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Lagrangian quantum theory (II). Several degrees of freedom
Nuclear Physics B59 (1973) 348-364. North-Holland Publishing Company
LAGRANGIAN QUANTUM THEORY (IlL SEVERAL DEGREES OF FREEDOM F.J. B L O O R E
and L. R O U T H
Department of Applied Mathematics, The University, Liverpool, L69 3BX, England Received 12 March 1973
Abstract: We present a consistent Lagrangian quantum theory for n degrees of freedom using Fr~chet calculus, by characterising the variations for which the action is stationary. If a set of algebraic quantities ql, Or, i = 1 . . . . . n are such that the q's commute, [qi, ilJ] is given and independent o f q and [qi, Of] is given and linear in ~ then the Hamiltonian H(q, it) satisfying qt = ilH..qt] is prescribed up to an additive function V(q). The commutators lq i, 4 ]] and [c~t, ~/] determine a metric tensor g and a closed two-form K R respectively. If V(q) is given, then there is a Lagrangian L, unique up to a gauge and a multiplicative constant X, having an action which is stationary with respect to n linearly independent variations of the form 6q(t) = l(t)a(q). The vector fields a(q) in these variations must be killing. All killing variations a and no others leave the action stationary. A vector a(q) is called generated if it is killing for g and if ,,," K R is conservative. Noether's theorem, which relates symmetries of the Lagrangian to conserved quantities, holds since all variations a which leave the Lag~angian invariant can be shown to be generated, If a vector a is generated then there is a gauge in which its generator is the Fr~chet derivative (aL/aO i, ai).
1. I n t r o d u c t i o n T h e s t a n d a r d way t o c o n s t r u c t a q u a n t u m t h e o r y f r o m a Lagrangian q l . . . tin) is to p o s t u l a t e c o m m u t a t i o n relations
[qi, q]] = [Pi' Pj] = O,
[qi, p]] = i~i,
L(q 1 . . . qn, (1.1)
w h e r e Pi is some f u n c t i o n o f q and q o b t a i n e d in analogy to the classical e q u a t i o n Pi = aL/Oq i. T h e i m p l e m e n t a t i o n o f this analogy p r e s e n t s a p r o b l e m , since in quant u m t h e o r y L a n d qZ b e l o n g to a n o n - c o m m u t a t i v e algebra for w h i c h the d e f i n i t i o n o f a derivative using c - n u m b e r calculus, 3L /}#i
-
lira [...] 6c)i ~ 0
,
is useless because the limit will d e p e n d o n the n a t u r e o f
6q i. F o r the same reason,
F.J. Bloore, L. Routh, Lagrangian quantum theory
349
the Euler-Lagrange equations need modification. The stationarity of the action fL dt with respect to one variation q ~ q + 6 lq is no guarantee of its stationarity for a different variation. Different variations lead to different Euler-Lagrange equations. So variations need to be selected; the question arises whether there exists for a given Lagrangian a natural and compatible set of variations to select. Then the question arises as to whether the canonical commutation relations (1.1) are the only possible choice of commutation relations for which the action based on this Lagrangian is stationary for some selected set of variations or whether one could postulate different commutation relations with the same Lagrangian L and still achieve a stationary action, but perhaps for different variations. It seems desirable to clarify the algebraic role of the commutation relations in the framework of the calculus of variations on a non-Abelian algebra. Are they an extra ad hoc postulate, as for example in Schwinger's action principle, or are they the expression of the only possible algebra for which the Lagrangian gives a stationary action ? In this paper we present a consistent formulation of Hamilton's principle of stationary action for quantum mechanics, based on the Fr~chet derivative, The work is an extension to n degrees of freedom of the work of ref. [ 1] * which treated a system with one degree of freedom. We adopt a more general starting point than that of paper I. In paper I we followed the standard procedure of "quantizing a given quadratic Lagrangian"
L = a(q) ~2 + b(q) q + c(q)
(1.2)
by postulating the commutation relation
[q, (OLIn#, 1)] = i,
i.e. [q, q] = i/2a(q)
(1.3)
where OL/Oq is a Fr~chet derivative. We assumed that the acceleration ~/"was such a function o f q and q as to preserve the equation (1.3) in time, that is to say
[q, 6"] = (i/2a) " , and asked for what variations q -+ q + eg(t) a(q) was the action stationary. In this paper we regard the equal-time algebra of q's and q's as given, (by eqs. (2.1)-(2.3)). These commutation relations are shown in sect. 2 to be mutually compatible so long as G is a certain function o f g and the real part K R of the tensor K is a closed two-form. In sect. 3 it is shown that this algebra already determines the Hamiltonian of the time development up to an arbitrary additive "potential" function V(q). When this function is specified, the time development is prescribed. In sect. 4 we present a formulation of the calculus of variations for this (non-Abelian) algebra, based on the Fr~chet derivative. We can then pose and solve the problem: Given the time development prescribed by the Hamiltonian, does there exist a Lagrangian L, quadratic in q,
L = qi aij(q) #j + ~ (bi(q), #i ) + c(q) • Ref. [ 1 ] referred to as I hereafter.
(1.4)
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350
and a set of allowable variations o~(q) such that the action is stationary with respect to these variations ? Iii sect. 6 we find that each allowable variation ot gives a number of conditions (eqs. (6.5)) on the Lagrangian and that if there are n independent allowable variations the Lagrangian is determined uniquely up to an arbitrary multiplicative constant X, namely ai/ =
)t gi/ '
:R = bj, i 2 X iV.i~
(1.5)
bi, j ,
(1.6)
and a third condition. The closure o f K R ensures the existence of" a solution b to eq. (1.6). The third condition is of the form C ( ~ g, c/X) = 0
(1.7)
and apparently prescribes c in terms of o~. Different variations ~ appear to require different terms c(q) in (1.4), if eq. (1.7) is to be satisfied. However, if (and only if) is a killing vector, the eq. (1.7) may be recast in the form
c~" V ( c / X + 2 V + d) = O,
(1.8)
where d is some function of q which involves g only. From this we may conclude that if c=
X(2V + d)
then the action for the Lagrangian (1.4) with a, b given by ( 1.5) and (1.6) is stationary for the class of killing variations ~(q) and for this class only. In sect. 7 we show that if we take ~. =-~1 in our discovered Lagrangian, then the commutation relations ( 2 . 1 ) - ( 2 . 3 ) are those obtained by the naive quantization procedure ( 1.1 ) with Pi = (OL/3~I i, 1). Thus in a sense, Schwinger's action principle is superfluous to Hamilton's. Since the commutation rules (2.1) (2.3) lead to a unique Lagrangian (except for the potential term V, or c) if we want a given Lagrangian to have an action which is stationary for a set of n linearly independent variations, then we are forced to choose the commutation rules found by the naive quantization procedure. At any rate, we cannot vary the right-hand sides of eqs. (2.1) (2.3), though presumably paracommutation rules or Fermi rules are still allowable. The properties of generated vectors are discussed in sect. 5. Briefly, a vector or(q) is generated if it is a killing vector of the m e t r i c g and 0~- K R is a conservative vector field. These conditions ensure that a function A(q, [l) exists whose commutators with qi and ~i are ~i and &i We show also that the class of generated vectors is closed under the Lie product of vector fields. In sect. 8 we relate the generator A of a generated variation ~ to the Fr~chet derivative (OL/O~t i, oJ). If o~ is a symmetry of the Lagrangian L then it is automatically generated by (OL/3ct i, oJ) and Noether's theorem is shown to hold. We show also
F.J. Bloore, L. Routh, Lagrangian quantum theory
35 l
that i f ~ is generated b y A , but is not a symmetry of the Lagrangian, then there is a solution b of eq. (1.6) for which A = (~/Oqi, ai).
2. The algebra of q's and q's In this section we derive properties of the algebra_~ generated over the complex field C by n "coordinates" qi and n "velocities" ~i with the commutation relations
[qi, qj] = 0 ,
(2.1)
[qi, ~tj] = i gij(q) ,
(2.2)
[qi, qJl = i( GO" (q) ~k + Ki/(q)) .
(2.3)
Repeated indices are summed. We assume that the matrix gi/is symmetric and has an inverse gi]" We show in sect. 3 that symmetry o f g is required if we are to interpret q as a time derivative of q, but in this section we simply regard q and el as abstract algebraic quantities. The functions g, G, K are not arbitrary but are constrained by the mutual compatibility of eqs. (2.11-(2.31 and by a Hermiticity condition, as explained below. Evidently,
Gi~" = -GJ~ ,
Ki/ = - K / i •
(2.4)
The relations (2.1) and (2.2) are compatible, regardless of the choice of the functionsg ij, in the sense that
j(qi, q/, qk) =j(qi, q/, ~k) = 0 ,
(2.5 t
where the Jacobi function J is defined by
J(x,y, z ) = [x, Iv, zll + [y,[z, x]l + [z,[x,y]] .
(2.6)
The eq. (2.3) is consistent with eqs. (2.1) and (2.2) if
j(qi, #/, •k) = 0 ,
(2.7)
J(q i, 4/, 4 k) = 0 .
(2.8)
Eq. (2.7) yields
Gi~" = gJa Piak _ g/a r ak i '
(2.9)
where P is the usual Christoffel symbol for the metricg. Eq. (2.8) yields the "zero rotation" condition [2] or the condition for a "closed two-form" [3],
Kij;k + K J k ; i + K k i ; J = o ,
ordK= 0,
(2.10)
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where the semicolon denotes covariant differentiation. In deriving eqs. (2.9) and (2.10) we use the result that for any flmction o~(q), it follows from eqs. (2.1) and (2.2) lhat
I~iq~.,)a]
= i u b e,h '
(2.1 l)
~ I~,.'~e a.,, denotes OoffOq b. Eq. (2.11) may be proved as follows. The equation holds when e = qi. Both sides ofeq. (2.11) are derivations on the function o~. Two dedva0ons agree on all functions o f q 1 . . . . . qn if they agree on q 1 . . . . , qn. The algebra_~ is "graded". By this we mean that all f l m c t i o n s f o f q and q have a unique expansion in "normal fornf', ,f(q,,;)= ~ ~ 3;....i.v(q)d i' d i2 . . . q / V , .\:=0 i I ..iN =1
(2.12)
,s here each coefficient 1')....i N is fully symmetric in its indices. We may thus equate Me coefficients of two expressions in normal form which are known to be equal. We denote the complex conjugate of the complex number X by X-. The "star" operation,
q*:q,
(~i)*=~i, ~k)*:Xq,
(#;#/)*=~i/# ;,
etc.,
(2.13)
is an involu/ion on _~' if and only if tile functions g and G / / + K are Hermitian, i.e. (g0 (q)), = g;k(q),
(2.14)
(a~" #k + :.;:),: -,;; D k qk + K i~ .
(2.15)
We assume eq. (2.14) to hold and examine whether (2.15) is possible. It is easy to verify, using (2.14), that the normal form
A(q, [l) = Aah(q) ~]a /t b + Aa(q) #a + A o ( q )
(2.16)
is Hermitian if and only if AI = 0
(2.17a)
ab
A a1 = 1
:l(j =
gt~C Aab.c .
(2.17b)
1 gab A a,b" R
(2.17c)
~
',~ here we denote tile real and imaginary parts of a function f(q) a s f R a n d f I. Eqs. (2.17) allow us to rewrite eq. (2.16) as ,,l(q. , ) ) = q a A R , ( q ) @ +½ ( A R ( q ) , q a } + A R ( q ) , where the curly brackets denote the anticommutator.
(2.18)
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By settingAab = 0 in (2.17) we obtain the result that the normal linear form Aa(q) 4 a +Ao(q) is Hermitian if and only if 1 gabAR A aI = 0 ~ A 0 = -~ a,b in which case the form can be written 1 {Aa(q) ,qa } +AR(q). Thus we see that (2.15) is equivalent to the conditions GI~{ = 0 ,
KI
q =
(2.19)
-~1 gab GRi/a,b "
(2.20)
Now eq. (2.19) is guaranteed by (2.9) and (2.14). We shall postulate (2.20). It may be verified that this is compatible with the imaginary part of (2.10). The commutation relation (2.3) may be now written
[4i, 4/] _, - 5 i { gia F/ak
i 4k } + iK RU , gla" Fak,
(2.21)
where g//is Hermitian, symmetric and nonsingular, K Rq is Hermitian and antisymmetric, and obeys the closure equation (2.10).
3. The Hamiltonian
In deriving the algebra M in sect. 2 we regarded q and 4 as independent elements of an abstract Lie algebra. We wish to make the physical interpretation that s4 is the (equal-time) algebra of observables of a system that the q's are the co(~rdinates and the (t's are the velocities of some system in the tteisenberg picture at time t. This motivate,: *he question whether a *-derivation (corresponding to differentiation with respect to time) exists ill s"~'.:,',dch takes qi into 4 i and which respects tile conmmtation relations (2. l), (2.2) 2,~,1 ~2.21 ). ('~ ~:-derivatmn is a derivation D such that (Dr)* = DO°*). ) If such a derivation, written d/dt or "dot", existed, then its application to eq. (2.1) would yield
[qi, 4j] + [di, q/] = 0 ,
(3.1)
which by eq. (2.2) implies that the matrix g is symmctric. Thus the symmetry o f g is necessary for such a derivation to exist. Conversely we shall show that if, as we have assumed, gq is symmetric, then we may exhibit a derivation with the required properties. A derivation D on .~ is called "inner" if there existsHE sff such that for all
13.2j
Dr= i Ill,fl.
Then H is said to "generate" the derivation D. Evidently, all inner derivations preserve the commutation relations (2.1)-(2.3); for example
D([q i, 4/] - i gi/(q)) = i[H, [qi, 41] _
i~/(q)]
= 0.
F.J. Bloore, L. Routh, Lagrangian quantum theory
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An inner derivation is a *-derivation if and only if its generator is Hermitian. We thus seek a Hermitian Hamiltonian function H(q, q) such that
[ti = i[H, qi] .
(3.3)
I f H exists we may write it in the unique symmetric form
tt : #a t i b(q) ~tt) + ½ {Ha(q) ' #a ) + V(q) ,
(3.4)
with
Hab:Hb
:Ha;,
Ha:H*
,
V:V*.
(3.5)
Eqs. (2.2) and (3.3) forbid the presence of terms higher than quadratic in c) in (3.4). If we substitute (3.4) into (3.3) and equate coefficients we obtain
H=½ c)a gab ~tb + V(q)
(3.6)
where V is arbitrary. Thus there exists a family of inner *-derivations which send
qi to c)i. It is straightforward to verify that dtd ~ ( q ) = d =
6 ) ,qi
= i [ H , ~ ] = ~1{ ~ , m '
)"
(3.7)
The tensor character of g6, G~, K ni/under a change of coordinates qf = qi'(ql . . . . qn) may be checked using eq. (3.7). We find
[qi', #]'1 = 1 [qi'(q), { qO'
#]l}
.,
.,
. .
: i q~i q(] gl!
(3.83
so that g is a second rank tensor. Similarly one finds that K R is a tensor (though G~' is not a tensor). We are now in a position to calculate the acceleration ~m. For a given choice of the potential V(q), we find
~7"m = i[H, ~1'"] :
Cli
r)7 #/+ 1 (K Rm/, #/}
+ 4I gbd (gaC gbe Gem a,c ),d "
V (3.9)
4. Lagrangian formulation of quantum mechanics In paper I we developed a Lagrangian formulation of quantum mechanics for a system with one degree of freedom, based on the Fr~chet derivative as the differential operator. In this section we extend those results of I which are easy to extend to systems with several degrees of freedom: in particular we define the Fr6chet derivative, state Hamilton's principle, obtain the Euler-Lagrange equations and derive Noether's theorem. The problems of what the Lagrangian might be and for what variations the action is stationary are solved in sects. 5 and 6.
F.J. Bloore, L. R outh, Lagrangianquantum theory
355
Consider the algebraA of functions J; g of variables x 1 . . . . . x n which do not necessarily commute. The Frechet derivative 3f/3x i is the linear map from A to A defined by
Of ~ --Zlim ~-1 [f(xl . -Of - " g ~ (~x 3x i i' e-~0 For fixed
. . . . . xi-l'xi+eg'xi+l
.
. . xn)--f(xl
xn)] (4. l)
x i and g the map
is a derivation on A. It was proved in ref, [4] that the commutator and the Fr6chetPoisson bracket are equal:
We now use the Frechet derivative to formulate Hamilton's action principle for quantum mechanics and derive the Euler-Lagrange equations of motion. For all t E [to., t 1 ] CC~ let q 1(t) . . . . . qn(t) be elements of an algebra with such structure that q~(t) is defined. We do not make this structure precise though this will be necessary in a rigorous formulation. Given a Lagrangian function L(q, q), Hamilton's action principle states that the action tl
WI0
= f L(q, if) at
(4,3)
to is stationary with respect to certain variations
qi(t) ~ qi(t) + 6qi(t)
(4.4)
such that
6qi(to) = &/i(tl) = 0 .
(4.5)
Then 0 = 5 W10 =
= ti to
~
~(~i
, 6q i + ~-~ ,
dtd 3L ,6q i) at , ~qi
(4.6)
where we define
8L
6qi)_ d
3L
6qi) -
3L
and drop the edge terms because they vanish.
6~i)
356
F.J. Bloore, L. Routh, Lagrangian quantum theory
To obtain a local equation from eq. (4.6) we must allow the 6q i to be sufficiently freely variable. We suppose we are able to choose 6q = l(t) or(q), i.e.
6qi(t) = l(t) ~i(q (t)) ,
(4.8)
where l(t) is an arbitrary C 2 function of time which vanishes at t O and t 1 and the are functions o f q and q (to be scrutinised later). We then obtain the EulerLagrange equation (p, ~) ~ (~1;~
d ~L a i ) = 0 dt 0~i '
(4.9)
where i is a summed suffix. Conversely the eq. (4.9) guarantees the stationarity of the action (4.3) with respect to the variation (4.8). A vector c~ = (c0(q) . . . . . o:(q)) is called "allowable" if it obeys eq. (4.9). We obtain one equation of motion from each independent allowable vector variation o¢. We finish this section with a comment on the validity of Noether's theorem. For any vector c~(q), define the "derivation by ~" of a function fl to be
Then the Euler Lagrange equation (4.9) may be written
Suppose now that L is invariant under the infinitesimal transformation
qi ~ qi + e oJ.
~ii _+ qi + e ~i
that is to say, 6eL = 0 .
(4.12)
Then by eq. (4.11) the quantity (iJL/Oq i, o:i) is conserved, dT
,~i
=0,
(4.13)
provided that the variation ~ is allowable. Thus Noether's theorem, which states that eq. (4.12) implies eq. (4.13) holds only i f ~ is allowable. In sect. 6 we obtain the criterion that a variation is allowable if and only if it is killing and in sect. 8 we prove that all variations which leave the Lagrangian invariant are indeed killing, so that Noether's theorem survives unscathed.
5. Killing vectors and generated vectors
Let L l( q, q), L 21L,' ,~}~be two Lagrange functions which can be proved equal to
F.J. Bloore, L. Routh, Lagrangian quantum theory
357
each other using the commutation relations (2.1), (2.2) and (2.21), so that L 1 - L 2 is a sum of terms which have the factors [qi, qJl, [qi, qj] _ igi/or [~i, ~/] _ i i ( G~ ~k ) iK Ri/. The commutation relations indicate that L 1 - L2 is zero, and it is natural to expect to obtain the same Euler-Lagrange equation from a given variation a from L 1 as from L 2. We proceed to show that this is not the case. Two ~/'s cannot be freely interchanged using (2.21); a Cl and a function of q may be commuted using (2.2) only when c~ is a killing vector. We first find for what vectors a, the derivations by ,~ of the commutation relations (2.1), (2.2), and (2.21) vanish. It is trivial that for all a(q) 8c~([qi, q / i ) = [d, q/] + [qi, aj] = O . It is easily verified [5] that
a( [q i , ql] - i gi]) = i (Ori'j + (if;i)
(5.1)
which vanishes if and only if c~ obeys the killing equation
Oti;f + aj'i = 0 .
(5.2)
Elementary algebra shows that the condition 6 ([# i, #J] - ~ i1( G i ~
", ~ k ) _
iKRij): 0
(5.3)
implies in addition the two conditions (V A 0¢' KR)ij = (o~ K~j); i - (ork KRi);j ,= O,
(5.4)
Further, eqs. (5.2) and (5.4) are necessary and sufficient to ensure the existence of a generator A(q, q) such that
o/= i [A, qi] ,
(5.6)
dei = i [A, ~i] .
(5.7)
Indeed if we set
A =I~ ~Ak(q) ' #k ) +Ao(q)
(5.8)
then eq. (.5.6) yields
A k = ak
(5.9)
and eq. (5.7) yields the killing equation again and the condition VA 0 = -o¢. K R .
(5.10)
Eq. (5.4) is necessary and sufficient to ensure the integrability ¢;f eq. (5.10). We shall therefore call vectors ~generated if they obey eqs. (5.2) and (5.4/. If ~ is gen-
F.J. Bloore, L. Routh, Lagrang&n quantum theory
358
erated byA, it follows that for any functionf(q, q), f a r = i [A, f] ,
(5.11)
since both sides of eq. (5. I 1) are derivations o f f which agree when f is qi or c/i. Hence eqs. (5.1) and (5.3) hold for generated vectors and so in fact eqs. (5.2) and (5.4) imply eq. (5.5). In sect. 8 we show that all variations which leave the Lagrangian invarian', are generated. This makes generated vectors worthy of closer study. A generator A may be characterised as a Hermitian function of q and el, linear in ci whose commutator with H is independent o f q . To see this, first observe that for such an A, ~i(q) = i [A, qi] is independent of q, and then that, by the Jacobi identity,
~ = i [ H , o~i] = - [ H , [A, qi]] = [,4, [qi, H]] + [qi, [ H , N ] I =i[A,#i] .
(5.12)
The other important fact about the class of generated vectors is that, like the class of killing vectors [6] it is closed under the Lie product operation: ifcx,/3 are generated b y A , B then -t = ( a . v ) P - (p. v ) 0 t
(5.13)
is generated by
G
i[A, B] .
=
(5.14)
To prove this we note that eqs. (5.8), (5.9) and (5.14) imply
i[G, qil =i[c~/B] +i[A,3il =~i[~i,{3/,#/)l
+li[{o~],~lj),{3i] =~i,
(5.15)
and then we use (5.15) to show that
i [G, ~ti] - ;/i = i 3 [ [.4, B] , [11, qi] ] _ i 3 [H, [ [,4, B ] , qi] ]
=i3[q i,
[H,
[A,B]]]
=0. The second line follows from the Jacobi identity, and the third from the fact that [H, A ] and [H, B] are independent of q while A and B are linear in ci. Now consider the difference D between two Lagrangians which may be converted into each other by use of the commutation relations. This difference will have the form
D=fl(q,q)([qi,~j] -
_ igJj)f2(q,~)+f3(q,d)([qi,~j
iKRi/)£ 4 (q, ~t),
]
x:rpij~t t o k , q k ) (5.16)
where the functions ft" are arbitrary. The two Lagrangians will yield the same Euler-
F.J. Bloore, L. Routh, Lagrangian quantum theory
359
Lagrange equation (4.11) for the variation a if
Now
~ D : fl 6a ( [qi, ~/] _ igiJ) f2 +f36c~ ( [c~i' q/] - ½i ( G~, qk } _ iKR if) f4 '
d(0D
d-t ~ '
aj)=id
(f3(o~i;J_otJ;i)f4) "
Thus i f D obeys eq. (5.17) for arbitrary fi then a is generated and o?';/= 0 , i.e. a is a parallel vector field. In the next section we see that in order to obtain n independent Euler-Lagrange equations of motion we need to assume that there are n independent allowable variations a. However, if a Riemannian manifold possesses n independent parallel vector fields then it is flat [6]. We do not wish to be so restrictive so we shall not assume a is parallel. Hence, in general we must conclude that arbitrary juxtaposition of pairs of q terms using the commutation relation (2.21) will change the Euler-Lagrange equation, though one may freely apply the commutation relations (2.1) and (2.2) so long as a is a killing vector. The relation between permitted interchanges of ~/'s and the scalarity of L requires further investigation. The results of this section show that one cannot, for example, add the zero quantity [~i, ~j] _ ½i {G~, ~k } _ iKRij to a Lagrangian without altering its Euler-Lagrange equation. However, this quantity is not a scalar so perhaps one should expect such a restriction. We shall not pursue this problem further.
6. The Lagrangian and the allowable variations In paper I, we assumed the Lagrangian was L = a(q) q2 + b(q) q + c(q) (for one coordinate) and the commutator was [q, q] = i/2a, and showed that the equation of motion dt 0q ' ~ (q
=0 1
gave an acceleration i/" compatible with the commutator only when ~(q) ~x a-~-. In this section we pose and solve the wide r problem: Given the algebra -~of sect. 2 with the commutators (2.1), (2.2) and (2.21), with the formula (3.9) for the acceleration, to find a Hermitian quadratic Lagrangian L and an allowable variation ~ (q) such that the Euler-Lagrange equations (4.9) hold. The only ingredient o f q 'm in (3.9) which was not implied by the commutation relations o f ~ was the term in V V. We shall find the Lagrangian and variation are prescribed almost completely by this requirement of consistency with the algebra .~, and shall see that Schwinger's extension of the action principle to obtain corn-
F.J. Bloore, L. Routh, Lagrangian quantum theory
360
mutation relations is superfluous; the commutation relations fix the Lagrangian by consistency; conversely for a given Lagrangian only one set of commutators is algebraically consistent with a variational principle. We suppose the Lagrangian is Hermitian and quadratic in velocity, and shall work with the form
L = #i ai/(q) # / + ½ (bi(q), ~ti } + c (q),
(6. l)
where aij = aji = a~. , b i = b* , c = c*. We examine the restrictions which the EulerLagrange eq. (4.9) places on a, b and c. Eq. (4.9) is ,
+
,
,
- {@'i, aikoLk}
---;
,c~ k
#i } + c k
qi "
--o~k " q --~' {[)k,a k } .
(6.2)
When we substitute eq. (3.9) for q"i, use eq. (3.7) for a and/~, and rearrange we obtain
qi ak Ai/k 6 / + 1
{ o~ Bik, Cli } + C = 0 ,
(6.3)
where
Aijk(q) = aiLk - aik,/ Bik(q) = bi, k - bk.i C(q) = ~
_
a/k,i + 2akl
[,/
i~ '
Zakl K R l i '
(6.4a) (6.4b)
a bd, ac Gei ~ • (C,k + 2aik [gia V, a _ ~ g LK- gbe a,c),d I )
+ gbd (gac [abk,a~, c + I abk,acak _ (amkO~), c P~]),d "
(6.4c)
Equation (6.3) implies
a k Aqk = 0 ,
(6.5a)
~k Bi k = 0 ,
(6.5b)
C = 0.
(6.5c)
These are the conditions for the action (4.3) with L given by (6. l) to be stationary for the variation (4.8). Eqs. (6.5) are the main general result of the paper. We obtain interesting consequences from eqs. (6.5) if we assume there are n linearly independent variations (a)ot(q), o = 1. . . . . n, for which the action is sta-
F.J. Bloore, L. R outh, Lagrangian quantum theory
361
tionary. It then follows that
Aij k = 0 ,
(6.6a)
Bik = 0 .
(6.6b)
The equation
Aij k +Aik j = 0 yields
aij;k = 0 .
(6.7)
The general solution of this equation was found by Eisenhart [7]; ifg is indecomposable then
ai/= X gi/"
(6.8)
If the metric tensor decomposes into a sum of tensors g =g(1) eg(2) • . . .
e g (k)
acting on parallel distributions (subspaces of the tangent spaces which are invariant under parallel transport) then a0 = X(I) ~6-(1)e . . . • X(k) ~6-(k) , where the h.(i) are arbitrary constants. We shall suppose that the metric is indecomposable so that eq. (6.8) holds. We conclude that the only possible quadratic Lagrangian consistent with the commutation relations (2.1), (2.2)..and (.2..21) has the form L = X ~tigi/~7/+ . . . . Conversely, given the Lagrangian q~ g6 ql + . . . , the only commutation relation of the form [qZ, ql] = i f if(q) which is preserved by the Euler-Lagrange equations of motion is eq. (2.2) withg given by (6.8). There is no need for any extra action principle of Schwinger's type to specify the commutation rule - it is implicit in the non-Abelian calculus of variations. Using eq. (6.8) the condition (6.6b) gives VAb=2XK
R.
(6.9)
This specifies the vector b in terms of the tensor K R ; the condition (2.10) ensures the existence of a solution b to eq. (6.9) by the converse of the Poincar~ lemma [3]. This solution is arbitrary to the extent that the gradient of any scalar field may be added to b without violating eq. (6.9). Using eq. (6.8) the remaining condition (6.5c) is a complicated equation for the coefficient e in the Lagrangian. It expresses the scalar product a • V c as a function of a and the metric. Now the specification of the Lagrangian cannot depend on the particular variations a for which its time integral is stationary. The only possible way in which the eq. (6.5c) can give c as a
F.J. Bloore, L. R outh, Lagrangian quantum theory
362
function which is independent of c~ is for the eq. (6.5c) to reduce to the form or" V ( c + 2 X V+ Xd) = 0 ,
(6.10)
where d(q) is a function depending on g but not on c~. It may be verified that if and only ifc~ is killing the eq. (6.5c) reduces to the form (6.10) with _
1
d - -g gbd(4 gab,cd
gac
c
- g~,~gac,d
+ 6
ac
g,d gab,c )
.
Thus by choosing c in (6.1) to obey V (c + 2 X V+ Xd) = 0 ,
ai/to obey (6.8) and b i to obey (6.9) we produce the only Lagrangian quadratic in q for which the action is stationary with respect to a set of variations. No other quadratic Lagrangian gives an action which might be stationary for a set of n independent variations and even this Lagrangian has action stationary only for killing variations c~(q). There may be other variations of the form a(q, q) for which the action is stationary, These will be required in a correct formulation of Noether's theorem for dynamical symmetries, but they will not be investigated here. The reduction of eq. (6.5c) to eq. (6.10) by means of the killing equation involved tile authors in considerable elementary tensor algebra: no doubt the result may be achieved more shortly and transparently using intrinsic methods. As noted in sect. 5, the fact that the variations are killing enables us to rearrange functions o f q with the factors q in the Lagrangian using the commutation relation (2.2) without altering the corresponding Euler-Lagrange equation.
7. Canonical quantization The usual quantization procedure is to start with a Lagrangian, say (6.1) and postulate the commutation relations
[qi, qj] = [pi, p] ] = 0 ,
[qi, pj] =i~5{l.
(7.1)
There is an immediate difficulty in defining Pi as OL/3q i, since this can only be meaningfully defined as a Fr&het derivative, and not an element of the algebra -~. If we define the momentum to be p/=
,
1)
,
(7.2)
(effectively the usual definition), with L given by (6, 1), then eqs. (7.1) yield eqs. (2.1), (2.2), and (2.21) w i t h E / = 1 at?• so that X in (6.8) is ½, and with K R given in terms of b by eq. (6.9) with X -- - 1~-. Thus the usual quantization procedure leads to the only possible algebra M, (up to the factor X), for which the action could be stationary.
F.J. Bloore, L. R outh, Lagrangian quantum theory
363
8. Noether's theorem and generators One may verify by elementary algebra that 6a(X ~i gi/ q l + 1 { bi ' #i } + c) = X ~i [°ti;/ + %';i] q] + ~ ( ~ b i , . k + bk °tk,.i , #i } + C ,kOtk ---~,
gbd c gbe seac) , , d"
(8.1)
Thus if 5~L = 0, then the vanishing of the quadratic term implies that a is killing, whence the vanishing of the linear term implies the vanishing of the Lie product of a and b,
a-Vb
b" V a = 0 .
(8.2)
This ensures that eq. (5.4) is satisfied, where 2LK R = VAb by eq. (6.9). Hence 6crL = 0 only if,v is generated. It may be verified that the generator of e, is A - ~a { a k, qk ) + ~" b = ( o r / O q i, a i ) ,
(8.3)
provided that ~. - ~1 in eq. (8.1). Thus a is allowable and by eq. (4.11) the generator of a is conserved. Thus Noether's theorem holds in quantum mechanics. The question now arises: If a is a generated variation, with generator A, which does not leave the Lagrangian invariant, is it true that =
,a
?
(8.4)
The answer is provided by the following theorem. T h e o r e m : If a(q) is generated by A then there is a gauge, that is to say a choice of b obeying K R = VA~ for which eq. (8.4) holds, where L ---~ ' qi gi/ q/ +'gx { ~ i ' # i ) + c " Proof.' We must show there exists b such that ~7 (a. b) = -or. K R .
(8.5)
We know from eq. (5.10) that (8.6)
or. K R = - ~ 7 A o ( q ) .
Let b be any vector potential for K R, K R = ~7Ab. Then since a is killing, ot " K R = - ~7 ( a " b ) + a " ~7 b
b . V ot .
(8.7)
Eq. (8.7) will still hold if we replace b by b=b+V~b where ~ is any function of q, so that, using (8.6), u'Vb
b'Va=ol.
KR+v(oe.b)=V(-Ao+Ot.b+ot.Vga).
(8.8)
364
F.J. Bloore, L. Routh, Lagrangian quantum theor),
We may choose ~ to be a solution of the first order differential equation ~ - V~b=A 0 - t ~ ' b .
(8.9)
This is always possible since the field lines of the killing field have no singularities. With this choice of ~beq. (8.8) implies eq. (8.5). We thank Dr. J. Underhill, Dr. N.B. Backhouse, Dr. T.A.S. Jackson and Professors J.G. Oldroyd and A.G. Walker for very helpful conversations and suggestions. L.R. thanks the Science Research Council for a studentship.
References [1] F.J. Bloore, L. Routh and J. Underhill, Nucl. Phys. B55 (1973) 637, referred to as I. [21 K. Yano, Differential geometry on complex and almost complex spaces(Pergamon Press, Oxford, 1965). [3] H. Flanders, Differential forms (Academic Press, New York, 1968). [41 F.J. Bloore, J. Phys. A6 (1973) L7. [5] T. Kimura, T. Ohtani and R. Sugano, Prog. Theor. Phys. 48 (1972) 1395, [6] L.P. Eisenhart, Riemannian geometry (Princeton University Press, 1926). [7] L.P. Eisenhart, Trans. Am. Math. Soc. 25 (1923) 297.
LAGRANGIAN QUANTUM THEORY (IlL SEVERAL DEGREES OF FREEDOM F.J. B L O O R E
and L. R O U T H
Department of Applied Mathematics, The University, Liverpool, L69 3BX, England Received 12 March 1973
Abstract: We present a consistent Lagrangian quantum theory for n degrees of freedom using Fr~chet calculus, by characterising the variations for which the action is stationary. If a set of algebraic quantities ql, Or, i = 1 . . . . . n are such that the q's commute, [qi, ilJ] is given and independent o f q and [qi, Of] is given and linear in ~ then the Hamiltonian H(q, it) satisfying qt = ilH..qt] is prescribed up to an additive function V(q). The commutators lq i, 4 ]] and [c~t, ~/] determine a metric tensor g and a closed two-form K R respectively. If V(q) is given, then there is a Lagrangian L, unique up to a gauge and a multiplicative constant X, having an action which is stationary with respect to n linearly independent variations of the form 6q(t) = l(t)a(q). The vector fields a(q) in these variations must be killing. All killing variations a and no others leave the action stationary. A vector a(q) is called generated if it is killing for g and if ,,," K R is conservative. Noether's theorem, which relates symmetries of the Lagrangian to conserved quantities, holds since all variations a which leave the Lag~angian invariant can be shown to be generated, If a vector a is generated then there is a gauge in which its generator is the Fr~chet derivative (aL/aO i, ai).
1. I n t r o d u c t i o n T h e s t a n d a r d way t o c o n s t r u c t a q u a n t u m t h e o r y f r o m a Lagrangian q l . . . tin) is to p o s t u l a t e c o m m u t a t i o n relations
[qi, q]] = [Pi' Pj] = O,
[qi, p]] = i~i,
L(q 1 . . . qn, (1.1)
w h e r e Pi is some f u n c t i o n o f q and q o b t a i n e d in analogy to the classical e q u a t i o n Pi = aL/Oq i. T h e i m p l e m e n t a t i o n o f this analogy p r e s e n t s a p r o b l e m , since in quant u m t h e o r y L a n d qZ b e l o n g to a n o n - c o m m u t a t i v e algebra for w h i c h the d e f i n i t i o n o f a derivative using c - n u m b e r calculus, 3L /}#i
-
lira [...] 6c)i ~ 0
,
is useless because the limit will d e p e n d o n the n a t u r e o f
6q i. F o r the same reason,
F.J. Bloore, L. Routh, Lagrangian quantum theory
349
the Euler-Lagrange equations need modification. The stationarity of the action fL dt with respect to one variation q ~ q + 6 lq is no guarantee of its stationarity for a different variation. Different variations lead to different Euler-Lagrange equations. So variations need to be selected; the question arises whether there exists for a given Lagrangian a natural and compatible set of variations to select. Then the question arises as to whether the canonical commutation relations (1.1) are the only possible choice of commutation relations for which the action based on this Lagrangian is stationary for some selected set of variations or whether one could postulate different commutation relations with the same Lagrangian L and still achieve a stationary action, but perhaps for different variations. It seems desirable to clarify the algebraic role of the commutation relations in the framework of the calculus of variations on a non-Abelian algebra. Are they an extra ad hoc postulate, as for example in Schwinger's action principle, or are they the expression of the only possible algebra for which the Lagrangian gives a stationary action ? In this paper we present a consistent formulation of Hamilton's principle of stationary action for quantum mechanics, based on the Fr~chet derivative, The work is an extension to n degrees of freedom of the work of ref. [ 1] * which treated a system with one degree of freedom. We adopt a more general starting point than that of paper I. In paper I we followed the standard procedure of "quantizing a given quadratic Lagrangian"
L = a(q) ~2 + b(q) q + c(q)
(1.2)
by postulating the commutation relation
[q, (OLIn#, 1)] = i,
i.e. [q, q] = i/2a(q)
(1.3)
where OL/Oq is a Fr~chet derivative. We assumed that the acceleration ~/"was such a function o f q and q as to preserve the equation (1.3) in time, that is to say
[q, 6"] = (i/2a) " , and asked for what variations q -+ q + eg(t) a(q) was the action stationary. In this paper we regard the equal-time algebra of q's and q's as given, (by eqs. (2.1)-(2.3)). These commutation relations are shown in sect. 2 to be mutually compatible so long as G is a certain function o f g and the real part K R of the tensor K is a closed two-form. In sect. 3 it is shown that this algebra already determines the Hamiltonian of the time development up to an arbitrary additive "potential" function V(q). When this function is specified, the time development is prescribed. In sect. 4 we present a formulation of the calculus of variations for this (non-Abelian) algebra, based on the Fr~chet derivative. We can then pose and solve the problem: Given the time development prescribed by the Hamiltonian, does there exist a Lagrangian L, quadratic in q,
L = qi aij(q) #j + ~ (bi(q), #i ) + c(q) • Ref. [ 1 ] referred to as I hereafter.
(1.4)
F.J. Bloore, L. Routh, Lagranlzian quantum theoo:
350
and a set of allowable variations o~(q) such that the action is stationary with respect to these variations ? Iii sect. 6 we find that each allowable variation ot gives a number of conditions (eqs. (6.5)) on the Lagrangian and that if there are n independent allowable variations the Lagrangian is determined uniquely up to an arbitrary multiplicative constant X, namely ai/ =
)t gi/ '
:R = bj, i 2 X iV.i~
(1.5)
bi, j ,
(1.6)
and a third condition. The closure o f K R ensures the existence of" a solution b to eq. (1.6). The third condition is of the form C ( ~ g, c/X) = 0
(1.7)
and apparently prescribes c in terms of o~. Different variations ~ appear to require different terms c(q) in (1.4), if eq. (1.7) is to be satisfied. However, if (and only if) is a killing vector, the eq. (1.7) may be recast in the form
c~" V ( c / X + 2 V + d) = O,
(1.8)
where d is some function of q which involves g only. From this we may conclude that if c=
X(2V + d)
then the action for the Lagrangian (1.4) with a, b given by ( 1.5) and (1.6) is stationary for the class of killing variations ~(q) and for this class only. In sect. 7 we show that if we take ~. =-~1 in our discovered Lagrangian, then the commutation relations ( 2 . 1 ) - ( 2 . 3 ) are those obtained by the naive quantization procedure ( 1.1 ) with Pi = (OL/3~I i, 1). Thus in a sense, Schwinger's action principle is superfluous to Hamilton's. Since the commutation rules (2.1) (2.3) lead to a unique Lagrangian (except for the potential term V, or c) if we want a given Lagrangian to have an action which is stationary for a set of n linearly independent variations, then we are forced to choose the commutation rules found by the naive quantization procedure. At any rate, we cannot vary the right-hand sides of eqs. (2.1) (2.3), though presumably paracommutation rules or Fermi rules are still allowable. The properties of generated vectors are discussed in sect. 5. Briefly, a vector or(q) is generated if it is a killing vector of the m e t r i c g and 0~- K R is a conservative vector field. These conditions ensure that a function A(q, [l) exists whose commutators with qi and ~i are ~i and &i We show also that the class of generated vectors is closed under the Lie product of vector fields. In sect. 8 we relate the generator A of a generated variation ~ to the Fr~chet derivative (OL/O~t i, oJ). If o~ is a symmetry of the Lagrangian L then it is automatically generated by (OL/3ct i, oJ) and Noether's theorem is shown to hold. We show also
F.J. Bloore, L. Routh, Lagrangian quantum theory
35 l
that i f ~ is generated b y A , but is not a symmetry of the Lagrangian, then there is a solution b of eq. (1.6) for which A = (~/Oqi, ai).
2. The algebra of q's and q's In this section we derive properties of the algebra_~ generated over the complex field C by n "coordinates" qi and n "velocities" ~i with the commutation relations
[qi, qj] = 0 ,
(2.1)
[qi, ~tj] = i gij(q) ,
(2.2)
[qi, qJl = i( GO" (q) ~k + Ki/(q)) .
(2.3)
Repeated indices are summed. We assume that the matrix gi/is symmetric and has an inverse gi]" We show in sect. 3 that symmetry o f g is required if we are to interpret q as a time derivative of q, but in this section we simply regard q and el as abstract algebraic quantities. The functions g, G, K are not arbitrary but are constrained by the mutual compatibility of eqs. (2.11-(2.31 and by a Hermiticity condition, as explained below. Evidently,
Gi~" = -GJ~ ,
Ki/ = - K / i •
(2.4)
The relations (2.1) and (2.2) are compatible, regardless of the choice of the functionsg ij, in the sense that
j(qi, q/, qk) =j(qi, q/, ~k) = 0 ,
(2.5 t
where the Jacobi function J is defined by
J(x,y, z ) = [x, Iv, zll + [y,[z, x]l + [z,[x,y]] .
(2.6)
The eq. (2.3) is consistent with eqs. (2.1) and (2.2) if
j(qi, #/, •k) = 0 ,
(2.7)
J(q i, 4/, 4 k) = 0 .
(2.8)
Eq. (2.7) yields
Gi~" = gJa Piak _ g/a r ak i '
(2.9)
where P is the usual Christoffel symbol for the metricg. Eq. (2.8) yields the "zero rotation" condition [2] or the condition for a "closed two-form" [3],
Kij;k + K J k ; i + K k i ; J = o ,
ordK= 0,
(2.10)
F.J. Bloore. L. R outh, Lagrangian quantum theory
352
where the semicolon denotes covariant differentiation. In deriving eqs. (2.9) and (2.10) we use the result that for any flmction o~(q), it follows from eqs. (2.1) and (2.2) lhat
I~iq~.,)a]
= i u b e,h '
(2.1 l)
~ I~,.'~e a.,, denotes OoffOq b. Eq. (2.11) may be proved as follows. The equation holds when e = qi. Both sides ofeq. (2.11) are derivations on the function o~. Two dedva0ons agree on all functions o f q 1 . . . . . qn if they agree on q 1 . . . . , qn. The algebra_~ is "graded". By this we mean that all f l m c t i o n s f o f q and q have a unique expansion in "normal fornf', ,f(q,,;)= ~ ~ 3;....i.v(q)d i' d i2 . . . q / V , .\:=0 i I ..iN =1
(2.12)
,s here each coefficient 1')....i N is fully symmetric in its indices. We may thus equate Me coefficients of two expressions in normal form which are known to be equal. We denote the complex conjugate of the complex number X by X-. The "star" operation,
q*:q,
(~i)*=~i, ~k)*:Xq,
(#;#/)*=~i/# ;,
etc.,
(2.13)
is an involu/ion on _~' if and only if tile functions g and G / / + K are Hermitian, i.e. (g0 (q)), = g;k(q),
(2.14)
(a~" #k + :.;:),: -,;; D k qk + K i~ .
(2.15)
We assume eq. (2.14) to hold and examine whether (2.15) is possible. It is easy to verify, using (2.14), that the normal form
A(q, [l) = Aah(q) ~]a /t b + Aa(q) #a + A o ( q )
(2.16)
is Hermitian if and only if AI = 0
(2.17a)
ab
A a1 = 1
:l(j =
gt~C Aab.c .
(2.17b)
1 gab A a,b" R
(2.17c)
~
',~ here we denote tile real and imaginary parts of a function f(q) a s f R a n d f I. Eqs. (2.17) allow us to rewrite eq. (2.16) as ,,l(q. , ) ) = q a A R , ( q ) @ +½ ( A R ( q ) , q a } + A R ( q ) , where the curly brackets denote the anticommutator.
(2.18)
F.J. Bloore, L. R outh, Lagrangian quantum theory
353
By settingAab = 0 in (2.17) we obtain the result that the normal linear form Aa(q) 4 a +Ao(q) is Hermitian if and only if 1 gabAR A aI = 0 ~ A 0 = -~ a,b in which case the form can be written 1 {Aa(q) ,qa } +AR(q). Thus we see that (2.15) is equivalent to the conditions GI~{ = 0 ,
KI
q =
(2.19)
-~1 gab GRi/a,b "
(2.20)
Now eq. (2.19) is guaranteed by (2.9) and (2.14). We shall postulate (2.20). It may be verified that this is compatible with the imaginary part of (2.10). The commutation relation (2.3) may be now written
[4i, 4/] _, - 5 i { gia F/ak
i 4k } + iK RU , gla" Fak,
(2.21)
where g//is Hermitian, symmetric and nonsingular, K Rq is Hermitian and antisymmetric, and obeys the closure equation (2.10).
3. The Hamiltonian
In deriving the algebra M in sect. 2 we regarded q and 4 as independent elements of an abstract Lie algebra. We wish to make the physical interpretation that s4 is the (equal-time) algebra of observables of a system that the q's are the co(~rdinates and the (t's are the velocities of some system in the tteisenberg picture at time t. This motivate,: *he question whether a *-derivation (corresponding to differentiation with respect to time) exists ill s"~'.:,',dch takes qi into 4 i and which respects tile conmmtation relations (2. l), (2.2) 2,~,1 ~2.21 ). ('~ ~:-derivatmn is a derivation D such that (Dr)* = DO°*). ) If such a derivation, written d/dt or "dot", existed, then its application to eq. (2.1) would yield
[qi, 4j] + [di, q/] = 0 ,
(3.1)
which by eq. (2.2) implies that the matrix g is symmctric. Thus the symmetry o f g is necessary for such a derivation to exist. Conversely we shall show that if, as we have assumed, gq is symmetric, then we may exhibit a derivation with the required properties. A derivation D on .~ is called "inner" if there existsHE sff such that for all
13.2j
Dr= i Ill,fl.
Then H is said to "generate" the derivation D. Evidently, all inner derivations preserve the commutation relations (2.1)-(2.3); for example
D([q i, 4/] - i gi/(q)) = i[H, [qi, 41] _
i~/(q)]
= 0.
F.J. Bloore, L. Routh, Lagrangian quantum theory
354
An inner derivation is a *-derivation if and only if its generator is Hermitian. We thus seek a Hermitian Hamiltonian function H(q, q) such that
[ti = i[H, qi] .
(3.3)
I f H exists we may write it in the unique symmetric form
tt : #a t i b(q) ~tt) + ½ {Ha(q) ' #a ) + V(q) ,
(3.4)
with
Hab:Hb
:Ha;,
Ha:H*
,
V:V*.
(3.5)
Eqs. (2.2) and (3.3) forbid the presence of terms higher than quadratic in c) in (3.4). If we substitute (3.4) into (3.3) and equate coefficients we obtain
H=½ c)a gab ~tb + V(q)
(3.6)
where V is arbitrary. Thus there exists a family of inner *-derivations which send
qi to c)i. It is straightforward to verify that dtd ~ ( q ) = d =
6 ) ,qi
= i [ H , ~ ] = ~1{ ~ , m '
)"
(3.7)
The tensor character of g6, G~, K ni/under a change of coordinates qf = qi'(ql . . . . qn) may be checked using eq. (3.7). We find
[qi', #]'1 = 1 [qi'(q), { qO'
#]l}
.,
.,
. .
: i q~i q(] gl!
(3.83
so that g is a second rank tensor. Similarly one finds that K R is a tensor (though G~' is not a tensor). We are now in a position to calculate the acceleration ~m. For a given choice of the potential V(q), we find
~7"m = i[H, ~1'"] :
Cli
r)7 #/+ 1 (K Rm/, #/}
+ 4I gbd (gaC gbe Gem a,c ),d "
V (3.9)
4. Lagrangian formulation of quantum mechanics In paper I we developed a Lagrangian formulation of quantum mechanics for a system with one degree of freedom, based on the Fr~chet derivative as the differential operator. In this section we extend those results of I which are easy to extend to systems with several degrees of freedom: in particular we define the Fr6chet derivative, state Hamilton's principle, obtain the Euler-Lagrange equations and derive Noether's theorem. The problems of what the Lagrangian might be and for what variations the action is stationary are solved in sects. 5 and 6.
F.J. Bloore, L. R outh, Lagrangianquantum theory
355
Consider the algebraA of functions J; g of variables x 1 . . . . . x n which do not necessarily commute. The Frechet derivative 3f/3x i is the linear map from A to A defined by
Of ~ --Zlim ~-1 [f(xl . -Of - " g ~ (~x 3x i i' e-~0 For fixed
. . . . . xi-l'xi+eg'xi+l
.
. . xn)--f(xl
xn)] (4. l)
x i and g the map
is a derivation on A. It was proved in ref, [4] that the commutator and the Fr6chetPoisson bracket are equal:
We now use the Frechet derivative to formulate Hamilton's action principle for quantum mechanics and derive the Euler-Lagrange equations of motion. For all t E [to., t 1 ] CC~ let q 1(t) . . . . . qn(t) be elements of an algebra with such structure that q~(t) is defined. We do not make this structure precise though this will be necessary in a rigorous formulation. Given a Lagrangian function L(q, q), Hamilton's action principle states that the action tl
WI0
= f L(q, if) at
(4,3)
to is stationary with respect to certain variations
qi(t) ~ qi(t) + 6qi(t)
(4.4)
such that
6qi(to) = &/i(tl) = 0 .
(4.5)
Then 0 = 5 W10 =
= ti to
~
~(~i
, 6q i + ~-~ ,
dtd 3L ,6q i) at , ~qi
(4.6)
where we define
8L
6qi)_ d
3L
6qi) -
3L
and drop the edge terms because they vanish.
6~i)
356
F.J. Bloore, L. Routh, Lagrangian quantum theory
To obtain a local equation from eq. (4.6) we must allow the 6q i to be sufficiently freely variable. We suppose we are able to choose 6q = l(t) or(q), i.e.
6qi(t) = l(t) ~i(q (t)) ,
(4.8)
where l(t) is an arbitrary C 2 function of time which vanishes at t O and t 1 and the are functions o f q and q (to be scrutinised later). We then obtain the EulerLagrange equation (p, ~) ~ (~1;~
d ~L a i ) = 0 dt 0~i '
(4.9)
where i is a summed suffix. Conversely the eq. (4.9) guarantees the stationarity of the action (4.3) with respect to the variation (4.8). A vector c~ = (c0(q) . . . . . o:(q)) is called "allowable" if it obeys eq. (4.9). We obtain one equation of motion from each independent allowable vector variation o¢. We finish this section with a comment on the validity of Noether's theorem. For any vector c~(q), define the "derivation by ~" of a function fl to be
Then the Euler Lagrange equation (4.9) may be written
Suppose now that L is invariant under the infinitesimal transformation
qi ~ qi + e oJ.
~ii _+ qi + e ~i
that is to say, 6eL = 0 .
(4.12)
Then by eq. (4.11) the quantity (iJL/Oq i, o:i) is conserved, dT
,~i
=0,
(4.13)
provided that the variation ~ is allowable. Thus Noether's theorem, which states that eq. (4.12) implies eq. (4.13) holds only i f ~ is allowable. In sect. 6 we obtain the criterion that a variation is allowable if and only if it is killing and in sect. 8 we prove that all variations which leave the Lagrangian invariant are indeed killing, so that Noether's theorem survives unscathed.
5. Killing vectors and generated vectors
Let L l( q, q), L 21L,' ,~}~be two Lagrange functions which can be proved equal to
F.J. Bloore, L. Routh, Lagrangian quantum theory
357
each other using the commutation relations (2.1), (2.2) and (2.21), so that L 1 - L 2 is a sum of terms which have the factors [qi, qJl, [qi, qj] _ igi/or [~i, ~/] _ i i ( G~ ~k ) iK Ri/. The commutation relations indicate that L 1 - L2 is zero, and it is natural to expect to obtain the same Euler-Lagrange equation from a given variation a from L 1 as from L 2. We proceed to show that this is not the case. Two ~/'s cannot be freely interchanged using (2.21); a Cl and a function of q may be commuted using (2.2) only when c~ is a killing vector. We first find for what vectors a, the derivations by ,~ of the commutation relations (2.1), (2.2), and (2.21) vanish. It is trivial that for all a(q) 8c~([qi, q / i ) = [d, q/] + [qi, aj] = O . It is easily verified [5] that
a( [q i , ql] - i gi]) = i (Ori'j + (if;i)
(5.1)
which vanishes if and only if c~ obeys the killing equation
Oti;f + aj'i = 0 .
(5.2)
Elementary algebra shows that the condition 6 ([# i, #J] - ~ i1( G i ~
", ~ k ) _
iKRij): 0
(5.3)
implies in addition the two conditions (V A 0¢' KR)ij = (o~ K~j); i - (ork KRi);j ,= O,
(5.4)
Further, eqs. (5.2) and (5.4) are necessary and sufficient to ensure the existence of a generator A(q, q) such that
o/= i [A, qi] ,
(5.6)
dei = i [A, ~i] .
(5.7)
Indeed if we set
A =I~ ~Ak(q) ' #k ) +Ao(q)
(5.8)
then eq. (.5.6) yields
A k = ak
(5.9)
and eq. (5.7) yields the killing equation again and the condition VA 0 = -o¢. K R .
(5.10)
Eq. (5.4) is necessary and sufficient to ensure the integrability ¢;f eq. (5.10). We shall therefore call vectors ~generated if they obey eqs. (5.2) and (5.4/. If ~ is gen-
F.J. Bloore, L. Routh, Lagrang&n quantum theory
358
erated byA, it follows that for any functionf(q, q), f a r = i [A, f] ,
(5.11)
since both sides of eq. (5. I 1) are derivations o f f which agree when f is qi or c/i. Hence eqs. (5.1) and (5.3) hold for generated vectors and so in fact eqs. (5.2) and (5.4) imply eq. (5.5). In sect. 8 we show that all variations which leave the Lagrangian invarian', are generated. This makes generated vectors worthy of closer study. A generator A may be characterised as a Hermitian function of q and el, linear in ci whose commutator with H is independent o f q . To see this, first observe that for such an A, ~i(q) = i [A, qi] is independent of q, and then that, by the Jacobi identity,
~ = i [ H , o~i] = - [ H , [A, qi]] = [,4, [qi, H]] + [qi, [ H , N ] I =i[A,#i] .
(5.12)
The other important fact about the class of generated vectors is that, like the class of killing vectors [6] it is closed under the Lie product operation: ifcx,/3 are generated b y A , B then -t = ( a . v ) P - (p. v ) 0 t
(5.13)
is generated by
G
i[A, B] .
=
(5.14)
To prove this we note that eqs. (5.8), (5.9) and (5.14) imply
i[G, qil =i[c~/B] +i[A,3il =~i[~i,{3/,#/)l
+li[{o~],~lj),{3i] =~i,
(5.15)
and then we use (5.15) to show that
i [G, ~ti] - ;/i = i 3 [ [.4, B] , [11, qi] ] _ i 3 [H, [ [,4, B ] , qi] ]
=i3[q i,
[H,
[A,B]]]
=0. The second line follows from the Jacobi identity, and the third from the fact that [H, A ] and [H, B] are independent of q while A and B are linear in ci. Now consider the difference D between two Lagrangians which may be converted into each other by use of the commutation relations. This difference will have the form
D=fl(q,q)([qi,~j] -
_ igJj)f2(q,~)+f3(q,d)([qi,~j
iKRi/)£ 4 (q, ~t),
]
x:rpij~t t o k , q k ) (5.16)
where the functions ft" are arbitrary. The two Lagrangians will yield the same Euler-
F.J. Bloore, L. Routh, Lagrangian quantum theory
359
Lagrange equation (4.11) for the variation a if
Now
~ D : fl 6a ( [qi, ~/] _ igiJ) f2 +f36c~ ( [c~i' q/] - ½i ( G~, qk } _ iKR if) f4 '
d(0D
d-t ~ '
aj)=id
(f3(o~i;J_otJ;i)f4) "
Thus i f D obeys eq. (5.17) for arbitrary fi then a is generated and o?';/= 0 , i.e. a is a parallel vector field. In the next section we see that in order to obtain n independent Euler-Lagrange equations of motion we need to assume that there are n independent allowable variations a. However, if a Riemannian manifold possesses n independent parallel vector fields then it is flat [6]. We do not wish to be so restrictive so we shall not assume a is parallel. Hence, in general we must conclude that arbitrary juxtaposition of pairs of q terms using the commutation relation (2.21) will change the Euler-Lagrange equation, though one may freely apply the commutation relations (2.1) and (2.2) so long as a is a killing vector. The relation between permitted interchanges of ~/'s and the scalarity of L requires further investigation. The results of this section show that one cannot, for example, add the zero quantity [~i, ~j] _ ½i {G~, ~k } _ iKRij to a Lagrangian without altering its Euler-Lagrange equation. However, this quantity is not a scalar so perhaps one should expect such a restriction. We shall not pursue this problem further.
6. The Lagrangian and the allowable variations In paper I, we assumed the Lagrangian was L = a(q) q2 + b(q) q + c(q) (for one coordinate) and the commutator was [q, q] = i/2a, and showed that the equation of motion dt 0q ' ~ (q
=0 1
gave an acceleration i/" compatible with the commutator only when ~(q) ~x a-~-. In this section we pose and solve the wide r problem: Given the algebra -~of sect. 2 with the commutators (2.1), (2.2) and (2.21), with the formula (3.9) for the acceleration, to find a Hermitian quadratic Lagrangian L and an allowable variation ~ (q) such that the Euler-Lagrange equations (4.9) hold. The only ingredient o f q 'm in (3.9) which was not implied by the commutation relations o f ~ was the term in V V. We shall find the Lagrangian and variation are prescribed almost completely by this requirement of consistency with the algebra .~, and shall see that Schwinger's extension of the action principle to obtain corn-
F.J. Bloore, L. Routh, Lagrangian quantum theory
360
mutation relations is superfluous; the commutation relations fix the Lagrangian by consistency; conversely for a given Lagrangian only one set of commutators is algebraically consistent with a variational principle. We suppose the Lagrangian is Hermitian and quadratic in velocity, and shall work with the form
L = #i ai/(q) # / + ½ (bi(q), ~ti } + c (q),
(6. l)
where aij = aji = a~. , b i = b* , c = c*. We examine the restrictions which the EulerLagrange eq. (4.9) places on a, b and c. Eq. (4.9) is ,
+
,
,
- {@'i, aikoLk}
---;
,c~ k
#i } + c k
qi "
--o~k " q --~' {[)k,a k } .
(6.2)
When we substitute eq. (3.9) for q"i, use eq. (3.7) for a and/~, and rearrange we obtain
qi ak Ai/k 6 / + 1
{ o~ Bik, Cli } + C = 0 ,
(6.3)
where
Aijk(q) = aiLk - aik,/ Bik(q) = bi, k - bk.i C(q) = ~
_
a/k,i + 2akl
[,/
i~ '
Zakl K R l i '
(6.4a) (6.4b)
a bd, ac Gei ~ • (C,k + 2aik [gia V, a _ ~ g LK- gbe a,c),d I )
+ gbd (gac [abk,a~, c + I abk,acak _ (amkO~), c P~]),d "
(6.4c)
Equation (6.3) implies
a k Aqk = 0 ,
(6.5a)
~k Bi k = 0 ,
(6.5b)
C = 0.
(6.5c)
These are the conditions for the action (4.3) with L given by (6. l) to be stationary for the variation (4.8). Eqs. (6.5) are the main general result of the paper. We obtain interesting consequences from eqs. (6.5) if we assume there are n linearly independent variations (a)ot(q), o = 1. . . . . n, for which the action is sta-
F.J. Bloore, L. R outh, Lagrangian quantum theory
361
tionary. It then follows that
Aij k = 0 ,
(6.6a)
Bik = 0 .
(6.6b)
The equation
Aij k +Aik j = 0 yields
aij;k = 0 .
(6.7)
The general solution of this equation was found by Eisenhart [7]; ifg is indecomposable then
ai/= X gi/"
(6.8)
If the metric tensor decomposes into a sum of tensors g =g(1) eg(2) • . . .
e g (k)
acting on parallel distributions (subspaces of the tangent spaces which are invariant under parallel transport) then a0 = X(I) ~6-(1)e . . . • X(k) ~6-(k) , where the h.(i) are arbitrary constants. We shall suppose that the metric is indecomposable so that eq. (6.8) holds. We conclude that the only possible quadratic Lagrangian consistent with the commutation relations (2.1), (2.2)..and (.2..21) has the form L = X ~tigi/~7/+ . . . . Conversely, given the Lagrangian q~ g6 ql + . . . , the only commutation relation of the form [qZ, ql] = i f if(q) which is preserved by the Euler-Lagrange equations of motion is eq. (2.2) withg given by (6.8). There is no need for any extra action principle of Schwinger's type to specify the commutation rule - it is implicit in the non-Abelian calculus of variations. Using eq. (6.8) the condition (6.6b) gives VAb=2XK
R.
(6.9)
This specifies the vector b in terms of the tensor K R ; the condition (2.10) ensures the existence of a solution b to eq. (6.9) by the converse of the Poincar~ lemma [3]. This solution is arbitrary to the extent that the gradient of any scalar field may be added to b without violating eq. (6.9). Using eq. (6.8) the remaining condition (6.5c) is a complicated equation for the coefficient e in the Lagrangian. It expresses the scalar product a • V c as a function of a and the metric. Now the specification of the Lagrangian cannot depend on the particular variations a for which its time integral is stationary. The only possible way in which the eq. (6.5c) can give c as a
F.J. Bloore, L. R outh, Lagrangian quantum theory
362
function which is independent of c~ is for the eq. (6.5c) to reduce to the form or" V ( c + 2 X V+ Xd) = 0 ,
(6.10)
where d(q) is a function depending on g but not on c~. It may be verified that if and only ifc~ is killing the eq. (6.5c) reduces to the form (6.10) with _
1
d - -g gbd(4 gab,cd
gac
c
- g~,~gac,d
+ 6
ac
g,d gab,c )
.
Thus by choosing c in (6.1) to obey V (c + 2 X V+ Xd) = 0 ,
ai/to obey (6.8) and b i to obey (6.9) we produce the only Lagrangian quadratic in q for which the action is stationary with respect to a set of variations. No other quadratic Lagrangian gives an action which might be stationary for a set of n independent variations and even this Lagrangian has action stationary only for killing variations c~(q). There may be other variations of the form a(q, q) for which the action is stationary, These will be required in a correct formulation of Noether's theorem for dynamical symmetries, but they will not be investigated here. The reduction of eq. (6.5c) to eq. (6.10) by means of the killing equation involved tile authors in considerable elementary tensor algebra: no doubt the result may be achieved more shortly and transparently using intrinsic methods. As noted in sect. 5, the fact that the variations are killing enables us to rearrange functions o f q with the factors q in the Lagrangian using the commutation relation (2.2) without altering the corresponding Euler-Lagrange equation.
7. Canonical quantization The usual quantization procedure is to start with a Lagrangian, say (6.1) and postulate the commutation relations
[qi, qj] = [pi, p] ] = 0 ,
[qi, pj] =i~5{l.
(7.1)
There is an immediate difficulty in defining Pi as OL/3q i, since this can only be meaningfully defined as a Fr&het derivative, and not an element of the algebra -~. If we define the momentum to be p/=
,
1)
,
(7.2)
(effectively the usual definition), with L given by (6, 1), then eqs. (7.1) yield eqs. (2.1), (2.2), and (2.21) w i t h E / = 1 at?• so that X in (6.8) is ½, and with K R given in terms of b by eq. (6.9) with X -- - 1~-. Thus the usual quantization procedure leads to the only possible algebra M, (up to the factor X), for which the action could be stationary.
F.J. Bloore, L. R outh, Lagrangian quantum theory
363
8. Noether's theorem and generators One may verify by elementary algebra that 6a(X ~i gi/ q l + 1 { bi ' #i } + c) = X ~i [°ti;/ + %';i] q] + ~ ( ~ b i , . k + bk °tk,.i , #i } + C ,kOtk ---~,
gbd c gbe seac) , , d"
(8.1)
Thus if 5~L = 0, then the vanishing of the quadratic term implies that a is killing, whence the vanishing of the linear term implies the vanishing of the Lie product of a and b,
a-Vb
b" V a = 0 .
(8.2)
This ensures that eq. (5.4) is satisfied, where 2LK R = VAb by eq. (6.9). Hence 6crL = 0 only if,v is generated. It may be verified that the generator of e, is A - ~a { a k, qk ) + ~" b = ( o r / O q i, a i ) ,
(8.3)
provided that ~. - ~1 in eq. (8.1). Thus a is allowable and by eq. (4.11) the generator of a is conserved. Thus Noether's theorem holds in quantum mechanics. The question now arises: If a is a generated variation, with generator A, which does not leave the Lagrangian invariant, is it true that =
,a
?
(8.4)
The answer is provided by the following theorem. T h e o r e m : If a(q) is generated by A then there is a gauge, that is to say a choice of b obeying K R = VA~ for which eq. (8.4) holds, where L ---~ ' qi gi/ q/ +'gx { ~ i ' # i ) + c " Proof.' We must show there exists b such that ~7 (a. b) = -or. K R .
(8.5)
We know from eq. (5.10) that (8.6)
or. K R = - ~ 7 A o ( q ) .
Let b be any vector potential for K R, K R = ~7Ab. Then since a is killing, ot " K R = - ~7 ( a " b ) + a " ~7 b
b . V ot .
(8.7)
Eq. (8.7) will still hold if we replace b by b=b+V~b where ~ is any function of q, so that, using (8.6), u'Vb
b'Va=ol.
KR+v(oe.b)=V(-Ao+Ot.b+ot.Vga).
(8.8)
364
F.J. Bloore, L. Routh, Lagrangian quantum theor),
We may choose ~ to be a solution of the first order differential equation ~ - V~b=A 0 - t ~ ' b .
(8.9)
This is always possible since the field lines of the killing field have no singularities. With this choice of ~beq. (8.8) implies eq. (8.5). We thank Dr. J. Underhill, Dr. N.B. Backhouse, Dr. T.A.S. Jackson and Professors J.G. Oldroyd and A.G. Walker for very helpful conversations and suggestions. L.R. thanks the Science Research Council for a studentship.
References [1] F.J. Bloore, L. Routh and J. Underhill, Nucl. Phys. B55 (1973) 637, referred to as I. [21 K. Yano, Differential geometry on complex and almost complex spaces(Pergamon Press, Oxford, 1965). [3] H. Flanders, Differential forms (Academic Press, New York, 1968). [41 F.J. Bloore, J. Phys. A6 (1973) L7. [5] T. Kimura, T. Ohtani and R. Sugano, Prog. Theor. Phys. 48 (1972) 1395, [6] L.P. Eisenhart, Riemannian geometry (Princeton University Press, 1926). [7] L.P. Eisenhart, Trans. Am. Math. Soc. 25 (1923) 297.