June 93

Landau Theory of Fermi-Liquids

Abha SoodDepartment of Physics, University of OldenburgD-2900 Oldenburg, Fedral Republic of Germany

First Referee: Professor Dr. E. R. HilfSecond Referee: Dr. P. Rujan

Contents

1 Introduction 2

2 Landau’s Theory of Fermi Liquid 3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Phenomological Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Microscopic Verification using Green’s Function Theory . . . . . . . . . 9

2.4 Application: Hard Core Potential . . . . . . . . . . . . . . . . . . . . . . 20

2.4.1 Galitskii’s integral equations . . . . . . . . . . . . . . . . . . . . . 20

2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Discussion and Conclusion 26

3.1 Suggestions for further study and calculations . . . . . . . . . . . . . . . 26

3.2 Results and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1

1 Introduction

The phase transition of a nucleon gas to nuclear matter still remains to be complete-ly understood. One attempt made in the past is on the basis of an approximation ofnormal fermi liquids suggested by Landau (1957) which has been further developed bymany authors using modern field theoretical methods [AGD63] [Noz64] [GM58]. Thisapproximation is applied to a nucleon gas such that a phase transition to nuclear matteris obtained.

The goal of this study was to find a method to extend this technique for higher tempe-ratures using a relativistic treatment. The first step was to estimate and study quantita-tively the magnitude of error made as a result of the approximation and the calculationtechniques applied. However before the quanitative study may be made the qualitativeunderstanding of the theory is deemed necessary. Thus the first part of this thesis isan exhaustive review study of the Landau approximation for Fermi liquids, presentingthe theory in a systematic and concise form. The application to nuclear matter of thisapproximation is also sketched here. Such a treatment was necessary since the aim ofthis study is also to understand the error occuring in the determination of physical quan-tities such as the density and energy of the system due to the approximation undertaken.

Since the knowledge of the basic modern many particle theory is absolutely necessa-ry for the understanding of the basic concepts, it is presented briefly in the appendices.It is felt necessary to introduce the field theoretical methods such that a motivated phy-sics graduate student is able to understand this text. For a proper study, however, it isstill necessary to refer to the original texts cited there. Most of the calculations leadingto the results given are not included since they are either straight forward or may beregarded as good exercises.

2

2 Landau’s Theory of Fermi Liquid

2.1 Introduction

Landau’s theory of Fermi liquids has a wide range of applications. The basic conceptsand assumptions are therefore presented in this chapter. Its application to matter ofshort range two body forces, with some modifications, is then considered and the relati-ve advantages of this method with respect to some others are discussed.

The phenomenological theory of Landau based on his original ideas [Lan57] is presentedin the next section [Noz64]. These ideas are then explored more thoroughly using the me-thods of quantum field theory [GM58] [Lan59] [AGD63] [Noz64] in the section 2.3. Thetechnical information necessary for the understanding of the text is presented separate-ly in the appendix. The basic assumption of Landau’s theory is that the weakly excitedstates of a Fermi liquid greatly resemble those of a weakly excited Fermi gas. These statescan be described with a set of elementary excitations with spin 1

2and momenta close to

the Fermi surface. It is then assumed that there is a one to one correspondence betweenthe number of states of a perfect gas to that of a normal Fermi liquid. This may be phy-sically realized by adiabatically switching on the two-particle interaction. The conceptof quasi-particles (elementary excitations) is thus introduced. In this manner, however,the important low lying collective states of the liquids are lost which are necessary forthe description of (for example) superconductivity.The quasiparticles thus obtained are not the exact stationary states of the system, buta superposition of a large number of exact stationary states of the system [AGD63] witha narrow spread of energy. This leads to the damping of the states. This damping maybe explained as the interaction between the quasiparticles, with conserved laws of mo-mentum and energy. This can occur through processes where the excitation decays intoseveral others or where the quasiparticles are scattered by each other. The decay of exci-tations plays a role only at higher temperatures. On the other hand, for sufficiently hightemperature, any system tends to behave like a noninteracting system. If the tempera-ture is sufficiently low, there are only a few low energy quasiparticles which rarely scatterand thus the interaction between the quasiparticles is weak. It is then regarded to be anacceptable approximation for many applications and a perfect gas of quasiparticles onlymay be considered.

In applying the above formalism to the specific problem of nuclear matter, further appro-ximations will become necessary for the calculation of the physically measurable quan-tities. It is observed that since the nuclear interparticle forces can not be described bya perturbative expansion, the general approach is to calculate the scattering amplitudewhich remains finite and then using appropriate approximations (such as the ladder ap-proximation). The expansion thus obtained is the ordinary perturbation for the divergentpart of the graphs in the small parameter kFa, where a is the range of the interactionand kF corresponds to the Fermi momentum. One of the manners of bypassing the pro-blem created by the assumed singular nuclear potentials has been has been suggested

3

and developed by Galitskii (1958) using the corresponding finite quantities of scatteringtheory. One alternative method of calculation will be suggested and developed here andwill be dealt with in an ensuing study.

2.2 Phenomological Derivation

1 Consider a system of N identical fermions in a volume V assumed to be large, at zerotemperature. In case of a simple gas of noninteracting particles, i.e. a perfect gas, theeigenstates are antisymmetric combinations of single particle states, to be taken as planewaves2. The plane waves are characterized by their wave vector k. To define the eigenstateof the whole system, it is sufficient to indicate which plane waves are occupied with thedistribution function n(k). However, the phase information is lost through this process,and the transformation back to the original single particle states before superposition isnot unique. Let the ground state of the system correspond to an isotropic distributionn0(k). The cutoff level in the ground state kF is called the Fermi level and is given bygk3

F/(6π2) = N/V (Appendix A).

If the distribution function is changed by an infinitesimal quantity δn(k) the total energyof the system changes by an amount

δE[n] =∑k

h2k2

2mδn(k). (1)

The functional derivative of the energy with respect to the distribution function is thekinetic energy of a particle with wave vector k,

δE[n]δn(k)

=∑k′

h2k′2

2mδkk′ where δkk′ :=

δn(k′)δn(k)

. (2)

For zero temperature, δn(k) is necessarily positive for k > kF and negative for k < kFsince all the energy levels are occupied below kF and are empty above it in the absenceof interaction.

Since it is assumed that the excitation spectrum of a real fermion system (i.e. Fermiliquid) has the same structure as the excitation spectrum of a perfect Fermi gas. In orderto extend these ideas to the Fermi liquid, the interaction is switched on adiabatically, thatis without exchange of energy with the surroundings. It is assumed that the states of anideal gas are transformed to those of a real gas as the interaction gradually increases; thetime development of each state can then be studied by means of perturbative treatment3.

1The major part of this derivation is based on the notation used in reference [Noz64].2ψ1 is the single particle wave function of the first particle and ψ2 is the wave function of the second

particle. Then the corresponding two particle wavefunction obeying the Pauli principle is given byψ(x1, x2) = ψ1(x1)ψ2(x2) − ψ1(x2)ψ2(x1).

3A system with a non-degenerate stationary state cannot make a transition to another state underthe action of an infinitely slow perturbation [Lan58].

4

n0(k)

1 1

0.5

nk

kF kFk k-

6

-

6 $

&

Figure 1: Landau Distribution Function a) n0k of noninteracting fermions and b) nk of

interacting fermions at 0 K temperature

It is further assumed that the interaction is only repulsive because the ground state ofan attractive system may be radically different from that of the perfect gas. The netrepulsive interaction is also assumed to be weak.Since the states in the eigenstate basis vectors of the noninteracting system of the realsystem are in general unstable and damp out after a certain time period τ , the adiabaticswitching on of interaction should require a time much shorter than τ . However, if theinteraction is turned on too fast, the final state is no longer an eigenstate of the system.It is therefore necessary that the time period τ should be large which implies that theexcited states have a long lifetime. This thus limits the excited states to lie in a low levelclose to the ground state.When an additional particle of wave vector k with k > kF is added to a perfect gas inthe ground state and then the interaction is turned on, an eigenstate of the real gas isobtained. A quasi-particle4 of wave vector k is thus added to the system. The lifetime ofthe particle defined in this manner is only long near the Fermi surface. Thus the conceptof quasi-particle is only valid in the neighbourhood of k = kF . Similarly, a quasi-hole isthe removal of a particle of wave vector k with k < kF . Though the same distributionfunction n(k) now characterizes the real states, it gives the distribution of quasi-particlesand not of real particles. It is assumed that the distribution of quasiparticles is spatiallyhomogeneous.The distribution of the quasiparticles in the ground state is thus still given by n0(k) andthe concept of Fermi surface is retained (Fig. 1).The excitation of the system is measured by δn(k) = n(k)− n0(k). As δn(k) is appre-

ciable only near the Fermi surface the quasi-particles are well defined only in this regionand poorly defined elsewhere. In other words, a quasi-particle is an elementary excitationin the neighbourhood of k = kF and gives no information about the ground state of thereal particles in the interior of the Fermi surface. The energy of the real system is nowa functional of n(k); i.e. E[n(k)]. As shown above, this functional reduces to the sum ofenergies of the individual particles in case of an ideal gas. If n0(k) is now altered by an

4refer to Appendix B.

5

amount δn(k), the variation of energy to first order will be given by

δE =∑

εkδn(k), (3)

with εk := δE/δn(k) being interpreted as the energy of the quasi-particle.This relation is only valid when the number of quasi-particles added or removed is smallas compared to the total number N of particles in the system. The energy of the wholesystem is, in particular, not the sum of the energies of the quasi-particles.Let εkF be the energy required for adding an additional particle to the system at theFermi surface

εkF = E0(N + 1)−E0(N) = µ. (4)

Thus µ := δE/δN is called the chemical potential of the system and the ground state ofa system with N + 1 particles is obtained. If the second order effects are not neglectedas in the above considerations, the variation of the total energy is given by

δE =∑k

ε0kδn(k) +∑k

∑k′

f(k, k′) δn(k) δn(k′) (5)

and thus

εk = ε0k +∑k′f(k, k′) δn(k′). (6)

Then f(k, k′) is the second functional derivative of E and so the variational derivative ofεk with respect to n(k).Using the classical methods of statistical mechanics, from the condition that the entropyshould be conserved when the number of particles and energy are conserved5, we obtainfor the quasi-particle of energy ε the distribution function

n(ε) =1

exp (ε− µ)β + 1. (9)

5The following relation of the entropy, derived using purely combinatoric considerations for an assu-med perfect gas, may be used to derive the distribution function of the quasi-particles [Hua64] of thismodel

S

V= −

∫[n lnn+ (1− n) ln(1− n)]

dk(2π)3

. (7)

Again this could be achieved by skipping the collective states. Since the energy levels of the quasi-particles are assumed to be similar to that of an ideal fermi gas this equation may also be appliedto obtain the quasi-particle distribution (refer to equation 3). Applying the condition for maximumentropy, that the number of particles and the energy are conserved (microcanonical ensemble), thedesired distribution is obtained

n(ε) =1

e(ε−µ)β + 1. (8)

6

This relation is valid because a quasi-particle obtained from fermions is itself a fermion.The probability of finding a particle with energy ε is n(ε) and µ is the chemical potentialadjusted such that the total number of quasi-particles is normalized to N . When T → 0then n(ε) tends to the step function in Fig. 1 a) and thus the distribution of the quasi-particles is taken to be a Fermi distribution.Including the spin of the particles the above formulation may be restated as

E = E0 + δE,

where E is the total energy of the system and E0 is the energy of the unperturbed systemand on redefining k = (k, σ) as

f(k, k′) = f(kσ, k′σ′) = f0(k, k′) + fe(k, k′) δσ,σ′ , (10)

where fe(k, k′) appears only when the spins are parallel and expresses the exchange inter-

action between the two quasi-particles. It is clear that f now depends upon the momentaand spin of two quasi-particles. It will be seen later that f is connected with the forwardscattering amplitude of two quasi-particles. In an isotropic case, e. g. in the absence ofmagnetic field the quasi-particle energy ε does not depend upon spin.

The function ε0k depends only on k in equation (6) and may be expanded in the se-ries at k ≈ kF as follows:

ε0k − µ ≈ v(k − kF ) where v ≡ kFm∗

; ε0kF ≡ µ. (11)

The relation between effective mass m∗ and the interaction term f will now be derived.The sum of momenta in a unit volume is equal to a flow of mass. The momentum ofa unit volume of the Fermi liquid is the same as the momentum of the quasiparticle inthis volume. The current of the particles in the Fermi liquid is equal to the current ofquasi-particles.

∫ dk(2π)3

kn(k) = m

∫ dk(2π)3

vn(k), (12)

where v = ∇kεk. Substituting this definition in the above equation and then varyingwith respect to n where ε = ε[n(k)], we obtain

∫ dk(2π)3

kmδn =

∫ dk(2π)3

δn∇kεk +12

∫ dkdk′

(2π)6(∇kf(k, k′))nδn′. (13)

which on integration by parts and permution of k to k′ is

∫ dk(2π)3

kmδn =

∫ dk(2π)3

δn∇kεk −12

∫ dkdk′

(2π)6(∇k′n

′)f(k, k′)δn. (14)

7

The average over spin indices is taken since n and ε do not depend upon spin here. Sinceδn is arbitrary, it follows that

km

= ∇kεk −12

∫ dk′

(2π)3f(k, k′)∇k′n′. (15)

An estimate for ∇k′ at k′ ≈ kF is given by the expression, obtained by integration byparts,

∇k′n′ ≈ −k′

k′δ(k′ − kF ). (16)

Since f depends only upon the angle ϕ between k and k′ we obtain the following relationbetween the effective mass m∗ of the quasiparticles and the mass m of the fermion. Thetranslation invariance property of an isotropic liquid is also used here.

1m∗

=1m

+kF

2(2π)3

∫f(ϕ)cosϕdΩ, (17)

where f(ϕ) is the value of f(k, k′) at | k |=| k′ |= kF and dΩ is the infinitesimal solidangle. This equation only holds for sufficiently low temperatures.It is additionally assumed that the potential may be described in a self consistent manner(i.e. the interaction is described as a self-consistent field experienced by one quasi-particledue to the presence of all others).

In the momentum space, the occupied states form a sphere with radius kF which istermed the Fermi sphere.

kF = (3πNV

)3. (18)

The number of states with kF < k should be equal to the number of particles.

8

2.3 Microscopic Verification using Green’s Function Theory

The modified version of Green’s functions theory applied to the many body problemhas been introduced and presented in a compact form in Appendix C. All the results,which have been derived there, will now only be referred to and will be used to derivethe Landau theory explicitly. It will be shown that this derivation is equivalent to hisoriginal phenomenological derivation presented in the last section6.

The general equation of motion of a system of N elementary particles which experiencetwo particle interaction is formulated. The poles of the propagator is then calculatedand interpreted as quasiparticles. The vertex part of the two particle Greens function isrelated to the interaction term f(k, k′) in equation (6). The effective mass can then becalculated as shown in the last section.

The general form of the N particle Hamiltonian used in the following calculations consistsof the free part corresponding to the kinetic energy of the system and the two particleinteraction (refer to equation (91) in Appendix C.). It is possible to describe the freepart of the Hamiltonian 7 H0 by two field operators with which a single particle Green’sfunction may be defined whereas the two particle interaction V (x1, x

′1) may be descri-

bed by the two particle Green’s function to obtain the equation of motion of a particleadded to the N particle ground state. The following equation may be derived using theHeisenberg equation of motion for the field operator ψ(x) and its hermitian conjugateand then applying the definition of the Green’s functions8.

[iδt −H0(x)]G(x, x′)−∫

d4x′1V (x− x′1)G(x′1, x;x′1 + ε, x′) = δ(x− x′). (19)

Analogous equations for 2- and 3- particle Green’s function etc. may be derived, which arein this approximation not necessary, since we only consider the two particle interaction.The two particle Green’s function may be split into a sum of an uncoupled product oftwo single particle Green’s functions which also takes the mean Hartree-Fock energy intoaccount and the interacting part which describes the interaction between the two particles(147), (148). The uncoupled part includes not only the zeroth order of the perturbativeexpansion but all orders to some extent. This statement will be made precise and justifiedin a later discussion. Substituting (147) in the above equation and separating the freepart of the two particle Green’s function we get

iδt −H0(x)G(x, x′)− i∫

d3r′1V (r − r′1)G(x, x′)G(r′1t′, r′1t

′+)

−G(x, r′1t′+)G(r′1t

′, x′) + δG(x, r′1t′; r′1t

′+, x

′) = δ(x− x′)

iδt −H0(x)G(x, x′)−∫

d4x′′1Σ(x, x′′1)G(x′, x′′1) = δ(x− x′), (20)

6This section follows the presentation of [Noz64] and [Eco83] unless otherwise stated.7The free Hamiltonian is defined as H0 = p2/2m where p is the quantum mechanical momentum

operator. The total Hamiltonian is then the sum of the free part and the interacting part H = H0 + V .8In this equation, the ε term adds only an infinitesimal time interval to x′1. In the later equation

somewhat different notation is used.

9

-

6

γk

ak

6

? ω

Figure 2: The Lorentzian peak and its cha-racteristic parameters.

where Σ is the self energy or mass operator. It may be also split into an uncoupled partand an interacting part; Σ = ΣF + δΣ withΣF = −i ∫ d3r′1V (r − r′1)G(r′1t

′, r′1t′+)δ(x− x′′1)−G(x, r′t′+)δ(t− t′)δ(r′1 − r′′1) and

δΣ =∫

d4r′1∫

d4x′1∫

d4x′2∫

d4x′3G(r′1, x′1)G(x, x′2)Γ(x′1, x

′2; x′3, x

′′1)G(x′3, r

′1)

The more complicated second term may now be neglected. This is justified with theobservation that the Fourier transform of δΣ with respect to the space variables for alarge interval in time becomes incoherent and the result goes to zero. However, it mustbe noted that in case the interaction is strong and long ranged, this term will dominateover the effect of the first two terms. The observation of the effect of the uncoupled termsin this case can not be made. The Fourier transform of the resulting equation is

(ε0k − ω − ΣL(k, ω))G(k, ω) = 1. (21)

ΣL is the effective self energy or the effective mass operator which has been calculatedin equation (155) and (156) to the lowest order in the perturbation series. A betterapproximation of the self energy is possible using self consistent equations to obtain asolution of the Green’s function (19) for a chosen singular potential (refer to the followi-ng section). ΣL(k, ω) is thus merely the average Hartree-Fock energy felt by the particlewith wave vector k as mentioned before. Thus the only effect of the interaction in thisapproximation, is a correction to the kinetic energy of the particles. Since ΣL(k, t− t′) islocalized in time the resultant Fourier transform is a smooth function of ω. The poles ofthe equation (21) may now be determined and it is assumed that there is only a singlesolution for each value of k. This is the basic Landau assumption.

The Lorentz curve (refer to Fig. 2) is the natural extension of the line in the line spectrumto the normed curve in the band spectrum for exponential damping. It is assumed thatthe states of the interacting particles experience an exponential damping when expandedwith respect to the orthonormal basis of the states of the noninteracting particles9. These

9refer to the Lehmann representation of Green’s function presented in Appendix C.

10

states of the interacting particles are not the eigenstates and thus not stable but tend towander away from the original states in a finite time interval. They thus have only a finitelifetime. If the states are damped by a factor eγkt then the inverse of γk is the measu-re of their lifetime. The characteristic features of the Lorentz curve are sketched in Fig. 2.

In order to define the position and the characteristics of the poles the following pro-cedure is adopted, analogous to the development in [Eco83] and [Wyl86]. The functionG(k, ω) is defined as

G(k, ω) := GR(k, ω)−GA(k, ω), (22)

where GA,R are defined in the equation (136) which are the advanced and the retaredGreens functions respectively and the term A(k, ω) is defined as

A(k, ω) := iG(k, ω). (23)

The purpose of this definition is obtain a function A such that its analytic continuationin the complete complex may be obtained. It may be observed now that A(k, ω) is realif ω is real, and for fermions it is always nonnegative. It can be shown that

∫ dω2πA(k, ω) = 1. (24)

Γ(k, ω) is defined by the equation

Γ(k, ω) =∫( dω′

2πA(k, ω′)ω − ω′ . (25)

One can obtain a relation between GR,A and the Γ using the complex analysis methodsof integrating over the poles as follows:

GR(k, ω) = lims→0+

Γ(k, ω + is).

GA(k, ω) = lims→0+

Γ(k, ω − is). (26)

Γ(k, ω) is an analytic function in the complex plane except for the region on the realaxis where A(k, ω) 6= 0. It follows directly from the equation (25) that

Γ(k, ω + is)− Γ(k, ω − is) = −iA(k, ω) when s −→ 0+. (27)

This quantifies the jump over the real axis. It is useful to define the above quantities inorder to obtain a suitable function Γ(k, ω) which may be analytically continued over thebranch cut onto the next Riemann sheet using Cauchy integral formula. Since A(k, ω)

11

may be regarded as the evaluation of an analytic function on the real axis10, it is possibleto obtain the required analytic continuation onto the next Riemann sheet rather simplyas11

a.c.Γ(k, ω) = Γ(k, ω) + a.c.iAb(k, ω)+ iA′(k, ω) for Imω > 0,

a.c.Γ(k, ω) = Γ(k, ω)− a.c.iAb(k, ω) − iA′(k, ω) for Imω < 0, (28)

where the function A(k, ω) is assumed to have the form of a Lorentzian peak over asmooth background (Fig. 2) given by

A(k, ω) = Ab(k, ω) + A′(k, ω)

where A′(k, ω) =2 | γk | ak

(ω − εk)2 + γ2k

=2 | γk | ak

(ω − εk + iγk)(ω − εk − iγk)(29)

and Ab(k, ω) is the smooth background which is a contribution of the incoherent part ofthe mass operator and will be neglected in the calculations.Using equation (26) and the above results, the following analytic continuation of the theretarded and the advanced Greens functions may be obtained.

a.c.GR(k, ω) =

Γ(k, ω) for Im ω > 0

Γ(k, ω)− a.c.iA(k, ω) for Im ω < 0(30)

a.c.GA(k, ω) =

Γ(k, ω) for Im ω < 0

Γ(k, ω) + a.c.iA(k, ω) for Im ω > 0(31)

Thus we see that the poles lie only in the lower half plane for GR(k, ω) and in the upperhalf plane for GA(k, ω).A further simplification may be considered at zero temperature for fermions, since thepoles (if any exist) of the analytic continuation of G(k, ω) lie only in one quadrant of thecomplex plane. Thus it is seen that

a.c.G(k, ω) =

Γ(k, ω) for Im ω > 0, Re ω > µ

a.c.Γ(k, ω) for Im ω > 0, Re ω < µ

a.c.G(k, ω) =

Γ(k, ω) for Im ω < 0, Re ω < µ

a.c.Γ(k, ω) for Im ω < 0, Re ω > µ(32)

Thus, analogous to the reasoning presented above and using the equations (29) and (32),we see that the poles lie only in the lower half plane for Re ω > µ and in the upperhalf plane for Re ω < µ. It may thus be inferred that there exists a branch cut for theanalytic function G(k, ω) at the line µ = Reω and the poles cross the real axis with theReω increasing from 0 to ∞ at µ = Reω. This surface defines the Fermi surface ofthe interacting particles. The peaks of the poles on the positive/negative complex planeof the analytic continuation of the Green’s function which appear, are the quasiparticles.The quantity 2πi(A(k, ω) +B(k, ω)) corresponds to A(k, ω) where A(k, ω) and B(k, ω)are the spectral norm density in the equation (136). Thus it can be seen, that the peak

10A prerequisite for analytic continuation of a function over a branch cut in a straightforward manner.In general, an analytic continuation over a branch cut for an arbitrary function is not a trivial problem.

11a.c. is the abbreviation for analytic continuation.

12

6

-

Ab(k, ω)

A(k, ω)

ω

B(k, ω)| k |> kFfor | k |< kF

Figure 3: A qualitative presentation of the shape and the positionof the poles of the analytic continuation of the Γ(k, ω).

of A(k, ω) appears as a peak in A(k, ω) if εk > µ which is interpreted as a quasiparticlewhereas for εk < µ it appears as peak in B(k, ω), which is then interpreted as a quasihole.Inside the Fermi surface are the quasiholes, whereas outside it are the quasiparticles fora given value of Reω (Fig. 3).

Thus it is assumed that the pole of the Green’s function is not a singular Dirac deltafunction but contain a sharp Lorentzian peak12. γk is the width of the curve (at halfmaximum) which defines the lifetime of the state, τk = 1/γk. The weight of the peak canbe shown to correspond to the residue of the pole ak. It is the measure13 of the extentto which the concept of independent particle, i.e. the bare part of the quasiparticle, isretained; and (1 − ak) is the proportion of the particle participating in the interaction,i.e. the dressed part of the quasiparticle. As the interaction tends to zero the quasiparticletends to the corresponding bare particle, as required, and the Lorentian peak convergesto Dirac delta peak. The position of the peak curve given by εk corresponds to the ave-rage energy of the state [Eco83]. The behaviour of the poles is diagramatically presentedin Fig. 3.It is crucial to observe that the peaks of A(k, ω) described in this manner are well de-fined only close to the real axis, if their lifetime is well defined (refer to equation (142)in Appendix C. and its discussion). Due to the short lifetime the peak of the quasipar-ticle will dissipate before they can develop. This restricts the position of the poles onlyclose to the Fermi surface. Thus, for the physically interesting properties of interactingfermions which are described through the quasiparticle, it is reasonable to study onlythe peaks rather than the complete Green’s function. A qualitative picture of the polesis illustrated in Fig. 4.

12in the sense of mass distribution13not in the mathematical sense

13

-

6

Reωzk µ

Im ω

Figure 4: The poles of the analyticcontinuation of Γ(k, ω).

-

6

k

ak

kF

1

n(k)

Figure 5: The distribution functi-on n(k) of the interacting fermi-ons - quasiparticles in the inde-pendent particle eigenfunctions.

It is further assumed, that the interaction γ(k, t) is continuous at the surface of disconti-nuity of Γ and thus the self-energy Σ too is continuous. This then assures the “continuity”of the pole of G(k, ω). It can be shown14 that the energy of the quasi-particle and quasi-hole at the Fermi surface is equal to the chemical potential µ. The residue ak of G canbe written as

ak = − 11 + ∂Σ/∂ω |ω=εk

. (33)

The figure 4 concisely presents the results. The slope of the curve on which the poles lie,at the point of intersection with the real axis (ωk = µ), is zero. This allows the physicalinterpretation of the residue of the pole as the magnitude of jump experienced at theFermi surface. This quantifies the extent to which the concept is retained. The disconti-nuity at the Fermi surface of the distribution of noninteracting fermions is exactly one,whereas that of the interacting particles can be shown to be equal to the residue ak. Asthe interaction increases the discontinuity tends to zero and the concept of quasi-particlesis no longer valid [Noz64].

The distribution of the interacting fermions can be understood as if the next unoccupiedfree state above the Fermi surface (the lowest excited state) is now occupied throughinteraction by a quasiparticle and at the same time a quasihole appears below the Fermisurface (Fig. 5). The distribution of the particles15 is calculated as follows

N( 12

)(k) = N(− 12

)(k) = 〈a†k, 1

2

, ak, 12〉 = −2i lim

t→0+

∫ ∞−∞

G(k, ω)eiωtdω2π. (34)

14by the Van Hove Theorem [Noz64].15The index 1

2 refers to the spin of the particles.

14

k1 k1 + k

k2 − kk2

k1 k2 − k

k2k1 + k

k1 k1 + k

k2k2 − k

q

k1 + k2 − q

q + k

q

q

k1 − k2 + k + q

U

- I7

Us

w

-

/M

Figure 6: The vertex pert of a two particle Greens function in the lowest order of pertur-bation theory.

The difference16 between N 12(k) for | k |< kF and | k |> kF is exactly due to the presence

of the pole and is equal to the residue a.

N( 12

)(kF + 0)−N( 12

)(kF − 0) = a. (35)

Thus the jump in the momentum distribution of the interacting particles takes place atthe same place as for the non-interacting particles. Since 0 ≤ N( 1

2)(k) ≤ 1 it follows that

the size of the jump also lies in this range.

The consequences of the Landau assumption will now be precisely formulated17. Withthe help of the quasiparticle description of the interacting particles, the single particleconcept and the Fermi surface concept are still partially valid, though their lifetimes andthe number are restricted.

The two particle interaction is described through the vertex part of the two particleGreens function (refer also to equations (147) and (148)). The vertex part has fermionpermutation symmetry18 since the particles under consideration are fermions. Interme-diate states can occur in the formulation of the vertex part, which corresponds to differentnumber of particles in the system19 in accordance with the appropriate time ordered pro-duct of the Ψ20 operators in the Greens functions.The vertex function of a pair of nearly equal wave vectors k1, k3 and k2, k4 with a smallmomentum tranfer21 k can be written as k3 = k1 + k and k4 = k2 − k and the vertexfunction22 is written as Γαβ,γδ(k1, k2; k1 + k, k2− k). In the lowest order of the perturba-tion theory the diagrams in Fig. 6 are obtained.The internal part of the vertex function correspond to the following pairs of propagators.

• G(q)G(k1 + k2 − q)16Since the rest of the integral may be made small for a proper choice of t17The derivation contained in this section is based on the presentation found in the references [AGD63]

and [Lan59].18Γαβ,γδ(k1, k2; k3, k4) = −Γβα,γδ(k2, k1; k3, k4)19i.e. N or N + 2 or N − 220These Ψ are defined as the field operators in Appendix B (refer to equation (84).21k := (| k |, ω) with | k | kF and ω µ22In short we can write Γαβ,γδ(k1, k2; k1 + k, k2 − k) = Γαβ,γδ(k1, k2; k).

15

• G(q)G(k + q)

• G(q)G(k1 − k2 + k − q)

It is observed that the second diagram is divergent for small k. This has a singularcontribution to the vertex function whereas the other two contribute a nonsingular term.The vertex part can now be calculated with the ladder summation of all the diagramswhich consist partly of singular and partly of nonsingular interaction. This summationcan be implicitly done for the second term such as to include ladder diagrams to allorders.

Γαβ,γδ(k1, k2; k) = Γ(1)αβ,γδ(k1, k2)− i

∫d4q

(2π)4Γ(1)αξ,γη(k1, q)G(q)G(q+k)Γηβ,ξδ(q, k2; k), (36)

where in Γ(1) k has been safely set to 0.The estimate for the propagator has already been calculated as

G(p) ≈ a

ε/h− v(| p | −kF ) + iδ sign(| k | −kF ), (37)

where | p ≈ kF and ε is nearly 0. The term v(| p | −kF ) is obtained from equation (11).The integral in equation (36) may be split to give a contribution of the region closeto the point | p |= kF and the region far away from it. The first region contains thesingularities and thus the corresponding integral determines the contribution of thesesingularities. For small k only a small region close this point comes into consideration.Since the arguments of the two Greens function lie close together, the other quanititiesin the integrand may be assumed to vary only slowly with respect to q. There is thus acontribution from the poles only if they lie on the opposite sides of the real axis. Thisimplies either

• if | q |< kF then | q + k |> kF or

• if | q |> kF then | q + k |< kF

Since k is assumed to be small, the product G(q)G(q + k) can be replaced byAδ(εq)δ(| q | −kF ) for the part of the integral with respect to q obtained by going aroundthe poles. The coefficient A can be determined by integrating the estimate of the productG(q)G(q + k) (36) which is given by

A =∫

d4qa

εq/h− v(| q | −kF ) + iδ sign(| q | −kF )

× a

εk/h+ ω − v(| k + q | −kF ) + iδ sign(| k + q | −kF )

=2πia2

v

vkω − vk

, (38)

16

where v is in the direction along q with | v |= v. The contribution of the rest of theregion is regular and is simply represented by φ(q). Thus the total product G(q)G(q+k)is

G(q)G(q + k) =2πia2

v

vkω − vk

δ(εq)δ(| q | −kF ) + φ(q). (39)

Substituting the above expression in the equation (36) and in the singular part carryingout the integration with respect to dqdε where d4q = q2 dqdεdo and do is the solid anglein the direction of q we get

Γαβ,γδ(k1, k2; k) = Γ(1)αβ,γδ(k1, k2; k) − i

∫d4q

(2π)4Γ(1)αξ,γη(k1, q)φ(q)Γηβ,ξδ(q, k1; k)

−i∫

d4q

(2π)4Γ(1)αξ,γη(k1, q)

2πia2

v

vkω − vk

δ(εq)δ(| q | −kF )Γηβ,ξδ(q, k1; k) (40)

= Γ(1)αβ,γδ(k1, k2; k) − i

∫d4q

(2π)4Γ(1)αξ,γη(k1, q)φ(q)Γηβ,ξδ(q, k1; k)

+2πia2kF

v

∫ do(2π)4

Γ(1)αξ,γη(k1, q)

vkω − vk

Γηβ,ξδ(q, k1; k). (41)

The limit k → 0 of the vertex part is not unique. There are two different ways of obtainingthe limit as are shown below

Γωαβ,γδ(k1, k2) = limk→0;k/ω→0

Γαβ,γδ(k1, k2; k)

Γkαβ,γδ(k1, k2) = limk→0;ω/k→0

Γαβ,γδ(k1, k2; k). (42)

It is possible to relate the two quantities to each other. Taking the first limit of equation(36), we obtain

Γωαβ,γδ(k1, k2) = Γ(1)αβ,γδ(k1, k2)− i

∫ d4q

(2π)4Γ(1)αξ,γη(k1, k2)φ(q)Γωξβ,ηδ(k1, k2). (43)

It is simple to eliminate Γ(1) from equation (36) using the above equation to obtain

Γαβ,γδ(k1, k2; k) = Γωαβ,γδ(k1, k2) +a2k2

F

(2π)3v

∫Γωαξ,γη(k1, q)Φ(q)Γξβ,ηδ(q, k2)

vkω − vk

do, (44)

where iΦ denotes the second term of the product G(q)G(q + k) in equation (39).

Now taking the second limit of the above equation we obtain the desired relation betweenthe two quantities.

Γkαβ,γδ(k1, k2) = Γωαβ,γδ(k1, k2) +a2k2

F

(2π)3v

∫Γωαξ,γη(k1, q)Φ(q)Γkξβ,ηδ(q, k2)do. (45)

17

Now the vertex part will be shown to be equal to the scattering amplitude f(k, k′) whichis given in the orignal phenomological paper of Landau [Lan57].

Since for small k and ω, i.e. close to the poles of Γ, it can be shown that Γ Γω

and thus Γω may reasonably be neglected in the equation (44). Γ may be representedby the product χαγ(k1; k)χ′βδ(k2; k) close to its pole. Substituting this expression in the

equation (44), and using the definition ναγ(n) = nkω−vnkχα,γ(n); n = unit vector along q,

we get

(ω − vnk)ναγ(n) = nkk2Fa

2

(2π)2

∫Γωαξ,γη(n, l)νηξ(l)do. (46)

It can be made plausible that the scattering amplitude of quasi-particles for small momen-tum transfer k is a2Γαβ,γδ(k1, k2; k) and the scattering amplitude for forward scatteringis a2Γ(k) [AGD63]. The quantity a2Γω itself does not have any physical significance but itis related to a2Γk (45) and | k1 |=| k2 |= kF . This is explained as the collision of the twoparticles involved when the momentum transfer tend to zero at the Fermi surface, theenergy exchange must also be strictly 0 (i.e. ω = 0). The other limit Γk allows a smallenergy tranfer even when the momentum transfer is strictly equal to zero. This impliesthat the momentum remains on the energy shell. Thus the process of the first limit isnon-physical at the Fermi surface and no scattering processes take place. The k and ωrefer to the energy and momentum difference respectively.

The effective mass can now be determined which has been defined at the end of thelast section. Some relations of the behavior of the system near the pole necessary for thederivation of this result have been calculated and listed below [AGD63].If the system is in an infinitely weak field δU , which is homogenous in space and slowlyvarying in time, the first order expansion of the Green’s function of the system in thepower series of δU can be used to derive the relation

∂G−1(p)∂ε

=1a

= 1− i

2

∫Γωαβ,αβ(p, q)G2(q)ω

d4q

(2π)4. (47)

If the particles are assumed to have an infinitesimally small charge δe and the systemis assumed to be in a weakly spacially homogeneous magnetic field which is constant intime, in the limit where δe→ 0 and k→ 0 the following relation may be obtained nearthe pole.

∇pG−1 = −v

a= − p

m∗a= − p

m+i

2

∫Γkαβ,αβ(p, q)

qmG2(q)k

d4q

(2π)4. (48)

The change in the Greens function when the system moves as a whole with a small andslowly varying velocity δu in the limit where ω → 0 and δu can be calculated and thefollowing relation be derived near the pole,

p∂G−1

∂ε=

pa

= p− i

2

∫Γωαβ,αβ(p, q)qG2(q)ω

d4q

(2π)4. (49)

18

The last important relation can be derived observing the change of the Greens functionin an infinitesimally small field δU which is constant in time and weakly homogeneousin space and using the equilibrium condition µ + δU =const. in the limit where k → 0and δU → 0.

∂G−1

∂µ= 1− i

2

∫Γkαβ,αβ(p, q)G2(q)k

d4q

(2π)4. (50)

Sustituting the relation between Γk and Γω (45) into the equation (48) and then usingformula (39) we obtain the desired result

1m

=1m∗

+kF

2(2π)3

∫a2Γωαβ,αβ(ϕ)cos(ϕ)do. (51)

With some what more effort and using the results derived above it is possible to obtain[AGD63] the relation

N

V=

8π3

k3F

(2π)3, (52)

which is however well known from the noninteracting limit.

19

2.4 Application: Hard Core Potential

The distant goal of the Landau theory is to describe the nuclear matter. Nuclear matteris assumed to consist of 4 (approximately) degenerate nucleons and there exists a shortrange two-body interaction in the first approximation which contains a repulsive as wellas an attractive component (for example, a hard core repulsion modelling the influenceof a strongly interacting meson cloud over a very short range and a Yukawa attractionwhich rapidly diminishes with increasing distance). The first step towards this goal is tosolve the hard core problem.

This problem is not trivial to solve since the first order of the perturbation theoryof an infinite hard core potential is divergent in momentum space. Thus an alternativeapproach has been developed relating the appropriate terms developed with this methodto their corresponding terms in the scattering theory [Gal58]. This formalism is brieflypresented here and further developed to calculate the physical quantities of interest. Thenuclear matter is only one of the many possible applications of this method. Thus thetreatment here will be kept quite general such that the results may be directly appliedelsewhere.

2.4.1 Galitskii’s integral equations

This is an attempt to calculate the energy spectrum of a non-ideal Fermi gas in a radiallysymmetric, short-ranged (i.e. na 1 where n is the density of the particles and a is therange), positive (i.e. repulsive) but strong hard-core potential V (r) [Gal58], [FW71]. Thisapproximation may be justified as follows [Mig77]. The case where na 1 correspondsto the “gas approximation”. The gas approximation refers to the case where the densityof particles is such that the interaction between more than two particles at one time maybe safely neglected. This approximation may not be directly applied to nuclear matter.In the case where the temperature is low, the number of excitations is small and insteadof a gas of interacting particles, a gas of elementary excitations or quasi- particles maybe considered and the gas approximation can be applied. For nuclear mater kFa ≈ 1/3[Eco83].The interaction between the particles is assumed to be not retarded23 and localized intime: V (x − x′) = U(r− r′)δ(t − t′). The excitation of the system is εs = ε(k1) − ε(k2)where k1 > kF > k2, kF being the momentum at the Fermi surface and k1 and k2 aremomenta of a particle being excited from k2 to k1 or vice versa. In comparision with thelast section, it is noted that this interaction is included in the vertex part Γ (127).

The effective self energy of the quasiparticle (quasihole) will now be determined. Inthe first order of approximation of the perturbation series for one praticle Greens functi-on, it is seen that the self energy Σ(k) corresponds to the two diagrams of the equations(155) and (156). For a singular potential these diagrams diverge and a simple perturba-

23i.e., the particles and holes are not allowed to interact. This corresponds to the non-relativisticapproximation where c =∞

20

tion expansion does not lead to a reasonable answer. Some of the higher terms can beobtained by either increasing the lines of interaction always only between the particles -the ladder approximation; or by adding lines of interaction between particles and holes;and/or by increasing the number of closed particle-hole loops (156). The ladder approxi-mation is the simplest approximation for two particle Greens function. This correspondsto the higher order Born approximations for forward scattering. For the strong repulsiveinteraction, it is necessary to include all orders of this series. This is possible by solvingan implicit equation for the self energy.Since the coupling constant of the interaction is not small, it is not necessary that theperturbation series converges. However, the parameter f0kF is assumed to be small, whe-re f0 is the real part of the scattering amplitude24 for small momenta which is relatedto the range of the potential25 a and kF corresponds to the Fermi wavenumber (refer toAppendix D and [FW71]).Since the two body scattering amplitude in presence of a medium remains finite eventhough the potential in the momentum representation is infinite to all orders of the per-turbation series, it is justified to make an expansion in terms of a parameter which canbe made small. The first three terms of the expansion are explicitly given below whichdo not correspond to the orders of a normal perturbative expansion but may be fullyaccounted by using only ladder diagrams. In the limit kF → 0 we get f(k,k′)→ −a andthe following expansion is used (refer to Appendix D).

E

N=k2F h

2

2m[A+BkFa+ Ck2

Fa2] (53)

where the constants A,B and C are to be determined.The summation of all the ladder diagrams is necessary since the higher order diagramsaccount for a change in the wave function in case of singular potential. The ladder dia-grams are kept and summed over since this corresponds to the picture that the repeatedinteraction between particles produces a change in the wave function. The interpretationof the equation (53) can now be given. The first term evidently is the energy of the non-interacting system; the second term which is linear in scattering length, is the forwardscattering (both direct and exchange) from the particles of the medium; and the thirdterm takes into account the Pauli priciple which reduce the number of free intermediatestates available to the fermion and only appears if the fermion is first excited and thendeexcited. [FW71] It is assumed that the higher order collisions arising from the particle-hole interaction may be safely neglected in the approximation used so far.It is convenient to represent the summation of the self-energy part of the ladder diagramsas a block which is the effective interaction of the vertex part Fig. 7.The diagram representation of the equation (20) is shown in Fig. 8. It is necessary now

to obtain a good estimate for the self energy. The self energy can be written as follows[Gal58]

Σ(k) = −2i∫

d4k′G0(k′)Γ(k, k′; k, k′) + i

∫d4k′G0(k′)Γ(k, k′; k′, k) (54)

24f0 := f(k,k′) where | k |=| k′ |25a may be also regarded as the s-wave scattering length. For | k |→ 0 ⇒ f0 → −a (refer to the

Appendix D.)

21

p p p

Figure 7: The Feynman diagrams for ef-fective interaction (vertex part) in lad-der approximation

Figure 8: The diagrammatic represen-tation of the equation (148) which ta-kes the two-particle interaction into ac-count.

Figure 9: The Feynman diagrams forproper self energy in ladder approxima-tion

for particles of spin 12, which corresponds to Fig. 9 in the diagram representation. The

effective interaction (vertex part) Γ which is required in the above equation is firstcalculated and then expressed in terms of the scattering function f(k,k′). Since thecalculations are rather tricky they are not presented here (refer to the orignal literatureby Galitskii [Gal58] or [FW71]). The final result obtained is

Γ(k1, k2; k3, k4) ≡ Γ(k,k′; g) = −4π

mf(k,k′) +

(4π)2

m

∫ d3q

(2π)3f(k,q)f ∗(q,k′)

× N(q)

mE − q2 + iδN(q)+

1

q2 − k′2 + iδ (55)

This equation is written in the center of mass coordinate system where k = k1− k2; k′ =k3−k4 and g = (k1 +k2)/2 = (k3 +k4)/2. The quanitity N(q) := 1−n0

g+q−n0g−q where

n0k := θ(kF − k) implies that N(q) is 1(−1) if both the state g ± q are outside (inside)

the Fermi sphere and zero otherwise.In the first approximation, the effective interaction Γ is equal to the scattering amplitudef in the above equation. Since one integration must still be done, the value of f(k,k′) forall k,k′ must be known (i.e. values of f off the energy shell26 must also be calculated.)The Dirac equation may be used in order to calculate if | k |=| k′ | and the following result

26This point still remains unclear and must be studied in more detail.

22

may be obtained by using the generalized optical theorem for the scattering amplitude.

Γ(k,k′; g) = Re4π

mf(k,k′) +

(4π)2

m

∫ d3q

(2π)3f(k,q)f ∗(q,k′)P 1

q2 − k′2

+(4π)2

m

∫ d3q

(2π)3f(k,q)f ∗(q,k′) N(q)

mE −mεq + iδN(q)+

1

q2 − k′2 + iδ (56)

The proper self energy for relative coordinates may be written as follows where k :=(k − k′)/2:

Σ(k) = −2i∫

d4k′G0(k′)Γ(k, k; g) + i

∫d4qG0(q)Γ(k, k; g) (57)

which may be calculated by substituting the equation (56) into (57) to the second order.The partial wave expansion is used using the long wavelength expansion δ0 ≈ −ka (referto Appendix D) and we get

f(k,k′) ≈ −4πa+ 4πika2 +O(k2a3) (58)

for | k |=| k′ | → 0.Thus the “exact” Green’s function in this approximation may be calculated and is givenby the following equation

G−1(k, ε) ≈ ε′ − ε0k − Σ(2)(k, ε′) (59)

where ε′ := ε− Σ(1).Σ(1) and Σ(2) can be calculated now. The first term is trival and only gives a shift inthe ground state energy. The second term is however very complicated and may only becalculated in an approximation close to the Fermi surface. The energy level which arethe poles of the Greens function (59) have been calculated by Galitskii [FW71]. Theseresults are used in the Chapter 3. This brief treatment is only attempted such that allthe concepts used are so documented.

23

2.5 Discussion

The Landau theory is an ingenious simplification of an extremely complex and almostincalculable many body system in an elegant manner such that a relatively good appro-ximation to all orders of perturbation may be obtained. It still remains to be discussedhow accurate this method actually is and therefore all the approximations used in theoryare listed and discussed here.

The Hartree-Fock method fails to properly include the correlations between the particles.Thus, the 2-particle Greens functions are included only partially without the vertex part.It is the vertex part which accounts for the correlations. The ladder approximation is asimple but self consistent manner to sum a subclass of Feynman diagrams to all orders ofthe perturbation theory. In the ladder diagrams repeated interactions are allowed beforethe particles leave the region where the interaction can take place. These diagrams alsocorrespond to the Born series in the order of the interaction V (r − r′). This method isuseful for obtaining a rough estimate of the binding energy for high density, stronglyinteracting fermion systems. Thus this method is an inprovement to the Hartree-Fockmethod.The two particle interaction is used here. It is well known that the cluster expansiondoes not converge in general for strong interaction. Thus considering only this simplecase here may be problematic. But in the low density approximation, this truncation isjustified.The potential is assumed to be retarded and thus instantaneous. It is thus denoted byhorizontal lines in the Feynman diagrams. The higher temperatures where the energyof the particles is relativistic, this approximation no longer holds, since all calculationshere are in the nonrelativistic limit. This limit is taken instead of the relativistic oneas appropriate for low densities and low temperature. A proper extension to this theoryto the description of relativistic phenomena requires the implementation of relativisticpropagators and an analogue calculation. However, it must be made compatible to theapproximations used here and the restrictions imposed on the density and excitationenergy of the system. Some useful preliminary references on the subject are [Eli62],[AP60] and [BC76].

In the microscopic description of the many body systems, Greens functions are nor-mally used instead of the wavefunction. This does not give us a complete descriptionof the system but allows us to concentrate on most essential features of the system forfinding the states and marcoscopic properties.The weakly excited states of a system of interacting particles can be described as anaggregrate of elementary excitations - quasi-particles. The excited state of the systemmay so be described by fewer parameters than is necessary for an exact description.The elementary state is not a stationary state but a packet of stationary states with anarrow energy spread. If the energy of the spread is small compared to the excitationenergy then the description of the states by means of elementary excitations is possible.These excitations are analogous to phonons. In contrast to phonons, which are bosons,

24

these quasi-particles arising from fermions also obey the Pauli principle and hence arefermions.The simple quasi-particles are one-particle excitations which may be considered to origi-nate due to a transition of a particle from a state under the Fermi momentum surface toa state above the Fermi surface. This may be interpreted as a particle-hole creation in abackground Fermi sea. These quasi-particles have a mass different from the free particlemass. The one particle excitation in a real fermi system is equivalent to the excitation ofan ideal gas composed of quasi-particles with a Fermi distribution with respect to energy.

Another unsolved problem as yet is that of including the degeneracy of the system expli-citly, which hasn’t been attempted due to lack of time but should be relatively straightforward. In principle, since we have now a weakly interacting gas of quasi-particles itshould be possible to include the degeneracy in the same manner as is done for the Fer-mi gas.

Thus the quasi-particle method is very interesting due to its conceptual simplicity andas it is applicable to a wide range of phenomena such as electrons in a conductor (jelliummodel), superfluidity and clusters. Thus it is still an interesting challenge not only touse this method to obtain physical results for specific systems but also to improve thistechnique further.

25

3 Discussion and Conclusion

The Landau approximation with one application has been presented so far which has beenstudied using the cited literature. It was the aim of this study to quantitatively determinethe errors of this method. Therefore it is necessary to compare it with other methodswhich are based on nonperturbative calculations to obtain an estimate of deviation fromthe expected results. However, this further goal has not been accomplished due to thelack of time.

3.1 Suggestions for further study and calculations

Consider a model of a many body system consisting of N nucleons which interact witheach other [Men92]. The nucleon interaction is assumed to consist of a repulsive hardcore potential27 and an attractive scalar Yukawa potential with a σ boson obtainable in asimilar fashion to the Nambu-Jona-Lasino model. This model has been solved in the firstorder perturbation theory and then using the path-integral formalism nonpertubativelyin the first quantization [Men93]. The results are consistent and may be compared to thelattice gauge calculations using first principles, also made by the author. The degeneracyof the nucleons is set to be 40 in this model in order to compare it with the results ofthe lattice calculations28, and degeneracy = 4 is used to make a comparison with theextrapolated values to the ones obtained in the nature.

This model may perhaps also be calculated using the Landau approximation revisedin this thesis. The first step will be to apply the results of the hard core problem obtai-ned by Galitiskii as roughly sketched in Section 2.4. The states lying close to the Fermisurface have been calculated by Galitskii [Gal58]) and the following result is obtainedupto the second order of the expansion parameter where k :=| k | with k ≈ kF .

ε(k) =h2k2

2m+

43π

(kF a)(h2k2

F

2m)+(kF a)2[

415π

h2k2F

2m(11−2 ln 2)− 16

15π2

h2k2F

2m(7 ln 2−1)(hk−hkF )](60)

At k = kF we obtain the result

ε(kF ) =h2k2

F

2m+

43π

(kFa)(h2k2

F

2m) + (kFa)2[

415π

h2k2F

2m(11 − 2 ln 2)] (61)

which is equal to the chemical potential µ29. Since kF may be written as a function ofdensity, ρ, we obtain a relationship between the energy and ρ. The plot of this graph maybe directly compared with a similar graph of the model calculated in the same manner

27which is included through excluded volume28This degeneracy is an effect of using staggerd fermions and is only an artifact of the upto date lattice

calculations.29This statement is only exact for T = 0

26

as in [Men93] above setting the attractive part of the potential equal to 0.

The Taylor expansion may be made close to the Fermi surface

εk = εkF +∂εk∂k|kF (k − kF ) (62)

and so it follows comparing the equation (60) and (61) that the effective mass m∗ isgiven by

m∗

m= 1 + [

815π

(11− 2 ln 2)(akF )2] (63)

The next step would be to calculate the propagator for the complete problem whichis described above, i. e. now the attractive Yukawa potential would be included in theHamiltonian, and a self consistent Dyson equation can be obtained in the momentumspace (refer to Appendix C. equation (158)). The equation (59) is the “exact” Green’sfunction G′0 for the hard core potential. Now the exact Green’s function G′ for the com-plete problem may be solved using equation (159) where Σ∗(k) may be calculated usingequation (160) to the first order. The poles of the Greens function again give the newdispersion relation with the help of which the energy levels of the complete system canbe obtained.This is however only a rough sketch of the proposed method. The details such as how toobtain the energy levels far from the Fermi surface (Galitskii equations) the extention tofinite temperature and the legitimacy of using an attractive interaction while using theLandau approximation have to be dealt with before attempting the calculations.

The problem solved so far, is applicable only for non-degenerate case. It is proposedto include the degeneracy in a consistent manner and then to the compare the resultswith those presented above. In this manner, some quantative control over magnitude ofthe error due to the approximation is sought. Even for the non-degenerate case it maybe checked whether the two methods converge to the same result.

One more control over the results can be achieved by comparing the above results withthose determined by independent pair approximation (Brueckner’s Theory) [FW71]. Thepotential is modelled here by a repulsive hard core and an attractive square well. Insteadof Galitski’s integral equations, the Bethe-Salpeter equations are used to obtain the solu-tion of the hard core potential which give the same ground state energies at low density.The effect of the square well is assumed to be small, thus it does not produce a changein the wave function. For the hard core solution the Born approximation is used in theregion of the square well. This last approximation, however, is not strictly necessary andthe coupled problem could be solved exactly [MS19].The more realistic physical problem may be solved using the derivation for the nucleon-nucleon interaction constructed as follows. Let the potential is constituted of a shortrange repulsive meson exchange, an intermediate attractive scalar meson exchange and a

27

long range light π-meson exchange. These contituents will not be discussed here [BBN85]but the effective potential for two interacting nucleons may be used in order to obtainthe quasi-particle energy spectrum and thus the statistics and relevant quantities.

3.2 Results and Conclusion

A preliminary study was undertaken using the Landau approximation and the ThomasFermi model for nuclear matter [KWH74]. A simple extension to relativitic energies anddensities was undertaken without closer scrutiny. The results obtain were not comparableto those of the model presented in the last section and would therefore not be discussedhere. This result was however the motivation for a closer examination if the Landautheory.

The main aim of the study was to gather a better understanding of the nuclear many-body problem and the phase transition of a nucleon to nuclear matter at low densityand temperature. Eventually, the further development of the methods learnt here tothe more complicated problem of finite temperature, higher density and degeneracy wasalso planned which too has been partially solved. The latter has been omitted hereand left as suggestion to further study. The complexity of the problem comples one touse complex and difficult field theorectical methods a few of which are described here.A direct application of these specific methods has also been developed but due to thelack of time, could not be calculated. An ensuing study would deal with these calculation.

It was however observed that these methods are very general and may be applied withsuccess to other fields, one of the best examples of which is cluster physics. The discus-sion 2.5 makes it clear that much better results in this field are to be expected thanwith other methods since all the problem concerning the strong interactions are absentthere. Thus this treatement is a useful introduction for students to the application ofthis method to the related fields.

28

Appendix A. Perfect Gas: a statistical mechanical view point.

30 Statistical mechanics relates the thermodynamical functions to the Hamiltonian ofthe many particle system. In the grand canonical ensemble, where the total energy andthe total number of particles of the entire system, the average energy and the averagenumber of particles of a large number of identical subsystems are conserved, the partitionfunction is given by

ZG :=∑N

∑j

e−β(E(j,N)−µN), (64)

that is, the summation overt all states j and for each number of particles N present inthe subsystem (from 0 to infinite). Here j is the j-th state for a fixed number of particlesN of the subsystem and β is the inverse of temperature T times Boltzmann constant kB.The partition function ZG is related to quantum mechanics with the relation

ZG =∑N

∑j

〈jN | e−β(H−µN) | jN〉 = Tr e−β(H−µN). (65)

Furthermore, the thermodynamical potential Ω is defined as

Ω(T, V,N) = −kB T ln ZG. (66)

Any thermodynamical quantities may now be calculated with the thermodynamical po-tential. For a non-interacting Fermion gas in occupation number representation, thepartition function for the ground state H = H0 may be written as

ZG = Tr e−β(H0−µN) =∑n1,···〈n1 · · · | e−β(H0−µN) | n1 · · ·〉. (67)

Since these states are the eigenstates of the Hamiltonian and the occupation numberoperator, the operators may be replaced by their eigenvalues. The exponent is now ac-number with respect to the state vectors. The sum of the expectation values wouldfactorize to the product of the trace of single particle expectation value. This is, however,only possible because there is no interaction between the particles and the total energyof the subsystem is the sum of the energy of individual particles.

ZG =∑n1

〈n1 | e−β(E1−µ)n1 | n1〉 · · · = Πi

∑ni

〈ni | exp−β(Ei−µ)ni | ni〉. (68)

30The notation used in this section by and large corresponds to the one found in Fetter and Walecka,1971 [FW71]

29

Since the occupation number of fermions is either 0 or 1 this reduces to

ZG = Πi[1 + e−β(Ei−µ)]. (69)

Now the thermodynamical potential and the average number of particles may be calcu-lated using equations (66) and (69).

Ω = −kB T∑i

ln (1 + exp−β(Ei−µ)) (70)

and

〈N〉 =∑i

n0i =

∑i

11 + e−β(Ei−µ)

. (71)

It is assumed that the ensemble is contained in a large volume with periodic boundaryconditions. The sum over single fermion levels can be replaced by an integral over wavenumber

∑i

in continuum limit where L is large7→ gV

(2π)3

∫d3k, (72)

where g is the degeneracy in the single particle momentum state. The maximum wavenumber kF (Fermi surface) for T = 0 is determined by computing the expectation valueof the number operator in the ground state G, where σ is the spin index

Nave = 〈G | N | G〉 =∑k,σ

〈G | nk,σ | G〉 =∑k,σ

θ(kF − k)

in continuum limit=

gV

(2π)3

∑σ

∫dk3θ(kF − k) =

gV

6π2· k3

F

⇒ k3F =

Nave(6π2)

g V. (73)

The density of states, that is the number of states of energy between k+dk, may now bewritten as

D(ε) =gV

(2π)34πk2 dk =

gV

4π2

(2mh2

)3/2

ε1/2dε. (74)

D(ε) may more precisely be called the density of orbitals because it refers to the solutionsof single particle problem and not to the states of the N particle system. D(ε) · n(ε) isthen the density of occupied orbitals [KK80].

30

Appendix B. Quasi-Particle Method: Second Quantization

This theory may be restated in the language of second quantization where the obser-vables are expressed in the occupation number representation of quantum theory. Thealgebra of the wave functions is now transformed to the observables which now obey thequantization rules. This is realized in the following formalism.

A system consists of N particles, where the states of the k-th particle are describedby the space Lk of square integrable functions on set Mk, and the states of the systemare described by square integrable functions of N variables x1, x2, · · · , xN where xk ∈Mk,form a Hilbert space, denoted by HN , where the scalar product is given by

(φ1, φ2) :=∫φ1(x1, x2, · · · , xN ) · φ2(x1, x2, · · · , xN ) dNx, (75)

where dNx := dx1. · · · .dxN is a differential measure on (Mk)N . [Ber66] HN is the direct

(tensor) product of N single particle Hilbert spaces H1,

HN = H11 ⊗H1

2 ⊗ · · · ⊗ H1N , (76)

and the states31 | α1 · · ·αN) := | α1 > · · · | αN >, that is, the product of N single-particlestates, which form a complete orthonormal basis [BR86].

(α1 · · ·αN | α′1 · · ·α′N ) = < α1 | α′1 > · · · < αN | α′N > = δα1,α′1· · · δαN ,α′N . (77)

If the system consists of N identical particles, the sets Mk coincide with each other, andit is unnecessary to consider the whole space HN [Ber66].Let the subspace of the system HN

B ⊂ HN consist of all totally symmetric states andthe subspace HN

F ⊂ HN consist of all antisymmetric states in HN . Then the particleslying in HN

B are called bosons and those lying in HNF are called fermions [Ber66].

The state of the system consisting of an indefinite number of particles are describedby vectors of the space32 H which is the direct sum of all HN and the one-dimensionalspace H0 which corresponds to the vacuum state33. The subspace HF ⊂ H is now forexample defined as

HF :=∞⊕0

HNF where H0

F = H0

31| αi >:= ψαi(·) and αi := (ni, li,mi) , i.e., the set of quantum numbers describing the i-th particle32called Fock space33the state in which no particles are present

31

These completely symmetric or antisymmetric states may be written in terms of creationaα† and annihilation aα operators acting on the vacuum state [BR86].

| α > = a†α | 0 >< α | = < 0 | aα. (78)

The algebra of these operators is derived from the physical restriction of permutationsymmetry imposed upon the wave function. The algebra of fermions thus obtained isgiven below [BR86]

aα, aα′ := aαaα′ + aα′aα (79)

aα, aα′† = δα,α′ = aα′†, aαaα, aα′ = 0 = aα†, aα′†. (80)

The state of an isolated fermion in an external field, created by other particles, is deter-mined by the Hamiltonian34 H(x). If εα and φα are the eigenvalue and the eigenfunction

respectively of the single particle Hamiltonian H1(x) in the state α, then the correspon-ding N particle Hamiltonian in the non-interacting case [Dav65] may be calculated asfollows. We have

HN(x1, · · · , xN ) =N∑i=1

H1(xi). (81)

The N particle wave function in the same representation is φ(x1, · · · , xN). The operatorof the number of particles in the state α is of the form

nα = a†αaα (82)

which is the operater for the wave function in the occupation number representation.Thus the eigenvalue equation (H1(x) − εα)φα = 0 is the equation for determining thesingle particle states | α >, the total Hamiltonian of the system of independent fermionscan be written in the form

H :=∫

Ψ†(x)H1(x)Ψ(x) dx. (83)

The field operators Ψ, in the occupation number representation, are expressed in termsof the operators aα

Ψ(x, t) =∑α

aαφα(x)e−iεαt. (84)

34x := (x, σ), where x is the spatial component and σ is the spin component.

32

The algebra for field operators is derived from the corresponding algebra of the creationand annihilation operators and by using the conditions of orthonormality and comple-teness of φ’s

Ψ(x), Ψ†(x′) = δ(x− x′)δσ,σ′ = Ψ†(x), Ψ(x′)Ψ(x), Ψ(x′) = 0 = Ψ†(x), Ψ†(x). (85)

The free Hamiltonian for the wave equation in the occupation number representationmay now be written with equation ( 84) ( 85) as

H0 =∑α

εαa†αaα (86)

and the number operator as

N =∑α

a†αaα. (87)

In general, any observable in the occupation number representation is obtained from thecoordinate representation with

F =∫

Ψ†(x)F(x)Ψ(x) dx =∑α,β

a†αaβ < α | F | β > (88)

where < α | F | β > =∫φ∗α(x)Fφβ(x)dx (89)

and where F in coordinate representation∑∞i=0 F(x1, · · · , xi) is a sum of the operators

F(x) acting upon the coordinates of N fermions [Dav65].

In the case of interaction between particles the Hamiltonian of the N particle systemmust be extended with the interaction term35 V

HN(x1, · · · , xN ) =N∑i=1

H1(xi) +∑i < j

V (xi, xj) (90)

which in second quantization is

H =∫

Ψ†(x)H1(x)Ψ(x) dx +∫

Ψ†(x)Ψ†(x′) V(x− x′)Ψ(x)Ψ(x′)dx dx′

=∑α

εαa†αaα +

∑α,β,α′,β′

a†αa†βaα′aβ′〈αβ | V | α′β′〉 (91)

〈αβ | V | α′β′〉 =∫ ∫

φ∗α(x)φ∗β(x′)V(x, x′)φα′(x)φβ′(x′) dx dx′. (92)

35two particle interaction

33

and correspondingly

N =∫

Ψ†(x)Ψ(x) =∑α

a†αaα. (93)

The chemical potential µ may now be defined again in order to avoid the auxiliarycondition that the total number of particles N is constant. Adding the term −µ∑α a

†αaα

to the operator defined in equation (91) we get

K = H− µN =∑α

(εα − µ)a†αaα +∑

α,β,α′,β′a†αa

†βaα′aβ′ < αβ | V | α′β′ > (94)

Thus all single particle states with energies below the Fermi level will be occupied andthose with energies above the Fermi level will be empty if there are no interactions ( orV = 0 ) and so µ = εF [Dav65] and T = 0.

The problem of solving the resulting Schrodinger equation can be simplified by usingcanonical transformations. They are defined as the transformations which define thetransition from one set of independent variable to another. Bogolyubov’s canonical trans-formations are used to bring the (main part of) the Hamiltonian to diagonal form. Thecomplete (or partial) diagonalisation of a Hamiltonian through a canonical transforma-tion leads to a new definition of the creation and annihilation operators which in turnleads to a new indepedent (or nearly independent) single particle (or quasi-particle) state[Dav65]. The canonical transformations preserve the algebra of the creation and anni-hilation operators.

The space of single particle states is truncated to a finite dimension n [BR86] to ensurethe convergence of the method. The creation and annihilation operators are combinedinto a column vector

η =

(a

a†

)ηi = ai; ηn+i = a†i ; i = 1, · · · , n. (95)

The hermitian conjugate of η is then the line vector

η† = ( a† a ). (96)

It is trivially36 possible to exchange the positions of creation and annihilation operatorsin η. The anti-commutation operators take the compact notation

η, η† =

(1 00 1

)= I. (97)

36If γ =(

0 11 0

)then γη =

(a†

a

)and η†γ = (a a†)

34

Now the inhomogeneous transformations of fermion creation and destruction operatorsars defined by37 (

b

b†

)=

(U VY X

) (a

a†

)+

(c

d

)(98)

or equivalently ζ = T η + χ (99)

It follows that

ζ† := ζγ = η†γT γ + χγ where T and χ are the transpose of T and χ resp. (100)

In general b and b† are not necesarily hermitian conjugate. This transformation is cano-nical if the following equivalent relations are also satisfied:

ζ, ζ† = I or if T satisfies (101)

T γ T = γ (102)

The matrices satisfying these conditions can be shown to form a group which for fermi-ons is isomorphic to the group of orthogonal complex matrices. These transformationsare called Bogolyubov transformations and the operators b and b† are the quasi-particlecreation and annihilition operators.The inverse of the transformation is obtained from the following relation

T−1 = γT γ =

(XV

Y U

)(103)

For finite degrees of freedom, there exists an operator S acting in the Fock space for theabove transformation such that

bi = S aiS−1 and b†i = S a†iS

−1 (104)

The operator S may be obtained by applying the identity

eABe−A = B + [A,B] +12!

[A, [A,B]] + · · · (105)

where B is a creation or an annihilation operator and A is the general quadratic form ofcreation and annihilation operators A := 1

2ηKη + lη with T =: e−γK and χ =: γl.

Applying the identity 105 to the following expression the relation

e12ηKη+lη ηie

−( 12ηKη+lη) =

∑j

(e−γK)ijηj + ljγji (106)

37where U,W,X and Y are n × n matrices and c and d are the anti-commutating elements of theGrassman algebra for fermions

35

is obtained. Let K = −K (without loss of generality)38). Then the expression (106) is alinear canonical transformation. It follows that

S = e12ηKη+lη (107)

Thus any transformation of the form 104 may be generated by an operator S of theform 107 described above. However, not all canonical transformations take this specialform [BR86].

Under the additional condition where b and b† are hermitian conjugate further restricti-ons on the free parameters have to be placed.

U = X∗, V = Y ∗, d = c∗

and the transformation 99 reduces to(b

b†

)=

(X∗ Y ∗

Y X

) (a

a†

)+

(c

c∗

)(108)

and the inverse transformation is given by

T−1 = IT †I =

(X Y †

Y X†

)(109)

The operator S defining the transformed operators b and b† is unitary when b and b† arehermitian conjugate and we obtain the analog transformation

e12η†Kη+l†η ηie

−( 12η†Kη+l†η) =

∑j

(e−iγK)ijηj + iljγji. (110)

Then T = e−iγK and χ = −iγl [BR86].

38If K contains a part such that Q = Q, then ηQη = 1/2Tr Qγ, Q adds only a constant to theexponentials, which does not change the transformation (106

36

Appendix C. Some Important Relations from the Green’s FunctionTheory

The Green’s function method has been developed in quantum field theory where it isused to describe the propagation of particles. It is possible with little modification touse this technique in the many body theory thus allowing a natural generalisation tofinite temperatures and a perturbative treatement of the system. The propagation of thebare particle (i.e. without interaction) is precisely formulated with this method and thequasi-particles states (i.e. particles with interaction) are constructed generally of fini-te lifetimes, where the first excited state consists of “configurations” of quasi-particles,whose components to first approximation are finite [Noz64].The single particle Green’s function 39 G(x, x′) is constructed as follows from the functionsatisfying the equation [BD66] [Eco83]

[H(r) − E ]G(r, r′;E) = −δ3(r − r′)⇒ G(r, r′;E) =∑n

φn(r)φ∗n(r′)E −En

, (111)

where the φn’s are normalized and satisfy the stationary Schrodinger equation. However,for the time dependent Schrodinger equation the relation

[ ih∂t − H(x) ]G(x, x′) = ihδ4(x− x′)

⇒ G(x, x′) = G(r, r′; t− t′) =i

2π

∫dE G(r, r′;E)e−iE

t−t′h (112)

has to be fulfilled. If we consider that the Hamiltonian H(x) = H0(x) + V (x), whereV (x) is a localized two particle interaction acting only at the “position” and H0 is theHamiltonian for a free particle, then the equation, corresponding to the Schrodingerequation, which has to be solved to determine the Green’s function is

[ ih∂t − H0(x) ]G(x, x′) = ihδ4(x− x′) + V (x)G(x, x′)

= −ih∫d4x′′δ4(x− x′′)[δ4(x′′ − x′) +

1

−ihV (x′′)G(x′′, x′)]

⇒ G(x, x′) = G0(x, x′) +∫d4x′′G0(x′′, x)V (x′′)G(x′′, x′) (113)

which in matrix notation may be written as G = G0 +G0V G

This iterative equation is exact and is called Dyson equation. If the interaction V is smalla perturbative expansion of the wavefunction Ψ in terms of the Green’s function may beformulated as follows40 by using the equation

Ψ(x) = Ψ0(x) +∫d4x′ G0(x, x′)V (x′)Ψ(x′)

39as before r := (r, σ) and using the notation defined in Appendix B and from now on x := (r, σ, t)40The wavefunction Ψ(n) is expanded in the powers of G0

37

with (i∂t −H0)Ψ0 = 0. Then we get

=⇒Ψ(0) = Ψ0

Ψ(1) =∫d4x′ G0(x, x′)V (x′)Ψ0(x′)

Ψ(2) =∫d4x′ G0(x, x′)V (x′)

∫d4x′′G0(x′, x′′)V (x′′)Ψ0(x′′)

(114)

In matrix notation this expression simply reduces to [Eco83]

Ψ = Ψ0 +G0 V Ψ0 + G0 V G0 V Ψ0 + · · · =∞∑n=0

(G0V )nΨ0 =1

1−G0VΨ0 (115)

This equation may also be expressed in implicit form as Ψ = Ψ0 +G0VΨ.

The function G(r, r′; t) are not analytic on the real axis and thus the equation 112is not directly solvable. The functions GR and GA are introduced as follows

GR(r, r′;E) =∑n

φn(r)φ∗n(r′)

E − (En − iε)

GA(r, r′;E) =∑n

φn(r)φ∗n(r′)

E − (En + iε). (116)

The corresponding solutions in the limiting case of ε→ 0 are then

GR(r, r′; t− t′) =∑n

φn(r)φ∗n(r′)e−iEn(t−t′)θ(t′ − t)

GA(r, r′; t− t′) = −∑n

φn(r)φ∗n(r′)e−iEn(t−t′)θ(t− t′) (117)

Thus an equivalent definition of the single particle Green’s function may now be formu-lated as follows:

G (r, r′; t− t′) = −i(GR +GA)⇒ G(r, r′; t− t′) = −i < φ | T (Ψn(r) ·Ψ†n(r′)) | φ >(118)

where T (Ψn(r) ·Ψ†n(r′)) :=

Ψn(r)Ψ†n(r′) if t > t′

−Ψ†n(r′)Ψn(r) if t < t′

and | φ > is the actual normalized ground state of the system. It is clear that if t > t′ thenthe function G may be interpreted as the probability amplitude of the system such thatwhen a particle is added to it in the ground state (r′, t′) it is found later at (r, t) withoutchange of spin σ, again in the ground state. Thus the analogous formulation for G if t < t′

is then that of the probability amplitude of the system where a particle is annihilated at(r, t) in the ground state and the system is found at (r′, t′) without change in the state.Thus G is the propagator of the additional particle or hole. It isolates the informationrelative to the ground state without being concerned about the exact structure of theground state [Noz64]. The method of obtaining the solutions to the Schrodinger equationhere is by first deriving the equations using appropriate Green’s functions and then usingthe correct boundary conditions.

38

It is possible to extend the definition of the time-ordering operator such that it operateson an arbitrary number of field operators as follows [AGD63]

T (O1O2 · · ·ON ) := (−1)POP1(tP1)OP2(tP2) · · ·OPN (tPN )

such that tP1 > · · · > tPN .

Before this formalism can be applied to a many particle system the notation of theinteraction picture has to be introduced in the second quantization notation. The inter-action picture is defined by isolating from the Hamiltonian41 Hint, and then transformingthe Schrodinger wavefunction into the interaction picture as follows [AGD63]

Φ(I) = eiH0t Φ(S). (119)

We obtain on differentiating with respect to time

i∂Φ(I)

∂t= HintΦ(I),

Hint := eiH0tHint e−iH0t. (120)

Every observable F in this representation is obtained from the corresponding Schrodingeroperator using 120., and hence it satisfies the following equation

∂F∂t

= i [H0, F ] (121)

As the Hamiltonians Hint(t) do not commute for different times, the solution is not simply

Φ(I)(t) = const. exp−i∫ t

Hint(t′)dt′Φ(I)(t = 0),

but the time-ordered product of Hamiltonians42 and the solution is now written as

Φ(I)(t) = S(t, t0)Φ(I)(t0) where S(t, t0) := T exp (−i∫ t

t0Hint(t′)dt′). (123)

41H = H0 + Hint where H0 is the Hamiltonian of the free particle and Hint is the Hamiltonian forthe interaction between particles.

42The integral form of the equation 120 is Φ(I)(t) = Φ(I)(t0) + i∫ tdt′Hint(t′)Φ(I)(t′) and the

solution for this may be written in the form Φ(I)(t) = Φ(0)(I) + Φ(1)

(I) + Φ(2)(I) + · · · where Φ(0)

(I) =

Φ(I)(t0) is the zeroth approximation, Φ(1)(I) = −i

∫ tt0dt1Hint(t1)Φ(I)(t0) is the first approximati-

on, Φ(2)(I) = (−i)2

∫ tt0dt1Hint(t1)

∫ t1t0dt2Hint(t2)Φ(I)(t0) is the second approximation and Φ(n)

(I) =

(−i)n∫ tt0dt1Hint(t1) · · ·

∫ tn−1

t0dtnHint(tn)Φ(I)(t0) is the n-th approximation where t > t1 > t2 > · · · >

tn. Now the solution can be written in the compact form [AGD63] using the relation

Φ(n)(I) = (−i)n

∫ t

t0

dt1Hint(t1) · · ·∫ tn−1

t0

dtnHint(tn)Φ(I)(t0) = T(−i)nn!

∫ t

t0

dt1Hint(t1) · · ·∫ t

t0

dtnHint(tn)Φ(I)(t0)

as Φ(I)(t) = const. T exp−i∫ t

t0Hint(t′)dt′Φ(I)(t0) (122)

39

In the Heisenberg picture the wave functions ΦH are time independent and we get

Φ(S)(r, t) = e−iHt Φ(H)(r). (124)

Thus Φ(I)(t) = Q(t)Φ(H) where Q(t) is an unitary operator. Then

Φ(t) = S(t, tα) P ΦH where P := eiH0αe−iHα. (125)

43If the interaction is now turned on adiabatically 44, (i.e. H(t)→ H0 for t→ ±∞), weobtain 45

Φ(I)(t) = S(t,−∞)ΦH .

The representation of observables in the Heisenberg picture F with respect to the obser-vables in the interaction picture F is then

F (t) = S−1(t)F (t)S(t). (126)

The Heisenberg operators averaged over the ground state may be calculated as follows:

< Φ0H | T (O1(t1) · · · ON(tn)) | Φ0

H >

=< Φ0H | S−1(t1)O1(t1)S(t1) · · ·S−1(tN)ON (tN)S(tN) | Φ0

H >

=< Φ0H | S−1(∞)S−1(∞, t1)O1(t1)S(t1, t2) · · ·ON (tN)S(tN ,∞)S(∞) | Φ0

H >

=< Φ0H | S−1(∞)T (O1 · · ·ON S(∞)) | Φ0

H > (127)

with t1 > t2 > · · · > tN , where this last condition is only required for the calculationsteps. The equation (127) holds in general, but the permutation symmetry must normallybe taken into account. S(∞) Φ0

H is the function Φ(I)(∞) obtained from the ground statefunction Φ(I)(−∞) by adiabatically turning on the interaction between the particles.Since the ground state of the system is assumed to be non-degenerate, and a stationarynon-degenerate state cannot make a transition to another state under the action of aninfinitely slow perturbation. It may be concluded that only a contribution to the phasefactor L can be made. This may be generalized to all non-degenerate states. Thus

Φ(I)(∞) = S(∞)Φ0H = eiLΦ0

H

Thus 〈Φ(I)(∞) |=< Φ0H | S−1(∞) =

< Φ0H |

< Φ0H | S(∞) | Φ0

H >

(127)=⇒< Φ0

H | T (O1 · · · ON) | Φ0H >=

< Φ0H | T (O1 · · ·ONS(∞)) | Φ0

H >

< Φ0H | S(∞) | Φ0

H >(128)

43whereΦ(I) = eiH0tΦ = S(t, α)PeiHtΦ⇒ eiH0t = S(t, α)PeiHt44where there is no interaction at t = −∞, which then implies that for α→ −∞ we get P → 145using the relation S(t2, t1)S(t1, t0) = S(t2, t0) where t2 > t1 > t0 ⇒ S(t2, t1) =

S(t2,−∞)S−1(t1,−∞) =: S(t2)S−1(t1) and it can be shown that S(t2, t1) = S−1(t1, t2) and defineS(t) := S(t,−∞)⇒ S−1(t) := S(∞, t)

40

This relation (128) is only valid for the ground state of the system, since any other energylevel if multiple degenerate can in general make a transition to another state as a resultof collision between particles.

The definition (118) is also the starting point of the physical interpretation of the Green’sfunction by means of the Lehmann representation. The purpose of using this representa-tion is to obtain the elementary spectrum of system consisting of an arbitrary number ofparticles. Since some of the features of the single particle Greens functions follow directlyfrom the quantum mechanical principles thus remaining independent of the specific formof the interaction, it is useful to exploit them to obtain the desired spectrum [FW71].A complete set of Heisenberg states46 is inserted between the (Heisenberg) field operators.These states are eigenfunctions of the Hamiltonian plus 3-momentum (or 4-momentum).

Gαβ(x, x′) = i∑n

[< φ | ΨαH(x) | φn >< φn | Ψ†βH(x′) | φ > θ(t− t′)

− < φ | Ψ†βH(x′) | φn >< φn | ΨαH(x) | φ > θ(t′ − t)]= i

∑n

[e−i(En−Eh

)(t−t′) < φ | ΨαS(r) | φn >< φn | Ψ†βS(r′) | φ > θ(t− t′) (129)

−ei(En−Eh)(t−t′) < φ | Ψ†βS(r′) | φn >< φn | ΨαS(r) | φ > θ(t′ − t)]

Since the momentum of the system is taken to be a constant of motion this inserted setof eigenstates may be regarded as a set of eigenfunctions with respect to the momentumoperator. Thus from the relations

Ψσ(r) = e−iph·rΨσ(0)ei

ph·r and pφ = 0

we obtain the result

Gαβ(x, x′) = i∑n

[e−i(En−Eh

)(t−t′)e−ipnh·(r−r′) < φ | Ψα(0) | φn >< φn | Ψ†β(0) | φ > θ(t− t′)

−ei(En−Eh)(t−t′)ei

pnh·(r−r′) < φ | Ψ†β(0) | φn >< φn | Ψα(0) | φ > θ(t′ − t)] (130)

and thus the Green’s function depends only upon the differences r − r′ and t− t′ .This result may be used to calculate the Fourier transform Gα,β(k, ω) of the Green’sfunction Gαβ(x− x′).

Gαβ(k, ω) =∫d3(r − r′)

∫d(t− t′)e−ik(r−r′)eiω(t−t′)Gαβ(x− x′) (131)

It follows with substitution of equation (130) in equation (131) and including the term±iη in order to ensure the convergence of the integral.

Gαβ(k, ω) = V∑n[<φ|Ψα(0)|n,k><n,k|Ψ†β(0)|φ>

ω−(En−E)/h+iη+

<φ|Ψ†β(0)|n,−k><n,−k|Ψα(0)|φ>ω+(En−E)/h−iη ] (132)

46In Heisenberg picture the states eiHt/hφS(t) = φH are time independent whereas the operators aretime dependent according to the relation OH(t) = eiHt/hOSe−iHt/h

41

where V is the volume of the system obtained on integration. The denominators in theabove expression in the first sum written in detail is

ω − En(N+1)−E(N)h

= ω − En(N+1)−E(N+1)h

− E(N+1)−E(N)h

= ω − εn(N+1)h− µ

h

where εn is the excitation energy of the N + 1 particle system and µ is the chemicalpotential of the system and an analogous term is obtained for the second denominatorfor an N − 1 particle system.

ω + En(N−1)−E(N)h

= ω + εn(N−1)h− µ

h

On substituting these term in the expression for G(k, ω) we obtain the desired Lehmannrepresentation.

Gα,β(k, ω) = hV∑n[<φ|Ψα(0)|n,k><n,k|Ψ†β(0)|φ>

hω−µ−εnk(N+1)+iη

+<φ|Ψ†β(0)|n,−k><n,−k|Ψα(0)|φ>

hω−µ+εn,−k(N−1)−iη ] (133)

At this stage a simplification of the matrix structure of Gαβ may be undertaken in caseof elementary particles with spin 1

2using the Pauli matrices σ and the identity matrix

as a basis. Since G must be scalar under spatial rotations, it necessarily takes the form

G(k, ω) = aI + bk · σ.and a and b may only be functions of k2 and ω. If in addition, the Hamiltonian is invariantunder reflections then G must have this property too, and b vanishes because k · σ is apseudoscalar and G has the following simple structure proportional to the unit matrix.

Gαβ(k, ω) = δαβG(| k |, ω). (134)

It is observed that the function G(k, ω) is a meromorphic function of hω , with simplepoles at the exact excitation energies of the interacting system corresponding to momen-tum hk. Thus the singularities of the Green’s function immediately yield those excitedstates for which the numerator does not vanish. In an interacting system, the field ope-rator connects the ground state with the numerous excited states of a system of N+1particles [FW71].In the infinite volume limit, i.e. the thermodynamic limit, the analytic structure ofG(k, ω) is completely altered because the discrete poles merge to form a branch line.This limit is taken by going from the discrete case (133) to the continuous case andthrough change of variable from dn to dω. The resulting expression is

G(k, ω) =∫ ∞

0dω′

[A(k, ω′)

ω − µ/h− ω′ + iη+

B(k, ω′)

ω − µ/h+ ω′ − iη

](135)

where A(k, ω) = const. V |< n,k | Ψ†(0) | Ψ0 >|2dn

dω

and B(k, ω) = const. V |< n,−k | Ψ(0) | Ψ0 >|2dn

dω. (136)

and GR,A(k, ω) =∫ ∞

0dω′

[A(k, ω′)

ω − µ/h− ω′ ± iη +B(k, ω′)

ω − µ/h+ ω′ ± iη

](137)

42

The real and the imaginary part of the Green’s function in this representation wouldthen be given using Dirac’s equation as

ReG(k, ω) = P∫ ∞

0dω′

[A(k, ω′)

ω − µ/h− ω′ +B(k, ω′)

ω − µ/h+ ω′

]

ImG(k, ω) =

−πA(k, ω − µ/h) for ω > h−1µπB(k, ω − µ/h) for ω < h−1µ

(138)

If the spacing between adjacent energy level is of the order of ∆ε, the discrete levels canonly be resolved in time scale τ which is long compared to h/∆ε. Inversely the condition∆ε ≈ h/τ has to be fulfilled for discrete resolution which is not possible for a macroscopicsample and only a level density may be determined47 [FW71].

The time evolution of the elementary excitation spectrum may now be studied usingthe fourier transformation of G(k, ω) with respect to t.

G(k, t) =∫( dω

2πG(k, ω)e−iωt (139)

Since G(k, ω) has a complex analytic structure the integral is split into two parts

G(k, t) =∫ ⌊h−∞

dω

2πG(k, ω)e−iωt +

∫ ∞µ/h

dω

2πG(k, ω)e−iωt (140)

These integrals can be evaluated using complex analysis techniques in the case where asimple pole48 exists at ω = εk − iγk with εk ≈ µ and εk > µ and the following expressionis obtained

G(k, t) =∫ ⌊h−i∞µ/h

dω

2πe−iωt[GA(k, ω)−GR(k, ω)] − iae−iεkt/he−γkt (141)

Since the integral in the above equation can be made arbitarily small with an appropriate

choice of t (| t | h/(εk − µ) and | t | γk <∼1), the remaining expression

G(k, t) ≈ −iae−iεkt/he−γkt (142)

can be interpreted as the propagation of an additional particle in an approximate eigen-state (quasiparticle) with the frequency εk/h, attenuation γk and the residue a of thepole corresponds to the amplitude of the wave packet [FW71].An estimate of the integral in equation (141) is now made in order to obtain an estimate

47Since it is assumed that ∆ε h/τ , where τ = t− t′ with t′ = 048If in case there are more than one simple poles, this analysis has to be made for each case separately.

When εk < µ an analogous analysis may be made with different signs and the case of the propagationof a quasihole is obtained.

43

of the maximum lifetime of such a quasiparticle. For k close to the Fermi surface GR(k)has the form

GR(k) ≈ a

εk/h− µ/h− v(k− kF ) + iγk. (143)

Thus this expression has a pole at εk − µ = hv(k − kF ). The estimate of the integralleads then to the following condition where the above approximation is valid.

−γkh2ae−iµt/h

πt[εk − µ]2 −iae−iεkt/he−γkt. (144)

It is also possible to obtain the particle density and the ground state energy of the systemdirectly from the definition of the Green’s function

N

V= 〈Ψ†α(x)Ψα(x)〉 = −i lim

r → r′

t′ → t+ 0

Gαα(x− x′) (145)

and by solving the above expression for µ(N) and using the formula

µ =

(∂E0

∂N

)V

respectively.

The definition (118) can now easily be extended to the N particle case as follows:

GN (x1, · · · , xN ;x′1, · · · , x′N ) := −i < T (Ψ(x1) · · ·Ψ(xN )Ψ†(x′N ) · · ·Ψ†(x′1) > . (146)

The thermodynamical properties including the density of states in the energy of thesystem can be related to single particle Green’s function. However, 2-particle interactionrequires the use of 2-particle Green’s function. This is clear observing equations (92) and(146).In general, the two particle Green’s function contains three parts:

G(x1, x2;x′1, x′2) = G(x1, x

′1)G(x2, x

′2)−G(x1, x

′2)G(x2, x

′2) + δG(x1, x2;x′1, x

′2) (147)

where the first two terms represent the free propagation of two particle and two holeswhich is thus called the “free” part of G. The last term is called the “bound” part ofG as it describes the interaction between the particles. This last term can explicitly bewritten as follows

δG(x1, x2;x′1, x′2) =

∫ ∫ ∫ ∫dx1dx2dx

′1dx′2G(x1, x1)G(x2, x2)Γ(x1, x2; x′1, x

′2)G(x′1, x

′1)G(x′2, x

′2)(148)

44

where the interaction is localized, that is, restricted to a small region in space and time.The free part in comparison has an infinite range. The function Γ describes the effectiveinteraction of two elementary excitations which is called the vertex part or proper vertexand the rest of the integrand is the propagation from r1, r2 to r′1, r

′2 of two particles or

two holes or a particle-hole pair [Noz64].This result can also be made clear on considering all feasible time ordering in the two-particle Green’s function. In the first case where the given time is either t1, t2 > t′1, t

′2 or

t1, t2 < t′1, t′2, we obtain the two cases of propagation of particle and hole pairs. If, howe-

ver, the observed time is t1, t′2 > t2, t

′1 or vice versa then the propagation of a particle-hole

pair is observed49.

In general, the time order of an arbitrary number of operators can be calculated usingWick’s theorem (149). It states that the time ordering of an arbitrary number of ope-rators in interaction picture is equal to the sum over the normal order of all possiblecontraction of the operators. The normal order of the operators is obtained when allthe creators are ordered to the left and all the annihilators are ordered to the right.This result can be obtained after appropriate number of permutations of the operatorsin accordance with the anticommutator rules.

N(O1 · · ·ON) := (−1)pOAP1· · ·OAPiO

CPi+1· · ·OCPN .

The contraction of 2 arbitrary operators is defined as︷ ︸︸ ︷O1O2 := T (O1O2)−N(O1O2),

and it follows

T (O1O2 · · ·ON ) =∑

all possible contractions

N(O1O2 · · ·ON ),

= N(O1O2 · · ·ON) +∑i<j

N(O1O2 · · ·︷ ︸︸ ︷Oi · · ·Oj · · ·ON) + · · · . (149)

Using equation (128) and applying it to the single particle Green’s function we obtainthe corresponding interaction representation

G(x, x′) =−i < T (Ψ(x)Ψ†(x′)S(∞) >

< S(∞) >(150)

=−i

< S(∞) >

∞∑n=0

(i)n

n!

∫ ∞−∞· · ·

∫ ∞−∞

dt1 · · ·dtn < T (Ψ(x)Ψ†(x′)Hint(t1) · · ·Hint(tn)) >

Let Hint be a spin-independent interaction for acting between a pair of identical fermions.Then Hint has the form

Hint =1

2

∫ ∫dxdx′Ψ†(x)Ψ†(x′)V (x− x′)Ψ(x′)Ψ(x); V (x− x′) = U(r − r′)δ(t− t′)

49refer to the defintion in equation (146) of the two particle Green’s function

45

Wick’s theorem can be applied to calculate the above expression. It is observed that theaverage of a T product of any number of field operators can be expressed as the sum ofthe products of free Green’s functions ( here without the spin indices) as follows

G(x, x′) = G0(x, x′) + δG1(x, x′) + δG2(x, x′) + · · ·

with

δG1(x, x′) =−1

2 < S(∞) >

∫d4x1d4x2V (x1 − x2)[iG0(x, x1)G0(x2, x2)G0(x1, x

′)(151)

−iG0(x, x1)G0(x1, x2)G0(x2, x′) + iG0(x, x2)G0(x1, x1)G0(x2, x

′)

−iG0(x, x2)G0(x2, x1)G0(x1, x′)− iG0(x, x′)G0(x1, x1)G0(x2, x2)

+iG0(x, x′)G0(x2, x1)G0(x1, x2)]

This result can be represented systematically using the Feynman diagram technique.This is an attempt to facilitate calculations and to obtain a graphical representation ofthe complicated integral equations [BR86]. This formalism would be demostrated here onthe example of a two body propagator which would later be used in the Landau theory.The first step is to draw all topologically distinct diagrams for n vertices, where n=0, 1, 2, ....One-to-one correspondence between a diagram and a system of contractions is referredto as the faithful representation. The diagrams of the faithful representation are referredto as the labeled Feynman diagrams. However, since the integrals corresponding to thelabeled diagrams are all integrated over the same time interval, they all contribute thesame amount to the summation. Thus for each n-th order expansion, there are 2n (inter-nal) vertices to which correspond 2nn! different diagrams, some of which with the samecontribution. The corresponding representative diagrams, where diagrams contributingthe same amount are omitted is termed the unlabeled Feynman diagram. It is, however,necessary to take into account those diagrams which are topologically invariant under achange of time label, which defines the symmetry factor S. Thus, the total number oftopologically distinct diagrams for 2n vertices are n!/S. (This treatement is only meantto present a brief simple introduction to the topic which is much too complex and richto be comlpetely presented here.)The second step is to determine the contribution of vertices. For a two-body interactionin the second quantization notation of the form in equation 92

V (t) =∑

α,α′,β,β′a†αa

†βaα′aβ′ < αβ | V | α′β′ >

< αβ | V | α′β′ > =∫ ∫

φ∗α(x)φ∗β(x′)V(x, x′)φα′(x)φβ′(x′) dx dx′,

each vertex has the contribution

< αβ | V | α′β′ > =

α′

β

AAK

JJ]

β′

α V(t)

Contraction of a pair of operators are represented by oriented lines called propagators.A normal propagator g contributes

46

< α, t | g | α′, t′ > =︷ ︸︸ ︷aα(t)a†α′(t

′) =α′, t′

α, t

and the abnormal propagators g and g contribute

< 0 | g | α, t;α′, t′ >=︷ ︸︸ ︷a†(t)αa

†α′(t

′) =α′, t′

α, t

< α, t;α′, t′ | g | 0 >=︷ ︸︸ ︷aα(t)aα′(t

′) =α, t

α, t

to the diagram.The single particle state labels are then summed and integration over time interval, fromtinitial to tfinal is undertaken.As the next step one-to-one correspondence between a diagram and a system of contrac-tions is achieved by multiplying the contribution of the diagram by (−1)nL where nL isthe number of closed fermion loops.In general, the diagrams may consist of many disconnected parts, where each of thedisconnected parts are a connected fermion loop. The expectation value in (151) is equalto the sum of the connected diagrams only and the disconnected diagrams are factorizedand cancelled with the denominator.The calculation of the many particle Green’s function is completely analogous to the abo-ve formulation. The external lines are replaced by thick lines, which denote the completeGreen’s function. They implicitly include the vacuum perturbation and the diagramswhere a two particle interaction between the four external legs of the propagator occurscan be represented by a square box. The disconnected parts of the diagrams are alreadyexcluded. The first and second order terms of a two particle Green’s function constructedwith the help of the prescription thus described are diagrammatically represented below.

(

()

)

(

()

)

@

@@@@ (

n)(

()

)

(

()

)

(

n)@

@@@@ (

()

)

(

()

)

@

@@@@

In the particle-hole case, where the particle may not be regarded as identical to the holes,the following diagrams for the interaction upto second order are obtained

@@@@

(

()

)

(

()

)

(

()

)

(

n)@

@@@@ (

()

)

(

()

)

@

@@@@

It is, however, necessary to simplify the calculation of the propagator by first calcu-lating the Fourier transforms with respect to the time coordinate. The advantage of thistransformation is that the different cases of time ordering are automatically consideredand G(x, x′;ω) has a simple form. In the case of uniform and isotropic systems, thatis, where the condition Gαβ(x, y) = δαβG(x − y) holds, the spatial and temporal inva-

47

riance allows a full Fourier transformation of the Green’s function and the two particleinteraction as follows

Gαβ(x, y) = (2π)−4∫

d4ke−ik·(x−y)Gαβ(k) (152)

Uλλ′,µµ′(k, k′) = Vλλ′,µµ′(k− k′)δ(t− t′)

= (2π)−4∫

dk3e−ix·(k−k′)Vλλ′,µµ′(x− x′)δ(t− t′)

The Feynman rules for momentum space may be now derived from the rules defined inthe coordinate space. The basic differences are that the energy and momenta at eachvertex has to be conserved∫

d4x ei(q−q′+q′′)·x ∼ (2π)4δ4(q − q′ + q′′) =

qAAKq′′q′

,the free Green’s function corresponds to the factor

G(0)αβ(k) = δαβ G(k, ω) = δαβ

θ(| k | −kF )ω − ωk + iη

+θ(kF− | k |)ω − ωk − iη

(153)

and the two particle interaction corresponds to

Uλλ′,µµ′(q) = Vλλ′,µµ′(q). (154)

The interaction in momentum space would be frequency independent since it is the Fou-rier transform of a time variable which occurs in the form δ(t − t′). The closed loopscontribute a factor (−1)F where F is the number of closed loops contained in the dia-gram. Each closed loop is associated with eiωηG(k, ω) where η → 0+ at the end of thecalculation.The topological structure of the diagrams remains identical to the structure in the coor-dinate space, but the interpretation and the labeling is naturally quite different. Thusthe first order of the perturbative expansion gives the terms

_ G(1a)αβ (k) =

i

h · (2π)4G0(k)

∫d4k1 − U(0)αβ,µµG0(k1)eiω1ηG0(k) (155)

G(1b)αβ (k) =

i

h · (2π)4G0(k)

∫d4k1U(k − k1)αµ,µβG(0)(k1)eiω1ηG0(k) (156)

Since the interaction in this case is assumed to be spin independent we get Uαα′,ββ′(q) =U(q)δαα′δββ′. It implies in spin space Uαβ,µµ(q) = 2U(q)δαβ and Uαµ,βµ(q) = U(q)δαβ .The Dyson equation in momentum space is now to be calculated, where Σαβ(k) is theself energy of the system (or the mass operator). The self energy Σ may be split intoseveral parts each of which are irreducible, i.e., the self energy cannot be separated into

48

parts by cutting only a single free Green’s function. The total roper self energy is thenthe sum of all orders of the proper self energies.

Σ∗ = Σ∗0 + Σ∗1 + Σ∗2 + · · · (157)

The corresponding Dyson equation for the proper self energy operator would be

Gαβ(k) = G(0)αβ(k) +G

(0)αβΣ∗(k)αβGαβ(k) (158)

This self consistent equation which may be explicitly solved for exact Green’s functionas follows

Gαβ(k) =1

[G(0)(k)]−1 − Σ∗(k)(159)

where [G(0)(k)]−1 = ω − h−1ε0kThe perturbative expansion of the proper self energy corresponding to the the first orderterms is

Σ∗ (1)(k) =i

(2π)4

∫d4k1[−2V (0) + V (k − k1)]G0(k1)eiω1η (160)

It is observed that the first term of the first order is only a shift in energy whereas thesecond term of the first order corresponds to the Born approximation for forward scat-tering (refer to Appendix D).

In order to investigate the thermodynamical properties of the system at finite tempe-ratures we require the grand canonical partition function ZG and the thermodynamicalpotential Ω of the system50.

ZG = e−βΩ = Tre−β(H(λ)−µN)

Ω(λ) = −(β)−1 ln Tre−β(H(λ)−µN), (161)

where a new operator K := H0 + µN may be defined to obtain the modified Heisenbergand interaction pictures.

OHK (r, τ) = eKτ/hOSK(r, τ)e−Kτ/h (162)

As long as τ is real.

Thus an analogous finite temperature Green’s functions theory may be formulated and

50H(λ) = H0 + λU

49

all the results obtained above for zero temperature can now be extended to finite tem-peratures for the new Hamiltonian (which is now no longer time dependent).

The first derivative of Ω is the expectation value of the potential energy U of the system.

∂Ω(λ)∂λ

= TrUe−βH(λ)

Z= 〈U〉 (163)

50

Appendix D. Scattering Theory

A short range two particle interaction may be described as a scattering process betweentwo particles. This theory is especially appropriate to describe the hard core interaction.The main features of this theory for this particular application would be briefly presentedhere.The Hamiltonian of a particle may be written as H = H0 + V , where H0 stands for thekinetic energy operator and V for the potential energy operator. If | φ〉 is eigenstate ofthe particle in case the potential tends to zero then the equation

H0 | φ〉 = E | φ〉

may be obtained, where E is the eigenvalue of H0. Similarly, if | Ψ〉 the is the eigenstateof the complete Hamiltonian H, then the equation

(H0 + V ) | Ψ〉 = E | Ψ〉

is obtained. The solution of the above equation under the boundary condition the Ψ → φin case V → 0 would be

| Ψ〉 =1

E −H0| Ψ〉+ | φ〉 (164)

since the operator 1E−H0

is singular, a further appropriate condition is supplied by makingthe denominator slightly complex.

| Ψ(±)〉 =1

E −H0 ± iεV | Ψ(±)〉+ | φ〉 (165)

This result is known as the Lippmann-Schwinger equation. The physical interpretationis that the positive or negative solutions are the sum of the incident unperturbed planewave and an outgonig or incoming spherical wave obtained through scattering. The sy-stem which is experimentally easy to prepare is the case of the positive solution where anincident plane wave is scattered and an outgoing spherical wave plus the incident planewave is obtained.The Lippmann-Schwinger equation is independent of the basis. The coordinate and themomentum representations are easily obtained as follows:

〈x | Ψ(±)〉 =∫

d3x′〈x | 1

E −H0 ± iε0| x′〉〈x | V | Ψ(±)〉+ 〈x | φ〉

⇐⇒ 〈x | Ψ(±)〉 =2m

h2

∫d3x′G(±)(x′,x)〈x | V | Ψ(±)〉+ 〈x | φ〉 (166)

〈p | Ψ(±)〉 =1

E − (p2/2m)± iε〈p | V | Ψ(±)〉+ 〈p | φ〉 (167)

51

The Born approximation consists of substituting the plane wave solution for outgoing orincoming spherical waves. A transition operator T may be defined as

V | Ψ(+)〉 = T | φ〉 (168)

for the purpose of obtaining the higher order Born approximations (also using theLippmann-Schwinger equations) which become necessary in the presence of more thanone scatterer. The implicit equation in the case of denumerably infinite number of scat-teres is

T | φ〉 = V | φ〉+ V1

E −H0 + iεT | φ〉. (169)

It can be shown in the coordinate representation, in the limit where | x || x′ | andwhere it is assumed that the potential is local, that

〈x | Ψ(±)〉 ' eik·x +eik·|x|

| x | f(k′,k) (170)

where the first term corresponds to the incident plane wave and the second term corre-sponds to the outgoing spherical wave observed asymptotically. The scattering amplitudef(k′, k) is the equivalent to

f(k′,k) ≡ − 14π

2mh2 〈k

′ | V | ψ(+)k 〉. (171)

The physically measureable quantity, the differential crosssection dσdΩ

, is the ratio of thenumber of particles scattered into an infinitesimal area dσ of the substended solid angledΩ per unit time to the number of incident particles crossing the unit area per unit timeintegrated over the whole solid angle. It is related to the scattering amplitude as follows

dσ

dΩ=| f(k′,k) |2 . (172)

The optical theorem describes a relation between scattering amplitude and the totalcrosssection. The physical interpretation is that the shadow cast in the forward scatte-ring is the attenuation of the the intensity of the incident beam so that the scatteredparticles are removed from in it in an amount proportional to the total crosssection σtot.This theorem is obtained as a direct consequence of the conservation of the probabilityflux.

σtot =4π| k |Imf(θ = 0) (173)

52

The partial wave expansion, i. e. the expansion of a spherical wave in an infinite superpo-sition of plane waves, is the most effective when only a small number of the partial wavescontribute which is the case at low energies. Thus if a partial wave expansion of planewaves with respect to spherical wave is made, the scattering process can be understoodas the change in coefficients of the expansion of the outgoing spherical wave. In case ofa singular potential this method is useful since the wave function of the non-interactingsystem are no longer a good approximation of the wave function after the scatteringprocess. Through a change in coeffiecients of the expansion, the new wave function isobtained. The scattering amplitude51 remains finite even though the scattering potentialis singular.

f(| k |) = f(θ) =1| k |

∞∑l=0

(2l + 1)al(| k |)eiδl(i|k|)Pl(cosθ) (174)

where l is the angular momentum, | k | the absolute value of the wave propagation vector,

Pl the Legendre polynome under the condition that m = 0, al(| k |) := eiδl(|k|)sinl(|k|)|k|

and

δ0(| k |)→ 0 for | k |→ 0.In the low energy limit only the s wave makes a contribution to the total cross sectionsince only the first term of the expansion is relavent where l = 0. Then the scatteringamplitude f0(:= f(k)), is related to the scattering length52 a [Joa79].Thus using the above equations we get

f0 → −adσ

dΩ→ a2 σtot → 4πa2 if | k |→ 0 (175)

51| k |:=| k,k′ |52where a := − lim

|k|→0

tan δ0(|k|)|k|

, which corresponds to the radius of the hard sphere in case of the

hard core potential.

53

References

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[Men92] E. Mendel. Finite baryon density in lattice simulations and nuclear matter.Phys. Rev., B (387):485, 1992.

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Acknowledgement

I take this oppertunity to express my gratitude to Prof. E. Hilf for giving me this topicand encouraging me at each phase of this work. I also thank Dr. E. Mendel for inspiringand encouraging discussions and Dipl. Phys. G. Nolte for very helpful and constructivesuggestions. I would also like to acknowledge Dr. B. Kleihaus, Dr. L. Polley and Dr. P.Rujan for reading and critizing this thesis.

I especially thank M. Thiele for encouraging me to study physics and giving me mo-ral and financial support throughout this period of study.

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Hiermit versichere ich, dass ich diese Arbeit selbststandig verfasst and keine anderenals die angegebenen Quellen benutzt habe.

Unterschrift

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