cluster-Expansion 614 Second Part — The Relativistic Case 6. Belativistic Kinematics and Dynamics 7. Relativistic Thermodynamics and Operator Traces 8. Microcanonical Volumes and Cluster Level-Densities 9. The Equation of State of the Relativistic Gas and its Cluster-Expansion 10. Conclusions References

622 627 635 642 646 647

1. Introduction The cluster expansion is a very much studied and central problem of both classical and quantum statistical mechanics in t h a t it is the most natural way of setting-up a computational method for the equilibrium properties of interacting matter, in particular in the infinite volume limit (thermodynamical limit) (see any text-book in statistical mechanics [32, 33] for an elementary approach and RITELLE'S book [49] for rigorous results). T h e present review is a simple and hopefully, a self-contained presentation of the ¿'-matri x approach to the cluster expansion, both for relativistic and non-relativistic kinematics. This approach to the cluster expansion is an outgrowth of the FS-matrix formulation of statistical mechanics by DASHEN, MA and BERNSTEIN [1], subsequently developed by D A S H E N a n d M A [2,

2)

43

3],

This work has been supported by D.F.G. with the Research Contract E R 76/1. On leave of absence from Istituto di Fisica Teorica dell' Università Torino (Italy). Zeitschrift „Fortschritte der Physik", Heft 10

580

P . FRE

The emphasis is on the construction of a full theory of the cluster expansion directly in terms of ¿-matrix elements applicable both to relativistic and non-relativistic quantum systems. The motivations and the purposes of this work are exposed in the following discussions. The richness of the spectrum of hadronic states found in high energy collision experiments and the difficulties in the dynamical description of these collisions have led many physicists to consider the description of hadronic processes by means of statistically averaged quantities. The main argument in such models is that due to the large number of intermediate states produced in the collision one can assume local equilibrium in the small interaction region where all the scattering energy is concentrated. The thermodynamical models [34—38], the hydrodynamical model [39—41] and various types of jet-models [42—44] have grown out of this kind of argument. In this context two classical problems of statistical mechanics present themselves, firstly the correct formulation of relativistic thermodynamics and secondly the relativistic generalization and interpretation of the cluster expansion. The need for the first is obvious. As to the second the presence of interactions rules out a naive ideal gas description and one is forced to consider structures typical of real gas situations, namely, the cluster expansion. The problem of relativistic thermodynamics goes back to the early days of the theory of relativity [45, 46] and has been discussed ever since by various authors. The main question has been how to define relativistic transformations of thermodynamical quantities, like temperature and heat, or how to realize the Lorentz group in the canonical phasespace in order to give an invariant meaning to the statistical ensembles, (see ref. [26] for a review of these topics and ref. [27, 28] for more detailed information on the Lorentz group realizations in phase space). In the application to high energy physics what is relevant is not so much the transformation of temperature from one frame to another, but how equilibrium states are realised in the presence of a relativistic interaction. The interaction does not just distort the phase space of free particles but through particle creation and annihilation complicates hopelessly the counting of states. There is no simple way of getting to the equation of state of such a system. On the other hand a relativistic generalization of the cluster expansion is more easily achieved and is directly expressible in terms of the ¿-matrix. This paper is subdivided into two parts. The first part deals with the non-relativistic formulation while the second deals with the relativistic formulation of the cluster expansion. To fix the notation we give in Section 2 the general definitions of thermodynamical functions and cluster expansion for quantum systems. In Section 3 we develop the mathematical machinery necessary to set-up the cluster expansion in quantum statistical mechanics both for the ideal and the interacting gas and its graph-theoretical interpretation. The material contained in this section is partly well known but some of the results are new and come from recent work in the applications of gas theory to particle physics. In Section 4 we review the ¿'-matrix formulation of statistical mechanics and in Section 5 we show how the ¿'-matrix formula of the previous section amounts to a resummation of the graphs contributing to the cluster expansion and discuss the graphical structure of the cluster expansion in the ¿'-matrix formulation. In Section 6 we present the problem of relativistic statistical mechanics and discuss the modifications implied by Lorentz invariance of the ¿-matrix in the formalism of section 4. In Section 7 we discuss the peculiarities of the relativistic partition function and deduce the relativistic equivalent of the ¿-matrix formula of DASHEN, MA, B E R N S T E I N . The

The S-Matrix Formulation of the Cluster Expansion in Statistical Mechanics

581

explicit structure of the traces involved is explained and the problem of the invariant measure of relativistic phase-space is solved in favour of the Touschek prescription. Section 8 is devoted to the study of the relativistic microcanonical volumes and cluster level-densities. We reobtain the results of CHAICHIAN, H A G E D O R N and H A Y A S H I [18] on the structure of the microcanonical volumes of relativistic ideal quantum gases from our graphical formalism and compare these with their non-relativistic analogues. The generalization of these results to the interaction gas case by F R É and SERTORIO [ 4 ] are also reobtained in the graphical formalism and shown to be equivalent to the microcanonical formulation of the cluster expansion. I n Section 9 the relativistic ^-matrix form of the cluster expansion is deduced and the general structure of the equation of state discussed. Section 10 contains our conclusions.

First Part: The Non Relativistic Case 2. General Definitions of Thermodynamical Functions for Quantum Systems

I n this first section we give the general definitions and establish the notation we shall use after on: The system in which we are interested is an assembly of quantum particles in thermal equilibrium in a box, of volume V. The total number of particles will be denoted by N. I n the thermodynamical limit the volume V will become infinite. The total hamiltonian of the system is then & * = ¡=i Z ¿m,

3

nuii

(2-1)

where ) Pi = -Wi

(2-2)

are the three-momentum operators of the individual particles and the potential F(r i j ) is a function of the distance between two particles: = IXi - Xj\. The canonical partition function

(2.3)

V) is defined as follows:

QnOS, V) = T r w r ) [exp ( - ^ y ) ]

(2.4)

P = 1 IT

(2.5)

where is the inverse temperature and the trace is performed over a complete set of wave-functions of the N particles normalized in the volume V. More explicitly if we label the wavefunctions of the complete set with the index « we can write QAP, V) = E f a r

. . . / d*xNy*(x,

...xN)

exp

...xN).

Depending on whether the particles are bosons or fermions the wave-functions ...,xN) 3

) We use units in which h = c = kB — 1 ; kB is Boltzmann's constant.

43*

(2.6)

582

P.

FEE

will be completely symmetric or completely antisymmetric functions of their arguments. We shall also consider, for purposes of reference the hamiltonian of the free system: Hn^

i=i

(2-7)

2m

and the corresponding free-gas partition function QA°Kß,

V)

= Tr(JV;F) [exp

.

(2.8)

The operator H

n

(I)

H

=

s

-

=

£

Viru)

(2-9)

¿=t=;

is the interaction hamiltonian or simply the interaction. Next we introduce the grand-canonical ensemble described by the grand-canonical partition function. We recall (see, for instance [32, 33]) that the connection between the canonical ensemble and thermodynamics is given by the following identification: F(N, T, V) = - / H lg QN(ß, V)

(2.10)

where F(N, T, V) is the Helmholtz free-energy of the -particles in the volume V at temperature T. If the thermodynamical limit exists and is well-defined then when both N and V are very large we have F(N, T, V)

^ N,V l a r g e

where the density g is defined by

Nf(ß, e)

g = lim

(2.11)

(2.12)

oo V

In this limit the chemical potential defined by fi(ß, q) = 8F(N, T, V)/8N (2.13) is a finite function of temperature and of density. We can therefore introduce the fugacity defined by z = exp [ßfi] (2.14) which also exists in the thermodynamical limit. Making use of this variable we define the grand-canonical partition function: oo Z(ß, V , z ) = 2 J z N Q N ( ß , V). (2.15) N=0,

I t is by definition the normalization factor of the particle-number distribution p

m

PN(ß,

v

A

V, z) =

F m

v>

>•

z)

19 1 fi^

(2.16)

The most probable N, that is that number N which maximizes the probability (2.16) is given by the condition \ g z + - ^ \ g Q

N

( ß , V ) \

N =

i = 0

(2.17)

583

The iS-Matrix Formulation of the Cluster Expansion in Statistical Mechanics

which, recalling (2.10) and (2.14) becomes t*

8F(T, =

V, N)

(2.18)

8N

showing that the most probable number of particles and hence the most probable density is determined by the value of the chemical potential. The average number of particles is on the other hand given by

The equivalence between (N) and N is ensured by the condition Zifi, V, z) « ZsQS{p, K-»oo

(2.20)

V)

which can be proved if the thermodynamical limit exists and if the hard-core condition of the interaction potential is satisfied (see Ref. [32]). The hard-core condition which is rather natural at room temperatures breaks down in a relativistic system due to free particle production and annihilation and eq. (2.20) becomes meaningless in the relativistic region. This fact will be discussed in the second part. For the time being we recall the physical interpretation of the grand-canonical partition function which is given by the following identification (P is the pressure): ^

= l g Z ( T , V,z).

(2.21)

The most interesting equation derivable from statistical mechanics which embodies the thermodynamical structure dictated by the microscopic dynamics of the system is the equation of state, namely the equation of the surface of all possible state-points of the system in the space (P, q, T) : /(P, e, T) = 0. (2.22) This equation can be obtained in parametric form if in the thermodynamical limit we have lim i

V->oo V

lg Z(,3, V, z) = Qtf, z)

(2.23)

where Q(f}, z) is a finite function of /? and z. In this case in fact we obtain from (2.19) and (2.21)

Q = z — Q(P, Z)

(2.24a)

PP=£2(P,Z).

(2.24b)

oz

The cluster expansion correspond to the series expansion of i2(/?, z) in powers of z oo N=o

!

(2.25)

The existence and analyticity properties of this expansion are difficult to establish. For rigorous results we refer to R u e l l e ' s book [49\ The coefficients bN(T) are the cluster functions and the study of their structure will be the basic aim of the subsequent sec-

584

P. FRÉ

tions. In the case of classical statistical mechanics their structure can be analysed with the help of linear graph theory and eq. (2.25) is the Mayer series. We refer to Ref. [21, 47, 49] for the discussion of this case. Once the cluster functions t>N (T) have been obtained we can construct the virial expansion, namely the expansion of the pressure in powers of the density by eliminating z in the power series e=ZNbs(T)w, N=0

i V

!

oo „N pp = z tAT) W ! r N=0

(2.26 a) (2.26b)

One gets (2.27) where the virial coefficients /?„(T) are combinations of the cluster functions [21,

47,40].

3. The Cluster Expansion in Quantum Statistical Mechanics In this section we study the general structure of the cluster expansion formalism. In subsection a) we introduce the second quantization formalism and the graphical techniques for analyzing scalar-products in Fock-space. In subsection b) we apply these techniques to the Ideal Gas case and we make it clear that, in the graphical formalism, the cluster expansion of the Ideal Bose or Fermi Gas is just a consequence of the general theorem of graph-theory which yields the cluster expansion of the classical gas and which is sometimes called first Mayer theorem. As a by-product we deduce the non-relativistic quantum level-densities whose relativistic analogues, worked out in [15], is discussed later in section 8. In subsection c) we derive the graphical structure of the Cluster E x pansion of the Interacting Gas and we show its perturbative character (high temperature expansion). a) Mathematical Preliminaries In order to study the cluster-expansion in quantum statistical mechanics and in preparation for its extension to relativistic system it is convenient to introduce the second quantized formalism. We introduce the particle field-operator cp± (x, t) ( ( + ) for Bose and (—) for Fermi fields) with equal time commutation or anticommutation rule [*>«?# ... D^y*) {f) = f d3xDJi J f ± projecting antisymmetric subspace -ifjV®±: let us define the projector r"v± =

( W * =

/ 1 \2

Fi£

onto its symmetric or

( ± ) i p j P

(3>18a)

( + )

E E

6

« K

PR = E ^ j —

=

•

(3.18b)

The last equation follows from the group-property of the permutations and the fact that, if K = PR, the number of transposition in K is equal to the number of transpositions in P plus the number of transpositions in R. To each element A 6 we associate an element 6 -JfA-®± by application of the projector F N ± : « = {A, •••,/*} «± =rir±Qi = ^ E

(3.19a)

(±YP{fp(l)>--;fptN)}-

(3.19b)

We can now define a nonsymmetrized Fock-space F which associates to each element a € -Jif the corresponding state |«) in the following way |S) = (,3, F, z) = exp

»

E

k=i

-n h^iP,

V)

(3.51)

where the first term in the series (3.50) (N = 0) corresponds to the the counting of the vacuum state (no particle present.) This result is a particular case of a general Theorem of Graph Theory which also provides the basis for the Classical Mayer Expansion. We want to state without proof this Theorem in its general form in order to make the analogies and the differences between the expansion (3.51) and the Mayer Expansion clear. First we need to define a graph in abstract form and to show that the pictures introduced in eq. (3.23) are indeed graphs. Following Ref. [-53] we say that we have a graph

The /S-Matrix Formulation of the Cluster Expansion in Statistical Mechanics

593

whenever we have a pair (X, U) where 1. X is a set {a--!, x2, ..., xN) of elements called vertices or points, and 2. U is a family (%, w2> • • •> um) °f elements of the Cartesian product X X X called arcs or lines. A line of the form (x, x) is called a loop and for a line u = (x, y) x is defined to be the initial end-point and y is defined to be the final end-point. The pictorial representation of the graph is obtained drawing a point for each vertex and an arrowed line from x to y for any arc (x, y). Using this abstract definition it is easy to check that a permutation is indeed a graph. In fact it is a collection of objects and a collection of oriented pairs of these objects. I t follows that the pictures introduced in eq. (3.23) are representations of graphs. To see this explicitely we can draw the corresponding graphs in standard notation. We have for instance

(12 3 I I 12 3/

=

(3.52a)

(' 2 3 I = I 13 2/

(3.52 b)

(3.52 c)

3 ) = (I13 2 12/

The general definition of connectedness [5-3] states that a graph is connected if it contains a chain ¡j\x, y\ for each pair x, y of distinct vertices. (A chain is a sequence ¡x = (uly u2, ..., ug) of arcs of a graph such that each arc in the sequence has one endpoint in common with its predecessor in the sequence and its other endpoint in common with its successor in the sequence). I t is evident from eq. (3.52), and is easy to check in general, that our definition (3.30) of connectedness is equivalent to the general one of Graph Theory which we have just recalled. If we define a weight W{0N) which associates a complex number to every graph GN such that it satisfies Lemma 1, namely such that the weight'of a disconnected graph is the product of the weights of its connected parts, and if we define M

=

E N=I

an £

/Y!

GN

(3.53)

594

P. Fre

where a; is a formal variable and JT means sum over G» to a given class and

with ZV-vertices belonging

all graphs

CO

m = Z

n=i

where

means sum over all

connected

W,Z'W{Gn) jy • Gn

(3.54)

with N-vertices

graphs

belonging to the same

GAT

class, then the general theorem already mentioned states that : (Theorem 1°) +

F{x)

1 = exp (f{x))

(3.55)

(for a proof see Ref. [21, 52]). Now it is apparent that eq. (3.51) like its analogue in the classical case can be obtained as a consequence of Theorem (3.55). Application of formula (3.27) shows that the number of permutations with a cyclic structure of the type {vt — 0, v2 = 0, . . . vN = 1},is just: ¿Vf"i=o.'»=o

= E l = iN -

1)!

(3.56)

which, in graph-theoretical language, means that there are [N — 1)! connected graphs with jV-points. As Lemma 1 states that all connected graphs with iV-points have the same weight, we obtain:

k\

r

WfkiP,

V)

=

^ i k

-

1)!

(

^

M

)

=

g

V).

(3.57)

Taking the logarithm of equation (3.51) and performing the thermodynamical limit we obtain the cluster-expansion of the Ideal Gas. QWtf,

z)

1 = lim - lg V^co

y

oo Z,

Z - Z e x T ? ( - i PN \

i

i=l

Pi2)

f t

(Pi,Pi»)-

/ i= l

(3.60)

The iS-Matrix formulation of the Cluster Expansion in Statistical Mechanics

Making use of the orthonormality of the functions y>Pi(x) in the space L2^)

595

we obtain

where A is the thermal wave length: = ]/2nmT.

(3.62)

The cluster-expansion of the Ideal Bose or Fermi Gas is therefore achieved in its standard form: p oo zk oo (i\k—l„k

This is a very well known result, (i.e. [32]) and the graphical approach has been developed, in this case, mainly for pedagogical reasons, as a first step toward the full clusterexpansion of the interacting gas properties. Nevertheless also in the ideal gas case the graph formalism shows up to be a powerful tool in the solution of the problem of constructing the microcanonical volume of 2V-quantum particles. The microcanonical volume aN{E, V) of a system of -particles enclosed in a box of volume V and with total energy E is generally defined as follows: . ¿oo ON(E, V) =

J

dpe^QAP,

V)

(3.64)

— ¿oo whereQN(P, V) is the corresponding canonical partition function. Recalling formula (3.48) we can immediately write down the microcanonical volume ±aNi-°)(E, V) of the Ideal Bose or Fermi Gas: i +¿00 e^ dfi n—

[±bkN(Jj). 4. Decoupling of Thermodynamics from Dynamics: The S-matrix Structure of the Level-Density We start in this section the programme of resummation of the graphs contributing to each cluster function in order to cast them in ¿'-matrix, finite and non-perturbative form. 1. The first step in our strategy is to express the grand-canonical partition function as a Laplace transform of the trace of a suitable microcanonical density operator. In this way we decouple thermodynamics from dynamics. 2. The second step consists in expressing the density operator through the ¿»-matrix and its derivatives. 3. The third step is the analysis of the resummed graphs generated by the ¿/-matrix form for the level-density and the use of the fundamental theorem (3.55) in order to obtain the cluster expansion.

604

P. Frb

We start with the first point. Recalling eq. (3.78) we have Ztf, V, z) = Tr {exp [-p(Ét

+ É¡ + ¡uÑ)]}

F

(4.1)

where É0, Hj and ¡u,Ñ are all hermitean operators. We define therefore a new hermitean operator É ' = H 0 + Éj+

pÑ

(4.2)

which acts on the Fock-space F and we can write: Z{fi, V, z) = Tr {exp ( - £ # ' ) } . F

(4.3)

We consider É'. I t is an unbounded but self-adjoint operator. Then the spectral theorem for self-adjoint operators (see [29], page 253 and [30], page 118) guarantees that there exists a spectral measure with the following properties: 1.

P(A) is a projection operator for any A: V A 6 [0, oo] v A,

2. 3.

¡X

£ [0, oo]

P(A)2 = P(/)

P(A) P(,a) = P(A)

if

/ ^

¡X

lim P(A + £) = P(A) (In strong convergence topology) £—(e>0) P(0) = 0

lim P(/.) = 1 A—»00

(In strong convergence topology)

such that if for any \x) £ F we define the associated Stieltjes measure on the real axis as follows: fis([a, 6]) = (x\ P(6) \x) - (x\ P{a) |x>. (4.4) Then we can write oo

(x\ f(É') \x) = J fW fiJW (4.5) 0 where in (4.5) the integral is the Stieltjes integral associated with the Stieltjes measure (4.4) and / is any function. Making use of the spectral Theorem we can write down the trace of the grand-canonical density operator as follows: oo

Z = Tr (exp ( - £ # ' ) ) = £ f «H» ^ ( X ) . l«>er o

(4.6)

Now let us consider the purely numerical function exp (—0%): it admits a Cauchy integral representation: Hi)

where L{X) is a closed path which encircles X. So we can write 2 = Tr (exp (-pÉ'))

= £

J ps{dX) ¿

^

^

dz.

(4.8)

Now suppose that we are able to find a path L which encircles all the spectrum of H' and that we can commute the circular integral with the Stieltjes integral and the sum over

The iS-Matrix Formulation of the Cluster Expansion in Statistical Mechanics

605

states. Then we can write Z = ($e^dzE J L

(4.9)

m 3— A

l®>6F J

In this case the spectral theorem of self-adjoint operators guarantees that: £

f

r r r

= isyer ZM

z—H & =

_ (* - ^

T r

=

_

T r

(4.10)

+ ¡00

rig. l

where the have defined the propagator &{z) = (z - A')-1

(4.11) (also called Green function) Finally distorting the integration path L as in Fig. 1 and disregarding the contour at infinite we get: oo Zifi, V, z ) = ^ ( J ) e~pz where

L

T r

i) + — 1, , . E{ — H o + ie l^(out)')

=

!«£.) +

—a o —

W^') HjIW^')

(4.17a) (4.17b)

where E{ is the eigenvalue of corresponding to the free state \4>i). We define the free Green-function operator Cr0(E — ie):

&0(E - ie) = {E - A o - ie)-1

(4.18a)

and its correspondent &0'(E' — ie)

60'(E' - ie) = (E' - A0' - ie)- 1 .

(4.18b)

For i =|= j the transition amplitude between an initial state \4>i) and a final state '(z*) = 6'(z*) ^'(z*).

(4.41)

The definition of the off-shell o'(z)-matrix is now obvious. The scattering matrix-element for a state Ift) to end up in ¡ft) is:

=

(4.70)

'\X)

E

and the projector on these eigenspaces P'E,. Then we introduce the restrictions of and [ 7 " ( ¿ O P to the eigenspaces J(E') ¿-'(E')

= P'E,F\E')

¿F'I(E')

=

(4.71a)

P'E,

P'E,[T'(E')]T

(4.71b)

P'E,

and their unp rimed analogues f ( E ) = PET(E) =

(4.72a)

P£

PET\E)

T'(E')

PE.

(4.72b)

These operators are on-shell amplitudes. We also define an on-shell scattering matrix y'(E') [é"(E')]~i

= 1 + 2ni#'(E')

(4.73a)

= 1 -

(4.73b)

2m[#'(E')Y

& ( E ) = 1 + 2 ni#{E) = 1 -

&-HE)

(4.73c) (4.73d)

2ni^"t(E).

We have the following identity: (4.74)

{ ¿ ' { E ' ) ) - 1 ¿fi, Proof: We recall formulae (4.52) and insert them into eq. (4.69). We obtain: E{L>'(E')

=

¿ r T r |[1 -

2niò(E' -

É0')

T'^E')]

A . [1

¡¡M D(E' -

+

É0')

t'(2?')]}

(4.75)

which yields : (iY{E')

ff

= i

Tr

45

[Ò(E'

-

ÓA') (T'(E')

(2 *»)» T r | [Ò(E' -

Zeitschrift „Fortschritte der Physik", Heft 10

È0')

-

Ì"W)]J

T'^E')]

A.

'

[Ò(E

-

È0')

T'(E')]

j.

(4.76)

612

p. Pré

The first addend on the r.h.s. of (4.75) can be written as: 1 d J à r T/

' ~

{Ô[E

Ôo)

^'(V') ~ T'HE')]}

(4.77)

so that it contributes to the trace (4.75) only with derivatives of on-shell amplitudes (Matrix-elements of T'(E') between states 0 f energy E'). It is a little bit more lengthy, but straightforward to show that the same happens with the second term in the r.h.s. of (4.77). We recall that, because of our hypothesis (HI), the free states (eigenvector of and #o> #o') form a complete set. We denote the free-states by where « is a label to distinguish states in a given eigenspace:

Then the completeness relation can be written : 00

2; «

o

If we use the notation W

W

=1. =

(4.79)

I T'{E')

(4.80)

and suppress the indices « we can write : Tr J[(E,

F)

=

V, p)

Tr J(P,

F)

(4.91)

F)

(4.92 a)

yN (E)

(4.92b)

Equation [4.92b) can also be written: (4.93) 45*

P. Feé

614

5. The 8-Matrix Formulation of the Cluster-Expansion Summarizing the results of the previous section we have shown that the following expansion of the grand-canonical partition function holds: m

oo

V,z)=E

Z*>QnOS, V) =

V, Z) +

(5.1)

V, z)

N=0

where

(5.2)

Qytf, V) = Qn«>W, V) + QNa>(P, V).

F ) is the canonical partition function of the N free quantum particles defined in (3.42) and explicitely worked out in (3.48) and I 7 ), the interaction part, is defined in (4.92) 4 ). To complete our programme we must show that Qna){fi, V) is the sum over all graphs with N>-points of a certain type and that Lemma 1 is valid. From here we can use Theorem 1 to deduce the cluster expansion of the grand-canonical potential. We have the following theorem [2, 3], Theorem 3: The following identity holds: Tr W

1

!®) ^

&N{E)\ = i - A Im Tr In i f N ( E ) .

Proof: B y direct check. We denote by |«) a complete set of states in FN(E).

a11,1^ -sar" ¿ - ?

=

Ie

~iTr

ln

^{E)

{oi)

te

~(