Consistent classical and quantum mixed dynamics

Transcription

1 Consistent classical an quantum mixe ynamics Author J. W. Hall, Michael Publishe 2008 Journal Title Physical Review A DOI Copyright Statement 2008 American Physical Society. This is the author-manuscript version of this paper. Reprouce in accorance with the copyright policy of the publisher. Please refer to the journal's website for access to the efinitive, publishe version. Downloae from Griffith Research Online

2 Consistent classical an quantum mixe ynamics Michael J. W. Hall Theoretical Physics, IAS, Australian National University, Canberra ACT 0200, Australia A recent proposal for mixe ynamics of classical an quantum ensembles is shown, in contrast to other proposals, to satisfy the minimal algebraic requirements propose by Salceo for any consistent formulation of such ynamics. Generalise Ehrenfest relations for the expectation values of classical an quantum observables are also obtaine. It is further shown that aitional esirable requirements, relate to separability, may be satisfie uner the assumption that only the configuration of the classical component is irectly accessible to measurement, eg, via a classical pointer. Although the mixe ynamics is formulate in terms of ensembles on configuration space, thermoynamic mixtures of such ensembles may be efine which are equivalent to canonical phase space ensembles on the classical sector. Hence, the formulation appears to be both consistent an physically complete. PACS numbers: Ca,03.65.Ta I. INTRODUCTION Many proposals have been mae for mixing classical an quantum ynamics [1, 2]. These proposals may be broaly classifie into mean-fiel, phase-space an trajectory categories, an all fail some important criterion - such as conservation of energy an probability, back-reaction by the quantum component on the classical component, positivity of probabilities, correct equations of motion in the limit of no interaction, an escribing all interactions of physical interest [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]). Hence, while often of practical interest for calculations in molecular ynamics [10, 11, 12], the above proposals have not le to a formulation of mixe classical an quantum ynamics that may be regare as physically funamental. Such a formulation woul not only be of interest for making physically-consistent numerical calculations, but also for moelling classical measurement apparatuses an coupling quantum systems to classical spacetime metrics [4]. Recently, a new proposal has been mae that falls outsie the above categories [13]. It is base on the escription of physical systems by ensembles on configuration space, an satisfies all of the abovementione criteria. Moreover, it has been successfully applie to iscuss position an spin measurements, interacting classical an quantum oscillators, an the coupling of quantum fiels to classical spacetime [13, 14]. However, while this configuration-ensemble approach is therefore a promising formulation of mixe ynamics, only particular observable properties, such as position, momentum, energy an angular momentum, have been iscusse in any etail. Hence, further investigation is necessary to etermine whether it is fully self-consistent. In this regar, Salceo has recently specifie two minimal requirements for a consistent classical-quantum formulation [10]: (i) a Lie bracket may be efine on the set of observables, an (ii) the Lie bracket is equivalent to the classical Poisson bracket for any two classical observables, an to (i h) 1 times the quantum commutator for any two quantum observables. These have been previously justifie on physical grouns by Caro an Salceo [6], an are higly nontrivial: none of the abovementione proposals, in those cases where they have been sufficiently evelope to ientify the general classical an quantum observables of the theory, satisfy both requirements [6, 10, 11]. These minimal requirements therefore provie a critical test for the configuration-ensemble approach to mixe ynamics. In Sec. II it is shown that the configuration-ensemble formulation, unlike other proposals, oes pass the above test. In particular, a Lie algebra of observables may be efine that satisfies requirements (i) an (ii) above. It is further shown in Sec. II that the quantum Ehrenfest relations generalise to mixe systems, an in particular that the expectation values for the position an momentum observables of linearly-couple classical an quantum oscillators obey the classical equations of motion. It follows that the configuration-ensemble formulation also satisfies a efinite benchmark... for an acceptable classical-quantum hybri formalism propose by Peres an Terno [7]. In Sec. III it is emonstrate that the aitional reasonable requirement (iii) The classical configuration is invariant uner any canonical transformation applie to the quantum component, an vice versa is also satisfie. This requirement is weaker than a relate requirement, that the Lie bracket vanishes for all pairs of classical an quantum observables, previously propose by Caro an Salceo [6] (see equation (18) thereof), but is sufficient if the physically reasonable assumption is mae that only the configuration of the clas-

3 2 sical component is irectly accessible by measurement (eg, via a classical pointer). Finally, in Sec. IV it is shown that the configurationensemble approach is consistent with thermoynamics. In particular, a generalise canonical ensemble may be efine as a suitable mixture of istinguishable stationary ensembles on configuration space, which is equivalent to the usual canonical ensemble on phase space for any semi-ergoic classical system, an to the usual canonical ensemble on Hilbert space for any quantum system. This emonstrates that the formulation of the approach on configuration space is not a barrier to escribing all physically relevant systems (an also implies classical statistical mechanics may be given a Hamilton-Jacobi formulation). II. OBSERVABLES IN THE CONFIGURATION-ENSEMBLE APPROACH A. General observables The escription of physical systems by ensembles on configuration space may be introuce at quite a funamental an generic level [13, 15]. The starting point is simply a probability ensity P on the configuration space of the system, the ynamics of which is assume to satisfy an action principle. Thus, there is a canonically conjugate quantity, S, on the configuration space, an an ensemble Hamiltonian, H[P, S], such that P t = δ H δs, S t = δ H δp. (1) Here δ/δf enotes the appropriate variational erivative on the configuration space. For the particular case of a continuous configuration space, inexe by position coorinate ξ, an a functional L[f] of the form L[f] = ξ F(f, f, ξ), one has the useful formula [16] δl/δf = F/ f [ F/ ( f)]. As a simple example, the ensemble Hamiltonian H Q = q P [ S 2 2m + h2 P 2 ] 8m P 2 + V (q) escribes a quantum spin-zero particle of mass m, moving uner a potential V (q). In particular, the equations of motion (1) reuce in this case to the real an imaginary parts of the Schröinger equation i h( ψ/ t) = [ ( h 2 /2m) 2 + V ] ψ, where ψ(q, t) := P 1/2 e is/ h. Moreover, if the limit h 0 is taken in the ensemble Hamiltonian H Q, the equations of motion for P an S reuce to the Hamilton-Jacobi an continuity equations for an ensemble of classical particles [13]. Thus, in the configuration-ensemble approach, the primary ifference between quantum an classical evolution lies in the choice of the ensemble Hamiltonian. It may be note that the numerical value of the above ensemble Hamiltonian is just the quantum average energy, i.e., H Q [P, S] = q ψ (q) [ ( h 2 /2m) 2 + V ] ψ(q), as first emonstrate by Maelung in Eq. (5 ) of Ref. [17]. The fact that the quantum equations of motion can be generate by regaring P an S as canonically conjugate fiels appears to have first been note by Bohm, following Eq. (9) of Ref. [18]. Examples of mixe quantum-classical ensemble Hamiltonians have been given elsewhere [13, 14], an also in the following sections. While a number of general properties an applications of the configuration ensemble formalism have been previously consiere [13], the escription of observables has only been briefly allue to, with emphasis on particular quantities such as position, momentum, spin an energy. The general escription is therefore aresse here, to enable the properties of observables for mixe configuration ensembles to be compare against the two minimal requirements iscusse in the Introuction. Note first that the conjugate pair (P, S) allows a Poisson bracket to be efine for any two functionals A[P, S] an B[P, S], via [16] {A, B} := ξ ( δa δb δp δs δb δp ) δa δs (2) (where integration is replace by summation over any iscrete parts of the configuration space). Thus, the equations of motion (1) may be written as P/ t = {P, H} an S/ t = {S, H}, an more generally it follows that A/t = {A, H} + A/ t (3) for any functional A[P, S, t]. The Poisson bracket is well known to be a Lie bracket [16], an in particular is linear, antisymmetric, an satisfies the Jacobi ientity. Hence, the first minimal consistency requirement, given in Sec. I above, is automatically satisfie by choosing the observables to be any set of functionals of P an S that is close with respect to the Poisson bracket. It is important to note that an arbitrary functional A[P, S] will not be allowable as an observable in general. This is analogous to the restriction of expectation values to bilinear forms of linear Hermitian operators in stanar quantum mechanics, even though a commutator bracket can be more generally efine [19]. For example, the infinitesimal canonical transformation P P + ǫ δa/δs, S S ǫ δa/δp generate by any observable A must preserve the interpretation of P as a probability ensity, i.e., the normalisation an positivity of P must be preserve. This imposes the respective funamental conitions [13] A[P, S + c] = A[P, S], δa/δs = 0 if P(ξ) = 0, (4)

4 3 where c is an arbitrary constant. Note that each of the above conitions is consistent with the Poisson bracket. First, efining I[P, S] := ξ P, the normalisation conition is simply the requirement that I is invariant uner allowe canonical transformations, i.e., that δi = ǫ{i, A} = 0. Hence, if it hols for two observables A an B, then it automatically hols for {A, B} via the Jacobi ientity, since {I, {A, B}} = {A, {B, I}} {B, {I, A}} = 0. Similarly, the positivity conition may be rewritten as δp = ǫ{p, A} = 0 whenever P(ξ) = 0 (otherwise P(ξ) can be ecrease below 0 by choosing the sign of ǫ appropriately), which again hols for {A, B}, if it hols for A an B, as a consequence of the Jacobi ientity. It follows that the observables corresponing to a configuration-ensemble escription shoul be chosen as some set of functionals satisfying the normalisation an positivity constraints (4), that is close with respect to the Poisson bracket in Eq. (2). Uner any such choice, the first minimal requirement (i) in Sec. I is automatically satisfie, where the Lie bracket is ientifie with the Poisson bracket. B. Classical an quantum observables To etermine whether the secon minimal requirement (ii) in Sec. I is also satisfie, it is necessary to first efine the classical an quantum observables of a mixe quantum-classical ensemble. Consier, therefore, a mixe quantum-classical ensemble, inexe by the joint configuration ξ = (q, x), where q labels the quantum configuration an x labels the classical configuration. For example, q may refer to the position of a quantum system, or, more generally, label some complete set of kets { q } [15]. In contrast, x will always be taken here to refer to some continuous set of coorinates on a classical configuration space (to enable the iscussion of classical phase space relations). Hence, the mixe ensemble is escribe by two conjugate quantities P(q, x, t) an S(q, x, t). First, for any real classical phase space function f(x, k), efine the corresponing classical observable C f by C f := q xp f(x, x S) (5) (where integration with respect to q is replace by summation over any iscrete portions of the quantum configuration space). This is similar in form to a classical average, an hence the numerical value of C f will be ientifie with the preicte expectation value of the corresponing function f(x, k), i.e., C f f. It is easily checke that C f satisfies the require normalisation an positivity conitions (4). Note that, for classical observables, x S plays the role of a momentum associate with the configuration (q, x). The Poisson bracket of any two classical observables C f an C g follows, using Eq. (2) an integration by parts with respect to x, as {C f, C g } = q x [ f x (P k g) + g x (P k f)] = q xp ( x f k g x g k f) = C {f,g}, (6) where all quantities in the integrans are evaluate at k = x S, an {f, g} enotes the usual Poisson bracket for phase space functions. Hence, the Lie bracket for classical observables is equivalent to the usual phase space Poisson bracket, as require. Given that the observables are evaluate on a configuration space, rather than on a phase space, this is a somewhat remarkable result. Secon, for any Hermitian operator M acting on the Hilbert space spanne by the kets { q }, efine the corresponing quantum observable Q M by Q M := q xψ (q, x)mψ(q, x) (7) = q q x(pp ) 1/2 e i(s S )/ h q M q, where ψ(q, x) := P(q, x) 1/2 e is(q,x)/ h, P = P(x, q), P = P(x, q ), etc (an where integration with respect to q an q is replace by summation over any iscrete portions of the quantum configuration space). This is similar in form to a quantum average, with respect to the hybri wavefunction ψ(q, x), an hence the numerical value of Q M will be ientifie with the preicte expectation value of the corresponing operator M, i.e., Q M M. It follows immeiately from the secon equality in Eq. (7) that Q M satisfies the normalisation an positivity conitions (4). To evaluate the Poisson bracket of any two quantum observables Q M an Q N, it is convenient to first express the Poisson bracket in terms of the hybri wavefunction ψ(q, x) an its complex conjugate ψ (q, x). One has in particular for any real functional A[P, S] that δa δp = ψ δa P δψ + ψ P δa δψ = 1 ψ ψ Re { ψ δa δψ }, δa δs = ψ { δa S δψ + ψ δa = Im ψ S δψ 2 h δa }, δψ an hence, noting a + bc = Im{(a + ib)(c i)}, that {A, B} = 2 h { Im q x δa } δb δψ δψ. (8)

5 4 Recalling that M an N are Hermitian, so that ψ Mψ may be replace by (Mψ) ψ in equation (7), it immeiately follows that {Q M, Q N } = 2 h { } Im q x(mψ) Nψ = Q [M,N]/(i h), (9) where [M, N] enotes the usual quantum commutator MN NM. Hence, the Lie bracket for quantum observables is equivalent to the usual quantum commutator, as require. Eqs. (2), (6) an (9) are the main results of this section. They imply that any set of observables containing the quantum observables C f an the quantum observables Q M, that is close uner the Poisson bracket, will satisfy both of the minimal requirements (i) an (ii) in Sec. I for a consistent mixe quantum-classical formulation. In the following sections, it will be shown that further esirable properties can also be accommoate within the configuration-ensemble formulation. C. Example: generalise Ehrenfest relations Consier a mixe quantum-classical ensemble, corresponing to a quantum particle of mass m interacting with a classical particle of mass M via a potential V (q, x), where q an x enote the position configurations of the quantum an classical particles respectively. For simplicity, it will be assume that both q an x are oneimensional. The mixe ensemble is therefore escribe by a probability ensity P(q, x), a canonically conjugate fiel S(q, x), an an ensemble Hamiltonian of the form [13] [ ( q S) H 2 QC [P, S] := q xp 2m + h2 ( q P) 2 8m P 2 + ( xs) 2 2M ] + V (q, x), (10) where q an x enote the partial erivatives with respect to q an x respectively. Now, the expectation values of the classical position an momentum observables follow from Eq. (5) as x = C x = q xp x, k = C k = q xp x S, an the expectation values of the quantum position an momentum observables follow from Eq. (7) as q = Q q = q xp q, p = Q p = q xp q S. The evolution of these expectation values may be calculate via Eqs. (3) an (10), an, as shown below, one fins t x = M 1 k, t k = xv, (11) t q = m 1 p, t p = qv. (12) These results are a clear generalisation of the stanar Ehrenfest relations for quantum systems, an similarly imply that the centroi of a narrow initial probability ensity P(q, x) will evolve classically, for short timescales at least. Note that for the case of linearly-couple classical an quantum oscillators, with V (q, x) = 1 2 mω2 q MΩ2 x 2 + Kqx, Eqs. (11) an (12) simplify to give the close set of equations t x = M 1 k, t k = MΩ2 x K q, t q = m 1 p, t p = mω2 q K x. Thus, the centroi of the probability ensity obeys precisely the same equations of motion as two fully classical oscillators (or two fully quantum oscillators). As note in the Introuction, this corresponence principle has been previously propose by Peres an Terno as a benchmark for any acceptable classical-quantum hybri formalism [7]. The above result therefore further supports the consistency of the configuration-ensemble formulation of mixe ynamics. To emonstrate the generalise Ehrenfest relations (11) an (12), note first that x has no explicit epenence on S or t, an hence from Eqs. (2), (3) an (10) one has t x = q xx δ H QC δs = q xx [ m 1 q (P q S) + M 1 x (P x S) ]. Applying integration by parts with respect to q an x to the first an secon terms, respectively, yiels the first relation in Eq. (11). The first relation in Eq. (12) is obtaine in a similar manner. To obtain the remaining relations, note that δ H QC δp = ( qs) 2 2m + ( xs) 2 2M + V + h2 P 1/2 δ 2m P δp 1/2 q x( q P 1/2 ) 2. The final term simplifies to ( h 2 /2m)( q 2P 1/2 )/P 1/2, which may be recognise as the so-calle quantum potential [18], an Eqs. (2), (3) an (10) then yiel t k = + ( q x( x S) [ q P ) ( )] qs P x S + x m M [ ( q S) 2 q x( x P) 2m + ( xs) 2 ] 2M + V ( h 2 /2m) q x( x P)( qp 2 1/2 )/P 1/2.

6 5 Applying integration by parts to the first line, with respect to q an x for the first an secon terms thereof respectively, an applying integration by parts to the secon line with respect to x, leas to cancellation of all terms involving S, with a term qxp x V remaining. Since the integral in the thir line is proportional to q x( x P 1/2 )( 2 qp 1/2 ) = 1 2 q x x ( q P 1/2) 2 which vanishes ientically, the secon relation in Eq. (11) immeiately follows. Finally, the secon relation in Eq. (12) is obtaine by similar reasoning (replacing q by x in appropriate places). III. SEPARABILITY AND MEASUREMENT In contrast to other proposals for mixe ynamics, the configuration-ensemble approach has been shown to pass the critical test of meeting the two minimal consistency requirements (i) an (ii) in Sec. I. However, while the approach thereby gains a special status, there are a number of other requirements that may reasonably be expecte of a physical theory. Some of these, relate to local aspects of separability an measurement, are iscusse below. A. Configuration separability It is natural to expect that the classical configuration is invariant uner any canonical transformations applie solely to the quantum system, an vice versa. This means, in particular, that a measurement of the classical configuration cannot etect whether or not a transformation has been applie to the quantum system, an vice versa, when the components are noninteracting. This property of configuration separability correspons to requirement (iii) of the Introuction. To show that this property inee hols, consier first a canonical transformation generate by an arbitrary quantum observable Q M. Any classical observable epening only on the classical configuration x is of the form C g with g = g(x) (eg, C x ). One thus has C g = q xpg(x) = q xψ ψ g(x), an it follows immeiately via Eq. (8) that {C g(x), Q M } = 2 h { } Im q xψ g(x)mψ = 0, (13) since M acts only on the quantum component of the hybri wavefunction an so commutes with g(x). Hence, the expectation value of C g is not change by the transformation. Similarly, consier a canonical transformation of the classical component, generate by an arbitrary classical observable C f. Any quantum observable epening only on the quantum configuration q is of the form Q G with G = G(q). Thus, Q G = q xψ G(q)ψ = q xp G(q), an it follows immeiately via Eq. (2) that {Q G(q), C f } = xq G(q) x (P k f) = 0, (14) using integration by parts with respect to x. Eqs (13) an (14) show that configuration separability is satisfie. Further, if q m an p m label the mth position an momentum coorinates of a quantum particle, an x m an k m similarly label the mth position an momentum coorinates of a classical particle, then, noting that Q pm = q xp( S/ q m ) an using Eq. (2), ( {C km, Q pn } = q x S P + S ) P = 0, x m q n q n x m using integration by parts with respect to q an x on the first an secon terms, respectively (it can also be shown that {C km, Q M } = {C f, Q pm } = 0). Together with Eqs. (13) an (14), this yiels the properties {C xm, Q qn } = {C xm, Q pn } = 0, (15) {C km, Q qn } = {C km, Q pn } = 0, (16) for all m an n. Noting Eqs. (6) an (9), these properties correspon to conitions assume by Salceo [5] (in Eq. (1) thereof) for proving a no-go theorem for mixe classical an quantum ynamics. However, this theorem is inapplicable here, as a further require conition, that observables are generate by a prouct algebra, oes not hol (see also Sec. V below). B. Strong separability It is important to note that while configuration separability hols, as per Eqs (13) an (14), the stronger separability requirement {C f, Q M } = 0? (17) oes not hol for arbitrary classical an quantum observables. For example, for classical an quantum particles having masses m an m respectively, the Poisson bracket of the kinetic energy observables corresponing to f(x, k) = k 2 /(2m) an M = h 2 q 2 /(2m ) is given by {C f, Q M } = h2 2mm q xp( x S) x (P 1/2 2 qp 1/2 ), which oes not vanish ientically for arbitrary P an S. However, it will be argue here that the violation of strong separability oes not necessarily lea to any physical inconsistencies. First, it shoul be note that in the particular case where the mixe system escribes quantum matter couple to classical spacetime [4, 13], a failure of Eq. (17)

7 6 is irrelevant to separability issues, as there is no sense in which interaction between the systems can be switche off - matter bens space an space curves matter, an so a change in one component is fully expecte to rive a change in the other component. This correspons to the irect coupling of the metric tensor to the fiels in the corresponing ensemble Hamiltonian [13], an there is no sense in which this ensemble Hamiltonian can be reuce to a simple sum of a classical an a quantum contribution. Secon, it is important to note that Eq. (17) oes hol in the special case that the classical an quantum components are inepenent [13], i.e., when P(q, x) = P Q (q)p C (x), S(q, x) = S Q (q)+s C (x). (18) Thus, inepenent ensembles are fully escribe by two conjugate pairs (P Q, S Q ) an (P C, S C ) corresponing to the quantum an classical components respectively. To emonstrate Eq. (17) for this case, note first from Eq. (5) that C f = q xψ ψ f(x, k), k = h ( x ψ 2i ψ xψ ) ψ, an hence that δc f δψ = ψ f + ψ ψ ( k f) k ψ ( ) x ψ k ψ ( k f) ( x ψ) = ψ f h ψ 2i ψ ( kf x ψ) h 2i x (ψ k f). Moreover, from Eq. (7) one has δq M /δψ = Mψ. Since Eq. (18) is equivalent to a factorisation ψ(q, x) = ψ Q (q)ψ C (x) of the hybri wavefunction, implying that k x S C is inepenent of q, it follows that q x δc { f δq M δψ δψ = q ψq Mψ Q xψc ψ Cf h 2i } x [ψc ( kf x ψ C ) + ψ C x (ψc kf)] = q ψq Mψ Q xψc ψ Cf, where integration by parts has been use to obtain the final result. This expression is clearly real, implying immeiately from Eq. (8) that {C f, Q M } = 0, as require. Thus, strong separability is satisfie for inepenent ensembles. Further, since such ensembles remain inepenent uner ensemble Hamiltonians of the form H = C + Q (for some pair of classical an quantum observables C an Q), strong separability hols at all times for noninteracting inepenent ensembles. More generally, the violation of strong separability poses an apparent problem: a transformation acting solely on one component can lea to changes in the expectation values of observables in the other component. However, this problem may be resolve by imposing a physically reasonable restriction on the type of observables which are irectly accessible to measurement. For example, consier the assumption: (A) The only observables accessible to irect measurement are classical configuration observables. Here configuration observables are those which epen only on the configuration of the classical system, i.e., of the form C g(x). Uner this assumption, information about any other classical or quantum observables is obtainable only inirectly, by coupling them to such a classical configuration observable. Empirically, this assumption is a very reasonable one to make, as in practice all measurements o reuce to the observation of some position or configuration of a classical apparatus (such as the position of a pointer). It is of interest to note that a similar assumption (for ifferent reasons) is also mae in the ebroglie-bohm interpretation of stanar quantum mechanics [18, 20]. Now, uner assumption (A), only changes in the expectation values of classical configuration observables are irectly observable. However, from Eq. (13) above, such expectation values are invariant uner any canonical transformations that act solely on the quantum component. Hence, the assumption implies that violations of strong separability are simply not observable. C. Measurement aspects Examples of measurements of position an spin on quantum ensembles, via interaction with an ensemble of classical pointers, an the corresponing ecoherence of the quantum component relative to the classical component, have been escribe previously within the configuration-ensemble approach [13]. Here it is note there is a simple moel for escribing the inirect measurement of any quantum observable, via a irect measurement of the configuration of a classical measuring apparatus. The existence of such a moel inicates that assumption (A), iscusse in Sec. III B above, oes not restrict the types of information that can be gaine by measurement. In particular, the measurement of an arbitrary quantum observable, Q M, may be moelle by the ensemble Hamiltonian H := H 0 + κ(t) ( h q xψ (q, x) i x ) Mψ(q, x), where κ(t) vanishes outsie the measurement perio, an x enotes the position of a one-imensional classical pointer (with integration over q replace by summation

8 7 over any iscrete values). Note that the interaction term satisfies the normalisation an positivity constraints (4). As will be shown, this term correlates the eigenvalues of M with the position of the pointer, in a manner rather similar to purely quantum moels of measurement. It is convenient to assume that the measurement takes place over a sufficiently short time perio, [0, T], such that H 0 can be ignore uring the measurement. The equations of motion uring the interaction then follow via Eqs. (1) an (8) as being equivalent to the hybri Schröinger equation i h ψ t = κ(t) ( h i x ) Mψ. For an initially inepenent ensemble at time t = 0, as per Eq. (18), this equation may be trivially integrate to give ψ(q, x, T) = n c n ψ C (x Kλ n ) q n (19) at the en of the measurement interaction, where K = T 0 t κ(t), n enotes the eigenstate corresponing to eigenvalue λ n of M, an c n = n ψ Q (with ψ Q efine via q ψ Q := ψ Q (q)). It has been assume for simplicity here that M is nonegenerate (the egenerate case is consiere further below). The pointer probability istribution after measurement follows immeiately from Eq. (19) as P(x, T) = q ψ ψ = c n 2 P C (x Kλ n ). (20) n Hence, the initial pointer istribution is isplace by an amount Kλ n with probability c n 2, thus correlating the position of the pointer with the eigenvalues of M. In particular, choosing a sufficiently narrow initial istribution P C (x) (eg, a elta-function), the isplace istributions will be nonoverlapping (corresponing to a goo measurement), an eigenvalue λ n will be perfectly correlate with the measure pointer position. The above shows that the inirect measurement of any quantum observable may be moelle via interaction with a strictly classical measuring apparatus, followe by a irect measurement of the classical configuration. Note that collapse of the quantum component of the ensemble can also be moelle, if esire. Suppose in particular that the real position of the pointer is etermine to be x = a. This must correspon to just one of the nonoverlapping istributions P C (x Kλ n ), an upating P(q, x, T) via Bayes theorem implies, via Eqs. (19) an (20), that P a (q, x, T) = δ(x a)p(q, a, T)/P(a, T) = δ(x a) q n 2. Moreover, it is natural to upate the conjugate quantity S(q, x, t) via the minimal substitution S a (q, x, t) = S(q, a, T). Note that the collapse ensemble after measurement is thus inepenent as per Eq. (18) (with quantum component escribe by ψ a (q) = q n up to a phase factor). Hence, strong separability as per Eq. (17) is satisfie if the pointer an the quantum system o not interact after the measurement. Finally, for a egenerate operator M with eigenvalue ecomposition n λ ne n, similar results are obtaine, but with c n q n in Eq. (19) replace by q E n ψ Q, c n 2 by p n = ψ Q E n ψ Q, an the collapse quantum component by ψ a (q) = (p n ) 1/2 q E n ψ Q. IV. MIXTURES AND THERMODYNAMICS It has been emonstrate above that the configurationensemble approach provies a consistent formulation of mixe quantum an classical ynamics. However, given that this approach escribes classical ensembles via a configuration space, rather than a phase space, this raises a potential completeness issue: o all classical phase space ensembles have a counterpart in this approach? Further, given the lack of a natural phase space entropy, can the configuration-ensemble approach eal with thermal ensembles in a manner that is compatible with both classical an quantum thermoynamics? It is shown briefly below that both these questions have positive answers. The central concept require is that of a mixture of configuration ensembles. In particular, if a physical system is escribe by the configuration ensemble (P j, S j ) with prior probability p j, then it may be sai to correspon to the mixture {(P j, S j ); p j }. For quantum ensembles, such mixtures are conveniently represente by ensity operators. More generally, however, there is no similarly convenient representation. The average of any observable A[P, S] over a mixture is given by A = j p j A[P j, S j ]. (21) The first question pose above can now easily be answere, using the fact that any classical phase space point, γ = (x, k ), may be escribe by a classical configuration ensemble (P γ, S γ ), efine by P γ (x) := δ(x x ), S γ (x) := k x. In particular, the value of any classical observable C f, corresponing to the average value of f, follows via Eq. (5) as C f [P γ, S γ ] = f γ = f(x, k ). (22) It follows immeiately that any classical phase space ensemble, represente by some phase space ensity p(x, k ), may equivalently be escribe by the mixture {(P γ, S γ ); p} of classical configuration ensembles. To aress the secon question above, one further requires the notions of stationary an istinguishable

9 8 configuration ensembles. First, stationary ensembles are those for which all observable quantities are timeinepenent, an are characterise by the property [13] P/ t = 0, S/ t = E, (23) for some constant E. Secon, two configuration ensembles are efine to be istinguishable if there is some observable which can istinguish unambiguously between them, i.e., the ranges of the observable for each ensemble o not overlap. Thus, for example, two quantum ensembles are istinguishable if the corresponing wavefunctions are orthogonal, while two classical ensembles are istinguishable if the ranges of (x, k) over the supports of the ensembles are nonoverlapping (with k = x S). A thermal mixture may now be efine as a mixture of istinguishable stationary ensembles {(P, S); p(p, S H)} such that p(p, S H) e β H[P,S], β > 0. (24) This efinition is, for present purposes, justifie by its consequences, but it may also be motivate by appealing to properties of two istinct noninteracting systems in thermal equilibrium, escribe by joint ensemble Hamiltonian H T, for which one expects p(pp, S + S H T ) = p(p, S H)p(P, S H ), (25) for pairs (P, S), (P, S ) of stationary ensembles. For a quantum system with Hamiltonian operator H, the above efinition immeiately leas to the usual quantum canonical ensemble represente by the ensity operator proportional to e βh. It will be shown below that, for an ergoic classical system with phase space Hamiltonian H(x, k), the corresponing thermal mixture of configuration ensembles is equivalent to the classical canonical ensemble with phase space ensity proportional to e βh. Note this result generalises immeiately, via the factorisability of thermal mixtures in Eq. (25), to all semi-ergoic classical systems, i.e., to any classical system comprising noninteracting ergoic systems. Thus, for example, since a one-imensional oscillator is ergoic for any energy E, an higher imensional oscillators can be ecompose into noninteracting normal moes [16], any classical oscillator is a semi-ergoic system. In particular, the classical ensemble Hamiltonian is H[P, S] = C H = xp H[x, S], an it follows from Eqs. (1) an (23) that the stationary ensembles are then given by the solutions of the continuity an Hamilton-Jacobi equations x [P k H(x, x S)] = 0, H(x, x S) = E. Now, since H is time-inepenent, the general solution to the Hamilton-Jacobi equation for S is of the form W(x) Et, an generates a canonical transformation on phase space from (x, k) to a set of constants of the motion, which may be chosen as the intial values (x 0, k 0 ) at some fixe time [16]. One then has E = H(x t, k t ) = H(x 0, k 0 ), an a corresponing set of solutions S x0,k 0 (x, t) := W k0 (x) H(x 0, k 0 )t of the Hamilton- Jacobi equation (where x 0 = k0 W k0 [16]). Further, recalling that the classical velocity is ẋ t = k H, the above continuity equation simply requires that P(x) ẋ t is constant along any given trajectory (x t, k t ). One may therefore efine a corresponing set of solutions by P x0,k 0 (x) ẋ t 1 when x lies on the particular trajectory having initial values (x 0, k 0 ), an P x0,k 0 (x) = 0 elsewhere. The stationary ensembles (P x0,k 0, S x0,k 0 ) are all istinguishable, since they correspon to a set of istinct initial values (x 0, k 0 ). Hence, they are suitable for efining a thermal mixture. Further, by construction, one has via efinition (5) that C f [P x0,k 0, S x0,k 0 ] = x t ẋ t 1 f(x t, k t ) / x t ẋ t 1, where integration is along the trajectory efine by initial point (x 0, k 0 ). Changing the variable of integration to t then gives 1 T C f [P x0,k 0, S x0,k 0 ] = lim t f(x t, k t ), (26) T T 0 which is always well efine for ergoic systems. Further, for such systems the righthan sie is simply the microcanonical ensemble average of f(x, k), corresponing to constant energy E = H(x 0, k 0 ) [21]. Finally, noting H[P x0,k 0, S x0,k 0 ] = H[x 0, k 0 ] by construction, the associate thermal mixture in Eq. (24) is characterise by p(x 0, k 0 ) e βh[x0,k0], an it immeiately follows via Eqs. (21) an (26) that the average of f over the mixture is the usual canonical ensemble average, as require. V. DISCUSSION The configuration-ensemble approach satisfies the two minimal requirements for a consistent formulation of mixe ynamics, as shown in Sec. II. No other formulation appears to be known which passes this critical test. Moreover, while the approach oes not satisfy a strong separability conition for quantum an classical observables, it oes satisfy the weaker conition of configuration separability, as per Eqs. (13) an (14) of Sec. III. This conition is sufficient to avoi ifficulties in the case of quantum matter couple to a classical spacetime metric, an is also sufficient more generally if it is assume that only the classical configuration of a mixe ensemble is irectly accessible to measurement. It shoul further be note that the configurationensemble approach has a number of interesting applications outsie the omain of mixe ynamics. It also provies a basis for, eg, the erivation of classical an quantum equations of motion [22], a generalisation of

10 9 quantum superselection rules [15], an, as seen in Sec. IV above, a Hamilton-Jacobi approach to classical statistical mechanics. Hence, overall, the approach appears to be valuable in proviing a funamental tool for escribing physical systems, an merits further general investigation. It is of interest to remark on how the configurationensemble approach is able to avoi various no-go theorems on mixe ynamics in the literature [5, 6, 7, 9]. This is essentially ue to such theorems requiring a formal assumption that the set of observables can be be extene to form a prouct algebra, where the prouct A B is assume to satisfy C f C g = C fg, Q M Q N = Q MN, an some further property such as the Leibniz rule {A, B C} = {A, B} C + B {A, C}. However, the assumption of such a prouct algebra clearly goes beyon the omain of observable quantities (eg, the prouct of two Hermitian operators is not a Hermitian operator), an hence cannot be justifie on physical grouns. Thus, any import of such no-go theorems, for the configuration-ensemble approach, where no such prouct is efine or require, is purely formal in nature. For application to mixe ynamics, the set of observables must be chosen such that it contains the classical an quantum observables efine in Eqs. (5) an (7), with all members satisfying the normalisation an positivity conitions in Eq. (4). It must also, of course, contain the Poisson bracket of any two of its members - however, this can always be assure by replacing a given set by its closure uner the Poisson bracket operation. It is of interest to consier what further physical conitions might be impose on the set of observables. For example, for ensembles of interacting classical an quantum nonrelativistic particles, it is reasonable to require that the equations of motion are invariant uner Galilean transformations, leaing to the interesting property that the centre of mass an relative motions o not ecouple [13]. A more general conition that might be impose on observables is that they are homogenous of egree unity with respect to the probability ensity P, i.e., A[λP, S] = λa[p, S] (27) for all λ 0. Note that if this conition hols for two observables A an B, then it hols for the Poisson bracket {A, B}, as may be checke by irect substitution into Eq. (2). It is also easily verifie to hol for the classical an quantum observables efine in Eqs. (5) an (7). Differentiating Eq. (27) on both sies with respect to λ an choosing λ = 1 yiels the numerical ientity A[P, S] = ξ P (δa/δp) =: δa/δp, (28) i.e., A can be calculate by integrating over a local ensity on the configuration space. Thus, the homogeneity conition consistently allows observables to be interprete both as generators of canonical transformations an as expectation values. Finally, while the question of ecoherence has not been aresse in any etail here, it is worth noting that the configuration-ensemble approach provies at least two possibilities in this regar. First, for any mixe quantumclassical ensemble, one may efine a conitional quantum wavefunction ψ x (q) an corresponing ensity operator ρ Q C, which escribe the conitional ecoherence of the quantum component relative to the classical component [13]. Secon, while at any time the hybri wavefunction ψ(q, x, t) escribing a mixe quantum-classical system always has a ecomposition of the form ψ(q, x, t) = n pn (t) ψ C,n (x, t)ψ Q,n (q, t) (eg, a Schmit ecomposition), only at particular times (if at all, epening on the ensemble Hamiltonian), can it have such a ecomposition for which (i) the classical ensembles corresponing to ψ C,n (x, t) are classically istinguishable (see Sec. IV), an (ii) the quantum ensembles corresponing to ψ Q,n (q, t) are mutually orthogonal. Moreover, unlike a Schmit ecomposition, such a ecomposition woul be unique even for equal p n (t). Hence, ecoherence coul be moelle by imposing a spontaneous collapse of the ensemble at such well-efine times, similarly to the collapse moel in Sec. III C. Acknowlegments I am grateful to Marcel Reginatto for valuable iscussions, an for pointing out an error in an early raft. [1] For example, E.C.G. Suarshan, Pramana 6, 117 (1976); I.V. Aleksanrov, Z. Naturforsch. Teil A (1981); A. Anerson, Phys. Rev. Lett (1995); L. Diosi an J.J. Halliwell, Phys. Rev. Lett. 81, 2846 (1998); E. Ginensperger, C. Meier, an J.A. Beswick, J. Chem. Phys (2000); O.V. Prezho an C. Brooksby, Phys. Rev. Lett. 86, 3215 (2001); I. Burghart an G. Parlant, J. Chem. Phys. 120, 3055 (2004); V.V. Kisil, Europhys. Lett. 72, 873 (2005); O.V. Prezho, J. Chem. Phys (2006). [2] L. Diosi, N. Gisin, an W.T. Strunz, Phys. Rev. A 61, (2000). [3] T.N. Sherry an E.C.G. Suarshan, Phys. Rev. D 20, 857 (1979). [4] W. Boucher an J. Traschen, Phys. Rev. D 37, 3522 (1988); [5] L.L. Salceo, Phys. Rev. A 54, 3657 (1996). [6] J. Caro an L.L. Salceo, Phys. Rev. A (1999).

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