BRST invariance and the conical pendulum
S G K A M A T H
Department of Mathematics, Indian Institute of Technology, Madras 600 036. India MS received 23 April 1991; revised 13 September 1991
Abstract. The Hamiitonian formulation of the BRST method for quantizing constrained systems developed recently by Nemeschansky et al is applied to the well-known problem of the conical pendulum in classical mechanics. The similarity of the system to a gauge theory wherein the two constraints serve as generators of local Abelian gauge transformations is also pointed out. The definition of the physical states of the system as a gauge theory and also as a BRST invariant theory is then discussed in some detail.
Keywords. BRST invariance; conical pendulum.
PACS No. 11.15
1. Introduction
In a recent study by Nemeschansky et al (1988) (hereafter referred to as I) a Hamiltonian formulation of the BRST m e t h o d (Becchi et al 1974, 1975, 1976) for quantizing constrained systems was developed in a m a n n e r so as to render the subject accessible, in particular, to the pedestrian. The quantization of a simple classical system, namely, that of a free particle of mass m constrained to move on a circle of radius a was taken up to: (a) establish the equivalence between a theory with constraints and a gauge theory, and (b) rewrite the gauge theory as a q u a n t u m system with BRST symmetry.
An immediate generalization of the aforesaid example is to that of a free particle constrained to m o v e on a sphere. It is easily seen, however that this replacement o f the constraint ((x 2 + y2)1/2 _ a) by ((x 2 + y2 + z 2 ) 1 / 2 _ a) adds little-to the content o f
§ 2 in I apart, of course, from appropriate changes in the definitions. Thus, in the case of m o t i o n on a sphere one would define
p o = r x p
1 1
P' = 2rr (r-p) + (p-r) 2~ (1)
with r and p as 3-component vectors, instead of 2 - c o m p o n e n t vectors as in eqs (2.6) and (2.7) of I. In this context, the extra constraint (0 = constant) implied by the problem of m o t i o n o f a free particle on a cone (instead of a sphere) turns out to be particularly interesting. Thus, at the level of a local gauge theory, there would be two kinds of gauge transformations, with the generators given by the constraint equations (r = constant) and ( 0 = c o n s t a n t ) . Besides, within the framework of the BRST formalism, one would have instead of a doublet of generators (Q, Q) for the case of the sphere - - a pair o f doublets which we denote here by (Qi, Q~, i = 1, 2).
11
12 S G K a m a t h
The m o t i o n of a p e n d u l u m - - b e it the simple, spherical or the c o n i c a l - - i s a concrete e x a m p l e wherein the three types of m o t i o n s discussed a b o v e are realized physically. Because the extra potential energy term relevant to the p e n d u l u m m o t i o n in each case c o m m u t e s with the respective constraint equations, the q u a n t i z a t i o n p r o c e d u r e a d o p t e d for the free particle m o t i o n will go t h r o u g h without alteration when extended to include the case of the p e n d u l u m m o t i o n . In other words, one could, for example, include the potential energy t e r m m y ( r - x ) for the simple p e n d u l u m in the L a g r a n g i a n (2.1) of I a n d simply repeat the a r g u m e n t s given in that section. Likewise a separate discussion for the spherical p e n d u l u m - - w i t h the potential energy given by m o ( r - z ) - - i s also unnecessary because o f the trivial generalization referred to earlier in respect of free particle m o t i o n on a circle a n d a sphere. We therefore confine o u r attention here to the p r o b l e m o f the conical p e n d u l u m only.
T h e rest. of this p a p e r is organized as follows. In § 2 we use the m o m e n t a given in (1) to derive a H a m i l t o n i a n for the conical p e n d u l u m which c o m m u t e s with the constraints r = r o a n d 0 = 0o, ro a n d 00 being constants. Defining a physical state as one which is annihilated by the constraint o p e r a t o r s (r - ro ) a n d (0 -- 0o) the invariance of the H a m i l t o n i a n u n d e r local Abelian gauge t r a n s f o r m a t i o n s is then w o r k e d out.
In § 3 the gauge theory discussed in § 2 is rewritten following I as a q u a n t u m system which possesses B R S T symmetry. Because the B R S T i n v a r i a n t L a g r a n g i a n that we shall writo breaks gauge invariance, the definition o f a physical state a d o p t e d in § 2 will n o w be altered to a new one using the B R S T g e n e r a t o r s Qi, Qi, (i = 1, 2). Finally,
§ 4 concludes the p a p e r with a short discussion.
2. The conical pendulum as a gauge theory
T h e L a g r a n g i a n for the conical p e n d u l u m is t a k e n as
2 L = m(~ 2 -I- 3) 2 -I- 22) - - 2mg(r - z) - 2(121 ¢1 +/22 ~2) (2) where the point of suspension is taken as the origin with the z-axis pointing vertically downwards, a n d /21 a n d /22 are the L a g r a n g e multipliers, with ~1 = ( r - t o ) a n d
~2 = ( 0 - 0o). With the usual definitions of the canonical m o m e n t a , one can n o w o b t a i n the H a m i l t o n i a n as
E m i l = (p2 + p2 + p2) + 2m2g(r _ z) + 2m(#1 ~1 +/22 ~2)- (3) If we quantize the system by using the c o m m u t a t o r s (h = 1)
[X, px] = i, [_y, py] = i, [z, pz] = i
one finds that as in tl~e case of the rotor, the constraints* ~1 = 0 = ~2 no longer c o m m u t e with the H a m i l t o n i a n .
* The equations ~ ~ = 0 and ~ 2 = 0 have the status of a first class constraint in the terminology introduced by Dirac (1964). Such constraints, which Dirac denoted by the equation c~(q,p) = 0, i = 1 .... m, when there are m constraints, physically represent generating functions of infinitesmal contact transformations that lead to changes in the q's and p's but do not affect the physical state. In the present context, the contact transformation is given by (7) below.
To construct a Hamiltonian which commutes with the constraints, we use the following c o m m u t a t o r s
-(i) 1
[ r , p , ] = i , [ r , / , 0 j = 0 , [ 0 , p , ] = 0 , [ O . f f o l ) ] = - - i y , [ O . f f o 2 ) ] = i ~ X
[q~,pt03)]=-i, [0, p~03)]=0, [O, P o ] = i , [r, P o ] = O (4) where p~o i) is the ith component of the vector P0 given in (1), Po = (1/2)(xPto 2) -YP~o I)) and 2 = (x 2 + y2)1/2. Using (1), eq. (3) can be rewritten, after normal ordering, as
2 m n = p2 + ~ p2 + 2m2g(r _ z) + 2 m ( , 141 1 + , 2 42). (5) With the help of the commutators given in (4), it is clear that there are three terms in (5) which do not commute with the constraints. They are the first term, and the sum (1/r 2) (pro1)2 + pro2)2 ) contained in the second. Following the prescription of Dirac (1966), we drop these terms from the Hamiltonian (5), and arrive at
1 pt03)2 + mg(r -- z) + i~ 141 + "2~2. (6)
H - 2mr2
The constraints 41 and 42 n o w commute with the H in (6) and therefore the three operators can be simultaneously diagonalized. The physical states of the conical pendulum will then be states for which 411~b)=0 and ~21~')= 0. In the equations below we shall, for notational convenience, write P0 for p[3) which appears in (6).
Following I we shall now show, that the Lagrangian constructed from (6) is invariant under local gauge transformations whose generators are 41 and 42. Since ¢1 and ~2 commute, the group associated with the transformations will be a two-parameter Abelian group.
It is easy to see using (4) that under a gauge transformation represented by the unitary operator
U = exp { -- i ( f l (t)~l + f 2 ( t ) 4 z ) } , (7) with fi(t), i = 1, 2 being arbitrary c-number functions of time t, that,
U p, U -1 = p, +/fl(t).
U Po U - 1= p . +/f2(t)" (8)
A first order Lagrangian constructed from (6) will be given by
L = p,~ + P c 0 + P~q~ - ~ 2mr 1 p2 _ mg(r -- r cos 0) -- ("141 + "2 42)" (9) Clearly, under the increments defined by
6 p . = f x ( t ) . 6,1 = --f~(t)
6Po=
f2(t), 6 . 2 = - - / 2 ( t )6Po = 0 = 6r = &cp = 6pu ' = 6p~ 2 = 60 = 6po (10)
14 S G K a m a t h
the change in the Lagrangian in (9) will be d
6 L = ~ [ ( f ~ , + f,~2). (11)
Since 6 L is a total derivative the action is invariant. Similarly, under (10), one finds easily that for the Hamiltonian given by (6),
6 H = - (.fl(t)¢t + j'2(t)¢2).
(12)
Unlike (11), eq. (12) is not a total time derivative; however on the sector of physical states, as defined earlier, 6 H = 0. Before concluding this section, we note that the m o m e n t a canonically conjugate to the Lagrange multipliers/~ are gauge invariant, as seen from (10). In the restriction of the Hamiltonian (6) to the space of physical (or gauge invariant) states the pu, have had no role to play, as the time derivative of
#i does not appear in the Lagrangian. There is therefore an arbitrariness associated with the eigenvalues of pu,, as far as the physical states of the pendulum are concerned.
Following I therefore, we shall define physical states [ ~ ) as those for which p~, [ ~ ) = 0 = P~2 [~ ) in addition to ~ 11 ~ ) = 0 = ~2 ]~b).
In the next section we shall rewrite the gauge theory developed here as a q u a n t u m system which possesses a BRST symmetry. It m a y be apt to point out here that, as in the case of rotor, there is no compelling reason for this generalization. Still, following I we shall write a gauge non-invariant (under the gauge transformations whose increments are given as in (10)), but BRST invariant Lagrangian and obtain subsequently the BRST charge*, Qi, (~i(i = 1, 2) alluded to in the introduction.
3. BRST transformations
We begin with a brief but necessarily incomplete (for reasons of space) introduction to the subject of BRST transformations. These transformations were first proposed (Becchi et al 1974, 1975, 1976) in connection with the renormalization of gauge theories.
It is enough for our purposes here to restrict our attention below to renormalization in Q E D , though for the record, the utility of BRST is strongly evident for non-Abelian gauge theories (Becchi et al 1976).
The renormalization of any q u a n t u m field theory, as is well-known, involves the elimination of divergent diagrams associated with the initial Lagrangian density .~o for the field theory; it is done by redefining the initial Lagrangian i.e. by adding to (.~e) a finite number of counterterms. Naturally, these counterterms should preserve the symmetries, for example, local gauge invariance, present in ~ . In proving that renormalization does not violate local gauge invariance it is convenient to use the W a r d identities--whic/1 are essentially relations between Green's functions t. While
*Physically, the BRST charges Q~, (~ will have the same status as the ~1, ~2 dealt with in this section;
namely they are the generators of infinitesimal contact transformations on the coordinates and their canonical momenta (see (13) and (20) below) but without any changes on the physically realizable state.
t i n Q E D the Ward identities ensure that the gauge invariance of the Lagrangian density (.~) is preserved under renormalization by virtue of the equality of the renormalization constants Z t and Z 2 to all orders.
the m e t h o d of derivation of these W a r d identities is largely a matter of choice, the BRST technique for Q E D (as well as for non-Abelian gauge theories) consists ( R a m o n d 1983) in enlarging the number of terms in the Lagrangian density; because the extra term in Q E D is associated with G r a s s m a n n fields which do not interact with the fields originally present in . ~ , this procedure merely redefines the generating functional for the Green's functions, but with a new normalization constant. One can now show (Ramond 1983) that this new effective Lagrangian for Q E D is invariant under BRST t r a n s f o r m a t i o n s - - a characteristic feature of which is that it is a gauge transformation but with a mixing between operators having different statistics. This is exactly what we shall meet with while constructing a BRST invariant version ofeq. (9) further below.
F r o m the above remarks it is clear that BRST was basically an aid (albeit involving fictional G r a s s m a n n variables) to the renormalization of a gauge theory. The paper of Nemeschansky et al (1988) is important because the equivalence between local gauge theories and physical systems with first class constraints that was pointed o u t by Dirac (1964), has been extended there to establish the connection between constrained systems, local gauge invariance and BRST symmetry. In order words, I show that BRST symmetry is not a mere artefact of the renormalization procedure, but a s y m m e t r y that is intrinsic to constrained systems.
We shall now show below that the Lagrangian for the conical p e n d u l u m given in (9), can be reformulated so as to admit of BRST invariance. Under a BRST transforma- tion as mentioned earlier gauge transformation which shifts operators by c-number functions (e.g. (10)) is replaced by a transformation which mixes operators with different statistics. Additionally under BRST, 5 2 = 0. Consider, for example, the following infinitesimal increments involving new anti-commuting variables ci,6~, (i = 1, 2) a n d a new commuting variable bi(i = 1, 2) besides the operators contained in (10):
J P l = - - e l , J ~ 2 == - - c 2 , JP, = C l , JPo = C2
JP~I = ~P~ = JO = Jr = JP~2 = J~P = 0 = JPo
JC 1 = JC 2 = O, 6~ 1 : bl, 6C 2 = b2, fib 1 = 0 --- Jb 2. (13) Under (13), the change in the following gauge non-invariant L'agrangian obtained by the addition of a gauge-fixing term to (9)
1 2
Z = q i ~ i -I- p4jdp - ~ P o - mgr(1 - - c o s 0)
--~u,~, + / ~ 2 ¢ 2 ) - - 6 ( i ~ , 6 , ( I J , - q i + l b i ) ) with ql = P,, q2 = Po can be easily found*.
We get
(14)
d i L =
~- (ci ~i
d "~- C 2 ~ 2 )(15)
* I n (14) and below, there will be a s u m on the repeated index i over i = 1,2 unless otherwise indicated.
16 S G K a m a t h
thus implying that the action is invariant. By choosing the gauge fixing term as in (14) we are, as in the case of rotor, identifying/~1 and/~2 with the time c o m p o n e n t s of the electromagnetic potential in Q E D and p, and Po with V-A. The definition of J in (13) can be used to rewrite (14) as
L = q i ~ + P ~ - - - 1 p2 _ m g r ( 1 - cos 0) 2 m r 2 o
- - I ~ i ~ - b,(l~, - q , + ½ b , ) + 6 , d , - 6 , c , . ( 1 6 )
Using the Eu|er equation for b~ = q l - tii, we see that
- i = 1 = i= 1 2 (/ii - q i ) 2 . ( 1 7 )
With the identification referred a b o v e we see that the extra B R S T invariant term plays the role of adding a gauge fixing term of the form (doA0 -- V-A) 2 to the Q E D Lagrangian.
F r o m (16.), the m o m e n t a canonically conjugate to ci, ~ a n d / ~ can be easily found to be roe. = ~, roe, = 6 , and p,. = - bi respectively.
We therefore expect the following non-zero c o m m u t a t o r s : [ # i , Pu,] = iSij, {ci, c j } + = iSiy, {6i, 6j} + = it~ij besides
{ c , , c j ) = 0 = {e,, ej} = {c,, ~j}.
N o t e however, that the last a n t i c o m m u t a t o r above yields, {~,, ej} = - {~,, ~j}
(18)
which does not agree with the signs in (18). The correct sign, as in I, is fixed using the fermionic part of the Hamiltonian (see eq. (24) below) to get
[ U , c,] = - i ~ , = - {~j, c,}~j so that
{ci, c,} = i60 = - {ci, t?j} (18a)
instead of the (incorrect) a n t i c o m m u t a t o r s in (18).
T h e B R S T o p e r a t o r Qi which effects the increments in the o p e r a t o r s given in (13) can now be worked out using (18a) as (no sum over i below)
Qi = - i ci~ i - i dip~. (19)
T h u s there are two generators Q1, Q~ whose action o n the physical states should be given by Q1 I~b ) = 0 = Q2I~k ), so as to reflect the conditions ~i]~b) = 0 = p~,ld/), i = 1, 2 m e n t i o n e d earlier. While deferring this issue to the latter part of this section, we can readily check that instead of (13), if the set of increments were written as,
61~i = ci, 6qi = - ci, 6ci = bi, 6~i = 0 (20)
with the increments in all the other variables set to zero then, the following Lagrangian L = q,~i + P, d P - ~ 1 P~ -- myr(1 - cos0) -- #i~i
is changed under (20) by
(21)
6 L = - - ~ ( e l ~ d 1 -Jl- C 2 ~ 2 )
¢1[
so that the action is again invariant. T h e reader will notice that the gauge fixing term in (21) differs from that in (14). However, using (20) in (21), o n e can recover the Lagrangian in (16) easily.
E q u a t i o n (20) is associated with the anti-BRST operators (no sum over i below)
Q-.i "~" i c'i¢i-t- i ciP~. (22)
The Qi in (22) are the adjoints of the Qi in (19), and eq. (18a) yields the following anti-commutators:
{Qi, Qi} = 0 = {Q,, Q~} = (Qi, Qi}" (23)
Thus, unlike the case of the rotor, one now has a pair of doublets (Qi, Q~, i = 1, 2) associated with the B R S T transformations given in (13) and (20). T h e H a m i l t o n i a n resulting from the Lagrangian given in (16) can be w o r k e d out as (with i summed as before):
H = qi~i + P~,~ + ncci + cin,. + p,,,IJi- L 1
2mr 2 p2 + mgr(1 -- cos 0) +/~i~i + P~,,qi 2e~, + cici + cicl. (24) Using the BRST generators in (19) and (22) one can verify with the help of the c o m m u t a t o r s in (18a) that the Hamiltonian is both BRST and a n t i - B R S T invariant, i.e.,
[Q,, H ] = 0 = [Q/, n ] . (25)
We now discuss in some detail, for the remainder of this section, the conditions that the state vectors should obey so that they can be labelled as physical states.
Much of the discussion here will parallel that in I and will hence be brief.
F r o m (16) we notice that the Euler equation of m o t i o n for the c i will be given by
~i + c~ = 0 = ~i + c i. (26)
E q u a t i o n (26) has the solution given by ci(t) = f i e " + #ie -it and, gi(t) = f / + e -i' + #~- e ~'.
Imposing the conditions c 2 = 62 = 0 we obtain since ci(0 ) = f~ + #i, 6~(0) = f i + + a~-,
18 S G K a m a t h the relations
f ~ + 9 7 + { f i , 9 , } + = 0 ,
+ 2
f + 2 + 9 , + { f + , o + } + = 0 ,
so that f 2 . 0 . f + 2 . . 92 9i . Also +2 { f i , g l } = 0 = d l ,9i ~. ¢~+ +~ Similarly, from the a n t i c o m m u t a t o r s {ci, (j} = 0 = {ci, c~} and {ci, c~} = ifij, we obtain, as in I,
{.fi,~7 + } = 0 = { f + , o j }
{ f ~ , f + } = - ½6i~ = - {9,, g+ }- (27)
So, by defining the vacuum as the state for which f~t0} = 9110> = 0, we find that the states f + 10> and 9 + 10> have a norm equal to 1/2 and - 1/2 respectively if the n o r m of 10> is chosen to be ~ 1 as in I. In terms of the operators fi, f + , g ~ and g+
one can express the Q~, (~ as Qi = - i(fiA+i + •im-i)
Q, = i(f + A_, + g~+ A+,) (28)
with
A_+r= ~i ++- iPu."
F r o m (28), it is clear that the condition ~ l ~ , > = 0 = p ~ , r ~ , > used to describe the physical states I~'> in the gauge theory in §2 can, for instance, be rewritten as Qi]~'> = 0. But, because t h e f i and 9i commute with A±i, one also has Qi[~b> = 0 for all states J q~ > for which
fil~b > = 0 = g,l~b). (29)
In other words, as observed by Nemeschansky et al, although the set of states annihilated by the operator Qi describes the set of states I~'> for which ~!¢'> = 0 = p ~ l g ' > , it also contains additional states for which (29) holds. It is here that the anti-BRST symmetry of the Hamiltonian expressed by the second equality in (25) comes in handy.
Because, by requiring as in I that (~i also annihilate the physical states (in addition to Qi), one easily excludes those extra states for which (29) is satisfied.
Thus, by insisting that for states given by (29) Qilq~) = 0, we obtain, f + IqS> = 0 =g+l~b>.
The first equality in (30) yields, using (27) and (29), fif~+ Iq~) = 0 = ( - ½ - f/+ f / ) [ ¢ ) = - ½ I S > . Similarly,
a,a?l~> =0=½1~>.
(30)
(31)
Both (31) and (31a) show that there cannot be any free eigenstates of the fermionic part of the Hamiltonian for which (29) and (30) hold simultaneously. However with Qi and (~i defined by (28), for all states for which the bosonic operators ~ and p~,.
satisfy ~ii¢> =0=p~,,[~,>, the conditions
Qi[I]I>=O=O__~i[I]l>,
for i = 1,2, continue to hold undisturbed.(31a)
Thus, as in the case of a particle moving on a circle, the extra anti-BRST symmetry is necessary in order to impose enough constraints so as to recover only the physical states for the conical pendulum.
4. Conclusions
In this paper, we have second quantized the well-known classical problem (see, for example, Symon 1978) of the conical pendulum with the two constraints r - r o = 0 and 0 - 0o = 0. The Lagrangian for this system was first shown to be invariant up to a total time derivative under local gauge transformations, with the constraints serving as generators of the transformations. The gauge invariance of the Hamiltonian, however, required the identification of the physical states of the conical pendulum as those which are annihilated by the operators ~1 = ( r - ro) and ~2 = ( 0 - 0o). Then, following I, a BRST invariant (but gauge non-invariant) Lagrangian for the system was written down. A pair of generators (Q i, Q2) associated with the BRST transform- ations were then obtained. It was then pointed out that within the BRST formalism, the identification of the physical states for the conical pendulum is made with the aid of the generators (QI,Q2) of an extra anti-BRST symmetry possessed by the BRST invariant Lagrangian.
In conclusion, we ought to mention that:
(a) The first quantization of the simple pendulum was studied long ago by Condon (1928) and more recently by Pradhan and Khare (1973) and Aldrovandi and Ferreira (1980). The last mentioned authors, in particular, have obtained the energy spectrum of the Schribdinger Hamiltonian as well as its eigenfunctions in terms of a certain class of Mathieu functions.
(b) Abdel-Rahman (1983) has studied the classical, semi-classical and quantum aspects of the conical pendulum having a negative total energy in its rotating frame. A peculiarity of the system is that it is a classic illustration of the phenomenon of spontaneous symmetry breaking.
(c) Finally, as a sequel to this work, we propose to examine in a subsequent paper, the Ward identities for the Green's functions associated with the BRST invariant Lagrangian given in (16).
Acknowledgements
The author thanks R Anishetty and H S Sharatchandra Mathematical Sciences, Madras for useful discussions.
of the Institute of
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