NUCLEAR PHYSICS A
Nuclear Physics A570 (1994) 31c-3%~ North-Holland, Amsterdam
The application of the BRST supersymmetry involving collective coordinates
to nuclear problems
D.R .Bes’* and R.De Luca” “Departamento de Fisica, C.N.E.A., (1429) Buenos Aires, Argentina
Av. de1 Libertador
8250
In the first part of this contributions we review the BRST formalism as has been applied to the treatment of collective coordinates. Subsequently we discuss some applications in order to investigate angular momentum effects in deformed nuclei. 1. Introduction It is often convenient to describe physical systems from moving frames of reference, in such a way that the parameters determining the motion of the frame are considered new genuine coordinates (the collective coordinates), on the same footing that the original ones of the problem (the intrinsic coordinates). For instance, in the case of a rigid top, the Euler angles determining the motion of the frame of reference associated with principal axis are the collective coordinates of the problem. We must generalize this solution to the case in which the intrinsic motion is not rigid. The most straightforward treatment that singles out these specially relevant degrees of freedom is to isolate them from the remaining variables. Unfortunately, this path has proven to be a blind alley but for systems with very few particles. A second approach is to use an overcomplete set of collective plus all the original variables. For instance, in unified models in nuclear physics. In such treatments there remain problems associated with the overcompletness of variables, infrared singularities, etc. A solution to these problems using the BRST supersymmetry [1,2] was presented two years ago [3]. 2. Some classical considerations
For simplicity we discuss the case of an abelian transformation reference, which is given classically by T(4) = exp [ (614
I I
to the moving frame of
(1)
where J is the generator of the transformation; #, the parameter to be raised to the level of collective coordinate and { , }, a Poisson bracket. Since the transformation is an artifact introduced by us, the laboratory version of physical functions should be independent of 9. This imposes a condition on physical functions Q,,., in the moving frame, namely iophys,
(J-1)) = 0
‘Fellow of the CONICET, Buenos Aires, Argentina. 03759474/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved. SSDI 03759474(94)OCtl76-Y
(2)
where I is the variable conjugate to (f, ({d,l} = 1). Eq. (2) reflects the fundamental invariance of the problem: while I changes the rotation of the frame of reference, J changes the coordinates as to leave the description invariant. This invariance is independent of the symmetries that the original hamiltonian may display. The corresponding transformation is called the gauge transformation. The generator J-I maps a given trajectory in phase space to another trajectory representing the same physical trajectory, but described from a different moving system. This transformation leaves invariant the constraint
(3)
J-I=0
To choose a gauge means to select one among the equivalent physical trajectories. This is accomplished by imposing a supplementary relation between the coordinates and velocities, G=O
(4)
G is called the gauge-fixing function and it is arbitrary to a large extent. It is precisely this freedom in the choice of G which makes a description using collective coordinates more flexible than the original one. For instance, in the case of two-dimensional rotations, a possible gauge-fixing function is the non-diagonal component of the quadrupole operator, in which case (4) b ecomes equivalent to the condition of principal axis. As Dirac taught us, we may impose the constraint by introducing a Lagrange multiplier 62, which is also treated as a dynamical variable. H2:H-fi((J-I)
(5)
If B denotes the conjugate momentum ([Q, B] = l), we impose the new (primary) straint
(61
B=O 3. The BRST
con-
formalism
The generalization of the previous ideas to the case of a general Lie group of transformations and their corresponding quantum mechanical transcriptions can be performed without major difficulties. What is not so easy is to carry them over in actual calculations. A possible treatment is based on the formalism of Dirac brackets [4]. Although relatively few applications exist in nuclear physics, most of the discussions concerning, for instance, the quantification of solitons are based on such formalism. However, as emphasized by the use of the terminology associated with gauge theories, it is also possible to apply the techniques developed for solving these last problems. A very powerful one is based on the BRST supersymmetry [1,2]. In this first part of the talk, we outline the BRST formalism as applied to many-body systems. For more details see ref. [3]. 3.1. The BRST invariance A gauge theory has an underlying invariance under transformations generated by the charge Q. This is a hermitian and nilpotent operator which is linear in the constraints
D.R. Bes, R. De Luca I Application of the BRST supersymmetry and contains an odd number of fermion $u& variables f,ii, being K, 77. In the case of our abelian transformation,
their conjugate
Q=(J-1)~+Bii
33c momenta (7)
According with the previous requirements (3) and (6), physical states are invariant under BRST transformations. However, the subspace carrying zero BRST charge includes Similarly, physical operators Or,+ another subset of states ]x > having zero norm. commute with the charge Q. So they do the operators 0, mapping physical states into zero-norm states Q ]phys>
=
0;
Ix>-
Q]un > ;
=
0
; O,lpbs>= lx > [“dw~>Ql = 0; 0,= [OumQl~
(8)
By working within the subspace carrying zero charge, the constraints formally disappear. The family of (equivalent) states connected by a gauge transformation corresponds in the BRST formalism to the family of (equivalent) states lphys >+
lphys > + 1x >
In particular,
;
Ophys
+
Ophy.
hQl+ = H-n(J-I)+ ivr?i + w2(GB
HBRST
=
+
0,
we replace the original hamiltonian
(9)
H by the BRST
hamiltonian
PO)
H+
- &EJ2
- ?j [G, J] 7)
where p is a hermitian odd-Grassmann operator which is conveniently chosen. The constant w is arbitrary and S is latter identified with the inertial parameter associated with the collective motion. The function G should not commute with the generator J. Since its expectation value vanishes, it is interpreted as the gauge fixing function, In Hzns~ the collective degree of freedom is explicitly separated. Moreover, Hznsr displays an unbroken (collective) symmetry, since it commutes with the collective generator I. On the contrary, Hznsr no longer commutes with J (as H does), due to the presence of the gauge-fixing function G. Therefore, in cases in which there is a breakdown of symmetries, we can perform perturbation calculations, since infrared divergencies are eliminated. The construction of the spurious sector In the laboratory frame the spurious sector is made up from the quartets of the variables 4, Cl, 7, ii which mutually cancel their effects. In the moving frame the collective variable 4 is considered to be a real one and, as a tradeoff, some original degree of freedom must enter in the spurious sector. At the level of elementary modes of excitation, this is the RPA zero-frequency mode &J&.* where the subindex RPA denotes the component of the generator J which is included in the linearization approximation. The inertia parameter S and the RPA angles BR~A are determined through well known equations [5]. It is both possible and convenient to identify the gauge-fixing function G with the RPA angle. Thus the elementary degrees of freedom include (physical) finite frequency modes plus the quadratic boson and ghost terms of the spurious sector
3.2.
Hi;L
=
1 2s
=
w (Iy,
J2
IwA
- R JR~A + ivii + w2(-&B2
- r,+l?o + 7ia + Ib)
+ BG + i~q)
(11)
D.R. Bes, R. De Luca I’Application of the BRST supersymmetry
34c
where the transformation
to normal modes is given by
and satisfies
[I’,,I’:]
= -[ITo, I?,+] =
[~?,a]+ = [i;,b],
=
1
(12)
All the excitations of the spurious sector are positively defined (note the metric of the 0-boson). The supersymmetry of the spurious sector is made obvious in eq. (11). The vacuum state of the spurious sector,
r-,/o>= r,lo> = U]O> = blo> = 0
(13)
is also annihilated by the BRST charge &, at least to the quadratic order Moreover, it can be proved that it is the only normaliaable state in the spurious sector carrying zero BRST charge. 3.3. The transformation to a moving system So far we have obtained a static solution in the intrinsic frame. The coupling between the intrinsic and the collective motion is made through the spurious sector via the term QI. This coupling may be reduced via a transformation setting the intrinsic system in motion, namely 04)
(15) which explicitly displays the collective kinetic energy. The transformation of the remaining hamiltonian yields new intrinsic-collective coupling terms which however are smaller than RI provided that some prescription such as the one in ref. [5] is used to determine the inertial parameter. We emphasize the fact that both the existence of the BRST hamiltonian of the form (lo), the transformation to normal modes as in (ll), and the transformation (14) to a moving system are very general features of the formalism, and thus straightforward applicable to wide variety of situations involving collective coordinates, both in manybody problems or in field-theory.
4. Some
I-dependent
of the BRST treatment
consequences
of deformed
nuclei
As an example we treat axially-symmetric nuclei at low values of I [S]“. For simplicity, we disregard the collective motion along the gauge space, associated with the non-conservation of the number of particles. In such case the BRST hamiltonian reads
H + 0, (&-I,)
=
HBRST
- i ~,ji~
- w2 G.P, - &P.P, ( + s % (“8770- m%f
+ v&[Gs,
Al) (16)
where the spherical representation has been used (s = fl, 6 = -s and v = s,O). The last term in (16) arises from the non-abelian nature of the rotational transformation. It is natural to factorize the space into collective and intrinsic subspaces
P rbW>
=
21+1
w
(17)
7) @ lxintr>
&&P,
Note that while in the unified model ?is interpreted as the sum of the collective angular momentum R and the particle angular momentum .I (without however existing a clear prescription to correct for the overcounting), sent the same total angular momentum,
in the BRST formalism both iand
the particle operator J’acting
frepre-
on Ix~~>,
while
In fact physical states should satisfy the the collective I’ operates on the D-functions. constraints (3). The original hamiltonian H is treated within the Nilsson-BCS approximations and the residual interaction is expanded as a sum of separable terms. The label X represents the angular momentum carried by the one-body operator Oc,*i,x while e is a shorthand for additional necessary quantum numbers. Even and odd terms with respect to the time reversal operation are explicitly distinguished (il) and are denoted hereforth by “even” and “odd” terms. H
@“)
=
H(“)
w
E, (c&Y, + &Y~)
H rtll = OLrtl,Ar
=
+ H,,,
(18) If@9 z 0
;
-==Q*l&o;,*lxr 2 8’
8%
@&a,
f w%)
4 p) (&$
+ olt:),,x(a, b,CL) (&b
* +a)
Since the components of the angular momentum J’ commute with the hamiltonian H at the RPA level of approximation, there should be two zero-frequency roots, Using the eqs. of ref. [5], [@O) $2o)j(*e)
) t
;5!2O) _ $4 4
-_
@“)](20) =
-i &t I ;5!20) _ i [H(“), 49(20) 4
_ nex Orf)l,As = 0
‘The second of refs. [S] includes a detailed application to the exactly soluble W(3)
(19) model.
D.R. Bes, R. De Luca I Application
36c
of the BRST supersymmetry
where K(A
3
i
x4;-1pA [OQ,,_l,xI, &I(@‘)
(20)
one obtains both the moment of inertia 9 and the RPA angles
Without the residual interaction (JC~X= 0) the Inglis results are obtained. Therefore, the deviations from them arise only through the odd terms in the interaction3. The gauge fixing function is taken as G, E B!“l = i 0f2*l(o, b, s) (+;
+ a&a,)
(22)
After the transformation corresponding to (14) is made, one obtains the I-dependent vertices given in table 1. Eqs. (19) g uarantee the vanishing of the backward vertices which are linear in I,.
Table 1 The vertices with a linear and a quadratic dependence in the collective angular momentum G,], Ga]@‘l and K,(Ais given in (20). I,. Here rl~x= -XtC’
3Exchange
effects are considered
to be smaller.
D.R. Bes, R. De Luca I Ap~~i~ati~n of the BRST s~er~~et~
37c
The Coriolis interaction in the particle-rotor model becomes justified4 by the first vertex in table 1, if the ‘ECXvanish. Thus, for instance, a Galilean invariant pairing interaction affects the Coriolis coupling via the last term in the first vertex, in the same way that modifies the cranking moment of inertia [7,8]. unfortunately, these effects are associated with the relatively poorly known velocity-dependent interactions. Even if the Coriolis interaction remains as the predominant linear coupling term, it must compete with coupling terms proportional to 1(1+1) which are not present in the particle-rotor model (third and fourth vertices in table 1). The corresponding (first order) contributions to the moment of inertia of odd-systems may be of similar value as the (second order) contributions of the Coriohs interaction. Several sources may contribute to the effective spin-orbit term in the central potential. Let us assume that the main one is a two-body spin-orbit term satisfying all conservation requirements (including Galilean invariance).
(23) This interaction yields a central term,
where A = C, is ihe number of particles, plus six separable interactions of the form (18). Since all of them carry angular momentum 1, the index X may be dropped and [ may be replaced by the label (7, v), with T = p, n and 1 5 I/ 5 6 (see table 2) Table 2 The residual terms (18) arising from the two-body
OT*Y-l,P 1
-1
(~XQJ
+ P#&
3 5
+1 +1
Jr i (zxp3r - ire,
I
spin-orbit interaction (23).
0 T%%+l*I(
v
2
-1
i (ZXTJP - ip,
4 6
+1 +1
i (4 - 4 (W% + rfi
The only term contributing to the expectation value appearing in (20) is v = Eqs. (20), (19), (21) and the first vertex in table 1 yield “0
=
&3
641
3.
x.0
(25) 4However there are some differences. For instance, the formalism does not justify the so-called recoil term. If anything, this term should be taken with the negative sign as a rough way to eliminate the spurious degree of freedom from the intrinsic hamiltonian
D.R. Bes, R. De Luca I Application of the BRST supersymmetry
38c
In U235, the rotational parameter $ and the spin-orbit strength AxsO have the value 10 keV and 820 keV, respectively. Therefore the interaction strength of the spin-orbit force (x.0 = 3 keV) should produce a 30% reduction in the strength of the Coriolis coupling.
REFERENCES 1.
2. 3. 4. 5. 6. 7. 8.
C.Becchi, A.Rouet and BStora, Phys. Lett. B 52, (1974) 344; Ann. Phys. (N.Y.) 98, (1976) 287; B.L.Voronov and I.V.Tyutin, Theor. Math. Phys. U.S.S.R. 50, (1982) 218. M. Henneaux and C, Teitelboim, Quanttiation of Gauge Systems, Princeton University Press, Princeton, New Jersey (1992). D.R.Bes and J.Kurchan, The Treatment of Collective Coordinates In Many-Body Systems, Lecture Notes in Physics Vol. 34 (World Scientific, Singapore, 1990). P.A.M.Dirac, Lecture in Quantum Mechanics. Yeshiva University, New York (1964). E.R.Marshalek and J.Weneser, Phys. Rev. C2, (1970) 1682. D.R.Bes, R.De Luca, J.Kurchan, Phys. Lett. B248, (1990) 1; D.R.Bes and R.De Luca, Nucl. Phys. A535 (1991) 221 A.B.Migdal, Nucl. Phys. 13, (1959) 655. I.Hamamoto, Nucl. Phys. A232, (1974) 445.