We argue with claims of the paper [Agostini F., Caprara S. and Ciccotti G.,
Europhys. Lett. EPL, 78 (2007) Art. 30001, 6] that the quantum-classic bracket
introduced in [arXiv:quant-ph/0506122] produces "artificial coupling" and has
"genuinely classical nature".
  Keywords: p-mechanics, quantum, classic, commutator, Poisson bracket, mixing,
coupling, semi-classicalComment: 2 pages, EPL2 style; v2: numerous improvement

Kisil, Vladimir V.

English

arXiv

V. V. Kisil

EDP Sciences OAI-PMH repository (1.2.0)

Comment on “Do we have a consistent non-adiabatic  quantum-classical mechanics?” 
by Agostini F. et al.

We argue with claims of the paper [Agostini F., Caprara S. and Ciccotti G., Europhys. Lett. EPL, 78 (2007) Art. 30001, 6] that the quantum-classic bracket introduced in [arXiv:quant-ph/0506122] produces "artificial coupling" and has "genuinely classical nature"

Kisil, VV

White Rose Research Online

This is a repository copy of Comment on "Do we have a consistent non-adiabatic quantum-classical mechanics?" by Agostini F. et al..White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/111533/Version: Accepted VersionArticle:Kisil, VV orcid.org/0000-0002-6593-6147 (2010) Comment on "Do we have a consistent non-adiabatic quantum-classical mechanics?" by Agostini F. et al. EPL, 89 (5). 50005. p. 50005. ISSN 0295-5075 https://doi.org/10.1209/0295-5075/89/50005This article is protected by copyright. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. The following article appeared in EPL and may be found at https://doi.org/10.1209/0295-5075/89/50005. Uploaded in accordance with the publisher's self-archiving policy.eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. arXiv:0907.0855v2  [quant-ph]  29 Dec 2009epl draftCommentComment on ‘Do we have a consistent non-adiabatic quantum-classical mechanics?’Vladimir V. Kisil(a), kisilv@maths.leeds.ac.ukSchool of Mathematics, University of Leeds, Leeds LS2 9JT, UK. Web: http://maths.leeds.ac.uk/~kisilv/PACS 02.70.Ns – Molecular dynamics and particle methodsPACS 03.65.Sq – Semiclassical theories and applicationsPACS 31.15.Qg – Molecular dynamics and other numerical methodsIntroduction. – This is a comment on the paper [1],which evaluates the quantum-classical (QC) bracket:[K1, K2]qc =1ih[K1, K2] +12({K1, K2} − {K2, K1})− i∂h2 [K1, K2]|h2=0 , (1)introduced in paper [4, (26)]. The authors in [1] claimedthat the QC bracket (1) exhibits:• an artificial coupling property (i.e., coupling betweenthe subsystems in the absence of an interaction);• a genuinely classical nature (i.e., the apparent mixedquantum classical form reduces to a purely classicalform for both subsystems).The assessment in [1] oversaw the following points:1. QC bracket (1) is the image of the universalbracket [4, (22)]:{[k1, k2]} = (k1 ∗ k2 − k2 ∗ k1)(A1 + A2), (2)under QC representation [4, (20)]. The universalbracket consists of convolution commutator and an-tiderivative operators [4, (12)]. QC bracket requiresconsideration of the first jet space [4]: the bracket isdetermined not only by their values of observables ath2 = 0 but also by the values of their first derivativewith respect to h2 at zero (see the last term in (1)).2. The derivation QC bracket (1) is independent of p-mechanisation procedure introduced in [4, (23)]:qj 7→ Qj = δ′xj(g1; g2), (3)p1 7→ Pj = χ′sk(s1 + s2) ∗ δ′yj(g1; g2), (4)where j = 1, 2 and k = 3 − j.(a)On leave from Odessa UniversityHere p-mechanisation [3, § 3.3], as an analog of quantisa-tion, is a prescription how to build p-mechanical observ-ables out of classical ones. It may not be very explicitin [4], but the deduction of the bracket (1) is compati-ble with different choices of p-mechanisation, however thevalue of the bracket will be different, see Exs. 3 and 4below. To illustrate this in the present comment we usep-mechanisation given by the Weyl (symmetric) calculusbased on the following correspondence, cf. [4, (23)], [1,(19)] and (3)–(4):qj 7→ Qj = δ′xj(g1; g2), pj 7→ Pj = δ′yj(g1; g2), (5)Then the quantum-quantum image of the universalbracket (2) of the respective coordinate and momentumobservables is:[Qj , Pj ]qq =h1 + h2hkI, k = 3 − j. (6)Now we review the above two claims from the paper [1].Artificial coupling property. – There is the fol-lowing claim in [1, 3001-p3]: “It must be underlined thateq. (16) describes an artificial interaction even if the twosystems are not coupled by the Hamiltonian.” This cou-pling property is attributed to the fact that quantum-quantum bracket in [4, (25)] and [1, (16)] always containsboth Planck’s constants h1 and h2, which are generatedby the presence of both antiderivative operators in thedefinition of universal bracket (2).Example 1. In order to exam the claim let us consideran uncoupled Hamiltonian H(q1, p1, q2, p2) = H1(q1, p1)+H2(q2, p2). The p-mechanisation (as well as quantisation)is a linear map [3, § 3.3], thus this uncoupled structurewill be preserved. Let B̂ be an observable depending onlyfrom X̂2 and D̂2, thus it will commute with H1. Thereforep-1Commentthe commutator of B and H will be the same as B andH2. The QC bracket is an image under a representationof the usual commutator, thus the universal bracket (2) ofB and H will be the same as B and H2. Consequently theĤ1 will not affect the dynamics of such an observable B̂.Therefore there is no coupling in the following meaning:arbitrary change of the Hamiltonian H1 of the first sub-system will not affect dynamics of any observable buildfrom coordinates and momenta of the second system only.Genuinely classical nature. – The paper [1, 3001-p3] said “In ref. [8] it was suggested that the dynamicalequation (16), in the limit h1 = h and h2 → 0, yields aQC dynamics.” However the derivation in [4] of the QCbracket intentionally avoids any kind of semiclassic lim-its due to its potential danger, see such an attempt in [2]and Example 2 below. The actual method evaluates theimage of the universal bracket (2) under the QC represen-tation [4, (20)] of the group Dm.The paper [1, 3001-p4] “corrected” the original deriva-tion of QC bracket replacing the initial set of Planck con-stants h1 and to h2 by the new one heff defined by theexpression:1heff=1h1+1h2. (7)However this transformation is singular for h1h2 = 0 andneeds special clarifications how to proceed for such values.Example 2. Let us consider the transformation Uh :f(x, y) 7→ f(hx, 1hy), which is a unitary operatorL2(R2) → L2(R2) for any h > 0. However this does notallow us “to take the limit h → 0” through the straight-forward substitution h = 0.Furthermore the paper [1, 3001-p3] claims that “we haveshown that the equation of motion (16) does not lead toa non-trivial QC limit”. However, this is caused by p-mechanisation (3)–(4), cf. the next two examples.Example 3. Let B1 and B2 are squares of coordinate Qand momentum P observables (of the quantum subsys-tem) respectively. Under p-mechanisation (3)–(4) theyare represented by squares of corresponding convolutions.Then the commutator (first term of bracket (1)) of theirQC representations is zero, the second term in (1)) van-ishes since no classical observables present, and the thirdtermin (1)) is equal to QC image of the observable 4QP .Thus the total bracket is indeed the same as the Poissonbracket for those observables.Let us examine the above claim for the p-mechanisation (5) and assume that two p-mechanical ob-servables B1 and B2, that is two convolutions on the groupDn [4, p. 876], for any fixed g1 are multiples of the deltafunction in g2, e.g. as in Ex. 3. Under the QC represen-tation ρ(h;q,p) [4, (20)] those observables become opera-tors ρ(h;q,p)(B1) and ρ(h;q,p)(B2) on the state space for thequantum subsystem without any dependence from classi-cal coordinates p, q and the respective Planck constanth2. Correspondingly the second and the third terms ofthe bracket (1) vanish and this bracket is equal to the(quantum) commutator 1ih [ρ(h;q,p)(B1), ρ(h;q,p)(B2)].Therefore if we admit the claim [1, 3001-p3] that QCbracket (1) always coincides with the purely classic Pois-son bracket, then we have to accept that any quantumcommutator is always equal to the Poisson bracket.Example 4. Under p-mechanisation (5) the squares of mo-mentum and coordinates from Ex. 3 are represented byconvolutions with kernels δ′′x1x1(g1; g2) and δ′′y1y1(g1; g2).Their commutator on the group D1 is 4δ′′′x1y1s1 + 2δ′′s1s1.Thus the universal bracket (2) is{[B1, B2]} = 4δ′′x1y1+ 2δ′s1 + (4δ′′′x1y1s1+ 2δ′′s1s1)A2. (8)In the QC representation of D1 the last term of the sumvanishes and two first terms produce 4QP + 2ihI. This isthe quantum commutator of Q2 and P 2 times 1ih . Thereis no unitary representation to get rid of the purely imag-inary term 2ihI in order to reduce the QC bracket of B1and B2 to the value 4QP of their Poisson bracket.Conclusion. – In this paper we demonstrated thatthe QC bracket (1) does not possess itself two propertiesof “artificial coupling” and “genuinely classical nature” asclaimed in [1]. Unfortunately the claims [1] were uncriti-cally translated by some other authors, see [5, 6].We showed that for a decoupled Hamiltonian the dy-namics of observables localised in one subsystem is unaf-fected by the Hamiltonian of the other subsystem. The“classical nature” described in [1] is rooted in the p-mechanisation used in [4] and does not appear with otherchoice of p-mechanical observables.The main conclusion of the commented paper [1] is:“We suggest that a different Ansatz for the equations ofmotion, could indeed produce non-trivial QC equations”.This comment is aimed to clarify possible directions forsuch a search.Acknowledgements. – I am grateful to Dr. Freder-ica Agostini for useful discussions and providing me witha part of her unpublished thesis. Prof. O.V. Prezhdo andan anonymous referee made suggestions, which improvedpresentation in this comment.REFERENCES[1] Agostini F., Caprara S. and Ciccotti G., Europhys.Lett. EPL , 78 (2007) Art. 30001, 6 .[2] Prezhdo O. V. and Kisil V. V., Phys. Rev. A (3) , 56(1997) 162 arXiv:quant-ph/9610016.[3] Kisil V. V., J. Phys. A , 37 (2004) 183arXiv:quant-ph/0212101, On-line. .[4] Kisil V. V., Europhys. Lett. , 72 (2005) 873arXiv:quant-ph/0506122, On-line.[5] Hall M. J. W., Physical Review A, 78 (2008) 042104.[6] Zhan F., Lin Y. and Wu B., Journal of Chemical Physics,128 (2008) 315204.p-2

Comment on "Do we have a consistent non-adiabatic quantum-classical mechanics?" by Agostini F. et al.

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