We consider the symplectic structure of the collective manifold as it emerges in a path integral ... the Poisson bracket on the collective manifold is undefined.
PHYSICAL REVIEW C
DECEMBER 1989
VOLUME 40, NUMBER 6
Symplectic structure of the collective manifold Aurel Bulgac National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State UniuersIty, East Lansing, Michigan 48824 (Received 15 June 1989)
We consider the symplectic structure of the collective manifold as it emerges in a path integral transformation in order to integrate out the approach. We make use of the Hubbard-Stratonovich fermionic degrees of freedom. The effective action obtained in this way is expressed in terms of the single-particle components of the one-body density or potential, which play the role of phase space variables. We show that the term usually designed as the quantum nonintegrable phase is the only term responsible for the definition of the symplectic structure on this manifold. Without this term the Poisson bracket on the collective manifold is undefined.
Two of the most sensible questions in the quantum mechanics of the many-body system follow: How does one define the collective variables? What plays the role of collective coordinates and momenta? In spite of the tremendous effort during the past few decades and in spite of the obvious progress, it seems that there are a number of principal problems which are still unsolved. Nobody has ever considered the global properties of the collective phase space. If this space is endowed with a nontrivial topology or/and curved, the emerging equations of motions must have some nontrivial features, as the occurrence of "gauge" fields which are simply a consequence of such nontrivial geometrical properties. In this work we shall consider the construction of the effective collective Lagrangian starting from a path integral approach over single-particle densities in the adiabatic approximation. We show that the symplectic structure of the collective manifold is defined only when one takes into account the dynamically generated effective gauge field on the collective phase space. Let us consider a many-body fermion system with the Hamiltonian
~ = g T &a ta&+ , g (aP~ ap apy6 '
U
~y5)a a@—zaz,
and we wish to evaluate the trace of the evolution opera-
tor
K ( T) = Tr exp(
—iHT),
U
(T)=TrT,
&(t)=g +
f
—f 2
0
dt
g
o.
(
T),
t&(t)
d—
—2+(aylu
IMP)
ata&
(4)
y
g
(~PlUly5)atttas~.
,(t),
f
j„(t)=~(t) ij'j„(t),
Q„(0)=g„(T) exp[i5„(T)], It is already common that knowledge ' that one can define a new set of single-particle wave functions P„(t) such that
(t)(a13~v ~y5)
X o gs( t ) U
T
0
where T, is the time ordering operator and U ( T) is the single-particle evolution operator in the Hartree approximation corresponding to a time-dependent single-particle density o &(t), with a corresponding Hartree singleparticle Hamiltonian &(t). In Eq. (3) the symbol Xltr stands for the path integral over all periodic singleparticle densities cr &(t) It is well-known that such a representation of the evolution operator is not unique. ' One can rearrange the potential-energy term in different ways so as to obtain the Hartree-Fock, Fock, Hartree-Fock-Bogoliubov, or any other combination of the above. This nonuniqueness of the Hubbard-Stratonovich representation for the evolution operator does not seem to change our line of argument, and for the sake of simplicity, we shall assume the above form. Our next step will involve the evaluation of the single-particle evolution operator (4). As we argued in a previous paper the most natural way to do this is by considering the periodic solutions of the time-dependent Hartree problem for a given path cr„&(t). Let us introduce this set of periodic single-particle wave functions t &, i)
2)cr exp
&
f
a/3y6
where T is the period. Using the Hubbard-Stratonovich transformation, ' one can represent this quantity as
K(T)=
T
a/3
i
exp
P„(0)=P„(T), g„(t)=P„(t)exp[iy„(t)—i f dt'E„(t')],
(3)
where
40
2840
1989
The American Physical Society
SYMPLECTIC STRUCTURE OF THE COLLECTIVE MANIFOLD
E„(t)= ( p„(t)~&(t) g„(t))
that contains the time derivatives of these phase-space variables is the term due to the nonintegrable quantum phase, the term with the effective gauge potential A. ;(a ). The classical equations of motion corresponding to this Lagrangian are (they emerge in the stationary-phase approximation to the above path integral)
~
= (p„(t)~&(t) ~p„(t) is the instantaneous
single-particle
y„(t)=f'dt'(y„(t 0
energy and
a, y„,(t )) .
)~
P~(a )a) = Bo
BD
It is obvious that dt s„(t). 5„(T)=y„(T)f —
y„(
' 6"(o )= — —,
g
cr
&(ap~v~y6)o
aPy5
s+gs„(a), n
which as we shall show is conserved during the time evolution of the many-body system along the path a(t) this expression the first term corrects for the overcounting of the potential energy in the sum over the singleparticle energies. (When we suppress the subscripts we mean the whole set of the corresponding quantities and, where appropriate, the sum over those indices. The sum over n is only over occupied single-particle orbitals. ) We can also introduce the effective gauge field A as
In.
(12) where A is a vector in the cr; space and the partial derivative in this formula is a grad operation. Here we suppressed the subscripts for the coordinates o. , and from here on instead of two subscripts ap we shall use only one i or One can then definedt's(t) the effective collective Lagrangian as
j.
X = At(o
6(cr ), )a~ —
(15)
t
(10)
T) is known as the Berry's or quantum nonintegrable phase ' and the last term in Eq. (10) has been denoted by Berry as the dynamical phase. In this case, different components of the single-particle density a. &(t) play the role of external parameters in the same sense as Berry introduced them for simpler quantum mechanical systems. The quantity
2841
(13}
V;
(a)=
BA
(a)
Bo,
BA;(cr ) Ba(
(16)
is the intensity of the effective gauge field A;(cr) and which is a gauge invariant quantity. The gauge structure of such a theory is due to the fact that one can arbitrarily change the phase of the single-particle wave function by them with a phase factor, which depends multiplying solely on the collective variables o. ; at any point in the collective space. The only restriction is that this phase factor must lead to single-valued Slater determinants in the collective space. Consequently, any Slater determinant transported along any periodic trajectory a, (t) must not acquire a nontrivial phase after a complete cycle. In the collective space 0. there are singularities at the diabolical points. Hill and Wheeler used the name funnels for the same singularities. They correspond to level crossings in the single-particle spectrum and at such points one does not know how to define the partial derivatives appearing in Eq. (12). These singularities are responsible for the nonvanishing intensity of the efFective gauge field (16). It is straightforward to establish the conservation of the collective energy 6(a ) since
db(o)
/BE(a)
.
yp
(
).
j
K(T)=
f 2)o exp f i
0
(14)
All of the above transformation relations are more or less standard, except for the occurrence of the effective gauge field A(cr) in Eq. (13). In Eq. (14) one still needs to sum over all possible single-particle configurations. We shall assume that for at least a certain class of physical phenomena the one restriction to single-particle configuration is adequate, which is a standard approach. One can also consider several such configurations at once. Such an approach will lead in general to a nonabelian structure of the gauge field. We shall not consider this more general case here. The structure of the Lagrangian (13) is remarkable. The variables o. ; are the phase space variables on this collective manifold. The only term in the Lagrangian (13)
0
because of the antisymmetric character of the curl (16). As one can see from the above classical equation of motion for the collective variables o. there is motion only if such an effective gauge field exists in this phase space. In the absence of such a field, Eqs. (15) will give only the stationary points of the collective Hamiltonian in the phase space, i.e., minima, maxima, or saddle points of the collective energy 6(a ). Another remarkable thing about such an expression for a Lagrangian is the way a gauge field enters into the problem. We all know that a gauge field couples to the velocity, as in classical electrodynamics, QED, QCD, or any other gauge field theories. In this case the gauge field coupled to the generalized velocity o. ; in the phase space, i.e., loosely speaking, to both velocity and acceleration. Obviously such a coupling is possible and more general than the standard one. Since in the present case there is nothing like a preferred coordinate and momentum in the first place, one should not expect the occurrence of a minimal gauge coupling. At this level we do not yet know which variables (or combination of them} a,. play the role of "real" coordinates and such a choice, if any, is left to the envisaged system only. Will the coupling in „
the summation over the single-particle indices a, p= is implied in the first term. As a consequence the evolution operator (3) can be written as where
.
AUREL BULGAC
2S42
the end look like a minimal one? There is no way yet we can answer this question. Can one define a Poisson bracket on the collective manifold'? If the matrix V; (o ) has an inverse
yP, J((7 )QJk(CT ) =yQ, J(cr)PJ/, (CT ) =5;k, J
J
then the Poisson bracket between two arbitrary functions A(a ) and X(o ) on this manifold is
IA(o ), X(cr) IpE=QQ, , (o ) lJ
(19) E
1
and one can write the equation of motion (15) in the familiar way in Hamiltonian mechanics (20)
As a rule one will not be able always to do this. In such a case one will have to use the Dirac approach and, likely, introduce constraints. Obviously the collective space, which is a boson space, is much richer than the initial Hilbert space for fermions and such constraints will In Ref. 9 we describe a procedure emerge automatically. suited for "inverting" the matrix VJ(o ) in the case when constraints arise. In this case this matrix, strictly speaking, is not invertable, since it has zero eigenvalues. The emerging Hamilton equations of motion (20) then automatically conserve the constraints, irrespective of the symmetries of the Hamiltonian @(cr ). We shall describe a very simple example, in order to give the reader an idea about how such constraints emerge. Let us imagine that the fermionic indices a, P can take only two values. Equation (5) then has a very simple form ~11 12
iB,
.
V
22
Q V
where all the quantities are time dependent. Consequently, the collective variables are o. 1&, o.22, Reo. , 2, Imo. 1z, and one would be led to think of a collective Hamiltonian with two degrees of freedom (two momenta and two coordinates). It is easy to see that (o»+o22)/2 leads only to a trivial dynamical phase and therefore the gauge potential does not depend on this combination of "collective" variables. We are left then with only three phase space variables. However, out of three variables one can define only one pair of canonical conjugate variables and the third independent one will not undergo any time evolution. This simple example actually models the timeThe component o. 12 is the dependent pairing problem. analog of the pairing field and (o» — cr2z)/2 is similar to the single-particle potential for X particles in a single shell. It is known that this problem has a SU(2) algebraic structure. This fact leads to the conservation of the corresponding total quasispin. As a result, out of four potential collective variables only two have an actual dynamical meaning. Two of them will be conserved ir-
j
respective of the nature of the starting fermionic Hamiltonian. In this case we have a very simple example of 0. , 1+cr 22 = const constraints, ' and namely
(a» —o 22) +4o &zo z, =const,
40 which appear only because
of the gauge structure of the effective action mentioned above and not because of the structure of the effective Hamiltonian. This way of introducing constraints seems very attractive, being governed by the dynamics of the collective motion the single-particle wave-functions, (through which define the gauge structure of the collective phase space), and the Pauli principle is already built into the theory from the very beginning. We would like to mention that recently we showed explicitly that in the case of pairing the right quantization procedure for the pairing degrees of freedom can be achieved only when considering the above-mentioned gauge structure of the phase space. The main result of this paper, the fact that the dynamics on the collective space is fully determined by gauge potentials on the phase space, is quite surprising. In itself this result is an almost exact one, the only approximations-made being standard ones in the ATDHF theory of collective motion. Legitimate questions arise: What is the physical interpretation of this result? Why are there gauge potentials acting on the phase space? Why have they been overlooked so far? In a recent investigation of the classical limit of Lie algebras it was shown that the classical Lagrangian, corresponding to the physical systern, described in terms of generators of a Lie algebra, has exactly the same structure as Eq. (13). The occurrence of gauge potentials in that case was related to the fact that the classical phase space, corresponding to a Lie algebra, has a nontrivial topology, it is compact for a compact Lie algebra, and it is curved. In order to recover the correct quantum limit from such a classical description, the presence of the gauge fields was essential. The case we are dealing with here has exactly the same features. The collective variables o. & represent nothing else but classical equivalents of the quantum operators a ap. They form the basis of the shell model Lie algebra U(N), where N is the total number of single levels particle (a, P=1, . . . , N). Therefore, the effective collective Lagrangian (13) is nothing else but the corresponding classical Lagrangian of such an algebra. It is noteworthy that this type of classical limit for a many fermion system does not rely on the introduction of Grassmann variables (see, e.g. , Ref. 1). It is no wonder then that the classical dynamics is governed by a set of gauge fields and that the classical phase space has such nontrivial topological and geometrical properties. The presence of gauge fields is simply a consequence of the non-Euclidean geometry and topology of the phase space. It also refIects the fact that not all of the classical phase space variables o. & are independent. There are "topological" conserved quantities, which are nothing else but the Casimirs of the shell model algebra U(N). These Casimirs are conserved at both quantum and classical levels, and as a result there are no "quantum" fluctuations of these quantities and no corresponding integration over them in the path integral (3). These facts were overlooked in all previous treatments of collective motion. Unless one considers the global properties of the phase space (since locally every space looks fiat in the first approximation), its compactness is not evi-
40
SYMPLECTIC STRUCTURE OF THE COLLECTIVE MANIFOLD
2843
of derivation of the interacting boson approximation. ' IBA is a phenomenological theory of collective nuclear motion, expressed from the outset in terms of generators of certain "small" Lie algebras. In light of this paper and Ref. 9, one can think of it as of an already quantized version. of a classical theory of collective motion with a Lagrangian as the one derived here, Eq. (3), and restricted to a smaller number of degrees of freedom than the initial one. One can assume that restricting the number of classical variables o. & only to those corresponding to S and D bosons will lead to an almost decouThe corresponding reduced pled collective submanifold. Lagrangian will still have the gauge structure (3), which will assure a correct quantum limit, i.e. , in terms of a definite Lie algebra. Such a line of argument looks very promising and can lead us to a microscopic deviation of the IBA Hamiltonian.
dent and one can easily be misled to extend to local Aatness of the phase space to the whole space. If the considered collective trajectories have a relatively small extent, e.g. , as in the random-phase approximation, the fact that the phase space has a nontrivial topology is not relevant. The case of large amplitude collective motion, when a periodic trajectory can explore a large part of the phase space, is completely difFerent. The quantization rules then become nontrivial and not only the compactness of the phase space, but its connectedness start to play important roles. In such a context the appearance of nontrivial and "interesting" topological quantum numbers, which will govern the collective dynamics, is very likely to occur. Moreover, the correct quantum numbers cannot be retrieved unless these topological features are embedded from the very beginning into the efFective collective action. As a final remark, we would like to point to a possible
way
Quantum Many Particle Sy-stems (Addison-Wesley, Reading, Mass. , 1987), Chap. 7. 2A. Bulgac, Phys. Rev. A 37, 4048 (1988). M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984). 4Y. Aharonov and J. Anandan, Phys. Rev. Lett. 5S, 1593 (1987). 5D. L. Hill and J. A. Wheeler, Phys. Rev. S9, 1102 (1953). L. Faddeev and R. Jackiw, Phys. Rev. Lett. 60, 1692 (1988). 7A. Bulgac, Michigan State University Cyclotron Laboratory
Report No. MSUCL-665, 1988 (unpublished). ~P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, New York, 1980). A. Bulgac and D. Kusnezov, Michigan State University Cyclotron Laboratory Report No. MSUCL-687, 1989 (unpublished). F. Iachello and A. Arima, Phys. Rev. Lett. 35, 1069 (1975).
~J. W. Negele and H. Orland,
