Generalized random phase approximation and gauge theories

Annals of Physics 307 (2003) 452–506 www.elsevier.com/locate/aop

Generalized random phase approximation and gauge theories Oliver Schr€ odera,* and Hugo Reinhardtb a

b

Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Institut f€ ur Theoretische Physik, Eberhard-Karls Universita¨t zu Tu¨bingen, D-72076 Tu¨bingen, Germany Received 18 April 2003

Abstract Mean-field treatments of Yang–Mills theory face the problem of how to treat the Gauss law constraint. In this paper we try to face this problem by studying the excited states instead of the ground state. For this purpose we extend the operator approach to the Random Phase Approximation (RPA) well-known from nuclear physics and recently also employed in pion physics to general bosonic theories with a standard kinetic term. We focus especially on conservation laws, and how they are translated from the full to the approximated theories, demonstrate that the operator approach has the same spectrum as the RPA derived from the time-dependent variational principle, and give—for Yang–Mills theory—a discussion of the moment of inertia connected to the energy contribution of the zero modes to the RPA ground state energy. We also indicate a line of thought that might be useful to improve the results of the Random Phase Approximation.
2003 Elsevier Science (USA). All rights reserved. Keywords: Gauge theories; Hamiltonian formalism; Random Phase Approximation; Restoration of symmetries

1. Introduction and motivation Recently, variational calculations based on Gaussian wave functionals have stirred some interest when applied to Yang–Mills theory [1–7]. A problem constantly *

Corresponding author. Fax: 1-617-253-8674. E-mail addresses: [email protected] (O. Schr€ oder), [email protected] (H. Reinhardt). 0003-4916/$ - see front matter
2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0003-4916(03)00119-2

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encountered is the fact that Gaussian wave functionals are not gauge invariant, i.e., they are not annihilated by the Gauss law operator (except for the case of electrodynamics). Therefore, one faces the situation, that the mean-field treatment (to which the Gaussian wave functionals correspond) does not respect a symmetry of the Hamiltonian. This is a situation commonly encountered in nuclear physics where one often even wants to break as many symmetries as possible in order to store a maximum amount of correlations in a wave function of a very simple form. We therefore look to nuclear physics for a possible remedy of this problem. One possibility is the introduction of a projector which projects the Gaussian wave functional onto the subspace of gauge invariant functionals [1–6]. Another possibility is the Random Phase Approximation. In this framework one studies not the ground state but excited states, and one finds that—if the mean-field ground state has a broken symmetry—spurious excitations exist. Under certain conditions these spurious excitations decouple, however, and we have the pleasant situation that our excitations are as good as they would be if we had started from a gauge invariant mean-field state. The computation of the excited states of Yang–Mills theory is especially interesting for a variety of reasons: First, one has an alternative point of view from which to shed on light onto the confinement problem, compared to the usual approaches that try to compute the linear potential between static quarks. Second, one has quite accurate lattice data for some of the low lying glueballs for both two and three spatial dimensions, e.g. [8,9]. This allows to judge the approximations that usually occur in the process of analytical calculations. On the other hand, there is quite an amount of information that is very hard to obtain from the lattice, like masses of excited glueballs, and also regularities and patterns in the glueball spectrum which can only be observed but not explained within the lattice framework. Third, in recent years the glueball spectrum has also become a matter of interest for theories that claim to have a very non-trivial connection to Yang–Mills theory like, e.g., supergravity theories [10–12]. There are basically two approaches to the generalized Random Phase Approximation (gRPA). The first one starts from the time-dependent variational principle of Dirac [13–15]. The variational parameters are decomposed into one part that solves the static equations of motion and one ‘‘small’’ fluctuating part. In this context, the nature of gRPA as a harmonic approximation becomes clear; the general procedure is outlined in Appendix A, since the equivalence to the second approach to gRPA has up to now only been shown for the fermionic case [13]. The second approach is based on the formalism of creation and annihilation operators, well-known in nuclear physics [16], and has recently been applied also in the context of pion physics [17–19]. In contrast to these latter investigations, we consider here generic bosonic field theories; they are only restricted by the requirements that the kinetic energy shall have a standard form, and the remainder of the Hamiltonian shall be expressible as a polynomial in the field operators. The structure of the paper is as follows: We begin by introducing (in Section 2) the Hamiltonian formulation of generic bosonic theories which will be the subject of investigation in the main part of this paper. In Section 3 we rewrite the usual canonical formulation in terms of creation and annihilation operators thereby bridging the gap

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between canonical and many-body treatment. Especially, we will give an expression of the Hamiltonian for a generic bosonic system (with a standard kinetic energy term) in terms of creation/annihilation operators. This is followed by an alternative form of the Schr€ odinger equation useful in the context of gRPA. We then introduce the two crucial approximations needed to obtain the gRPA equations from the Schr€ odinger equation. The second of these approximations can—under certain conditions—be rephrased in terms of the so-called quasi-boson approximation which will allow for a simpler formulation. We then discuss the normal mode form of the Hamiltonian and see what kind of difficulties appear if the gRPA equations have zero mode solutions. In Section 4 we will discuss how conservation laws translate from the full theory to the theory approximated by gRPA, when zero mode solutions are implied, and also the connection to symmetry breaking in the mean-field treatment is indicated. We will also draw attention to the fact that the character of the symmetry under consideration can change from a non-Abelian to an Abelian symmetry. In Section 5 we demonstrate the equivalence of the gRPA formulation based on the time-dependent variational principle to the operator approach employed in this paper; this equivalence has so far been demonstrated only in the case of fermionic systems [13]. In Section 6 we discuss further how the normal mode form of the Hamiltonian is altered if the gRPA equations have zero mode solutions, and the special role that is played by the moment of inertia. We then restrict the further discussion from generic bosonic theories to SU ðN Þ Yang–Mills theory, where we discuss both the moment of inertia in general, and also give an explicit computation of the leading terms in a perturbative expansion. We end the section with a discussion of the possibility of interpreting the moment of inertia as the static quark potential. In Section 7 we give a critical evaluation of the generalized Random Phase Approximation and give an outlook to further applications. At the end of the paper a number of appendices is given. In Appendix A we give a short account of the gRPA as derived from the time-dependent variational principle in so far as it is needed for the purposes of this paper. In Appendix B we discuss shortly Yang–Mills theory in Weyl gauge, and how it fits into the general framework of bosonic theories as discussed in the main body of the text. In Appendix C a number of explicit expressions and computations can be found, in Appendix D the proof for bosonic commutation relations among gRPA excitation operators is given, and in Appendix E the proof of a theorem used in Section 3 can be found. The important issue of renormalization will not be addressed in this paper. This topic in the context of the gRPA is subject to further investigation. In the approach to the gRPA that is based on the time-dependent variational principle a discussion of renormalization issues for the /4 theory has been given in [14,15].

2. Hamiltonian formalism In this section the Hamiltonian formalism for fairly general bosonic theories is introduced. Since there are good references to the subject, e.g. [20,21], we will be very brief here.

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The basic variables in the Hamiltonian approach consist of the the field operators /i and the canonical momenta pi . In order to allow for sufficient generality we will use super-indices which can—besides the position coordinate x—also contain spatial indices or internal (e.g., color) indices. The Einstein summation convention is adopted, implying sums over all discrete and integrals over all continuous variables. These basic variables satisfy the basic commutation relations ½/i ; pj
¼ idij :

ð1Þ

We will work in the Schr€ odinger representation, i.e., we will not work with abstract states but with a field representation of the states (analogous to the position representation in quantum mechanics); the objects to be considered are thus wave functionals w½/
, given by w½/
¼ h/jwi

ð2Þ

for the state jwi. The states j/i are eigenstates of the field operators /i . The usage of j/i as basis states implies that we work in the position representation, i.e., in the remainder of the paper we will realize / multiplicatively and p as a derivative operator h/j/i jwi ¼ /i w½/
and

h/jpi jwi ¼

1 d w½/
; i d/i

ð3Þ

or more pragmatically we will read in all formulas where the field operator / appears this only as a multiplicative / and p as d=ðid/Þ. Since we work in the Schr€ odinger picture the operators considered are all timeindependent. The states, however, satisfy the Schr€ odinger equation iot jwðtÞi ¼ H jwðtÞi;

ð4Þ

where H is the Hamiltonian of the system. In the main section we will be mostly interested in stationary states, i.e., states that can be written as jwðtÞi ¼ eiEt jwð0Þi. These are then eigenstates of the Hamiltonian H jwi ¼ Ejwi:

ð5Þ

3. Formulation of many-body language for generic bosonic theories 3.1. Creation and annihilation operators In this section we will apply and generalize the operator approach used in nuclear physics [16,22–30] and recently also in pion physics [17–19]. We draw the connection between the canonical treatment and the many-body language as was already briefly indicated in [31]. Since for our purposes it is sufficient, and in order to allow close comparison to gRPA as obtained from the time-dependent variational principle, cf. Appendix A, we will consider Hamiltonians of the form 1 H ¼ p2i þ V ½/
; 2

ð6Þ

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where V ½/
is a functional of the field operator, in the following referred to as Ôpotential.Õ Since the basis of this approach is the stationary Schr€ odinger equation, we start therefore from the most general time-independent Gaussian state (i; j are superindices) !
 1 1 G  iR ð/  /
Þj þ i
w½/
¼ N exp  ð/  /
Þi pi ð/  /
Þi ; ð7Þ 4 ij where N is a normalization constant. A Gaussian state allows the explicit construction of creation and annihilation operators as linear combinations of /i and pi that satisfy the basic relations ai w½/
¼ 0 and

½ai ; ayj
¼ dij ;

ð8Þ

where the latter relation fixes the normalization. Using Eq. (7) as reference state to be annihilated by ai one obtains as explicit expressions for ay ; a: 
   1 1 y
G þ 2iRjk ð/  /Þk  i p  p
ai ¼ Uij ; ð9Þ j 2 jk 
   1 1
G  2iRjk ð/  /Þk þ i p  p
ai ¼ Uij ; ð10Þ j 2 jk where U is (implicitly) defined via the relation Uij Ujk ¼ Gik

ð11Þ

and could also be called the square root of G. A short excursion on the existence and the implicit assumption of reality of U is in order here: Since Gik is a real symmetric matrix (matrix is used in the generalized sense s.t. also continuous indices are allowed) one can always diagonalize it. Therefore, one can also always write down a U as given above. However, G has to satisfy another condition, namely all of its eigenvalues have to be strictly positive, since otherwise one will run into two kinds of problems: If G has a zero eigenvalue G1 does not exist and the matrix element of hp2 i will be infinite. This is certainly undesirable, and will be assumed not to be the case in the following. Moreover, if G has a negative eigenvalue, w½/
is not even normalizable, since in the direction of the negative eigenvalue w½/
will increase exponentially for increasing values of /. Thus, all eigenvalues of G must be strictly positive, therefore U can be chosen to be real. Since ai ; ayi are just given via linear combinations of / and p, one can invert these relations to obtain /; p in terms of a; ay and the parameters of the Gaussian wave functional we started with:   ð12Þ /i ¼ /
i þ Uij aj þ ayj ; 

  1 1 1 1 pi ¼ p
i þ 2i Gik  iRik Ukj ayj  Gik þ iRik Ukj aj : 4 4

ð13Þ

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One should note that the dependence of the canonical operators on the parameters of the wave functional we choose is only seeming, since the creation and annihilation operators depend implicitly on these parameters as well. If one inserts Eqs. (9), (10) into Eqs. (12), (13) one obtains an identity / ¼ /; p ¼ p. In a practical sense, however, we have transferred information that is contained in the wave functional to the operators, since in the following the only property of w½/
that we will use for practical computations is that ai w½/
¼ 0. All parameter dependence that usually comes about by calculating matrix elements now enters the formulas via normal ordering. Since we have now a representation of the canonical operators in terms of a; ay (referred to in the following as c/a representation), all operators permissible in a canonical system can be expressed in terms of a; ay , especially the Hamiltonian which for obvious reasons is central to the following calculations. 3.2. Form of Hamiltonian in the c/a representation We have required the Hamiltonian to have a certain structure [cf. Eq. (6)]. The Hamiltonian there resolves naturally into a kinetic energy T ¼ 12 p2 and a potential term V ½/
which is a functional of / only. It is very useful to normal-order these expressions to make further progress. In the course of this, one makes the useful observation that one can write the kinetic energy as  
1 2 1 2 1 pi ¼ p
i þ Tr ðG1 Þ þ 4Tr ðRGRÞ 2 2 4 (

!
)   y  d 1 2 i 1  y þ p U a  aj1 þ 2Rk1 l1 Ul1 j1 aj1 þ aj1 d
pk1 2 i 2 k 1 j1 j1

 
  y y  d 1 2 y p þ Uk1 j1 Uk2 j2 aj1 aj2 þ aj1 aj2 þ 2aj1 aj2 dGk1 k2 2 i

 
  y y  d 1 2 1 1 i p þ Uk1 j1 Uk2 j2 aj1 aj2  aj1 aj2 dRk1 k2 2 i 4 
   1 1 i  1 y 1 G þ U RU  U RU j j 2aj1 aj2 ; þ ð14Þ 1 2 4 j1 j2 2 where Tr is a trace over the super-indices. A similar result can be found for the potential part (except that it is independent of R and p
). In Appendix E we will demonstrate that the potential can be decomposed into c/a operators s.t. the prefactors can be written as functional derivatives of the expectation value of the potential between Gaussian states, and the c/a operators always appear in a fixed structure.1 In the following we will restrict ourselves to the contributions to V with 1

For example if in the potential there are terms that contain two creation operators, or two annihilation operators, or one creation/one annihilation operator, they can always be written as factor ðayi ayj þ ai aj þ 2ayi aj Þ, and similarly for all other terms that contain a fixed sum of creation and annihilation operators. For the kinetic terms things are a bit different, but we have given the decomposition of the only kinetic term allowed in Eq. (14).

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up to four c/a operators, since the terms containing higher numbers of c/a operators do not contribute2 to the gRPA matrices that will be introduced in Eq. (26): !   d hV ½/
i Uk1 j1 ayj1 þ aj1 V ½/
¼ hV ½/
i þ
d/k1
   d þ hV ½/
i Uk1 j1 Uk2 j2 ayj1 ayj2 þ aj1 aj2 þ 2ayj1 aj2 dGk1 k2 ! 1 d d þ hV ½/
i Uk1 j1 Uk2 j2 Uk3 j3 3 d/
k1 dGk2 k3    ayj1 ayj2 ayj3 þ 3ayj1 ayj2 aj3 þ 3ayj1 aj2 aj3 þ aj1 aj2 aj3
  1 d d hV ½/
i Uk1 j1 Uk2 j2 Uk3 j3 Uk4 j4  ayj1 ayj2 ayj3 ayj4 þ 6 dGk1 k2 dGk3 k4  þ 4ayj1 ayj2 ayj3 aj4 þ 6ayj1 ayj2 aj3 aj4 þ 4ayj1 aj2 aj3 aj4 þ aj1 aj2 aj3 aj4 : ð15Þ By adding the expressions Eqs. (14), (15) together, one observes that the Hamiltonian has a very simple schematic structure: !   d H ¼ hH i þ hH i Uk1 j1 ayj1 þ aj1 d/
k1 !
   y  d i 1  y U þ hH i a  aj1 þ 2Rk1 l Ulj1 aj1 þ aj1 d
pk 1 2 k1 j1 j1
   d þ hH i Uk1 j1 Uk2 j2 ayj1 ayj2 þ aj1 aj2 þ 2ayj1 aj2 dGk1 k2
  d i y y a a  aj1 aj2 þ hH i Uk1 Uk1 1 j1 2 j2 dRk1 k2 4 j1 j2
  1 1 i  1 1 G þ U RU  U RU j j 2ayj1 aj2 þ 1 2 4 j1 j2 2 !  y y y 1 d d þ hH i U U U aj1 aj2 aj3 þ 3ayj1 ayj2 aj3 k j k j k j 1 1 2 2 3 3 3 d/
k1 dGk2 k3  þ 3ayj1 aj2 aj3 þ aj1 aj2 aj3
 1 d d hV i Uk1 j1 Uk2 j2 Uk3 j3 Uk4 j4 þ 6 dGk1 k2 dGk3 k4    ayj1 ayj2 ayj3 ayj4 þ 4ayj1 ayj2 ayj3 aj4 þ 6ayj1 ayj2 aj3 aj4 þ 4ayj1 aj2 aj3 aj4 þ aj1 aj2 aj3 aj4 : ð16Þ One should note that in all save the last term, the derivatives are taken of hH i whereas in the last term the derivative is taken of hV i. The importance of this will 2 That this is true can be seen by usage of WickÕs theorem and the so-called second gRPA approximation aji ¼ 0 that will be introduced in Section 3.4.

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459

become clear in Section 5.2. Up to now, we have taken an arbitrary Gaussian as a reference state: we have defined our creation/annihilation operators relative to that state—nothing else. Eq. (16) is exactly the same as H given in Eq. (6). However, it is obvious from Eq. (16) that the Hamiltonian greatly simplifies if we choose a specific reference state: a state that is a stationary point of the energy functional hH i under variation of the parameters /
; p
; G; R—in other words, a state that satisfies the Rayleigh–Ritz variational principle.3 3.3. Alternative form of Schro¨dinger equation In the following we want to generalize the RPA known for many-body physics to the quantum field theory described by the Hamiltonian of Eq. (6). For this purpose we assume that the exact (excited) state jmi can be created from the exact vacuum j0i by an operator Qym , i.e., jmi ¼ Qym j0i: If Em denotes the corresponding eigenvalue of H we have HQym j0i ¼ Em Qym j0i:

ð17Þ ð18Þ

If we denote the vacuum energy by E0 , we can write this equivalently with a commutator: ½H ; Qym
j0i ¼ ðEm  E0 ÞQym j0i: ð19Þ We may multiply both sides of the equation with an arbitrary operator dQ, and obtain an expectation value by multiplying from the left with h0j h0jdQ½H ; Qym
j0i ¼ ðEm  E0 Þh0jdQQym j0i:

ð20Þ

By subtracting zero on both sides we obtain an equation that only contains expectation values of commutators h0j½dQ; ½H ; Qym

j0i ¼ ðEm  E0 Þh0j½dQ; Qym
j0i:

ð21Þ

Using this equation as a starting point to derive the (generalized) RPA is known in nuclear physics as the equations of motion method [16,32]. 3.4. First and second gRPA approximations The generalized Random Phase Approximation consists now of approximating Eq. (21) in order to obtain a solvable set of equations. Two obvious candidates for approximations suggest themselves: first, the excitation operator Qym , second, the vacuum state. In the remainder of the paper, we will call the approximation concerning the first topic the first gRPA approximation, the approximation concerning the vacuum state the second gRPA approximation (even though this may be doubtful 3

One should note that we require the same in the time-dependent approach, when we decompose the time-dependent parameters /
ðtÞ; p
ðtÞ; GðtÞ; RðtÞ into a static part and small fluctuations, cf. Appendix A. The equations for the static part are identical to the Rayleigh–Ritz equations.

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linguistically). The second gRPA approximation is actually easier to state, therefore we start with it: approximate in all expressions involving vacuum expectation values of commutators the true vacuum state by the reference state of the creation/annihilation operators, which in this context we will usually refer to as mean-field vacuum and which will be denoted by ji. Later on, we will be even more restrictive and require the reference state to really be a stationary state of the Rayleigh–Ritz principle as was indicated before, but we will state explicitly from when on this additional restriction will be necessary. As already noted, the first gRPA approximation deals with the class of allowed operators. In nuclear physics it is quite reasonable to assume that the lowest excited state above a Hartree–Fock ground state consists of a particle–hole excitation. This results in the so-called Tamm–Dancoff approximation. In the generalized RPA one assumes that one has a correlated ground state, s.t. not only the creation of a hole and a particle, but also their destruction is a possible excitation. In Yang–Mills theory—the theory we will be ultimately interested in—it is by far not so clear what structure the lowest excitation will have; we have thus taken the two following guiding principles (an argument similar to our second principle can be found in [17,19]): 1. One of the main differences between fermionic and bosonic systems is that in the latter there exist single-particle condensates; thus, one has at least to extend the ansatz for the excitation operator by linear terms allowing for fluctuations. Furthermore, since we can compare to the gRPA as derived from the time-dependent variational principle, we will see that the ansatz for Qym to be proposed leads to equations of motion that are identical to those derived from the time-dependent variational principle, thereby verifying the ansatz to be the correct one. 2. The principal goal of the Random Phase Approximation is the restoration of symmetries that are violated at the mean-field level. We give here a short outline, drawing from concepts that will be introduced further below, in order to motivate from this property of symmetry restoration the form to be allowed for the excitation operators. The first point is that the second gRPA approximation can be replaced (under certain conditions) by the so-called quasi-boson approximation (QBA). There, it will turn out that gRPA equations can be written as ½HB ; QyBm
¼ ðEm  E0 ÞQyBm ;

ð22Þ

where OB indicates that the QBA has been used. One the other hand, for a symmetry generator (in our case the Gauss law operator) C which is a one-body operator (i.e., an operator that can be written as a linear combination of the operators a; ay ; aa; ay ay ; ay a), one can derive that ½H ; C
¼ 0 ! ½HB ; CB
¼ 0:

ð23Þ

If the mean-field vacuum is annihilated by C this is a trivial statement, since in this case CB  0. However, if the symmetry is violated on the mean-field level, one obtains a non-trivial result, namely that CB is a solution of the gRPA equations with zero excitation energy provided that the class of excitation operators contains CB . Since all gRPA excitations are orthogonal this would be a very desirable state of affairs: effectively the spurious excitations caused by the symmetry violation on the mean-field level would not affect the physical excitations we are interested in.

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To put it all in a nut-shell: one looks at the Gauss law operator, determines its structure (cf. Appendix C.2), and keeps the class of excitation operators so large that the Gauss law operator belongs to this class. All the concepts and claims will become clear and will be proved in what follows. After this rather long motivation, we will consider excitation operators of the following form:    1 m y y Qym ¼ Xmi am ai  Ymim am ai þ Z~mm aym  Zmm am ; ð24Þ 2 where X ; Y ; Z; Z~ are called amplitudes. From the form of the Gauss law operator one would have expected a few more terms; the reasons for leaving them out will be explained in Section 3.6. With the ansatz (24) we derive from Eq. (21) a closed set of equations for the amplitudes X m ; Y m ; Z m ; Z~m . To this end, the (up to now arbitrary) operators dQ will be appropriately chosen, together with the second gRPA approximation. We choose dQ in such a way that the different amplitudes are extracted individually on the RHS of Eq. (21). It will be useful to consider dQ 2 fai ; ayi ; ayi aj ; ai aj ; ayi ayj g, and compute the commutators ½dQ; Qym
, approximating the real vacuum by the mean-field vacuum according to the second gRPA approximation. We obtain h½aj ; Qym
i ¼ Z~jm ;

h½ayj ; Qym
i ¼ Zjm ;

1 m ; h½ayn ayj ; Qym
i ¼ Yfjng 2 h½ayn aj ; Qym
i ¼ 0;

1 m h½an aj ; Qym
i ¼ Xfjng ; 2

ð25Þ

where we have used the notation hj    ji ¼ h  i and introduced the abbreviation Xfijg defined as Xfijg ¼ Xij þ Xji , and correspondingly for Yfijg . With the dQs defined above we now have to compute the LHS of Eq. (21). For this purpose we introduce a number of matrices Anjmi ¼ h½ayn ayj ; ½H ; aym ayi

i;

Dnjm ¼ h½ayn ayj ; ½H ; am

i;

Bnjmi ¼ h½ayn ayj ; ½H ; am ai

i;

Enm ¼ h½ayn ; ½H ; aym

i;

Cnjm ¼

h½ayn ayj ; ½H ; aym

i;

Fnm ¼

ð26Þ

h½ayn ; ½H ; am

i;

and study their properties under interchange of labels. This will allow to reduce the number of independent entries of the LHS of Eq. (21). The basic tools for this study are the Jacobi identity and the second gRPA approximation, i.e., aji ¼ 0. We also use frequently that ½a; a
¼ ½ay ; ay
¼ 0. It turns out that the matrices A and E are  symmetric in all indices, whereas F is a hermitian matrix, Fnm ¼ Fmn . In order to  establish that B is also hermitian in the sense Bnjmi ¼ Bminj (whereas it is symmetric under interchange n $ j; m $ i) we need the second gRPA approximation since only upon usage of this approximation h½H ; ½aym ayi ; an aj

i will be zero generally. Apart from one matrix, all other matrices that arise from inserting the different dQs into the LHS of Eq. (21) are trivially related to the matrices A; . . . ; F introduced above; this one non-trivial matrix is h½ayn ; ½H ; am ai

i:

ð27Þ

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It is related to the matrix D, cf. Eq. (26), via h½ayn ; ½H ; am ai

i ¼ din Km þ dnm Ki þ Dmin ;

ð28Þ

with Ki;m ¼ h½H ; ai;m
i. In Section 3.5, the second gRPA approximation will be replaced by the so-called quasi-boson approximation. In that formulation, it will be obvious that we have to neglect the terms K. From the explicit expression for the expectation values, given further below in Eq. (38), we see that, at the stationary mean-field point,4 K is indeed zero. We note two points: • The gRPA matrix [essentially the LHS of Eq. (21)] is hermitian iff K ¼ 0. • The second gRPA approximation and the quasi-boson approximation to be introduced below are compatible iff K ¼ 0; such a constraint does not appear in nuclear physics as is easily comprehensible in two different ways: First, the matrix under consideration (as well as the matrices C; D) are non-zero only if the Hamiltonian contains terms with in total three creation/annihilation operators, i.e., terms that violate particle-number conservation. Such terms are not allowed in the usual nuclear physics framework, and thus there is no possibility for the above consistency condition to arise.5 Second, if we replace (as we will do in Section 3.5) the operators containing two ordinary boson creation operators by a new boson operator, in the case of bosonic theories we have still to keep the original boson, in contrast to the RPA treatment of fermion systems, where it is not necessary to retain the original fermions once one has the formulation in terms of bosonic operators at hand. This whole discussion allows us now to write down a first form of the gRPA equations 10 1 m 1 0 1   X B  12 Aij;kl Dij;k Cij;k 2 fklg 2 ij;kl CB 1 m C B 1 B  2 Aij;kl Bij;kl Cij;k Dij;k CB 2 Yfklg C CB C B 1  C B 1D B  2 Ckl;i Fik Eik C A@ Z~km A @ 2 kl;i 1   12 Ckl;i D Eik Fik Zkm 2 kl;i 0 1 0 1 1Xm 1 2 fijg 1 m C B CB B 1 B CB 2 Yfijg C ¼ Xm B ð29Þ CB m C @ A@ Z~ C 1 A i 1 Zim with Xm ¼ Em  E0 . The factors 1=2 associated with some of the matrices C; D should not lead one to conclude that the gRPA matrix on the LHS is not hermitian. In fact, as we will see later on, we can write the gRPA equations in a very compact form resulting from a hermitian Hamiltonian (at the stationary point of the mean-field ! ! equations ðd=d/
i ÞhH i ¼ 0, and ðd=d
pi ÞhH i ¼ 0). 4 That is that point in parameter space where the energy expectation value is stationary with respect to variations of /
; p
; G; R; for K to be zero it is necessary that dhH i=d/
i ¼ 0, and dhHi=d
pi ¼ 0. 5 In the bosonic theories under consideration here, these terms with three creation/annihilation operators appear in the Hamiltonian usually due to a condensate.

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463

3.5. The quasi-boson approximation Sometimes, it is useful to have the second gRPA approximation at hand without having to take expectation values. This can be done with the help of the quasi-boson approximation. The name has its roots in nuclear physics [16,22], where two fermions are combined into one boson. This works only approximately,6 so that the result was christened a quasi-boson. Interestingly enough, we can do the same in a bosonic system: we replace a two boson operator (like aa or ay ay ) by a new operator that again has bosonic commutation relations. We construct the new boson-pair operators B; By s.t. the commutation relations are identical to the mean-field expectation values of commutators containing still the pair of original boson operators: ½am ai ; aj
¼ 0 ! ½Bmi ; aj
¼ 0; ½am ai ; ayj
¼ dij am þ dmj ai ! h½am ai ; ayj
i ¼ 0 ! ½Bmi ; ayj
¼ 0; ½aym ayi ; aj
¼ dij aym  dmj ayi ! h½aym ayi ; aj
i ¼ 0 ! ½Bymi ; aj
¼ 0: The only non-vanishing commutator involving B; By originates from ½am ai ; ayn ayj
:

 1 1 y y 1 pffiffiffi am ai ; pffiffiffi an aj ¼ din dmj þ dij dmn þ din ayj am þ dmn ayj ai þ dij ayn am þ dmj ayn ai 2 2 2  1 ! ½Bmi ; Bynj
¼ din dmj þ dij dmn : 2 We therefore replace 1 pffiffiffi ðam ai Þ $ Bmi ; 2 1  y y pffiffiffi am ai $ Bymi : 2

ð30Þ

The operators B; By have the following commutation relations: ½aj ; Bmi
¼ ½aj ; Bymi
¼ ½Bmi ; Bnj
¼ 0

ð31Þ

1 ½Bmi ; Bynj
¼ ðdin dmj þ dij dmn Þ: 2

ð32Þ

and

With these new operators, the excitation operators    1 m y y am ai  Ymim am ai þ Z~mm aym  Zmm am Qym ¼ Xmi 2 become    1  m y Bmi  Ymim Bmi þ Z~mm aym  Zmm am : QyBm ¼ pffiffiffi Xmi 2 6

ð33Þ

ð34Þ

That is, the ordinary bosonic commutation relations are only fulfilled if we take the mean-field expectation value of the commutator of the boson pairs, and thus in some sense employ the second gRPA approximation, by using the mean-field instead of the exact vacuum state.

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At this point a short comment on how one constructs quasi-boson approximations is in order, cf. also Eq. (35) below. Usually we will have to consider only one- and twobody operators, but the procedure should work for higher-body operators as well: (i) First, one writes down the operator O under consideration as a polynomial of a; ay operators. Then one tries to extract the coefficients of the different powers in analogy to Eqs. (25), (26) via (a) taking multiple commutators of O with up to two a; ay operators, and then (b) taking vacuum expectation values of these multiple commutators employing the second gRPA approximation. (ii) The quasi-boson approximation OB of this operator O is then given as a polynomial of B; By ; a; ay operators. The coefficients of the different powers are extracted by taking multiple commutators of OB with a; ay ; B; By without taking expectation values. One then requires that the coefficients of OB determined in this way are identical to those of O determined in (i) if one replaces pffiffiffi in the multiple of (i) O by OB and terms of the structure aa by 2B and ay ay by pffiffiffi commutators 2By as indicated in Eq. (30). This is precisely the procedure that led to Eq. (34). Using the procedure outlined, one can in fact express the QBA OB of a general one-body operator O in terms of meanfield vacuum expectation values of commutators 1 1 OB ¼ hOi  pffiffiffi h½ayi ayj ; O
iBij þ pffiffiffi h½ai aj ; O
iByij  h½ayi ; O
iai þ h½ai ; O
iayi : 2 2 ð35Þ Incidentally, one can at this point easily verify—by writing out the commutators—the claim made above: the QBA of a symmetry generator (which is generally hermitian) of one-body type that annihilates the mean-field vacuum vanishes. Now we will follow the above procedure to obtain the Hamiltonian in quasi-boson approximation. The coefficients mentioned are nothing but the matrices A; . . . ; F . We therefore have the requirements pffiffiffi pffiffiffi ! Anjmi ¼ h½ayn ayj ; ½H ; aym ayi

i ¼ ½ 2Bynj ; ½HB ; 2Bymi

; pffiffiffi pffiffiffi ! Bnjmi ¼ h½ayn ayj ; ½H ; am ai

i ¼ ½ 2Bynj ; ½HB ; 2Bmi

; pffiffiffi ! Cnjm ¼ h½ayn ayj ; ½H ; aym

i ¼ ½ 2Bynj ; ½HB ; aym

; ð36Þ pffiffiffi ! Dnjm ¼ h½ayn ayj ; ½H ; am

i ¼ ½ 2Bynj ; ½HB ; am

; !

Enm ¼ h½ayn ; ½H ; aym

i ¼ ½ayn ; ½HB ; aym

; !

Fnm ¼ h½ayn ; ½H ; am

i ¼ ½ayn ; ½HB ; am

; where HB denotes the Hamiltonian in quasi-boson approximation. At this point it seems in order to discuss a question that turned up in connection with Eq. (27): There we saw that one of the double commutators was not directly connected to one of the matrices A; . . . ; F but that some extra terms appeared, the ÔK-terms.Õ If we consider the same double commutator in the quasi-boson approximation, we obtain

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

pffiffiffi h½H ; ½ayn ; am ai

i ! ½HB ; ½ayn ; 2Bmi

¼ 0: |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

465

ð37Þ

¼0

Thus the quasi-boson approximation implies the vanishing ofK. An explicit expression can be given for K as well (cf. Appendix C.1)


 d d 1 1 U Ki ¼ hH i U þ hH i U þ 2R ; ð38Þ ki kl li d
pk 2i ki d/
k which vanishes if we choose the parameters of the reference state s.t. the energy is stationary under their variation (at least w.r.t. /
; p
). We obtain a consistency condition between the second gRPA approximation and the quasi-boson approximation: the latter is equivalent to the former only at the stationary point—at least w.r.t the condensates /
; p
—of hH i. Thus, in the following we will always assume that the state (which we often call the mean-field vacuum) relative to which our creation/annihilation operators are defined [cf. Eq. (7)] has its parameters chosen s.t. it satisfies the Rayleigh– Ritz principle—that this restriction would become necessary was already indicated when the second gRPA approximation was introduced. With this qualification, we can give the (now hermitian) Hamiltonian in quasi-boson approximation:  1 1 HB ¼ EMF  An1 j1 n2 j2 Bn1 j1 Bn2 j2 þ An1 j1 n2 j2 Byn1 j1 Byn2 j2 þ Bn2 j2 n1 j1 Byn1 j1 Bn2 j2 4 2  1  y  y  pffiffiffi Cn1 j1 n2 Bn1 j1 an2 þ Cn1 j1 n2 Bn1 j1 an2 2  1  þ pffiffiffi Dn1 j1 n2 Bn1 j1 ayn2 þ Dn1 j1 n2 Byn1 j1 an2 2  1 ð39Þ  En1 n2 an1 an2 þ En1 n2 ayn1 ayn2 þ Fn2 n1 ayn1 an2 ; 2 with EMF ¼ hH i. We have achieved writing the approximated Hamiltonian as a quadratic form, which can always be diagonalized. Using this Hamiltonian HB one can see that the gRPA equations, Eq. (29), can be written in the transparent form7 ½HB ; QyBm
¼ Xm QyBm :

ð40Þ

This form of the gRPA equations demonstrates that we have performed the approximation to the dynamics (Hamiltonian) and to the excitation operators consistently, since the form of the equations is identical to the form of the Schr€ odinger equation, Eq. (19), with the exact Hamiltonian and the exact excitation operators. 3.6. Restriction on excitation operators, part II Having introduced all the concepts of gRPA, it is time to look again at some terms that appear neither in Qyl nor in QyBl . 1. Qym does not contain an ay a term. The superficial reason for this is that the corresponding amplitude cannot be extracted in a way similar to the other amplitudes X ; Y ; Z; Z~, since due to the second gRPA approximation 7 One inserts Eqs. (39) and (34) into Eq. (40) and compares the coefficients of the different operators a; ay ; B; By . This then reproduces Eq. (29).

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O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

hj½dQ; ay a
ji ¼ hjdQay aji  hjay adQji ¼ 0: |{z} |{z} ¼0

ð41Þ

¼0

This point will become clearer in the context of the QBA, cf. item 3 below. 2. Qym does not contain an aay term. This is due to the fact that aay is distinguished from ay a only by a constant. However, this constant would lead to a non-vanishing expectation value of Qym between mean-field vacua, in view of the second gRPA approximation a situation certainly not desirable for an excitation operator (all other terms contained in Qym vanish between mean-field vacua!). One could also argue on a formal level that the way the generalized RPA is derived here does not allow to determine any constant parts of Qym , so if we cannot determine a constant, we should not put it into our ansatz in the first place. 3. Here we want to have a look at the fact that ay a does not appear in our excitation operator from the perspective of the QBA. The excitation operator is linear in all the different boson creation/annihilation operators that we construct as one can see in Eq. (34). Thus, the question arises whether we can construct a boson operator from ay a similar to how we construct one from ay ay . The problem is immediately apparent: if we consider Cmi ¼ aym ai , then its adjoint has the same structure y as Cmi , since Cmi ¼ ayi am ¼ Cim ; if we compute the commutator ½Cij ; Ckly
¼ y y djk ai al  dli ak aj we see that its mean-field expectation value is zero. Therefore, we cannot construct boson operators with the correct commutation relations from ay a in the same way as we did for ay ay , and thus they do not appear in the excitation operators Qym . In nuclear physics this is well-known. There one can see that the quasi-boson operators correspond to a creation of a particle–hole pair (or its annihilation), whereas the operator that creates and annihilates a particle is translated into an operator that creates and annihilates a quasi-boson [22]. 3.7. Normal mode form In Appendix D we consider the commutation relations of the normal mode operators. There we show that if all excitation energies Xm are distinct and non-zero, the normal modes have the usual bosonic commutation relations ½QBl ; QyBm
¼ dlm : Using this fact, we can conclude from Eq. (40) X Xm QyBm QBm : HB ¼ ERPA þ

ð42Þ

ð43Þ

m

The sum over m extends over all positive semi-definite Xm . The constant ERPA can be determined as usual, cf. [16], namely by requiring hHB i ¼ EMF : Using hQyBm QBm i

 2 X 1 X  1 m  2 Y ¼ þ jZim j ; 2 mi  2 fmig  i

ð44Þ

ð45Þ

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506 m with Yfmig ¼ Ymim þ Yimm this gives !  2 X X 1 X  1 m  2 Y Xm jZim j : ERPA ¼ EMF  þ 2 mi  2 fmig  m i

467

ð46Þ

A similar expression can be obtained in nuclear physics [16] but without the ap2 pearance of jZim j . The changes brought about by excitations with zero excitation energy will be considered in Section 6. 4. Conservation laws In this section we want to demonstrate that—under certain conditions—conservation laws from the full theory translate to conservation laws in the gRPA treatment. The proof follows the lines of the fermionic case as given in [22]. For the conservation laws to hold in the approximated theory, it is mandatory that the reference state minimizes the expectation value of the Hamiltonian, since then the terms of the Hamiltonian linear in a; ay ; aa; ay ay vanish, cf. Eq. (16). One should also keep in mind that the terms of H linear in a; ay have to vanish anyway, since otherwise a treatment using the QBA is not valid as was discussed before. The fact that conservation laws translate into the approximated theory has to be taken with a grain of salt, however, since it turns out (and will be discussed in Section 4.2) that the character of symmetries may change. Non-Abelian symmetries are usually reduced to Abelian symmetries which is not surprising since the gRPA is basically a small-fluctuation approximation which does not probe the group manifold. 4.1. General observation With these qualifications, let us now compute the commutator of HB , cf. Eq. (39), with a general one-body operator CB   pffiffiffi  y y 01 02 CB ¼ C0 þ C10 2 C20 ð47Þ n1 an1 þ Cn1 an1 þ n1 n2 Bn1 n2 þ Cn1 n2 Bn1 n2 ; where the coefficients have been chosen, s.t. the full operator before QBA reads y 01 20 y y 02 11 y C ¼ C0 þ C10 n1 an1 þ Cn1 an1 þ Cn1 n2 an1 an2 þ Cn1 n2 an1 an2 þ Cn1 n2 an1 an2 :

ð48Þ

Then the commutator gives   1 01 20 02 ½HB ; CB
¼  pffiffiffi Bn1 j1 Cn1 j1 m1 C10 C þ A C þ B C þ D n1 j1 m1 m1 n1 j1 m1 m2 m1 m2 n1 j1 m1 m2 m1 m2 m1 2   1 10 02 20    C þ A C þ B C þ pffiffiffi Byn1 j1 Cn1 j1 m1 C01 þ D m1 n1 j1 m1 m1 n1 j1 m1 m2 m1 m2 n1 j1 m1 m2 m1 m2 2   01 20 02   an1 En1 m1 C10 C þ C C þ C D þ F n m m m n 1 1 1 2 1 m1 m1 m1 m2 m1 m2 m1 m2 n1   y  01  10 02  þ an1 En1 m1 Cm1 þ Fn1 m1 Cm1 þ Cm1 m2 Cm1 m2 n1 þ C20 ð49Þ m1 m2 Dm1 m2 n1 : Let us now concentrate on the first line to make the principle clear; we use the definition of A; B; C; D as mean-field expectation values of double commutators

468

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

involving the full Hamiltonian, cf. Eq. (36). Then it is clear that—using also Eq. (48)—one can rewrite it as  01 y y Bn1 j1 ðh½ayn1 ayj1 ; ½H ; aym1

iC10 m1 þ h½an1 aj1 ; ½H ; am1

iCm1  y y 02 þ h½ayn1 ayj1 ; ½H ; aym1 aym2

iC20 þ h½a a ; ½H ; a a

iC m m 1 2 m1 m2 n1 j1 m1 m2   y y 11 y y y ¼ Bn1 j1 ðh½an1 aj1 ; ½H ; C

i  Cm1 m2 h½an1 aj1 ; ½H ; am1 am2

i : ð50Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ðÞ

02

The last term ðÞ is proportional to H , i.e., that term in the Hamiltonian that multiplies two annihilation operators, and is zero in the case we are considering here, i.e., our reference state minimizes the expectation value of the Hamiltonian. The procedure can be repeated for the other three terms in ½HB ; CB
. Eventually, one finds 1 1 ½HB ; CB
¼  pffiffiffi h½ayn1 ayj1 ; ½H ; C

iBn1 j1 þ pffiffiffi h½an1 aj1 ; ½H ; C

iByn1 j1 2 2  h½ayn1 ; ½H ; C

ian1 þ h½an1 ; ½H ; C

iayn1 þ terms that vanish at the stationary point:

ð51Þ

Two points should be noted: 1. In Eq. (35) we have given a formula of the QBA of a general one-body operator; if ½H ; C
is a one-body operator, then by comparing Eq. (51) to Eq. (35) we conclude that ½HB ; CB
is the QBA of this operator (apart from a possible mean-field expectation value). 2. The stationarity condition (H 20 ¼ H 10 ¼ 0) has been used several times; if we are not at the stationary point, some of the terms neglected above do not vanish, i.e., ½H ; C
¼ 0 does not imply ½HB ; CB
¼ 0 away from the stationary point. We want to assume that we have chosen the reference state s.t. indeed H 20 ¼ H 10 ¼ 0. Then Eq. (51) simplifies to ½HB ; CB
¼ 0:

ð52Þ

Comparing this with the gRPA equations as given in Eq. (40) we see that—since CB is exactly of the form of QyBm —CB is a solution of the gRPA equations with excitation energy zero, i.e., it is a zero mode. 4.2. Change of symmetry character It seems that by this construction we have precisely obtained what we wanted: the spurious excitation made possible by the deformed mean-field state is a solution by itself, and does not influence the other—physical—solutions inappropriately. However, this result has to be taken with a bit of caution: as has been indicated before, the character of the symmetry may change. This can be seen as follows: Compute the commutator of the QBA QB ; PB of two arbitrary one-body operators Q; P . Using Eq. (35) it is very simple to obtain

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

½QB ; PB
¼ h½Q; P
i:

469

ð53Þ

In the case of Yang–Mills theory, we obtain for the commutator of the Gauss law operators ½CaB ; CbB
¼ h½Ca ; Cb
i ¼ if abc hCc i:

ð54Þ

At the stationary point, however, we have for the bosonic theories under consideration p
¼ R ¼ 0, since hppi is quadratic in both p
and R, cf. Eq. (A.7). Thus, in the case of Yang–Mills theory, we obtain ½CaB ; CbB
¼ 0;

ð55Þ

using the expression given in Eq. (C.16) for hCa i. In other words, the QBA has reN 2 1 duced the non-Abelian SU ðN Þ symmetry to an Abelian U ð1Þ symmetry. This is a phenomenon also well known from perturbation theory. 4.3. Possibilities for improvement One main drawback of the gRPA is the fact that non-Abelian symmetries may be reduced to Abelian symmetries. There have been a couple of investigations in nuclear physics, e.g. [16,22], and other quantum field theories (where the fields were in the fundamental representation) [17–19] whether one can regard the gRPA as a certain order in a systematic expansion,8 symbolically complete result ¼ mean field þ gRPA þ higher orders:

ð56Þ

A first hint on how one may construct such higher orders for one-body operators like the Gauss law operator can be obtained from the following observation: if we calculate (as we have done above) the commutator of two quasi-boson approximated Gauss law operators, we obtain the mean-field value of the full commutator; apparently, in the expansion alluded to above, taking a commutator reduces the order in the expansion by one (for a similar observation in nuclear physics, cf. e.g. [16,34]). If we construct a next order of say the Gauss law operator, taking the appropriate commutator should give the quasi-boson approximation of the commutator, the latter being again a one-body operator. In formulas, if we write9 C |{z}

full operator

¼

hCi |{z}

mean-field

þ

CB |{z}

conventional gRPA

þ

C2B þ higher orders |{z}

ð57Þ

next term

we have ½CaB ; CbB
¼ h½Ca ; Cb
i

ð58Þ

and C2B would be correct if it would reproduce 8 Within the path integral approach one can show that the mean-field approximation and the gRPA correspond to the first and second, respectively, order in the loop expansion [33]. 9 One should note that this is a slight deviation from the conventions in the rest of the paper, e.g., in Eq. (35) the mean-field order hOi is assigned to OB .

470

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

½CaB ; Cb2B
þ ½Ca2B ; CbB
¼ ½Ca ; Cb
B :

ð59Þ

Such a C2B can indeed be constructed. If one requires that C2B reproduces the same commutators as HB in Eq. (36) [with H obviously replaced by C on the LHS of Eq. (36)] one obtains C2B as y y y 20 02 11 y C2B ¼ 2C11 kj Bkl Blj þ Cj1 j2 aj1 aj2 þ Cj1 j2 aj1 aj2 þ Ckj ak aj ;

ð60Þ

where the coefficients Cab ij are defined in Eq. (48). For the calculation of the commutator ½CaB ; Cb2B
we use a technique very similar to the one used in Section 4.1 by rewriting the coefficients of Cb2B as double commutators (note that the coefficients of Cb2B are called b C11 jk etc; note also that—as in the remainder of the paper—we use Cfijg ¼ Cij þ Cji ) b

y b C11 kj ¼ h½aj ; ½C ; ak

i;

b

b C20 fj1 j2 g ¼ h½aj1 ; ½C ; aj2

i:

b

y y b C02 fj1 j2 g ¼ h½aj1 ; ½C ; aj2

i;

ð61Þ

Using these results, one can rewrite y y a 01 b a 10 y a 01 ½CaB ; Cb2B
¼ aj1 h½ayj1 ; ½Cb ; ða C10 j2 aj2 þ Cj2 aj2 Þ

i  aj1 h½aj1 ; ½C ; ð Cj2 aj2 þ Cj2 aj2 Þ

i

1 þ pffiffiffi Bj1 j2 h½ayj1 ayj2 ; ½Cb ; ða C02 k1 k2 ak1 ak2 Þ

i 2 1 y y  pffiffiffi Byj1 j2 h½aj1 aj2 ; ½Cb ; ða C20 k1 k2 ak1 ak2 Þ

i 2 ¼ ½CbB ; Ca2B


aj1 h½ayj1 ; ½Ca ; Cb

i

þ

ayj1 h½aj1 ; ½Ca ; Cb

i

1 1  pffiffiffi Bj1 j2 h½ayj1 ayj2 ; ½Ca ; Cb

i þ pffiffiffi Byj1 j2 h½aj1 aj2 ; ½Ca ; Cb

i 2 2

ð62aÞ ð62bÞ ð62cÞ

Since ½Ca ; Cb
just again gives a one-body operator, we can compare Eqs. (62b), (62c) to Eq. (35), and see that it is indeed the QBA of10 ½Ca ; Cb
, i.e., ½Ca ; Cb
B . Thus we have fulfilled our goal: we have constructed an extension of CaB which improves the commutation relations as was desired. At this point one should keeppin ffiffiffi mind the following: We have constructed C2B by ! requiring that h½ayi ayj ; ½C; ak

i ¼ ½ 2Byij ; ½C2B ; ak

. This gave zero aspthe ffiffiffi coefficient of ! ay B. Had we chosen instead to require h½ak ; ½C; ayi ayj

i ¼ ½ak ; ½C2B ; 2Byij

we would have obtained a non-zero coefficient for ay B. This incompatibility of the two construction principles goes back basically to the problem that the commutation relations for a; B are constructed to work with one-body operators (as was noted in the context of HB where both coefficients are required to be equal. This can be achieved by going to the stationary point since the offending terms all vanish there; here we cannot follow the same route since the state is already specified).

10

Note again that in this section the expectation value h½Ca ; Cb
i is not incorporated into ½Ca ; Cb
B .

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

471

The ambiguity is here in some sense resolved, since the latter construction does not lead to the aspired commutation relations.11 5. P-Q formalism In Section 3.7 we have learned that the usual gRPA procedure encounters problems when the gRPA equations have zero mode solutions; in Section 4 we have learned that in the case we are interested in—namely a mean-field vacuum that breaks certain symmetries—zero modes appear necessarily. Therefore, it seems that we may have a problem here. This problem, however, was already encountered in nuclear physics and solved in [22]. We present the problem and its solution for the simplest case available, namely the simple harmonic oscillator. We then give an alternative formulation of the gRPA in QBA using—instead of creation/annihilation operators—operators with canonical commutation relations. This formulation will furthermore allow a simple comparison between the gRPA equations derived from the time-dependent variational principle and the operator approach. 5.1. Resolution of zero mode problem In the nuclear physics literature it is well known [22] that the appearance of zero modes, despite being welcome in our case, leads to all sorts of problems, especially that the solutions of the gRPA equations no longer form a complete set.12 Marshallek and Weneser [22] however pointed out that the problems one has are actually an artifact of the creation/annihilation operator formalism. We want to give a very primitive illustration of their point: One can write down a quantum mechanical oscillator in two equivalent forms (we have set the mass equal to one for simplicity):

11 There can be circumstances where even in the latter construction the correct commutation relations can be produced. We have investigated the case where the symmetry generators are the Gauss law operators, and found two instances where the correct commutation relations were produced even in the presence of the aBy ; ay B-terms. These examples were: G1 / 1—this is not a very likely mean-field vacuum
^ U 1 ¼ 0. For the since there is no correlation between different points in space—and the case where D definition of U, cf. Eq. (11). But this requirement on U 1 implies that G1 has zero modes, and is thus excluded in the gRPA framework, cf. Section 3.1. Thus, for practical purposes, at least in Yang–Mills theory, the required commutation relations resolve all ambiguities in the construction of C2B . 12 In nuclear physics this is a problem since one there cannot prove that the Hamiltonian can be written as a sum of Qym Qm . Completeness is lost in the following way: We see—in Eq. (D.1)—that from every solution of the gRPA equations we can form another solution by considering the adjoint operator; therefore, the solutions to the gRPA equations always come in pairs. In the case of zero modes, this is not necessarily true any more, since now the Ôexcitation operatorsÕ that we find may be hermitian; thus, by considering the adjoint, we do not get a new solution. In fact this situation is the usual case for zero modes, as we have seen in Section 4 the excitation operators for the zero modes are just the symmetry generators, which are usually hermitian. In our discussion in Appendix D, we hit upon the problem from a different direction; we see that only if there are no zero modes in the spectrum we can show that the excitation operators can be considered as independent oscillators.

472

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

p 2 x2 2 þ x; 2
2  1 ¼ x ay a þ ; 2

H¼

ð63Þ ð64Þ

where x denotes the frequency of the harmonic oscillator. If we now send the frequency to zero, something seemingly strange happens: p2 x2 2 x!0 p2 þ x ¼ ; ð65Þ 2 2 2
 1 x!0 x ay a þ ð66Þ ¼ 0: 2 If one now looks at the transformation from canonical to c/a operators, one begins to see more clearly: rffiffiffiffi x i i x!0 a¼ x þ pffiffiffiffiffiffi p ¼ pffiffiffiffiffiffi p; ð67Þ 2 2x 2x rffiffiffiffi x i i x!0 y x  pffiffiffiffiffiffi p ¼  pffiffiffiffiffiffi p ð68Þ a ¼ 2 2x 2x and if we take this small-x behavior into account, there is actually no paradox,13 since
2 x y x!0 x 1 p2 pffiffiffiffi p2 ¼ : aa ¼ ð69Þ 2 2 2 x This is actually the solution of the problems involving zero modes: one puts the system into an external field, s.t. there are no zero modes any more, since the external field14 breaks all symmetries. One diagonalizes the Hamiltonian into a set of harmonic oscillators using canonical coordinates, and at the end sends the external field 13 One should note that, in this limit x ! 0, no new information can be obtained by considering the adjoint of ay . 14 The external field can in fact be engineered just to do that, and this works as follows: We solve the gRPA equations and determine all modes available. The operator belonging to the zero mode will be known beforehand — it is just the quasi-boson approximation of the (known) symmetry generator, which is usually a hermitian operator, and so will be the operator associated with the zero mode. One then constructs a canonically conjugate variable, s.t. ½q; p
¼ i and q commutes with all other normal mode creation/annihilation operators, and adds a term x2 q2 to the Hamiltonian; this will lift the zero mode to a finite frequency. By this construction, only the zero mode will be lifted and nothing strange will happen to the other modes. Two points should be mentioned: First, the construction of a canonically conjugate coordinate to a such a generator is only possible in the gRPA context, since we have seen in Section 4 that the symmetry we have in mind is usually reduced to an Abelian symmetry; if we would still have the nonAbelian symmetry this would not be possible (the problems with the construction of Ôangle operatorsÕ in nuclear physics are well-known, for a recent discussion cf. e.g. [35]). As a second point, one should mention that in a quantum field theory there might be problems with breaking gauge symmetries in intermediate steps. Since we have indicated here only a formal construction, it is not easy to see where these kinds of problems might surface.

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

473

to zero. Then the would-be zero modes indeed become zero modes with the correct operator representation, namely p2 , and the non-zero modes can be rewritten using creation and annihilation operators. As a last comment on this topic, one should keep in mind the following: since we already have some prejudice about the zero mode operators (as the symmetry generators), we cannot just choose their mass to be one as we did in the example. We will rather have to allow for a mass tensor and this will be dealt with in Section 6. 5.2. Equivalence of operator and TDVP approach to gRPA We have seen that at least three formulations (using the tdvp, the two gRPA approximations, and the quasi-boson approximation) of the generalized Random Phase Approximation of bosonic systems are available. In this paragraph we want to add one more, since it simplifies the treatment of those Hamiltonians we set out to consider, cf. Eq. (6). Since we start from the gRPA formulation using quasi-boson operators, the whole treatment is only valid for the mean-field vacuum representing a solution of the Rayleigh–Ritz variational principle (at least w.r.t. variations of /
and p
). In this section we assume that the mean-field vacuum is a solution of the Rayleigh–Ritz variational principle w.r.t. all parameters /
; p
; R; G. We consider linear combinations of a; ay and B; By , s.t. 1 i qm ¼ pffiffiffi ðaym þ am Þ; pm ¼ pffiffiffi ðaym  am Þ; 2 2 1 i Qmi ¼ pffiffiffi ðBymi þ Bmi Þ; Pmi ¼ pffiffiffi ðBymi  Bmi Þ: 2 2

ð70Þ

They fulfill the usual canonical commutation relations ½qm ; pn
¼ idnm ;

i ½Qmi ; Pnj
¼ ðdmn dij þ dmj dni Þ 2

ð71Þ

and all other commutators vanishing. If we now rewrite the Hamiltonian HB in terms of these new operators, we obtain a complicated expression where the real and imaginary parts of the matrices A; . . . ; F are separated. We can now put to use that we have computed the matrices A; . . . ; F in Appendix C.1; we have found that 1. A; C; D are always real. 2. C ¼ D. 3. B; E; F may be imaginary; however, as this imaginary part is due to R, and in the theories under consideration15 at the stationary point R ¼ 0, the imaginary parts of B; E; F vanish. 4. As a matter of fact, E vanishes altogether at the stationary point. 5. A last simplification is that one may decompose B ¼ B11 þ B22 and, using this notation, we have A ¼ B22 . R This excludes the cranking Hamiltonian Hcr ¼ H  d3 xxa ðxÞCa ðxÞ—where H is the Yang–Mills a Hamiltonian and x ðxÞ is a Lagrange multiplier, cf. e.g. [5]—here, since it is not of the form given in Eq. (6). 15

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Using this information, the Hamiltonian simplifies considerably 1 1 HB ¼ E0 þ Qn1 j1 Qn2 j2 ðBn1 j1 n2 j2  An1 j1 n2 j2 Þ þ Pn1 j1 Pn2 j2 ðBn1 j1 n2 j2 þ An1 j1 n2 j2 Þ 4 4 pffiffiffi 1 þ 2Qn1 j1 qn2 Dn1 j1 n2 þ Fn2 n1 ðqn2 qn1 þ pn2 pn1 Þ 2 and can even be written in a nice matrix form

1 þ ðQn1 j1 qn1 Þ 2

1 ðBn1 j1 n2 j2 2

!


 Pn2 j2 pn2 Fn2 n1 !
pffiffiffi  Qn2 j2  An1 j1 n2 j2 Þ 2Dn1 j1 n2

1 ðBn1 j1 n2 j2 2

1 H ¼ E þ ðPn1 j1 pn1 Þ 2 0

þ An1 j1 n2 j2 Þ 0

pffiffiffi 2Dn2 j2 n1

0

Fn2 n1

ð72Þ

qn2

;

ð73Þ

where E0 is a constant that has to be chosen s.t. hHB i ¼ EMF . This is a viable starting point for proving the equivalence between the gRPA formulation originating from the time-dependent variational principle and the one using the quasi-boson approximation. We will rewrite Eq. (73) into a Hamiltonian that leads to the same eigenvalue equations as does the Hamiltonian of the small-fluctuation approach given in Eq. (A.33). From this we will conclude—since we have both times the same matrix to diagonalize—that the spectra of the Hamiltonians of the two approaches are identical, and we will see a correspondence in the eigenvectors. For this purpose we note the following: 1. We start with the term quadratic in p; P ; there we have

 1 1 1 1 1 1 1 ðB þ AÞn1 j1 n2 j2 ¼ U U U U ð74Þ ðG1Þm1 i1 ;m2 i2 2 2 n1 m1 j1 i1 2 n2 m2 j2 i2 and


Fn2 n1 ¼

1 pffiffiffi Un1 m 2 2 2




 1 pffiffiffi Un1 dm2 m1 ; m 2 1 1

ð75Þ

where we have used the abbreviation ðG1Þij;kl ¼ Gki dlj þ Gkj dli þ Gjl dki þ Gli dkj :

ð76Þ

This suggests that one should introduce new momentum operators 1 Pm1 i1 ¼ Un1 U 1 Pn j ; 2 1 m1 j1 i1 1 1

ð77Þ

1 pm1 ¼ pffiffiffi Un1 m pn1 : 2 1 1

ð78Þ

2. Now we consider the term quadratic in q; Q; there the observation B11 minj ¼ 8Umk1 Uik2 Unk3 Ujk4

d d hT i; dGk1 k2 dGk3 k4

ð79Þ

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

B22 minj  Aminj ¼ 8Umk1 Uik2 Unk3 Ujk4

d d hV i dGk1 k2 dGk3 k4

475

ð80Þ

(where T denotes the kinetic energy ð1=2Þp2i ) is useful. Then one can rewrite ðB  AÞminj ¼ 8Umk1 Uik2 Unk3 Ujk4

d d hH i: dGk1 k2 dGk3 k4

ð81Þ

Next, one can put to use the fact that the mean-field vacuum is a stationary point (s.p.) and thus dhH i=dG ¼ 0, the identity equation (E.19) from Appendix E and the fact that the kinetic energy hð1=2Þp2i i is independent of /
: ! ! 1 1 d d s:p: hT i Uk1 n1 Uk2 n2 ¼ 4 hV i Uk1 n1 Uk2 n2 Fn2 n1 ¼ Gn2 n1 ¼ 4 2 dGk1 k2 dGk1 k2 ! ! d2 d2 Eq:ðE:19Þ ¼ 2 hV i Uk1 n1 Uk2 n2 ¼ 2 hH i Uk1 n1 Uk2 n2 : ð82Þ d/
d/
d/
d/
k1

k2

k1

k2

This now suggests introducing new coordinates as well Qm1 i1 ¼ 2Un1 m1 Uj1 i1 Qn1 j1 ;

ð83Þ

pffiffiffi 2Un1 m1 qn1 :

ð84Þ

qm1 ¼

3. One should note that the newly defined coordinates and momenta still fulfill the canonical commutation relations Eq. (71); thus, the new definitions amount to a canonical transformation that leaves the dynamics unchanged. We therefore end up with a Hamiltonian where the matrices in the quadratic forms are identical to those appearing in the Hamiltonian of the small-fluctuation approach, Eq. (A.33),

1 1 H ¼ E0 þ ðp PÞ 0 2

0 ðG1Þ

!


0

1 d2 H d2 H  B

C

1 p B C q þ ðq QÞB d/2 d/ d/2 dG C ; P @ dH 2 dH A Q dG d/
dG dG

ð85Þ

since H ¼ hH i as defined in Eqs. (A.10), (A.17). As a last point, we want to show that the eigenvalue equations that result from the Hamiltonian given in Eq. (85) are identical to Eqs. (A.31), (A.32). This is in fact quite simple; since the excitation operator QyBm is a linear combination of B; By ; a; ay one can write it just as well as a linear combination of q; Q; p; P or q; Q; p; P. Thus, we may write QyBm ¼

X mi

~ m Qmi þ P~m Pmi Þ þ ðQ mi mi

X ð~ qmm qm þ p~mm pm Þ: m

ð86Þ

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O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

We can now use the gRPA equations, Eq. (40), to derive the eigenvalue equations: we insert Eq. (86) and Eq. (85) into ½HB ; QyBm
¼ Xm QyBm and compare the coefficients of the various operators q; p; Q; P. The resulting equations are then easily expressed as 0 2 1 dH d2 H !
m  ðG1Þ C B

q~ B d/ d/ d/
dG C q~m 2 ð87Þ ¼ X B 2 C 1 ~m m ~m Q @ dH A 12 Q d2 H s 2 s ðG1Þ dG d/
dG dG and

0

1 d2 H d2 H B C B d/
d/
d/
dG C B C 2 2 @ dH dH A ðG1Þ ðG1Þ dG dG dG d/


p~m 1 ~m P 2

s

! ¼

X2m




 p~m 1 ~m ; P 2

ð88Þ

s

~ m ; P~m respectively, e.g., ~ m ; P~m denote the symmetric part of Q where Q s s m m m m ~ Þ ¼Q ~ þQ ~ ¼Q ~ . These equations are now identical to Eqs. (A.31), (A.32), ðQ s ij ij ji fijg only the components of the vectors have acquired new names 1 1 ~m d/
! p~m ; dG ! P~sm and d
p ! q~m ; dR ! Q : ð89Þ 2 2 s Thus, we have proven that the two different approaches to the generalized RPA, namely the small-fluctuation approach from the time-dependent variational principle, and the operator approach, give in fact the same spectrum of possible excitations. Now the comments made at the end of Appendix A carry over to the operator formulation of gRPA: if the stability matrix of the mean-field problem is positive, all ÔeigenvaluesÕ X2m are larger than zero, and thus all Xm s are real; if we are not at a minimum of the energy with our choice of the mean-field vacuum, the stability matrix will also have negative eigenvalues, and thus we will obtain complex conjugate pairs ijXm j. As long as we have real ÔeigenvaluesÕ Xm in the gRPA problem, the minimum under consideration will be stable at least w.r.t. small fluctuations. One should note that this relation was of some importance in proving the bosonic commutation relations of the excitation operators. 6. The moment of inertia 6.1. General form of kinetic energy From the discussion in Section 5.1 we saw that we can in general write the Hamiltonian in quasi-boson approximation as X X1 HB ¼ ERPA þ Pm2 ; Xm Qym Qm þ ð90Þ 2 m2mþ m2m0 where fmþ g denotes the set of modes with positive Xm and fm0 g denotes the set of zero modes. However, quite often an alternative expression to Eq. (90) in terms of

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

477

symmetry generators is useful. For this we have to recall from Section 4 that if our reference state is a solution of the Rayleigh–Ritz equations, and if the symmetry generators under consideration are one-body operators, then the quasi-boson approximations CaB of the symmetry generators commute with the quasi-boson approximation of the Hamiltonian. This implies that they are zero mode solutions of the gRPA equations. If we now assume that there are no Ôaccidental zero modes,Õ i.e., a the whole space of zero modes is spanned B s, then we can write the Pm s, m 2 fm0 g, Pby the C a 2 as linear combinations of the CB s, and m2fm0 g Pm becomes a quadratic form in terms of the CaB s. The advantage of this alternative expression is twofold: First, in general the quasi-boson approximations of the symmetry operators are known. Second, the expression one obtains is (physically) more transparent, and can be more easily compared to other frameworks like, e.g., the Thouless–Valatin method [5,36] or the Kamlah expansion [6]. The general dependence of HB on the generators CaB will then be16 1 ab HB;zm ¼ CaB ðM1 Þ CbB ; 2

ð91Þ

where M is the moment-of-inertia tensor and EinsteinÕs summation convention is also used for the indices a; b which are employed to label the different symmetry generators CaB . We cannot (as we have done in preceding discussions where masses were set to 1) ÔnormalizeÕ M Ôaway,Õ since the normalization of CaB is fixed by its very nature of a known operator. Furthermore, if the original symmetry generators (before the quasi-boson approximation) are generators of a non-Abelian symmetry, their normalization is fixed by the commutation relations. Thus, in this section, we will see how one can actually compute the moment-of-inertia tensor, cf. [16]. We have seen in Section 5.1 that in cases of appearances of zero modes one has to pass to a description of the oscillators in terms of canonical coordinates17 and momenta (which we have done above). This actually allows to determine the moment-of-inertia tensor. We assume that we can construct a set of coordinates HaB , s.t. ½HaB ; CbB
¼ idab ;

ð92Þ

and which commute with all other normal modes. Since we already know the form of the part of HB that is not supposed to commute with HB , we also require18 ½HaB ; HB
¼ iMab CbB :

ð93Þ

We start by making an ansatz of HaB as a generalized one-body operator in quasiboson approximation19 HaB ¼ HaQ;mi Qmi þ HaP ;mi Pmi þ Haq;m qm þ Hap;m pm

ð94Þ

16 HB;zm means Ôthat part of the Hamiltonian in quasi-boson approximation that only contains the zero mode operators.Õ 17 Let us once again mention that the reduction of the non-Abelian to an Abelian symmetry (in the Yang–Mills case) in the quasi-boson approximation seems to be essential since otherwise one cannot construct canonically conjugate ÔangleÕ operators fulfilling the canonical commutation relations [35]. 18 We assume here that the moment-of-inertia tensor is symmetric, which is at least true for the case of Yang–Mills theory, since there ½CaB ; CbB
¼ 0. 19 The usage of the p-q formulation as in Section 5.2 will be useful in this context.

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O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

whereas in general CB reads CaB ¼ CaQ;mi Qmi þ CaP ;mi Pmi þ Caq;m qm þ Cap;m pm :

ð95Þ

In the latter case the coefficients are obviously known. We now insert Eqs. (94), (95) into Eq. (93), and compare coefficients. This gives in general four equations, and allows to determine HaQ;mi ; HaP ;mi ; Haq;m ; Hap;m in terms of M1 and the coefficients of CB . If we now insert the thus determined coefficients of HB into the equation that results from Eq. (92) we obtain an equation of the type ac

ðM1 Þ N

cb

¼ idab ;

ð96Þ

where N is a matrix given entirely in terms of the matrices A; . . . ; F and the coefficients of CB . Thus, one has to invert the matrix N in order to obtain the correct kinetic term for the zero modes. One should note that the logic is just the other way around from what one would expect, namely that HB is defined via its commutator with CB . However, for a proper definition of HB we also need the fact that it commutes with all the other normal modes. These general considerations find their application to the specific case of Yang–Mills theory in Sections 6.3 and 6.4 below. 6.2. Energy contributions of the zero modes We have discussed in some detail that zero modes cannot be treated via the ordinary creation/annihilation operator formalism; they have to be treated with the help of canonical coordinates and momenta. This changes their contribution to the gRPA vacuum energy. The gRPA energy in Eq. (43) was fixed s.t. hHB i ¼ EMF :

ð97Þ

Since now HB no longer contains oscillator modes only but rather has the form X 1 HB ¼ ERPA þ CaB ðM1 Þab CbB þ Xm Qym Qm ; ð98Þ 2 m2fm g þ

where fmþ g denotes the set of modes with positive Xm , Eq. (46) becomes in this context  2 ! 1  X X X 1 a 1  1 ab b m  m 2 Xm jZi j ; ERPA ¼ EMF  hCB ðM Þ CB i   Y  þ 2 2 mi  2 fmig  i m2fm g þ

ð99Þ which is again similar to the result obtained in nuclear physics [16]. In the special case of Yang–Mills theory to leading order in perturbation theory, we obtain an energy correction (see below) to the mean-field energy due to the zero modes20 1 DE ¼ Gab ðx; yÞhCa ðxÞCb ðyÞi 2 D 20

In this context, one should note that hCaB CbB i ¼ hCa Cb i.

ð100Þ

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

479

that is essentially identical in form to what one obtains in a second-order Kamlah expansion [6] and also to the Thouless–Valatin correction proposed in [5,36], but also with an important difference: the energy contribution which is due to the zero modes (and thus ultimately due to the deformation) is subtracted off after variation of the mean-field vacuum wave functional, and not before, as in the cases of [5,6,36]; therefore, the determination of the parameters of the mean-field vacuum is not influenced by the correction. 6.3. Special considerations for Yang–Mills theory We have repeatedly emphasized that the quasi-boson formulation works properly only if the reference state fulfills the Rayleigh–Ritz equations (the parameters are such that Ôthe energy functional is at its stationary pointÕ). In Yang–Mills theory without a cranking term21 this leads automatically to
¼ 0 and p

R ¼ 0:

ð101Þ

This simplifies the expression for the Gauss law operator considerably, since, when we insert these results into the expression given in Appendix C.2, we obtain22 CaB ¼ CaP ;mi Pmi þ Cap;m pm bc

b

¼ ðCaP ðxÞÞij ðx1 ; x2 ÞPijbc ðx1 ; x2 Þ þ ðCap ðxÞÞi ðx1 Þpib ðx1 Þ with b b ðCaP ðxÞÞn11 n22 ðz1 ; z2 Þ

b ðCap ðxÞÞn11 ðz1 Þ

1 ¼ 2

1 ¼ pffiffiffi 2

d

^ ab Pb i hD dRax11ax22;l1 l2 x;i ix

ð102Þ

! a b ;l1 n1

ðU 1 Þx11 z11

! d a b ;l n ab b ^ hD P i ðU 1 Þy11 z11 1 1 ; d
pal11y1 x;i ix

a b ;l2 n2

ðU 1 Þx22 z22

;

ð103Þ

ð104Þ

where we do not integrate over x. Further remarks on notation can be found in Footnote 22. One should note that all double indices are summed over except for x! Inserting Eq. (94) into Eq. (93), with the Hamiltonian given in Eq. (72), we obtain (apart from HP ¼ Hp ¼ 0) ab

Haq;m Fmi ¼ ðM1 Þ Cbp;i ;

21

As mentioned before the cranking Hamiltonian differs from the ordinary Yang–Mills Hamiltonian R by the term d3 x xa ðxÞCa ðxÞ, where xa ðxÞ is a Lagrange multiplier. 22 In this section we have to give up the super-index notation used until now in some places; instead of one super-index, the operators p will carry three indices (color, spatial, and position), i.e., pm ! pib ðxÞ where b is the color, i the spatial, and x the position index, and correspondingly for Pmi . In this context, also the index a carried by CaB has to be re-examined; in fact it is also a super-index, consisting of a color index a and a position index x: CaB ! CaB ðxÞ; the same applies to the super-indices that are carried by the moment-of-inertia tensor: Mab ! Mab ðx; yÞ. Having clarified this, in the more formal parts of this section, we will still stick to the super-index notation, since otherwise the formulas will become unreadable.

480

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506 ab

1 b HaQ;mi B11 minj ¼ ðM Þ CP ;nj ;

ð105Þ

where we have used the properties of the stationary point, i.e., the matrices A; . . . ; F are all real, etc. [cf. Section 5.2 and Eqs. (C.8)–(C.15)]. We invert the second line in a way that will be discussed below in some detail. Let us only mention at this point that ðB11 Þ1 is defined by 1 1 ððB11 Þ Þi1 m1 j1 n1 ðB11 Þj1 n1 i2 m2 ¼ ðdi1 i2 dm1 m2 þ di1 m2 dm1 i2 Þ: 2

ð106Þ

We insert the result into the normalization equation (92) and obtain   ðM1 Þac Ccp;m ðF 1 Þmi Cbp;i þ 2CcP ;mi ððB11 Þ1 Þminj CbP ;nj ¼ dab ;

ð107Þ

which bears some resemblance to the expression obtained in nuclear physics [16], especially if one notes that ðB11 Þminj / ðA þ BÞminj and Fij / ðE þ F Þij at the stationary point. 6.4. Explicit calculations ab

Unfortunately, we cannot read off an explicit expression for ðM1 Þ from this in general. However, we can do two things: First, we can try to evaluate the terms Ccp;m ðF 1 Þmi Cbp;i

and

1

2CcP ;mi ððB11 Þ Þminj CbP ;nj

ð108Þ

further. This will provide us with the moment of inertia—though for the energy correction due to zero modes we will need its inverse and this cannot be given in a general form. Second, we can go back to perturbation theory, where we in fact can give the inverse for the leading and next-to leading part in g2 for the moment of inertia. 6.4.1. Ccp;m (F 1 )mi Cbp;i It is a simple exercise to take Cbp;i and ðF 1 Þmi to calculate   ðM1 Þac ðx; yÞ ðCcp ðyÞÞbn11 ðz1 ÞðF 1 Þbn11 bn22 ðz1 ; z2 ÞðCbp ðzÞÞbn22 ðz2 Þ ^D ^ Þbc ðzÞðM1 Þca ðz; xÞ:

¼ ðD

ð109Þ

6.4.2. 2CcP ;mi ((B11 )1 )minj CbP ;nj The evaluation of this expression needs a bit more thought. The first step is to give 1 a practical expression for ððB11 Þ Þminj beyond its implicit definition in Eq. (106). For this purpose, it is useful to rewrite G1 , cf. Eq. (7), as X G1 kA PAij ; ð110Þ ij ¼ A

where kA are the eigenvalues of G1 and PA are the projectors onto the corresponding eigenspaces. The projectors are complete in the sense that X PAij ¼ dij : ð111Þ A

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

481

Then 1 1 1 1 ðG di m þ G1 i1 m2 di2 m1 þ Gi2 m1 di1 m2 þ Gi2 m2 di1 m1 Þ 2 i1 m1 2 2 1X ¼ ðkA þ kB ÞðPAi1 m1 PBi2 m2 þ PAi1 m2 PBi2 m1 Þ: 2 A;B

B11 i1 i2 m1 m2 ¼

ð112Þ ð113Þ

Using this expression for B11 it is simple to verify that 1X 1 ððB11 Þ1 Þi1 i2 m1 m2 ¼ ðPA PB þ PAi1 m2 PBi2 m1 Þ: 2 A;B kA þ kB i1 m1 i2 m2

ð114Þ

In order to give a compact expression, using super-indices, for CbP ;nj we introduce T^xa : with n; m being super-indices, where m stands for ðx1 ; b; iÞ and n stands for ðx2 ; c; jÞ , we define (no integration over x!) ðT^a Þbc dx1 x dxx2 dij ¼ ðT^xa Þmn ;

ð115Þ ^a bc

bac

bac

which inherits the property of anti-symmetry from ðT Þ ¼ f where f are the SU ðN Þ structure constants. Then we have g ðCaP ðxÞÞn1 n2 ¼ ððU T^xa U 1 Þn2 n1 þ ðU T^xa U 1 Þn1 n2 Þ: ð116Þ 2 This expression can now be cast in the same form as ðB11 Þ1 by using the eigenvalue decomposition X pffiffiffiffiffi X 1 pffiffiffiffiffi PAij ; Uij1 ¼ ð117Þ kA PAij and Uij ¼ kA A A namely one obtains ðCaP ðxÞÞn1 n2 ¼

g X kB  kA A ^a B pffiffiffiffiffiffiffiffiffiffi ðP Tx P Þn2 n1 : 2 A;B kA kB

ð118Þ

Thus, we obtain overall 2ðCcP ðxÞÞi1 i2 ððB11 Þ1 Þi1 i2 m1 m2 ðCbP ðyÞÞm1 m2 " # 2 g2 X ðkB  kA Þ Tr ðPA T^xc PB T^yb Þ : ¼ 2 A;B kA kB ðkA þ kB Þ

ð119Þ

Of course, the moment-of-inertia calculations ultimately have the goal to be connected to the static quark potential. It turns out to be interesting to compare the structure presented in Eq. (119) to the structures one obtains in a perturbative Coulomb gauge investigation into the static quark potential [37]. 6.4.3. A short excursion into Coulomb gauge perturbation theory In Coulomb gauge, the charge density consists of two parts, namely the external c part qaext and the gauge part qagauge ¼ f abc Abi ðPtr Þi , where Ptr denotes the transversal components of the momentum (the longitudinal component is eliminated using the

482

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

Gauss law constraint). We then decompose the Coulomb gauge Hamiltonian into a free part, independent of the coupling, and a rest. In Coulomb gauge, the Hamiltonian contains arbitrary powers of g. The free part (i.e., that part of the Hamiltonian which is left if we set g ¼ 0) defines what in the following is to be called gluons. Since we are in Coulomb gauge these are given in terms of the transversal fields and the corresponding momenta. In order to compute the energy corrections to the energy of the perturbative ground state by the presence of external charges to Oðg4 Þ in Rayleigh–Schr€ odinger perturbation theory, we need on the one hand the Oðg4 Þ contribution to the interaction Hamiltonian in order to compute h0jHint j0i

ð120Þ

and on the other hand the Oðg2 Þ contribution to the interaction Hamiltonian in order to compute X jh0jHint jN ij2 ; EN0 N 6¼0

ð121Þ

where the sum runs over all states except for the (perturbative) vacuum, and EN0 is the energy of this state (the energy of the perturbative vacuum has been set to zero). Since we are interested only in that part of the interaction energy that is proportional to qext ðx1 Þ  qext ðx2 Þ, we drop everything from Hint which is not quadratic in qext for the first term, and not linear in qext for the second. Since the term of the Hamiltonian which contains q at all is
ab 1 1 a D q qb ¼ qa Oab qb ð122Þ Dr Dr (which consequently has to be expanded in powers of g), for the second term we need qaext ðOab þ Oba Þqbgauge . Whereas the terms stemming from the first order contribution give the anti-screening part of the one-loop b-function, the second order contribution gives the screening part of the one-loop b-function (which is given by 1=12 times the anti-screening part). We will go into a little bit more detail of this latter contribution. What goes really into the computation of Eq. (121) is h0jqagauge jN i ¼ f abc h0jAbi ðPtri Þc jN i:

ð123Þ

Using the representation in terms of creation and annihilation operators it becomes clear that the matrix element h0jqagauge jN i is non-zero only if jN i contains two gluons. If the gluons carry energy xk1 ; xk2 (and color index m1 ; m2 ) one obtains xk  xk hN jqagauge j0i / f am1 m2 p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 : ð124Þ xk1 xk2 The comparison to Eq. (118) is striking, especially if one takes into account that in the case of absence of condensates the eigenvalues of G1 can be interpreted as single-particle energies, cf. Appendix A. The second quantity needed is 1=EN0 ; EN0 is

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

483

obviously the sum of the energies of the two gluons, i.e., EN0 ¼ xk1 þ xk2 , and 1 therefore the similarity of 1=EN0 to ðB11 Þ is obvious, cf. Eq. (119). Therefore, it appears that the ‘‘quantum part’’ of the moment of inertia contains only the (albeit generalized) screening component of the static interaction potential, and the (obviously dominant) anti-screening component has to be found elsewhere. This has to be subject to further investigations. 6.4.4. Evaluation to leading order in perturbation theory After this excursion, we go back to perturbation theory to make an evaluation of our expressions for the two components needed for the moment of inertia. We note23 that CaP is, in the perturbative scaling scheme given in Appendix B.2, of higher order in g than24 Cap . Thus, we just compute the leading order piece of the moment of inertia   ðM1 Þac ðx; yÞ ðCcp ðyÞÞbn11 ðz1 ÞðF 1 Þbn11 bn22 ðz1 ; z2 ÞðCbp ðzÞÞbn22 ðz2 Þ ^ ^
D
Þbc ðzÞðM1 Þca ðz; xÞ ¼! dba dxz : ¼ ðD

ð125Þ

From this we conclude that the leading order piece of the moment of inertia in a perturbative expansion is just the GreenÕs function of the covariant Laplacian in the
background field A ab

2 ðM1 Þ ðx; yÞ ¼ Gab D ðx; yÞ þ Oðg Þ

ð126Þ

ac ^ db bc ^ ad D

D x;l x;l GD ðx; yÞ ¼ d dxy :

ð127Þ

with

Obviously, in the case of a vanishing background field, we obtain CoulombÕs law ab

ðM1 Þ ðx; yÞ ¼

dab 1 : 4p jx  yj

ð128Þ

Let us now consider the perturbative evaluation of Eq. (119). If we solve the mean
, we field equation in a perturbative approximation (cf., e.g. [1,5,38]) for vanishing A obtain for G1
 Z d3 k ikðxyÞ ki kj ab 1 ab ðG Þij ðx; yÞ ¼ 2d e jkj dij  2 : ð129Þ 3 k ð2pÞ This shows immediately that we have a problem of building a gRPA treatment on top of the perturbative ground state, since G1 has zero modes which is (as we have argued before) unacceptable for the gRPA formalism. Since this section has mainly illustrative purposes, we proceed by simply ignoring the zero modes in what follows. The non-zero eigenvalues of G1 are given by k2jqj ¼ 2jqj, the corresponding projector onto this eigenspace is given by 23

This can be read off the expressions Eqs. (103), (104) together with the mean-field expectation value of Ca ðxÞ given in Eq. (C.16). 24 ^
is completely of Oðg0 Þ.
to be of Oðg1 Þ, s.t. D For the purpose of counting powers of g, we take A

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ab

ðP2jkj Þij ðx; yÞ ¼

Z

jkj2 dXk 3

ð2pÞ


 ki kj dab dij  2 eikðxyÞ ; k

ð130Þ

where dXk denotes the solid angle in k-space. Inserting these expressions for k2jkj and ab ðP2jkj Þij into Eq. (119), we obtain 1

2ðCcP ðxÞÞi1 i2 ððB11 Þ Þi1 i2 m1 m2 ðCbP ðyÞÞm1 m2 Z Ng2 d3 p 2 ipðxyÞ bc 2 2 ¼ lnðK =K Þ pe d þ finite terms; UV IR 3 48p2 ð2pÞ

ð131Þ

where finite terms are not determined explicitly. KIR is an IR-cutoff needed in the calculation due to the primitive way used to evaluate the loop integral. The main point is here, however, that the ‘‘quantum part’’ of the moment of inertia is just the classical part, multiplied by a divergent constant. Therefore, abbreviating ðNg2 =48p2 Þ lnðK2UV =K2IR Þ by a, we can now rewrite Eq. (107) as ðDz ðM1 Þab ðx; zÞÞð1 þ aÞ ¼ dab dxz ;

ð132Þ

where Dz is the Laplacian w.r.t. the coordinates z. Thus, ðM1 Þab ðx; zÞ ¼

dab 1 1 : 1 þ a 4p jx  zj

ð133Þ

If we introduce a renormalized coupling in the standard fashion [1] via 1 1 ¼ þ b0 lnðK2UV =l2 Þ gR2 ðlÞ g2

ð134Þ

the expression for M is made finite only for b0 ¼ a, thereby corroborating our earlier claim that the moment of inertia can only account for the screening contribution of the static quark potential. It is interesting to note that the moment of inertia obtained here is the same as in a cranking type calculation [5]. This also could have been anticipated, as it is wellknown in nuclear physics that the moments of inertia determined from RPA and those from cranking are in fact identical in nuclear physics, cf. [22]. 6.5. Interpretation of moment of inertia as static quark potential We have to conclude that we cannot interpret the moment of inertia as the static quark potential directly. However, this is not necessarily fatal for a gRPA treatment of Yang–Mills theory. Due to the approximations made to the Gauss law operator a a (e.g., making it Abelian) it is not necessarily clear that25 jci ¼ eic H jRPAi is a 25 The gRPA vacuum jRPAi is annihilated by all gRPA annihilation operators QmB and by the gRPA approximated symmetry generators, CB jRPAi ¼ 0. Whereas in the case of non-zero modes the excited states are generated by the creation operators, in the case of zero modes the excited states jci are given as plane waves and employ the canonically conjugate coordinate of the symmetry generator, b b jci ¼ eic H jRPAi. These excited states are obviously eigenstates of the symmetry generators: Ca jci ¼ ca jci. Their energy can be determined easily, too: HB jci ¼ ð1=2ÞðM1 Þab ca cb jci.

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

485

ab

correctly charged state, and thus hcjðM1 Þ CaB CbB jci is also not necessarily the energy of a state with a certain prescribed charge. This demonstrates, however, that further developments are necessary in two areas: First, one needs a systematic expansion where the quasi-boson approximation is the leading order. Second, a clear interpretation of gRPA states in terms of real physical states appears necessary. 6.6. Observations in electrodynamics We want to close this section with a few remarks and observations on electrodynamics. In electrodynamics the (full) Gauss law operator is given by CðxÞ ¼ rxi Pi ðxÞ:

ð135Þ

We guess immediately the form of the corresponding ‘‘angle’’ coordinate H as was discussed before in the context of gRPA; if we take HðxÞ ¼ GD ðx; yÞryi Ai ðyÞ

with Dx GD ðx; yÞ ¼ dxy

ð136Þ

then on the one hand ½HðxÞ; CðyÞ
¼ idxy

ð137Þ

and on the other hand ½HðxÞ; H
¼ GD ðx; yÞCðyÞ;

ð138Þ

thereby identifying the moment of inertia as ðM1 Þðx; yÞ ¼ GD ðx; yÞ:

ð139Þ

It seems interesting that the angle coordinate Eq. (136) shown here appears to be closely related to the dressing function used in [39,40], cf. also [41], to construct gauge invariant electron fields y

wc ðxÞ ¼ eieGD ðx;yÞri Ai ðyÞ wðxÞ:

ð140Þ

The Hamiltonian of electrodynamics furthermore allows for a decomposition very similar to the one made possible by the gRPA treatment for general bosonic theories, cf. Eq. (90). If we introduce the longitudinal and transversal projectors PLij ðx; yÞ; PTij ðx; yÞ as PLij ðx; yÞ ¼ rxi GD ðx; yÞryj and PTij ðx; yÞ ¼ dij dxy  PLij ðx; yÞ;

ð141Þ

we can decompose the Hamiltonian into one depending solely on transversal degrees of freedom, PTi ðxÞ ¼ PTij ðx; yÞPj ðyÞ, ATi ðxÞ ¼ PTij ðx; yÞAj ðyÞ , and one depending solely on the Gauss law operator (symmetry generator) Z Z 1 1 3 T 2 T T d xðPi ðxÞPi ðxÞ þ B ½A
Þ þ d3 x1 d3 x2 GD ðx2 ; x1 ÞCðx1 ÞCðx2 Þ: H¼ 2 2 ð142Þ The gRPA treatment, however, differs from the general treatment discussed above, though not in an unexpected manner. Since electrodynamics is eventually a free

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theory, the solution of the full Schr€ odinger equation gives the same result as the mean-field treatment, thus we can use as reference state the well-known ground state of electrodynamics [42]: "Z # ( ) Z pi pj ipðxyÞ 1 d3 p 3 3 d x d yAi ðxÞ Aj ðyÞ : w½A
/ exp  jpjðdij  2 Þe 3 2 p ð2pÞ ð143Þ This state, however, is annihilated by the Gauss law operator and consequently we expect CB to vanish, and indeed this is verified by explicit computation. It should be noted in passing, however, that an application of Eq. (125) for the moment of inertia gives the correct result (since the zeros in Cp , which come about since r acts upon the transversal U 1 , are canceled by F 1 ¼ G which is infinite in the longitudinal subspace, G1 being proportional to the transversal projector).

7. Summary and conclusion Let us shortly summarize what has been achieved in this paper, and which problems remain to be solved. We started by introducing shortly the canonical quantization and Schr€ odinger picture treatment of general bosonic theories with a standard kinetic term. We then considered the operator formulation of the generalized Random Phase Approximation that is the one common in nuclear physics, and which has recently also been investigated in the context of pion physics. We demonstrated that this approach can also be implemented for a general class of bosonic field theories, but that the class of excitation operators to be considered has to be larger than in nuclear physics; namely, one has to also allow for terms linear in creation/annihilation operators of the fundamental boson fields. It turned out to be possible (at least for a certain class of Hamiltonians with standard kinetic term) to prove the equivalence of the operator formulation to the formulation starting from the time-dependent variational principle. Then we demonstrated that, in the absence of zero modes, the gRPA Hamiltonian is just a collection of harmonic oscillators. The zero modes required special attention, but the problems could be solved along lines paralleling nuclear physics. We then considered the question of how conservation laws of the full theory translate into conservation laws of the theory in generalized Random Phase Approximation, and saw that, at least in the case of symmetries generated by one-body operators, existence of symmetries in the full theory implies existence of symmetries in the gRPA-approximated theory, although these symmetries need not be the same; in Yang–Mills theory, the generalized RPA only carries an Abelian symmetry. This also opened up a possible way of improving the gRPA by requiring a certain fulfillment of the commutation relations by the approximated operators. Then we investigated the difference between the mean-field energy and the energy of the gRPA ground state with special emphasis on the energy contribution of the zero modes. For this purpose we had to calculate the moment-of-inertia tensor which

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

487

(at least to lowest order in perturbation theory) turned out to be the GreenÕs function of the covariant Laplacian in the background field at the stationary point. Furthermore, we compared the quantum corrections to the moment of inertia, and found that their structure is very similar to the screening contribution in a Coulomb gauge perturbative calculation of the static quark potential. As a last point, we made some observations of electrodynamics where the concepts of moment of inertia and zero mode operators can be applied to the system without any form of approximation. In the gRPA treatment of electrodynamics some of the general observations can be verified explicitly; the most interesting point seems to be that—even though the Gauss law operator vanishes in the gRPA treatment, due to the fact that the reference state is gauge invariant,—it is still possible to determine correctly the moment of inertia. To put the method into perspective, let us summarize the main positive and negative aspects: On the positive side, we first have to mention that in the generalized RPA only energy differences w.r.t. the ground state are computed. This simplifies matters, since that part of renormalization that is usually done by normal-ordering is automatically taken care of. This brings us directly to the second point: energies of excited states can be computed. This is usually very difficult if one relies upon, e.g., the Rayleigh–Ritz principle. The most important point, however, is the effective implementation of the Gauss law constraint. Even though the gauge symmetry is broken in the mean-field treatment, the unphysical excitations generated by the (gRPA approximated) Gauss law operator (more precisely given by plane waves given in the coordinate conjugate to the Gauss law operator) are orthogonal to all the other physical excited states, which is almost as good as if they did not even exist. But there are also a number of drawbacks of the method presented. The first has to do with the ground state energy: Whereas one does obtain a lowering of the ground state energy w.r.t. the mean-field energy due to deformation, in a manner which is even formally quite similar to the lowering obtained, e.g., in the second order Kamlah expansion (at least to the order in perturbation theory considered) and also in other methods, the point of the calculation at which the energy correction due to the deformation is considered is fundamentally different. In the Kamlah expansion and also in the Thouless–Valatin method, the corrections are subtracted before the variation is carried out, whereas in the generalized RPA they are subtracted only after the variation. Therefore, the energy correction does not have any influence on the parameters of the mean-field. This brings us directly to the next problematic point: of course the mean-field vacuum is not the gRPA ground state. In nuclear physics, one was able, however, to construct the gRPA ground state from the mean-field ground state; but, in the presence of zero modes, this state has a divergent norm. This is due to the fact that the gRPA is a Ôsmall angleÕ approximation [22], or in other words, the compact nature of the non-Abelian symmetry is lost. We have seen explicitly that in the case of Yang–Mills theory, the compact SU ðN Þ symmetry N 2 1 is replaced by the non-compact U ð1Þ symmetry. In nuclear physics, at least in the case of two-dimensional rotations, the fact that RPA is only a small-angle approximation was not so much of a problem, since the global dependence on the angle

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is generally known and therefore one can extract enough information from the smallangle approximation to determine the whole wave function [22]. In Yang–Mills theory, we have no comparable knowledge that could be put to use practically and therefore, we cannot compute the ground-state wave functional. Lacking this knowledge, however, one needs other methods to evaluate matrix elements of operators, which up to now we have not developed. Therefore, the only quantities we currently can calculate are the energies of the excited states. As a last though very important point, we have to mention that we have not dealt with the problems of renormalization. However, it should be possible to deal with them, since (at least in the case of /4 theory) this problem has been faced already by Kerman and Lin [14,15] in the context of the generalized RPA derived from the time-dependent variational principle.

Acknowledgments The authors thank Michael Engelhardt, Jean-Dominique L€ange, Markus Quandt, Cecile Martin, and Peter Schuck for continuous discussion. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grants Mu 705/3, GRK 683, Re 856/4-1, Re 856/5-1, and Schr 749/1-1. This work is also supported in part by funds provided by the US Department of Energy (D.O.E.) under cooperative research agreement DF-FC02-94ER40818.

Appendix A. Generalized RPA from the time-dependent variational principle In this appendix we present the approach to the generalized Random Phase Approximation via the time-dependent variational principle as pioneered by Kerman et al. [13–15]. We show that it leads to equations of motion of coupled harmonic oscillators. The purpose of this appendix is to make this paper reasonably self-contained, so that the reader can follow our claim that (under the circumstances outlined in the main text) the operator approach and the time-dependent approach to the generalized RPA yield identical equations of motion. We do not want to deal with a specific theory at the moment, we will only require the Hamiltonian to be of the form 1 H ¼ p2i þ V ½/i
; ðA:1Þ 2 where /i are a set of fields, and pi are the canonical momenta conjugate to /i , in the field (coordinate) representation under consideration pi ¼

1 d i d/i

ðA:2Þ

and i is a super-index, containing a position variable x, and all other indices required (like color, spatial, etc). The Einstein summation convention is adopted, implying

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

489

sums over all discrete and integrals over all continuous variables. V ½/
is a functional of the field operators, in the following referred to as Ôpotential.Õ The states that we consider as trial states for the time-dependent variational principle are the most general time-dependent Gaussian states (we only indicate the time-variable explicitly, i; j are super-indices and N is a normalization constant): !
 1 1 w½/; t
¼ N exp  ð/  /
ðtÞÞi G ðtÞ  iRðtÞ ð/  /
ðtÞÞj þ i
pi ðtÞð/  /
ðtÞÞi : 4 ij ðA:3Þ

The meaning of the parameters becomes clear by considering expectation values: ðA:4Þ hwðtÞj/i jwðtÞi ¼ /
i ðtÞ; hwðtÞj/i /j jwðtÞi ¼ /
i ðtÞ/
j ðtÞ þ Gij ðtÞ;

ðA:5Þ

hwðtÞjpi jwðtÞi ¼ p
i ðtÞ;

ðA:6Þ

1 hwðtÞjpi pj jwðtÞi ¼ p
i p
j þ G1 ðtÞ þ 4ðRðtÞGðtÞRðtÞÞij : ðA:7Þ 4 ij We can now compute the action of the time-dependent variational principle [13] Z S ¼ dthwðtÞjiot  H jwðtÞi ðA:8Þ and obtain

 Z  i S ¼ dt p
i ðtÞ/
_ i ðtÞ  tr ðR_ GÞ þ tr ðG_ 1 GÞ  HðtÞ½/
; p
; G; R
4

ðA:9Þ

with HðtÞ ¼ hwðtÞjH jwðtÞi:

ðA:10Þ

We can now add a total time derivative26 that does not change the equations of motion, and obtain for the action Z ðA:11Þ S ¼ dtðRij ðtÞG_ ij ðtÞ þ p
i ðtÞ/
_ i ðtÞ  HðtÞÞ; which shows that R is to be considered as the canonical momentum conjugate to G, and p
that of /
. The parameters of the wave functional are now determined via HamiltonÕs classical equations of motion: dH /
_ i ðtÞ ¼ ; d
pi 26

dH p
_ i ðtÞ ¼ 
; d/i

ðA:12Þ

Its form can, e.g., be found in [14] as (here only symbolically) ðd=dtÞðði=4Þ logðG1 Þ  RGÞ.

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O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

dH ; G_ ij ðtÞ ¼ dRij

dH R_ ij ðtÞ ¼  : dGij

ðA:13Þ

However, in general it will be much too complicated to solve these equations, therefore we now consider a two-step procedure 1. look for static solutions to the equations of motion; 2. consider small fluctuations around these static solutions. The static solutions are (obviously) determined via /
_ i ðtÞ ¼ 0;

p
_ i ðtÞ ¼ 0;

G_ ij ðtÞ ¼ 0;

R_ ij ðtÞ ¼ 0:

ðA:14Þ

But these are nothing but the equations resulting from the Rayleigh–Ritz principle dH ¼ 0; d
pi

dH ¼ 0; d/
i

dH ¼ 0; dRij

dH ¼ 0: dGij

ðA:15Þ

Thus, for a static solution of the time-dependent variational principle, the parameters are those which minimize (or at least extremize) the energy. This should not really come as a surprise, since for a static state w½/
, the action reduces just to minus the energy times the respective time interval under consideration. For the next step, we decompose the general, time-dependent parameters into the static solution plus a time-dependent contribution that later on is considered to be small, e.g., for /
: /
i ðtÞ ¼ /
i;s þ d/
i ðtÞ;

ðA:16Þ

where /
i;s denotes the static solution,27 and d/
i ðtÞ the ÔsmallÕ time-dependent part. We insert this decomposition into the equations of motion, Eqs. (A.12), (A.13), and obtain d/
_ i ðtÞ ¼ ¼

dH
½/ þ d/
ðtÞ; . . .
d
pi s dH
d2 H
½/s ; p
s ; Gs ; Rs
þ ½/ ; p
s ; Gs ; Rs
d
pj þ    þ Oðd2 Þ: d
pi d
pj d
pi s

ðA:17Þ

Now the meaning of d/
, etc. being small is clarified: in the equations of motion terms of higher than linear order are neglected (in the action, it would be terms of higher than quadratic order). The first contribution ðdH=d
pi Þ½/
s ; p
s ; Gs ; Rs
vanishes by virtue of the static equations of motion, Eqs. (A.14), (A.15). The same construction can be carried out for all four parameter types (/
; p
; R; G), and the resulting equations of motion can be nicely summarized as follows:

27 In other words, if we evaluate the first derivative dH=d
pi for p
¼ p
s , it is zero, and correspondingly for /
; G; R.

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

0

d2 H d2 H d2 H B d
d
p d
p d
p dG p d/
0 1 B 2 2 2 B _ B d H  d H  d H d/
B dp
_ C B

d/
d
p d/
dG B C B d/2 d/ 2 @ dG_ A ¼ B B dH dH d2 H B B dRd /
dR_ dRd p
dR dG B @ d2 H d2 H d2 H    dG d
p dG dG dG d/


1 d2 H d
p dR C C0 1 d2 H C d/

C C d
pC d/ dR CB C; CB 2 @ d H C dG A C dR dR dR C C 2 A dH  dG dR

491

ðA:18Þ

where d2 H=dA dB ¼ ðd2 H=dA dBÞ½/
s ; p
s ; Gs ; Rs
and can be made even more transparent if one introduces an auxiliary matrix X: 0 1 0 1 B 1 0 C C: X¼B ðA:19Þ @ 0 1A 1 0 Then one obtains 0

d2 H d2 H d2 H B d/
d/
d/
d
p d/
dG B 0 1 2 B d2 H _ d H d2 H
B d/ B B dp
_ C p d
p d
p dG B d
p d/
d
B C @ dG_ A ¼ XB B d2 H d2 H d2 H B B dG d/
dG d
dR_ p dG dG B 2 @ dH d2 H d2 H p dR dG dR d/
dR d


1 d2 H d/
dR C C0 1 d2 H C C d/
CB d
p dR CB d
pC C: C@ 2 d H C dG A C dR dG dR C C d2 H A

ðA:20Þ

dR dR

At this point it becomes clear what determines the spectrum of small fluctuations around a static mean-field solution: it is the stability matrix of this static mean-field solution. Eqs. (A.20) are usually called the generalized RPA equations. Up to now, the only assumption that has been used was the assumption of w being a Gaussian state. But we have also restricted the choice of Hamiltonians that we want to consider by Eq. (A.1). This restriction will allow to carry the calculation a bit further. H depends only via hp2 i on both p
and R, V ½/
depends only on /
and G. Thus for all Hamiltonians that we are considering, we have the kinetic energy written as 

  1 2 1 1  w pi w ¼ p
i p
i þ Tr ðG1 Þ þ 2Tr ðRGRÞ; ðA:21Þ 2 2 8 where Tr denotes the trace over the super-indices. The static solutions p
i;s ; Rij;s are determined via dH ¼ 0; d
pi

dH ¼0 dRij

ðA:22Þ

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O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

and result in p
i;s ¼ 0;

Rij;s ¼ 0:

ðA:23Þ

This information, together with the knowledge that V ½/
does neither depend on p
nor on R, determines a number of second derivatives d2 H ¼ 0; d
p d/


d2 H ¼ 0; dR d/


d2 H ¼ 0; dG d
p

d2 H ¼ dij ; d
pi d
pj

d2 H ¼ 0; dR d
p

d2 H ¼ 0; d
p dG

d2 H ¼ 0; dR dG

d2 H ¼ ðG1Þ; dR dR

ðA:24Þ

where we have used the abbreviation28 ðG1Þij;kl ¼ 2

d d tr ðRGRÞ ¼ Gki dlj þ Gkj dli þ Gjl dki þ Gli dkj : dRij dRkl

These results can be used to simplify Eq. (A.18) 1 0 0 1 0 0 0 1 0 1 2 2 dH C d/
B dH d/
_ C B 0  0 CB d
C B dp
_ C B d/
d/
d/
dG CB p C B C B @ dG_ A ¼ B 0 0 0 ðG1Þ C@ dG A C B A dR @ d2 H d2 H dR_  0  0
dG dG dG d/

ðA:25Þ

ðA:26Þ

with ð1Þij ¼ dij . By taking the derivative of Eq. (A.26) with respect to time, and reinserting Eq. (A.26), one obtains two sets of partially decoupled equations 0 1 d2 H d2 H
  B C
€
B d/
d/
d/
dG C d/
; ðA:27Þ ðIÞ d/ ¼ B C € @ d2 H d2 H A dG dG ðG1Þ ðG1Þ dG dG dG d/
0


ðIIÞ

€
dp € dR



1 d2 H d2 H  ðG1Þ C
B

p B d/ d/ d/
dG C d
¼ B 2 : C @ dH A dR d2 H ðG1Þ dG d/
dG dG

If we now make the ansatz of a harmonic time-dependence !
 d/
ið0Þ d/
i ðtÞ ðIÞ cosðxt þ d1 Þ; ¼ ð0Þ dGij ðtÞ dGij
ðIIÞ

28

d
pi ðtÞ dRij ðtÞ



ð0Þ

¼

d
pi ð0Þ dRij

ðA:28Þ

ðA:29Þ

! cosðxt þ d2 Þ

In this context, one should note that dRij =dRkl ¼ ð1=2Þðdik djl þ dil djk Þ.

ðA:30Þ

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

493

we obtain the eigenvalue equations 0

ðIÞ x2




d/
dG



1 d2 H d2 H
 B
dG C B d/
d/
C d/
d / ; ¼B C @ d2 H d2 H A dG ðG1Þ ðG1Þ dG dG dG d/


1 d2 H d2 H 
 B ðG1Þ C
p d
p B d/
d/
d/
dG C d
ðIIÞ x2 : ¼B 2 C dR @ dH A dR d2 H ðG1Þ dG d/
dG dG

ðA:31Þ

0

ðA:32Þ

In analyzing properties of the generalized RPA equations, Eqs. (A.31), (A.32), it is often simpler to study the Hamiltonian they are derived from than to study the equations themselves. In this context one can imagine Eq. (A.26) to originate from a Hamiltonian29 H : 0


1 1 p dRÞ H ¼ ðd
0 2

1 d2 H d2 H 
  B

C
1
0 d
p B d/ d/ d/
dG C d/
þ ðd/ dGÞB 2 : C ðG1Þ dR @ dH 2 d2 H A dG dG d/
dG dG ðA:33Þ

This evidently is the Hamiltonian of a set of coupled oscillators. The important point is that the signs of the eigenvalues of the decoupled oscillators are determined by the reduced stability matrix30 containing only second derivatives w.r.t. to /
and G (We assume here that ðG1Þ is positive definite and since ðG1Þ is only multiplied by objects symmetric in their two indices, e.g., dRij ¼ dRji , this boils down to assuming that G is positive definite. This is a sensible assumption connected to the normalizabilty of w½/
and is discussed extensively in Section 3.1). An interesting observation can be made immediately: usually d2 H=d/
dG will be only non-zero if there is a condensate (i.e., /
6¼ 0) in the system. Thus if we do not have a condensate, the equations for the one- and two-particle content31 decouple, and, since in a system without condensate G will usually be translation invariant, we can see by considering the Fourier transformed quantities that 1=~ gðpÞ just describes the energy spectrum of

They originate by the canonical equations of motion, e.g., dp
_ ¼ dH =dðd/
Þ. The problem of coupled oscillators is well-known, cf. [43,44]. If the reduced stability matrix is a positive matrix (and thus our mean field vacuum is indeed a minimum), all the eigenvalues will be positive, and thus all oscillator frequencies will be real. 31 We see in Section 5.2 that in a creation/annihilation operator formalism d/
; d
p are connected to the amplitudes of operators containing one creation/annihilation operator, whereas dG; dR are connected to the amplitudes containing two creation/annihilation operators. 29 30

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single-particle excitations with momentum p, where g~ðpÞ is defined by the following ~ðp; qÞ by procedure. We introduce the Fourier transform G Z d3 p d3 q ipx ~ e Gðp; qÞeiqy : ðA:34Þ Gðx; yÞ ¼ 3 3 ð2pÞ ð2pÞ If we now require that G shall be translation invariant, i.e., Gðx; yÞ ¼ Gðx  yÞ, we ~ðp; qÞ obtain for G ~ðp; qÞ ¼ ð2pÞ3 dðp  qÞ~ G gðpÞ:

ðA:35Þ

This defines g~ðpÞ.

Appendix B. Hamiltonian treatment of Yang–Mills theory B.1. Yang–Mills theory There is not much to be said about the Hamiltonian treatment of Yang–Mills theory in particular, since there is an excellent treatment in [45]. The main points to be kept in mind are: First, in order to work in the Hamiltonian formalism at all one chooses the Weyl gauge, i.e., A0 ¼ 0. The price to be paid in this gauge ^ ab Eb ¼ 0 (E is the color-electric field, is that the classical constraint equation, D i i ^ cf. Eq. (B.4) below), i.e., the Gauss law equation, has for the definition of D to be implemented as a constraint on states: jwi is a physical state if it is anni^ ab Eb upon quanhilated by the Gauss law operator (which is constructed from D i i tization). Even in the presence of this constraint the canonical pair of A; P can be quantized straightforwardly without the appearance of non-trivial metric tensors [46]. To put our treatment of Yang–Mills theory in a nutshell: We consider it to be described as a canonical system, defined in terms of coordinates Aai ðxÞ and conjugate momenta Pai ðxÞ which satisfy ordinary commutation relations: ½Aai ðxÞ; Pbj ðyÞ
¼ idab dij dxy :

ðB:1Þ

The states in the physical subspace have to satisfy GaussÕ law, i.e., Ca ðxÞjwi ¼ 0;

ðB:2Þ

where we have introduced both the Gauss law operator ^ ab ðxÞPb ðxÞ Ca ðxÞ ¼ D i i

ðB:3Þ

and the covariant derivative in the adjoint representation ^ ab ðxÞ ¼ dab ri  ðgÞf acb Ac ðxÞ: D i i

ðB:4Þ

We have put the coupling constant in brackets since its appearance depends on whether we have chosen ‘‘perturbative’’ or ‘‘non-perturbative’’ scaling, cf. Appendix B.2. The SU ðN Þ structure constants are denoted as f abc . One should note that the

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495

Gauss law operator used here may differ from those in other publications by some proportionality factors, as it is not directly the generator of (small) gauge transformations. The reason for choosing this form of the Gauss law operator is that the only factors of g appearing during a change from perturbative to non-perturbative scaling appear inside the covariant derivative. The wave functional of the reference state taken for Yang–Mills theory is of Gaussian form  

 1 a a 1 ab ab
w½A
¼ N exp  Ai ðxÞ  Ai ðxÞ ðG Þij ðx; yÞ  iRij ðx; yÞ 4    n  o
b ðyÞ
a ðxÞ ;  Abj ðyÞ  A  exp i
pai ðxÞ Aai ðxÞ  A ðB:5Þ j i falling into the class of reference states taken in the main text for generic bosonic theories. The remaining defintions, like the Hamiltonian, can be found in the next section, Appendix B.2. B.2. Factors of g In Yang–Mills theory one has basically two options concerning where one wants to put the coupling constant, either in front of the action (here called Ônon-perturbative scalingÕ), or in front of the commutator term in the field strength (here called Ôperturbative scalingÕ). In Table 1 we give a short list concerning which convention leads to which placing of factors of g in other quantities of interest.

Appendix C. Explicit expressions C.1. Matrices A–F In this appendix, we want to give explicit expressions that are valid for the theories that have a Hamiltonian of the form Eq. (6). As we have already mentioned, terms in the Hamiltonian that contain more than four c/a operators do not contribute to any of the matrix elements due to the second gRPA approximation.32 Then the computation is straightforward: we simply insert Eq. (16) into Eq. (26), and compute the double commutators. The computation is simplified by the observation that only those terms of H contribute to the matrices where the number of creation operators together with the number of creation operators in the definition of the matrix elements match the respective number of annihilation operators. This is the reason 32

One can use WickÕs theorem to compute the double commutators that appear in the definition of the matrices A; . . . ; F . Then one realizes that for those terms of the Hamiltonian that contain more than four c/ a operators the evaluated double commutators still contain at least one c/a operator. Taking the vacuum expectation values—where according to the second gRPA approximation we use the mean-field vacuum state—of these expressions sets these terms then to zero.

496

Non-perturbative scaling

Perturbative scaling

Covariant derivative

Dl ¼ ol  iAl

Dl ¼ ol  igAl

Field strength

Flm ¼ ol Am  om Al  i½Al ; Am
¼ i½Dl ; Dm


Flm ¼ ol Am  om Al  ig½Al ; Am
¼ gi ½Dl ; Dm


Action

S ¼  4g12 Flma ðxÞF alm ðxÞ Eai ¼ F0ia Bai ¼  12
ijk F ajk ¼ ðr  AÞai  12 f abc ðAb Pai ¼ g12 Fi0a ¼  g12 Eai 2 H ¼ g2 Pai ðxÞPai ðxÞ þ 2g12 Bai ðxÞBai ðxÞ ¼ 2g12 ðEai ðxÞEai ðxÞ þ Bai ðxÞBai ðxÞÞ  12 AG1 A

S ¼  14 Flma ðxÞF alm ðxÞ

Electrical field Magnetic field Momenta

Pai

¼ ooL A_ a i

Hamiltonian Wave functional of ÔfreeÕ theory

w½A
 e

Generators of time-independent gauge trafos Finite gauge trafos (gluonic part)

½ðCax Þ; ðCby Þ
¼ idxy f abc ðCcx Þ R a a ei / C

g

Eai ¼ F0ia c

 A Þi

Bai ¼  12
ijk F ajk ¼ ðr  AÞai  g2 f abc ðAb  Ac Þi Pai ¼ Fi0a ¼ Eai H ¼ 12 Pai ðxÞPai ðxÞ þ 12 Bai ðxÞBai ðxÞ ¼ 12 ðEai ðxÞEai ðxÞ þ Bai ðxÞBai ðxÞÞ 1 A

w½A
 eAG

½1g ðCax Þ; 1g ðCby Þ
¼ gg idxy f abc 1g ðCcx Þ R a1 a ei g/ gC

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Table 1 Placement of the coupling constant g in the Ônon-perturbativeÕ and the ÔperturbativeÕ scaling scheme. We have used the shorter form Cax for Ca ðxÞ

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497

for the matrices A; C; D; E having contributions from one term of H only. B obtains contributions from two terms of H as is to be expected, whereas F obtains only one. Individually, the matrices read: • Matrix A fminjg Aminj ¼ 24H04 ;

ðC:1Þ

• Matrix B 22 Bminj ¼ B11 minj þ Bminj

ðC:2Þ

with ji jm ni nm B11 minj ¼ ðH11 dnm þ H11 djm þ H11 dni þ H11 dji Þ;

fnjgfmig

B22 minj ¼ 4H22

;

ðC:3Þ

• Matrix C fjmig

Cjmi ¼ 6H03

ðC:4Þ

;

• Matrix D ifjmg

Djmi ¼ 2H12

;

ðC:5Þ

fmig

ðC:6Þ

• Matrix E Emi ¼ 2H02 ; • Matrix F im ; Fmi ¼ H11

ðC:7Þ

where fijg means: symmetrize in the indices i; j (i.e., add all permutations and divide ... by the number of permutations) and Hab means Ôthat factor in the Hamiltonian that multiplies a creation and b annihilation operators.Õ We see that at the stationary point they simplify considerably: Aminj ¼ 4Umm1 Uii1 Unn1 Ujj1

d2 hV i; dGm1 i1 dGn1 j1

22 Bminj ¼ B11 minj þ Bminj ;

ðC:8Þ ðC:9Þ

B11 minj ¼ ðFmn dij þ Fmj din þ Fin dmj þ Fij dmn Þ;

ðC:10Þ

d2 hV i; dGm1 i1 dGn1 j1

ðC:11Þ

B22 minj ¼ 4Umm1 Uii1 Unn1 Ujj1

d2 hH i; Cjmi ¼ 2Ujj1 Umm1 Uii1
d/j1 dGm1 i1

ðC:12Þ

Djmi ¼ Cjmi ;

ðC:13Þ

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O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

Emi ¼ 0;

ðC:14Þ

1 : Fmi ¼ G1 2 mi

ðC:15Þ

C.2. Gauss law operator The Gauss law operator was defined in Eq. (B.3). Here, we give its decomposition into creation/annihilation operators. After normal ordering, we obtain the result b ^ ^ ab Pb ¼ D
ab p ^a D x;i ix x;i
ix þ 2gtr ðT Rxx1 Gx1 x Þ ! d ab b a1 b1 ;l1 n1 1 n1 ^ ðayb þ abz11 n1 Þ þ z1
a1 hDx;i Pix i Ux1 z1 dA l1 x1 !
d i 1 a1 b1 ;l1 n1 yb1 n1 b ab ^ P i ðU Þx1 z1 ðaz1  abz11 n1 Þ þ2 hD d
pal11x1 x;i ix 4  b1 n1 1 n1 þ Rax11cy11;l1 k1 Uyc11zb11 ;k1 n1 ðayb þ a Þ z1 z1
 d ^ ab Pb i U a1 b1 ;l1 n1 U a2 b2 ;l2 n2 þ hD x;i ix x1 z1 x2 z2 dGax11ax22;l1 l2 1 n1 yb2 n2 1 n1 b2 n2  ðayb az2 þ abz11 n1 abz22 n2 þ 2ayb a z2 Þ z1 z1 ! d ^ ab Pb i i ðU 1 Þa1 b1 ;l1 n1 ðU 1 Þa2 b2 ;l2 n2 hD þ x1 z1 x2 z2 dRax11ax22;l1 l2 x;i ix 4

i a a a b ;ln a1 b1 ;ln1 1 n1 yb2 n2  ðayb az2  abz11 n1 abz22 n2 Þ þ g ðT^a Þ 1 2 ðUxz ðU 1 Þxz2 2 2 2 z1 1 2 a b ;ln a2 b2 ;ln2 1 n1 b2 n2 þ ðU 1 Þxz1 1 1 1 Uxz Þayb a z2 : z1 2

ðC:16Þ

Here we have employed the notation ðT^a Þ

a1 a2

¼ f a1 aa2

ðC:17Þ

where f a1 aa2 denote the SU ðN Þ structure constants.

Appendix D. Commutation relations of normal modes In this appendix we want to consider the question Ôwhat are the conditions under which the eigenmodes QyBm can be treated as harmonic oscillators ?Õ We start with the observation that Eq. (40) implies by hermitian conjugation ½HB ; QBm
¼ Xm QBm :

ðD:1Þ

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499

Thus, to every eigenfrequency, the negative eigenfrequency also belongs to the spectrum. Thus, without loss of generality33 in the following we will assume that ðD:2Þ Xm 0; otherwise we just exchange the respective QBm ; QyBm . The next point is that all the commutators ðD:3Þ ½QyBm ; QyBl
½QBm ; QyBl
½QBm ; QBl
are pure numbers. This is due to the fact that all operators considered in this context are by construction linear in B; By ; a; ay . We denote these numbers as ½QyBl ; QyBm
¼ Mlm ; ½QBl ; QyBm
¼ Nlm ; ½QBl ; QBm
¼ Olm : ðD:4Þ Up to now these numbers are arbitrary; one can, however, put the equations of motion to some good use. Consider eq: ð40Þ

Xm ½QBl ; QyBm
¼ ½QBl ; ½HB QyBm

Jacobi id:

¼

¼

 ½HB ; ½QyBm ; QBl

 ½QyBm ; ½QBl ; HB



ðD:6Þ

½HB ; Nlm
þ½QyBm ; ½HB ; QBl



ðD:7Þ

|fflfflfflfflffl{zfflfflfflfflffl} ¼0

¼ In other words ðXm  Xl ÞNlm ¼ 0

ðD:5Þ

|fflfflfflfflffl{zfflfflfflfflffl} ¼Xl QBl

Xl ½QBl ; QyBm
: ðno sum over l; mÞ:

ðD:8Þ ðD:9Þ

If all eigenvalues are distinct and non-zero then Eq. (D.9) implies that Nlm is diagonal. Since the gRPA equations are homogeneous equations, we may now normalize the amplitudes in such a way that34; 35 33 We have seen in Section 5.2 that the assumption that the eigenenergies of the modes are real already implies that we are dealing with a stable mean-field solution. 34 In [16] it is stated that one can show that Nlm is positive if one is considering a positive definite Hessian at the stationary point of the mean-field problem; a possible proof of this statement works as follows: only if the Hessian is a positive matrix it is guaranteed that all eigenvalues X2 are positive. This has been assumed so far (e.g., how we concluded that for every positive frequency there is also a negative one, etc.); thus we know that the system is stable. Knowing this, we can argue as follows: we know that ½H; QyBm
¼ Xm QyBm ; ½H ; QBm
P ¼ Xm QBm and ½QBm ; QyBm0
¼ N m dmm0 . From this we conclude that the Hamiltonian has to look like H ¼ m ðXm =N m ÞQyBm QBm . Now we can study two different scenarios, namely N m can be positive (we call the set of m for which this is true mþ ) or negative (correspondingly m ). In order to have the usual creation/annihilation commutation relations we have—for m 2 m —to interchange ÔcreationÕ and ÔannihilationÕ operators; for clarity, we introduce new letters for them, i.e., for y m 2 m ; QBm ! PBm ; QyBm ! PBm . Now we normalize QB s and PB s s.t. for m 2 mþ ; N m ¼ 1 and for P P y m 2 m ; N m ¼ 1. The Hamiltonian then reads H ¼ m2mþ Xm QyBm QBm þ m2m ðXm ÞPBm PBm þ const. But here we see that we are dealing with a rather unstable system: the more modes are generated by PBy , the lower the energy becomes. This cannot be true, however, since we know that we started from a stable system (with all frequencies real)! Thus m has to be empty. 35 This relation has also as its consequence that the states generated by QyBl ; QyBm from the gRPA vacuum (defined to be annihilated by all gRPA annihilation operators, QBm jRPAi ¼ 08m) are indeed orthogonal: hRPAjQBl QyBm jRPAi ¼ hRPAj½QBl ; QyBm
jRPAi ¼ hj½QBl ; QyBm
ji ¼ dlm where we have used the second gRPA approximation in the next to last step [16].

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Nlm ¼ dlm :

ðD:10Þ

The same procedure can be carried out for Mlm ; Olm . We derive in analogy to Eq. (D.9): ðXm þ Xl ÞMlm ¼ 0; ðXm þ Xl ÞOlm ¼ 0;

ðD:11Þ

where again no sum over double indices is performed. From this we conclude, again if there are no zero modes, that Mlm ¼ 0; Olm ¼ 0:

ðD:12Þ

Thus, under the aforementioned conditions, the normal modes satisfy ordinary c/a commutation relations. To arrive at this, we have indeed used both gRPA approximations, since the basic ingredient was that Qy is a one-particle operator (first gRPA approximation) and that all the commutators of a; ay ; B; By are pure numbers (quasiboson approximation).

Appendix E. General potential In this appendix we will demonstrate that a potential that is an arbitrary polynomial in the field operators, can—once one decomposes the field operator into creation and annihilation operators—be written in normal ordered form s.t. all the coefficients appearing in front of the normal ordered products of creation/annihilation operators can be written as functional derivatives w.r.t. G and /
of the vacuum expectation value of the potential (the creation/annihilation operators are defined with respect to that vacuum). For simplicity we take the potential of the form V ½/
¼ Mx1 ...xn /x1    /xn ;

ðE:1Þ

where xi are super-indices. It is clear that, if the claim holds for this potential, it will hold for an arbitrary polynomial since it will be a sum of terms of type (E.1). Since the field operators commute, it is sufficient to consider an Mx1 ...xn that is symmetric in all indices. We have seen in Section 3.1 that one can write ðE:2Þ / ¼ /
þ Uxy ðby þ by Þ; x

x

y

where x; y are super-indices, U 2 ¼ G, and ½bx ; byy
¼ dxy . For our purposes, it will be more useful to define rescaled operators ayx ¼ Uxy byy ;

ax ¼ Uxy by

with ½ax ; ayy
¼ Gxy :

Sometimes we find it also useful to write /x ¼ /
x þ ux with ux ¼ ayx þ ax :

ðE:3Þ

ðE:4Þ

After all this notational introduction, let us come to the proof. We note that, similar to /, both /
and u commute; thus we can write equally36 instead of Eq. (E.1): 36 It is understood, obviously, that the index of x increases from left to right; if it should ever decrease, as in the case m ¼ 0, the /
s are to be considered absent.

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V ½/
¼ Mx1 ...xn

 n
X n m¼0

m

/
x1    /
xm uxmþ1    uxn :

501

ðE:5Þ

Now we normal order the terms with a fixed m: uxmþ1    uxn ¼ ðaxmþ1 þ ayxmþ1 Þ    ðaxn þ ayxn Þ ¼ ayxmþ1 ayxmþ2    ayxn þaxmþ1 ayxmþ2 ayxmþ3    ayxn þ    þ ayxmþ1 ayxmþ2 ayxmþ3    axn þaxmþ1 axmþ2 ayxmþ3    ayxn þ    þ ayxmþ1 ayxmþ2 ayxmþ3    axn .. . þaxmþ1 axmþ2    axn :

ðE:6Þ

We see that every row contains a fixed number of creation/annihilation operators and that it contains all possible permutations of types (creation or annihilation) among the possible indices. Especially for every term    ayxp    axq    ðE:7Þ there also exists a term    axp    ayxq   

ðE:8Þ

with all undenoted operators identical. We now use WickÕs theorem, cf. e.g. [16], to put every line into normal order. It is practical to deal with the whole line for the following reason: if we use WickÕs theorem naively we obtain the normal ordered expression plus the normal ordered expression of two less operators times their contraction, etc. However, a lot of these contractions are zero, since (we denote the contraction of two operators by C) Cðayi aj Þ ¼ 0: ðE:9Þ However, if we deal with the complete line at once, we can use that, since for every arrangement Eq. (E.7) there also exists a partner Eq. (E.8), we will always obtain contractions Cðayi aj þ ai ayj Þ ¼ Gij : ðE:10Þ It is clear that for k contractions present we need 2k partners to obtain a non-vanishing contribution. The important point is that they exist if we deal with the whole line at once. Remember that the contractions are multiplied by normal ordered terms, and that ½a; a
¼ ½ay ; ay
¼ 0. Since in addition Mx1 ...xn is symmetric in all its indices, all terms with a fixed number of contractions, creation, and annihilation operators, will give an identical contribution. Thus the question appears: assume we start out from the line where every term contains p creation operators, q annihilation operators and consider now terms with k contractions; how many terms will we obtain? The answer is simple,
 1 ðp þ qÞ! 1 p! q! number of terms ¼ k   ; ðE:11Þ 2 p!q! k! ðp  kÞ! ðq  kÞ!

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and comes about as follows:37 1. the line containing p cÕs and q aÕs contains ðpþqÞ! terms. p!q! 2. Out of the p cÕs and q aÕs we take k each, and put them together to form contractions: p! q! ways to do this. there are obviously ðpkÞ! ðqkÞ! 3. However, it does not matter in which order we perform the contractions since the contractions commute; thus in the step before we have overcounted by a factor of k!. 4. We now have to take into account what was said above: On the one hand, a lot of contractions are zero. On the other hand we can form pairs—this gives as argued above an additional factor of 2k . With this formula at hand, we can at first answer the following important question: assume that we have started from an expression with n field operators; upon normal ordering we obtain expressions with n; n  2; . . . ; n  2k c/a operators each; the question now is: is the relative number of P cÕs and Q aÕs (with P þ Q fixed) always the same, no matter from which n one starts and how many contractions one needs38 (provided n  ðP þ QÞ is even)? This question can be answered in the affirmative in the following way: We start out with an expression that contains p cÕs and q aÕs; after k contractions we will end up with P ¼ p  k cÕs and Q ¼ q  k aÕs. Since in the beginning p þ q ¼ n was fixed, and we end up with P þ Q ¼ n0 fixed we need for every term the same number of contractions, namely 2k ¼ n  n0 . With this we can rewrite Eq. (E.11) as n! 1 n! 1 : ðE:12Þ ¼ k k 0 2 k! P !ðn  P Þ! 2 k! P !Q! Thus, we have decomposed the number of terms into a factor that depends on n; n0 which is a constant for P þ Q ¼ n0 fixed and P varying, and a factor that depends on the number of creation and annihilation operators. If we go back to Eq. (E.5) we see that we have different possibilities to end up with P cÕs and Q aÕs (P þ Q ¼ n0 ): either start from n us, and perform k contractions, or start from two /
s, ðn  2Þus and perform k  1 contractions, etc. Thus the contribution to Mx1 ...xn /x1    /xn containing n0 cÕs and aÕs can be written as39 " 0 # n X 1 ay    ayxP axP þ1    axn0 Mx1 ...xn P !ðn0  P Þ! x1 P ¼0  ½12ðnn0 Þ

X n!

Gx x    Gxn0 þð2k1Þ xn0 þ2k /xn0 þ2kþ1    /xn ;  2k k!ðn  n0  2kÞ! n0 þ1 n0 þ2 k¼0 ðE:13Þ 37

In the following, we abbreviate Ôcreation operatorsÕ as cÕs and Ôannihilation operatorsÕ as aÕs. As an example: no matter where one starts—if one can get to the expression with, in total, two cÕs and aÕs, will they always come as aa þ ay ay þ 2ay a? 39 Note that the factors of the contractions and of the binomial decomposition of Eq. (E.5) can be put together in a practical manner: 0 1 ðn0 þ 2kÞ! @ n! n A¼ : n þ 1  ðn0 þ 2k þ 1Þ 2k k! 2k k!ðn  n0  2kÞ! 38

Note also that we have arranged here u; /
opposite to Eq. (E.5).

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

503

where the latter sum runs to ð1=2Þðn  n0 Þ if n  n0 is even, and to ð1=2Þðn  n0  1Þ if it is odd—we will deal with these details below. This expression allows us to write down the vacuum expectation value (VEV) of V ½/
since it corresponds to the case n0 ¼ 0. Distinguish • n ¼ 2N : the VEV reads  N
X n!

/ G    G    / hV ½/
i ¼ Mx1 ...x2N ðE:14Þ x x xð2k1Þ x2k x2kþ1 x2N ; 2k k!ðn  2kÞ! 1 2 k¼0 • n ¼ 2N þ 1: the VEV reads N
X hV ½/
i ¼ Mx1 ...x2N þ1 k¼0

 n!

Gx x    Gxð2k1Þ x2k /x2kþ1    /x2N /
x2N þ1 : 2k k!ðn  2kÞ! 1 2 ðE:15Þ

For the following treatment, we can treat both cases with the same formula if we realize that Mx1 ...x2Nþ1 /
x2N þ1 has the same properties as Mx1 ...x2N and that in the sum in Eq. (E.15) always at least one /
survives. Thus we only have to treat the case with n ¼ 2N . We now consider d d hV ½/
i ¼ Mx1 ...x2N dGy1 y2 dGy1 y2   N
X  n!

 Gx x    Gx /     / x2kþ1 x2N ð2k1Þ x2k 2k k!ðn  n0  2kÞ! n0 ¼0 1 2 k¼0   N
X  n! ðÞ

 ¼ My1 y2 x3 ...x2N kGx3 x4    Gxð2k1Þ x2k /x2kþ1    /x2N 2k k!ðn  n0  2kÞ! n0 ¼0 k¼1 NX N 0
1 n! G ¼ N 0 My1 y2 xn0 þ1 ...x2N     Gxð2k1Þ x2k k k!ðn  n0  2kÞ! xn0 þ1 xn0 þ2 2 2 k¼0   

 /x2kþ1    /x2N  : ðE:16Þ n0 ¼2¼2N 0

In ðÞ we have used the symmetry of M. Obviously, as has been indicated by the suggestive notation, one is not restricted to one functional derivative but one can also perform N 0 of them, and then the restriction N 0 ¼ 1 in the last line is rendered 0 unnecessary. We see clearly that, apart from the factor 2N the outcome of N 0 derivatives of the vacuum expectation value w.r.t. G is identical to the prefactor of the addend containing P þ Q ¼ n0 ¼ 2N 0 cÕs and aÕs in Eq. (E.13). Thus we can rewrite Eq. (E.13) as " 2

N0

n0 X P ¼0

# 1 d d y y a    ayP ayP þ1    ayn0  hV ½/
i: P !ðn0  P Þ! y1 dGy1 y2 dGyn0 1 yn0

ðE:17Þ

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The treatment we have presented up to here is valid for P þ Q even. If P þ Q is odd, we have to perform derivatives w.r.t. /
and thus we have to treat the n even/odd cases individually. Let us start with n even: d d hV ½/
i ¼
Mx1 ...xn1 xn
d/y1 d/y1 
N X  n!   k 0 2 k!ðn  n  2kÞ!  k¼0

n0 ¼0

Gx1 x2    Gxð2k1Þ x2k /
x2kþ1    /
xn1 /
xn



   n!

 Gx x    Gx /    / ðn  2kÞ x2kþ1 xn1 ð2k1Þ x2k 2k k!ðn  n0  2kÞ! n0 ¼0 1 2 k¼0   N 1
X  n!

 Gx x    Gx ¼ Mx1 ...xn1 y1 /    / x2kþ1 xn1 ð2k1Þ x2k 2k k!ðn  n0  2kÞ! n0 ¼1 1 2 k¼0   N 1
X  n!

 / ¼ My1 x2 ...xn G    G    / : x x x x 0 0 0 0 x x 0 n 0 n þ1 n þ2 n þð2k1Þ n þ2k n þ2kþ1 2k k!ðn  n0  2kÞ! n ¼1 k¼0

ðÞ

¼ Mx1 ...xn1 y1

N 1
X

ðE:18Þ The main point happened in line ðÞ where the fact that we started from even n played a role. If n is even, k ¼ N means that this addend does not contain a single factor of /
, thus its derivative vanishes. This is different in case of n odd, there the upper boundary is not affected by the first differentiation. It is different if one performs two derivatives w.r.t. /
since then the upper limit of the sum changes once altogether independent of whether one starts from n even or odd. Thus one obtains a relation between derivatives w.r.t. G and to /
1 d2 d hV ½/
i ¼ hV ½/
i; ðE:19Þ 2 d/
x d/
y dGxy which will be very useful in proving the equivalence between the generalized RPA using the operator approach and generalized RPA as derived from the time-dependent variational principle. To put this section in a nutshell, we have proved that a potential that is an arbitrary polynomial in the field operators can be decomposed into creation and annihilation operators, s.t. upon normal ordering one obtains a sum of subsums where each subsum contains a fixed number of cÕs and aÕs. The subsum consisting of the terms containing n0 cÕs and aÕs can be written as a standard polynomial in cÕs and aÕs " 0 # n X 1 y y a    axP axP þ1    axn0 ðE:20Þ P !ðn0  P Þ! x1 P ¼0 multiplied by 0 • if n0 is even, n0 =2 derivatives w.r.t. G times a factor 2n =2 , 0 0 • if n is odd, one derivative w.r.t. /
and ðn  1Þ=2 derivatives w.r.t. G times a factor 0 2ðn 1Þ=2 .

O. Schr€ oder, H. Reinhardt / Annals of Physics 307 (2003) 452–506

505

We have also shown that each derivative w.r.t. G may be traded for two derivatives w.r.t. /
.

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