Nuclear Physics B55 (1973) 6 3 7  6 4 7 . NorthHolland Publishing C o m p a n y
LAGRANG1AN QUANTUM THEORY F.J. B L O O R E ,
L. R O U T H a n d J. U N D E R H I L L Department o f Applied Mathematics, The University, Liverpool, L69 3BX, England Received 17 November 1972 (Revised 13 January 1973)
Abstract: We present a consistent Lagrangian q u a n t u m theory for one degree of freedom based on the Fr6chet derivative. We consider a q u a n t u m mechanical system with one independent coordinate q and Lagrangian L(q,q) = a(q)q 2 + b(q)Lt + c(q). In a consistent q u a n t u m theory, the canonical equaltime c o m m u t a t i o n relation (c.c.r.), [q,q] = i/2a(q) must be preserved in time; that is to say [q,d/] = (//2a)': Now the acceleration q is given in terms o f q and q by the EulerLagrange equation; the precise form of this depends on which variations 6q are assumed to leave the action stationary. We show that the constraint that q preserve the c.c.r. restricts the class o f allowable variations o f q for which the action integral is stationary to functions of the form 8q(q,t) = g(t)(a(q))~ where g(t) is a cnumber function which vanishes at the endpoint of the time interval. This specification retains its form under a change of variable q ~ q(r). T h u s unless the coefficient a(q) is a constant, cnumber variations are not permissible. This means that the Euler Lagrange equation of m o t i o n is not the naive form with cnumber variations which is usually assumed. Noether's theorem, which relates the invariance o f a Lagrangian under an infinitesimal coordinate transformation to an associated conserved quantity, only holds if the infinitesimal transformation is an allowable variation, that is to say one for which the action is stationary. It is proved that only variations of the form a~ can leave the Lagrangian invariant, so that the possibility of an invariance of L unaccompanied by a conserved quantity is removed. A Hamiltonian H(q,Lt) is given which generates the time transformations given by the EulerLagrange equation, Finally, the allowability condition is shown to be the condition for which rearrangement of terms in the Lagrangian by means of the c.c.r, does n o t alter the ultimate E u l e r  L a g r a n g e equation of motion.
1. Introduction The quantum theory of a system with one degree of freedom can be described abstractly as (a) an algebra of observables at time t. The algebra is generated, in some sense which we shall not make precise, by two elements q, q ("all observables are functions of q, q"), which satisfy a canonical commutation relation, plus (b) a derivation*, written "dot", which sends q into q. This requirement together * For footnote, see next page.
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F J. Bloore et al., Lagrangian quantum theory
with the equation which expresses q in terms of q and q (i.e. the equation of motion), determine the derivation. The equation of motion is assumed to determine the timedevelopment of observables. If the derivation is specified by a Hamiltonian H ( q , q ) , according to F(t) = i [ H , F ( t ) ] ,
(1)
then the timedevelopment will be given by the usual expression F ( t ) = e iHt F(O) e iHt .
in an alternative approach, the equations of motion are derived from a quarttum mechanical Hamilton's principle, i.e. from a requirement of the form tl Wlo = j L ( q ( t ) , ( l ( t ) ) d t to is stationary with respect to small variations of q, Cl which vanish at the endpoints. However, because L is a function of noncommuting operators it is impossible to ensure that the integral is stationary for arbitrary variations q(t) ~ q(t) + 8q(t) .
The problem then arises: Find a class o f a l l o w a b l e variations (i.e. for which the action integral is stationary) which is sufficiently numerous to allow one to deduce an equation of motion. It turns out that this can be done in many ways, each of which leads to a different equation of motion. In this paper we shall restrict ourselves to Lagrangians which are quadratic in el: L ( q , q ) = a(q)(t 2 + b(q)(l + c ( q )
(2)
and will consider only variations of the form q ( t ) ~ q ( t ) + g(t)c~(q)
where g is an arbitrary cnumber function which vanishes at t o and t 1. Thus q and q + 6q commute at each point of the orbit. With this restriction (and the assumption that a ' ( q ) 4: 0) we find that each of the following conditions uniquely determines the same class of allowable variations, viz that for which a(q) = (a(q))~ : (i) The canonical commutation relation has the same form at all times (sect. 5); (ii) The class of allowable variations is invariant under a change of coordinate q * q ( r ) (sect. 6); (iii) Noether's theorem holds: with any variation which leaves the Lagrangian invariant is associated a conserved quantity (sect. 7); * A derivation is a linear map from the algebra of observables into itself, which satisfies (AB)" = A B + A B .
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(iv) Rearrangement of terms in the Lagrangian by means of the c.c.r, does not affect the EulerLagrange equation of motion (sect. 9). One can also ask if the timedependence of observables determined by the resuiting equation of motion is generated by a Hamiltonian. This question is answered affirmatively by exhibiting a Hamiltonian H(q,q) with the property (1), see sect. 8. In order to find the EulerLagrange equation which corresponds to the Lagrangian L, one must define derivatives of the form OL/aq, d/dt(OL/Odl). We define these in sect. 2 as linear maps acting on variations o f q and q. We call them Fr6chet derivatives, by virtue of the obvious formal analogy between these objects and Frechet derivatives of maps between Banach spaces. Because of our vagueness on the topological properties of the algebra of observables, the similarity cannot be other than formal. However, since the action principle is for us a procedure designed to set up a quantum theory in the sense of (a), (b), we may regard it as justified by its success in this regard. We use these Frechet derivatives in sect. 2 to obtain the canonical commutation relation (c.c.r.) and then in sect. 3 to express the stationariness of the action integral with respect to a variation 6q as a Frechet differential equation involving 6q. In sect. 4 we show that the connection between the invariance of the Lagrangian under a certain infinitesimal transformation and the conservation in time of the generator of this transformation (Noether's theorem) depends crucially on the variation being allowable. Sects. 2  4 prepare the ground for the main results of sects. 5  9 , listed earlier.
2. Fr~chet derivative Consider a set of n variables X 1 . . . . . 3¢n which do not necessarily commute, and let f ( x 1..... Xn) be a function of them. We define the Fr6chet derivative af/Ox i to be the linear map:
6X i ~ lira ~ [f(xl...Xi_l,X i e6xi,xi+ 1..... Xn)f(xl...Xn) ] =~
,6x
e'~O
Thus, for example, ( ~qa (gt n )),~
= 4 nlol + qn2olq + ... + otq n1 ,
(~q (dtn), l) = n(t n1 . There is no ambiguity about the order of the factors. The momentum p canonically conjugate to q is classically 3L/aq. To obtain an operator like q or q from this object which in our formalism is a Fr~chet derivative we must apply aL/aq to some 6q; the natural one to take is 6q = 1. We define the momentum to be
F.J. Bloore et aL, Lagrangian quantum theory
640
The canonical commutation relation is then ,
~,
1
= i,
(3)
which with the Lagrangian (2) becomes
[q, 2aq +b ] = i whence
[q,q] = if(q)
(4)
where f ( q ) = (2a(q)) 1 . It then follows that for any function c~(q),
, ~,~
:is,
(5)
so that (3L/O(I,~) is the generator of the infinitesimal transformation q ~ q + a(q). Eq. (5) may be derived from Schwinger's action principle but this does not concern us here. The Frechet derivative gives a simple equation between the c o m m u t a t o r of two functions o f q and q and their Poisson bracket, l f F and G are functions o f q and q, then identically,
3. Hamilton's principle In this section we derive a Fr6chet derivative version of the EulerLagrange equation which results from the condition tl
6 W l o = f 6L(q,o)dt = 0 , to where 6L is induced by the variation q(t) ~ q(t) + 6q(t) with the condition that 6q(t0) = 6 q ( t l ) = 0. We may write eq. (7) as
(7)
F.J. Bloore et al., Lagrangian quantum theory tl
s/( L~q,6
(°L
+ ~,64
to
)/
dt=0.
641
(8)
If we define the map d/dt(8L/8q) by the equation d 8L d ~, 5 q ) = ~ (
[email protected], 6 q )  ( ~ , 6q) , then we may write eq. (8) as ti
to
where the vanishing of 6q at t o and t 1 makes zero the integral of the total derivative. We shall say that the variation c~(q) is allowable if, for all cnumber functions g(t) of time which vanish at t o and tl, the variation 6q(t) = g(t)a(q(t)) leaves W10 stationary. Evidently, the condition for c~ to be allowable is
dtOq' ~ = 0 "
(10)
The choice of c~in eq. (10) determines the expression for q in terms of q and q, but is not specified by Hamilton's principle. The usual assumption is that the function = 1 is allowable. This leads to the usual naive equation of motion for q, (.12,1) = 0
(1 1)
but we shall show in sect. 5 that if a'(q) 4: O, then c~= 1 is not usually allowable and eq. (11) does not hold.
4. Noether's theorem
Suppose the Lagrangian L(q,dl) is invariant under the infinitesimal transformation q >q + e~(q), dl > dt + e(~. Then
6L=e
,~
+e~q
(12)
or
(aL
d aL ) + d ( a L ) dt ad/' at ~ ' a = o.
(13)
Thus from eq. (10) the quantity (bL/~q,a) is conserved provided that ~ is an allowable variation. We saw at the end of sect. 2 that this quantity generates the variation
F.J. Bloore et aL, Lagrangianquantum theory
642
a regardless of whether a is allowable. It is clearly important to classify the allowable variations; this is done in sect. 5.
5. Allowable variations
For the c.c.r. (4) to be true for all times, its time derivative must vanish, [q,ql = i / .
(14)
We first ask what is the most general expression for 0 in terms o f q and q which satisfies eq. (14), and then what is the most general form of cz(q) for which the EulerLagrange equation ( 1 0) leads to this expression for//. We need expressions for the time derivative a o f a general function a of q, in terms of c((q) (defined as (d~/dq, 1)) and q. To evaluate a we observe that
[qn,cl] = ifnqn I and
d qqn 1 +qqqn 2 +... +qnlq d~(qn)= = (qn 14 if(n =nqn lq
1)q n2) + (qnlq
~ifn(n
if(n 2)q n2) +... + ( q n  l q  i o )
1)qn2.
Thus if a(q) has a power series expansion
a = ~ anqn then [~,q] = ~
an [qn,q] : ~ anifnqnI = ife/ ,
& = ~ an(qn)"
=
~an(nqnlq_½ifn(n
1)qn 2)
(1 5)
=
a'4
__
~lfot , 1
•
t¢
(16)
and in particular, [a,a] = ifa'a',
[a,q 2] = 2ifa'q + f ( f ~ ' ) ' .
Thus eq. (14) may be written
[q,q] = i ( f ' q ½if/'"). If we substitute the general expansion of q,
(17)
F.J. Bloore et al., Lagrangian quantum theory
cl : ~
643
(18)
An(q)q n
n=O
in eq. (17) and equate coefficients ofq n we obtain the equations n >12 : An+ 1 = 0 ,
n = 1 : 2irA 2 = i f ' ,
Iff,, . n = O " A 2 f f ' + ifA 1 = ~j.
Thus eq. (17) implies that 1 " • 2 + 31
f 
~q
ff ) f
f
+~
tt
d1 + Ao(q)
(19)
q+Ao(q)
where Ao(q ) is arbitrary. To find for what function a(q) the EulerLagrange equation (10) leads to the acceleration (19) we observe that (O~q ~ =a,otq2 + b,o~q + c, a
Thus a(q) is allowable if d
(
[email protected],,a)+ (
[email protected], &)
=a'c~q 2 + b'~q + c'a  a(&q+qo0
a(&q+q&)
boL.
(20)
If we substitute the form (19) for q into the expression (20), we obtain, after rearrangement, Oq 2 +ia'a a
where
(
+~
"
o,
\
The term in q vanishes only ifa' = 0 or Ot p
a I
a + ~ = 0 '
(21)
q  2ae~Ao + 7(q)
4a 2] 2a"
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F..L Bloore et al., Lagrangian quantum theory
i.e. a = Xa 1 ,
(23)
where X is a constant. Thus, if a'(q) 4= O, the only allowable variation a which gives an acceleration which preserves the c.c.r, in time is a = Xa '2. Using eq. (23), the formula (22) for 7(q) simplifies to 7(q) = ~ ( c ~ ib' 4a
a '2 ' ] ' _ 3~aJ] ar~(q).
Hence the term i n d e p e n d e n t of q in (21) can be made to vanish by choosing AO
2a"
This n o w specifies q in eq. (19) completely. The action WI0 of eq. (7) will n o t be stationary u n d e r any other variation/3 since the e q u a t i o n (.Q,/3) = 0 gives the same ~/, eq. (19), only if/3 = a 1 X const.
6. Change of coordinate If we transform from q to a new coordinate r = r(q) then the new variation is 6r = r ' ( q ) 6 q and the new velocity is t: = r ' ( q ) q 51 l.j.r. . . The quadratic term in the t l_ new Lagrangian is a*(r)t ~2 where a = a * r '2. T h e n 6 r = r a 2 = (a*){ so that the specification 6q = a ~ is invariant u n d e r change of coordinate, and of course so is the c.c.r., [r,r] = i f * ( r ) where f * = (2a*)  I .
7. Invariance of the Lagrangian It is conceivable at first sight that the Lagrangian (2) might be invariant under a transformation q ~ q + a, eq. (l 2) where ~ 4= a~ X const. In such a case the associated generator would n o t be conserved. In fact this c a n n o t happen, since =
,a
+
= (aa'+2aa'~
,a
= a ( a ' q 2 + b ' q + c ') +a(gld~+agl) + b a
2 + lower t e r m s .
The coefficient o f q 2 will vanish only i f a = a~ × const.
F.J. Bloore et al., Lagrangian quantum theory
645
8. Hamiltonian It is of interest tc exhibit a Hamiltonian which will generate the derivation
ql by commutation, i.e.
A = i[H,A]. It is sufficient to require the property for the algebraic basis {q,gl},
dt=i[H,q]=
f
,
g ] : i [ H , q ] =  [~q f ) ,
(24)
where we have used eq. (6) to express the commutator as a Frdchet derivative. If we set
H: ~
Hn(q)o n
n=O and compare coefficients o f q n in eq. (24) we find the unique form
ia' . ( ib ' H=aq 2  ~ a q  C~4a
a'2 1 32a3/'
9. Symmetrised Lagrangian It is not hard to verify that if, instead of the Lagrangian (2), we had taken the symmetrised Lagrangian
L1 = qotl(q)q + 1~bl(q),q) + c l ( q ) ,
(25)
where the curly bracket denotes the anticommutator, then eq. (20) becomes 
r

1
t
.
0 = (Z?l,ot) = qalot q + i~blot,q } + ClOt  ~lalot  otalC1 
qalot  otalq  l{bl,ot }
and eq. (21) becomes O q 2 + O q 
where
2alotA 0 + 71(q) = 0
(26)
F.J. Bloore et al., Lagrangian quantum theory
646
¢
71(q) =eCCl + 8 a l \
tt
al !
The term linear in q which in eq. (21) gave the crucial constraint on ~ has disappeared. Thus if we start with the symmetrised Lagrangian L1, we find, at first sight, that we may choose any function e~(q) to be the allowable variation, but for each choice of ~ we should obtain a different expression for A 0 and hence for q. However, the Lagrangian (2) can be rearranged into the form 1 b ,q] " L o = qaq + l { b + [a,q] ,q} + c + ½[[a,q] ,q] + 3[
,
(28)
and if we were to use the commutation relation (15), we would find L 0 was equal to L 1 with a 1 =a,
b I =b + i r a ' ,
c 1 =c  l f ( f a ' ) ' + ½ i f b ' .
(29)
Why should the use of (15) on L0, which has EulerLagrange equation (20), change it to a Lagrangian L 1 which has, apparently, a completely different Euler Lagrange equation? In the remainder of this section we resolve this conundrum. Given functions g(q,q) and ~(q), define the derivation 6~ by
Then 6 ~ ( [ q , q ]  i f ) = [ocq] + [q,a]
if'c~ = i(2)'b~'f'c 0
(31)
which vanishes if and only if ~ = X V ~ ) . Eq. (31 ) states that [q +ea,q +ea] = if(q +e~) + O(e 2) .
(32)
Evidently 6~ [q,q] = 8~ [q,(t] = O. Thus ifg 0 a n d g 1 are two functions o f q and q which can be proved equal to each other using the c.c.r, then their difference will have the factor [q,q] if, so that 6~(go g l ) = 0
(33)
if and only if a = Xx/(f). Now consider the EulerLagrange equation for the difference between L o given by (28) and L 1 given by (25) and (29), D = L 0  L 1 = } { [ a , q ]  i r a ' , q } + ½([ [a,q] , q ] + f ( f a ' ) ' + [ b , q ]  i f b ' ) . We have at 3q' Now
~
~'
~
•
(34)
F.J. Bloore et aZ, Lagrangian quantum theory
d(0 o) ,
=g
647
(( [a,a],q} + ( [ a , q ]  t f a ,c~} + [[a,a],(I]+[[a,q],c~]+[b,a]) = 0
for all a(q) by eq. (14) and, by eq. (33) 6~D = 0
iff
t~=~f~ .
(35)
Thus we may regard the result (23) of sect. 5 as a consequence of eqs. (25), (26) and (35). The Euler Lagrange equations for L 0 and L 1 agree provided that (34) vanishes, i.e. (23). In summary, the Lagrange equation (26) based on the symmetrised Lagrangian (25) does not explicitly restrict the choice of variation a, but different a's lead to different accelerations ~. The correct a is that for which the same Lagrange equation also holds when based on any other Lagrangian obtained from (25) by use of the commutation relations.
10. Discussion We have presented a formulation of quantum theory for one coordinate based on Hamilton's action principle and the Frechet derivative. There is no ambiguity or choice made in the order of any factors, once the Lagrangian is given. We have not discussed the problem of quantising a classical system, and our work does not resolve the ambiguities which arise there. Our results do not depend on the Lagrangian being Hermitian, although this is the only case of physical interest. It follows from eq. (15) that the Lagrangian (2) is Hermitian if and only if a = a*,
b I = Ca',
c I = ½f(bR)'
where b = b R + ib I. The main results are that the only allowable variation is a}, that this specification of variation is coordinatefree and that Noether's theorem always works. Further problems now open to attack are the extensions to more degrees of freedora, to anticommuting and mixed systems, to systems with constraints and to applications in the theory of non linear and gauge symmetries. We thank Dr. Nigel Backhouse for the argument leading to eq. (16). L.R. thanks the Science Research Council for a studentship.