Table Of ContentComment on “A quantum-classical bracket that satisfies the Jacobi identity” [J.
Chem. Phys. 124, 201104 (2006)]
∗
L. L. Salcedo
Department of Atomic, Molecular, and Nuclear Physics,
University of Granada, E-18071 Granada, Spain
(Dated: February 1, 2008)
The quantum mechanical description of microscopic actproperties(e.g.,symmetries,conservationslaws,uni-
systems is nowadays well established and cleanly formu- tarity, etc) of a theory, while internally consistent ap-
lated (at least above the very small Planck scale, where proximations tend to preserve them, and so they are, in
quantum gravity awaits a firmer foundation). However, principle,preferable. (Classicalmechanicsispreciselyan
7
0 inmanycases,evenifthewaveequationstobesolvedare exampleofaconsistentapproximation,namely,toquan-
0 known, they are not easily amenable to a full quantum tum mechanics.)
2 computation, in practice. A standard approach is then A simple case, often considered in the literature, is
n toresorttoapproximatedescriptionsofthesemiclassical that of a one dimensional system with two structureless
a type, where some of the degrees of freedom of the full particles,wherexandk arethepositionandmomentum
J system are treated quantum mechanically while others variablesof the classicalparticle,respectively, andq and
0 aretreatedataclassicallevel. Inthisapproachoneends p those of the quantum particle. x and k commute with
1
up with a mixed quantum-classical system. everythingwhile [q,p]=i¯h. Inthis case, the observables
Thecanonicalstructuresofbothclassicalandquantum are constructed with x, k, q and p,
1
v mechanics are based on the existence of a Lie bracket
4 between observables, (A,B); this is the commutator in A= X anmrtxnkmqrpt. (4)
5 thequantumcaseandthePoissonbracketintheclassical n,m,r,t
0
case. The dynamical bracket is such that if G is the
1 Minimal requirements for a consistent quantum-
infinitesimal generator a symmetry transformation, the
0 classical formulation would include the following:
7 variation of an observable A takes the following form
0 (i) Symmetrytransformationsarecarriedoutbyady-
/ δA=(A,G)δλ. (1)
h namical bracket between observables that must be
p a Lie bracket.
The Lie bracket properties
-
t
n (ii) The dynamical bracket between two purely quan-
(aA+bB,C)=a(A,C)+b(B,C) (linearity),
a tumobservablesshouldreducetothestandardone,
u (A,B)=−(B,A) (antisymmetry), (2)
q 1 1
′ ′ ′ ′
: ((A,B),C)+((B,C),A)+((C,A),B)=0 (Jacobi), (Q,Q)= [Q,Q]= (QQ −QQ), (5)
v i¯h i¯h
i
X encapsulate the group structure of symmetry transfor- and likewise, for two purely classical observables
mations, including the dynamical evolution,
r
a ∂C ∂C′ ∂C ∂C′
′ ′
d ∂ (C,C )={C,C }= − . (6)
A=(A,H)+ A (3) ∂x ∂k ∂k ∂x
dt ∂t
A standard proposal, made independently by several
(H being the Hamiltonian), and are therefore crucial for
authors [1, 2], is as follows,
a fully consistent physical description of the classical or
quantum system. 1 1
(A,B)= [A,B]+ ({A,B}−{B,A}). (7)
In view of this similarity between the classical and i¯h 2
quantum formulations, many authors have considered
Unfortunately, this definition fails to satisfy the Jacobi
the possibility of finding a consistent description for the
identity. An explicit counterexample is provided in [3],
abovementioned mixed quantum-classical systems, i.e.,
for A=xq, B =xqp, and C =k2p, since
systems composed of two interacting sectors, a quantum
one and a classical one. Note that such mixed systems
1
are perfectly legitimate as approximations to an exact ((A,B),C)+((B,C),A)+((C,A),B) = ¯h2. (8)
2
quantum-quantum system, the challenge is, however, to
findanautonomous andcloseddescriptionforthe mixed (Jacobiidentityviolationsalwayscomefromfinitenessof
case, rather than an intrinsically approximated one. As ¯h, since in the classical limit any bracket must revert to
is well known, ad hoc approximations tend to break ex- the Poissonbracket which does satisfy Jacobi.)
2
In a recent work [4], a new proposalis made, claiming are at most quadratic in the dynamical variables x, k, q,
to define a true Lie bracket. No mathematical proof is and p, [3].)
provided, but it is shown (correctly) that Jacobi is sat- Such violation of the Lie bracket property is not sur-
isfied for the three observables of the example just dis- prising;in[3]ano-gotheoremwasproven(seealso[5,6]),
cussed. Regrettably,asIshowbelow,the Jacobiidentity namely, if one requires the quantum-classical bracket to
is not satisfied by this new bracket either, if one takes fulfill the rather natural axioms
three generic observables.
1
′ ′ ′ ′
In the new proposal of [4] (CQ,C )={C,C }Q, (CQ,Q)= [Q,Q]C, (14)
i¯h
(A,B)=(A,B)q +(A,B)c, (9) (C, and C′ being purely classical and Q and Q′ purely
quantum observables), then Jacobi cannot be satisfied.
with
(Note that the bracket (9), with (10) and (13), satisfies
1
(A,B) = [A,B], (10) the axioms.)
q i¯h Finally, let me note that in addition to the require-
as in (7). However, the classical part differs from (7). ments (i) and (ii), another natural property (common
ThenewprescriptionistotakethePoissonbracketofthe to classical and quantum mechanics) is that the prod-
classical variables involved, while the quantum variables uct of two observables AB at t = 0 should evolve into
are ordered by moving (commuting) the q’s to the left the product A(t)B(t) and time t (and similarly for other
of the p’s and setting the h¯ so generated to zero. Of symmetry transformations). This implies the
course,thisisequivalenttoa“normalorder”prescription
(iii) Leibniz rule property for the dynamical bracket,
inwhich q is setto the leftof p byhand, :pq :=qp. (For
simplicity, I usethe standardnotation: :to denote this (AB,C)=(A,C)B+A(B,C). (15)
“normal order” of operators.) That is,1
As shown in [7], a bracket fulfilling (i-iii) is necessarily
(A,B) =:{A,B}:, (11)
c
either the Poisson bracket (no quantum sector) or the
where the Poisson bracket affects the classical variables. quantumcommutator(noclassicalsector). Noquantum-
More explicitly, for two observables, classical mixture is allowed.
In summary, the proposal in [4] does not actually de-
A = Xanm(x,k)qnpm, fine a Lie bracketsince it fails to satisfy the Jacobiiden-
n,m tity. Furthermore, any quantum-classical bracket must
B = Xbrt(x,k)qrpt, (12) have awkwardproperties,as it has to violate rather nat-
ural requirements, satisfied both in the purely classi-
r,t
cal or purely quantum cases. This just means that the
a (x,k) and b (x,k) being ordinary functions on the
nm rt quantum-classicalmixing remains as a useful but intrin-
phase space of the classical particle,
sically approximated approach.
This work was supported by DGI, FEDER, UE, and
(A,B)c =XX(cid:8)anm(x,k),brt(x,k)(cid:9)qn+rpm+t.(13) Junta de Andaluc´ıa funds (FIS2005-00810, HPRN-CT-
n,m r,t
2002-00311,FQM225).
As I said, this definition does not preserve the Jacobi
identity. An explicit counterexample is provided by the
new triple A = kp, B = xp, and C = q2. For these
observables, one easily finds
∗
Electronic address: [email protected]
(A,B) = −p2, ((A,B),C)=4qp−2i¯h, [1] I. V.Aleksandrov, Z. fuer Naturforsch. 36A, 902 (1981).
[2] W. Boucher and J. H. Traschen, Phys. Rev. D37, 3522
(B,C) = −2xq, ((B,C),A)=−2xk−2qp,
(1988).
(C,A) = 2kq, ((C,A),B)=2xk−2qp, [3] J. Caro and L. L. Salcedo, Phys. Rev. A60, 842 (1999),
[quant-ph/9812046].
and hence, [4] O. V.Prezhdo, J. Chem. Phys. 124, 201104 (2006).
[5] D. R. Terno, Found. Phys. 36, 102 (2006),
((A,B),C)+((B,C),A)+((C,A),B)=−2i¯h. [quant-ph/0402092].
Therefore the Jacobi identity is violated. (However, Ja- [6] D. Sahoo, J. Phys. A37, 997 (2004).
cobi is preserved by this triple with the original bracket [7] L. L. Salcedo, Phys. Rev. A54, 3657 (1996),
(7). The same is true whenever the observables involved [hep-th/9509089].
1 Moreprecisely,
(:A(x,k,q,p):,:B(x,k,q,p):)c=:{A(x,k,q,p),B(x,k,q,p)}: .
