Born Rule in Quantum and Classical Mechanics

PHYSICAL REVIEW A 73, 052109 ͑2006͒

Born rule in quantum and classical mechanics

Paul Brumer and Jiangbin Gong Chemical Physics Theory Group and Center for Quantum Information and Quantum Control, University of Toronto, Toronto, Ontario, Canada M5S 3H6 ͑Received 17 January 2006; published 19 May 2006͒

Considerable effort has been devoted to deriving the Born rule ͓i.e., that ͉␺͑x͉͒2dx is the probability of ﬁnding a system, described by ␺, between x and x+dx͔ in quantum mechanics. Here we show that the Born rule is not solely quantum mechanical; rather, it arises naturally in the Hilbert-space formulation of classical mechanics as well. These results provide insights into the nature of the Born rule, and impact on its understanding in the framework of quantum mechanics.

DOI: 10.1103/PhysRevA.73.052109 PACS number͑s͒: 03.65.Sq, 03.65.Ca

I. INTRODUCTION will not help predict any experimentally new physics. How- The Born rule ͓1͔ postulates a connection between deter- ever, understanding the origin of the Born rule is important ministic quantum mechanics in a Hilbert-space formalism for isolating this postulate from other concepts in quantum with probabilistic predictions of measurement outcomes. It is mechanics and for understanding what is truly unique in typically stated ͓2͔ as follows ͑without considering degenera- quantum mechanics as compared with the classical physics. ͒ ˆ ͕͉ ͖͘ For example, based on the above-mentioned efforts to derive cies : if an observable O, with eigenstates Oi and spec- the Born rule ͓2,4,6,9–11͔, it seems now clear that the physi- ͕ ͖ trum Oi , is measured on a system described by the state cal origin of Born’s rule is unrelated to the details ͑e.g., wave vector ͉␺͘, the probability for the measurement to yield the function collapse͒ of quantum measurement processes. ͦ͗ ͉␺ͦ͘2 value Oi is given by Oi . Alternatively, in the density The main purpose of this work is to show that Born’s rule matrix formulation used below, this rule states that the prob- is not solely quantum mechanical and that it arises naturally ability is Tr͓␳␺␳ ͔, where ␳␺ =͉␺͗͘␺͉ and ␳ =͉O ͗͘O ͉. Most Oi Oi i i in the Hilbert space formulation of classical mechanics. As familiar is the textbook example that the probability of ob- such, the Born rule connecting probabilities with eigenvalues serving a system that is in a state ␺ in the coordinate range x and eigenfunctions is not as “quantum” as it sounds. Indeed, to x+dx is given by ͦ͗x͉␺ͦ͘2dx. Born’s rule appears as a quantum-classical correspondence, which played no role in fundamental postulate in quantum mechanics and is thus far Born’s original considerations ͓1,12͔, can then be arguably in agreement with experiment. Hence, there is intense inter- regarded as an interesting motivation for the Born rule in est in providing an underlying motivation for, or derivation quantum mechanics. Similarly, exposing the Born rule in of, this rule. classical mechanics should stimulate new routes to under- Gleason’s theorem ͓3͔, for example, provides a formal standing the physical origin of this rule in quantum mechan- motivation of the Born rule, but it is a purely mathematical ics. In particular, new and interesting questions can be asked result about vectors in Hilbert spaces and does not provide in connection with the previous derivations of the Born rule. insight into the physics of this postulate. For this reason there To demonstrate that the Born rule exists in both quantum have been several attempts to provide a physical derivation and classical mechanics we ͑1͒ recall that both quantum and of the Born rule. For example, Deutsch showed the possibil- classical mechanics can be formulated in the Hilbert space of ity of deriving the Born rule from “the nonprobabilistic axi- density operators ͓13–18͔, that the quantum and classical ␳ ␳ oms of quantum theory” and “the nonprobabilistic part of systems are represented by vectors and c, respectively, in ␳ ␳ classical decision theory” ͓4͔. Deutsch’s approach was criti- that Hilbert space, and that and c can be expanded in cized by Barnum et al. ͓5͔ but was recently reinforced by eigenstates of a set of commuting quantum and classical su- Saunders ͓6͔. Hanson ͓7͔ and Wallace ͓8͔ analyzed the con- peroperators, respectively; ͑2͒ show that the quantum- nection between possible derivations of the Born rule and mechanical Born rule can be expressed in terms of the ex- Everett’s many-worlds interpretation of quantum mechanics. pansion coefficient of a given density associated with Zurek recently proposed a significantly new approach, the eigendistributions of a set of superoperators in the Hilbert so-called envariance approach ͓9͔, for deriving the Born rule space of density matrix; and ͑3͒ show that the classical in- ␳ from within quantum mechanics. This approach, totally dif- terpretation of the phase-space representation of c as a prob- ferent from Deutsch’s method, was recently analyzed in de- ability density allows the extraction of Born’s rule in classi- tail by Schlosshauer and Fine ͓2͔ and by Zurek ͓10͔. Zurek’s cal mechanics, and gives exactly the same structure as the envariance approach has also been analyzed and modified by quantum-mechanical Born rule. Barnum ͓11͔. All these studies have attracted considerable These results suggest that the quantum-mechanical Born interest in deriving Born’s rule by making some basic as- rule not only applies to cases of large quantum numbers, but sumptions about quantum probabilities or expectation values also has a well-defined purely classical limit. Hence, inde- of observables. pendent of other subtle elements of the quantum theory, the The Born rule is not expected to violate any future experi- inherent consistency with the classical Born rule for the mac- ments. In this sense, even a strict derivation of the Born rule roscopic world imposes an important condition on any

eigenvalue-eigenfunction-based probability rule in quantum pure quantum state, e.g., ␳=͉␺͗͘␺͉͑the extension to mixed mechanics. states is straightforward͒. Then the Born rule gives that P ͑KЈ͒ = ͉͗␺͉KЈ͉͘2 =Tr͓͉KЈ͗͘KЈ͉͉␺͗͘␺͉͔ = Tr͓͉KЈ͗͘KЈ͉␳͔, II. THE QUANTUM-MECHANICAL BORN RULE Q IN DENSITY MATRIX FORMALISM ͑5͒ Consider first quantum mechanics in the Hilbert space of where ͉KЈ͘ is a common and normalized eigenfunction of ͓ ͔ ˆ ϵ ˆ operators Kˆ , i=1,2,...,N. However, the density matrix 16,17 . Given an operator O KN for a i ͑ system of N degrees of freedom, we first consider the clas- 1 ͒ ͓ ˆ ͉ Ј͗͘ Ј͉͔ Ј͉ Ј͗͘ Ј͉ ͓ ˆ ͉ Ј͗͘ Ј͉͔ ͑ ͒ sically integrable case where there exist N independent and Ki, K K + = Ki K K ; Ki, K K =0, 6 ˆ 2 commuting observables Ki, i=1,...,N. Another extreme, the chaotic case, will be discussed in Sec. IV. For convenience so that ͉KЈ͗͘KЈ͉ is seen to be the common eigendistribution ˆ 1 ͓ ˆ ͔ Ј we also assume that the Ki, i=1,2,...,N, have a discrete of superoperators 2 Ki , + with eigenvalues Ki and of super- spectrum, but the central result below applies to cases with a 1 ˆ operators ប ͓K , ͔ with eigenvalue zero. That is, continuous spectrum as well. The complete set of commuting i ͉ ͗͘ ͉ ␳ ͑ ͒ superoperators in the quantum Hilbert space can be con- KЈ KЈ = KЈ,0. 7 structed as Equations ͑4͒, ͑5͒, and ͑7͒, then lead to 1 ˆ 1 ˆ P ͑ ͒ ͑ ͒ ͓K ,͔, ͓K ,͔ ͑i =1,2, ...,N͒, ͑1͒ Q KЈ = DKЈ,0. 8 ប i 2 i + ͑ ͒ ͓ ͔ ͓ ͔ Equation 8 is a general restatement of the quantum- where , denotes the commutator and , + denotes the mechanical Born rule based on the Hilbert-space structure of ͓ ˆ ˆ ͔ ˆ ˆ ˆ ˆ ͓ ˆ ˆ ͔ ˆ ˆ ˆ ˆ anticommutator, i.e., A,B =AB−BA, A,B + =AB+BA. The the density matrix. simultaneous eigendensities of the complete set of superop- Note that the multidimensional result of Eq. ͑5͒ has care- ␳ erators are denoted ␣,␤. That is, fully accounted for possible degeneracies associated with KЈ. That is, the total probability of observing KЈ would be 1 N N ͓ ˆ ␳ ͔ ␣ ␳ P ͑KЈ͒ KЈ i Ki, ␣,␤ + = i ␣,␤, obtained by summing Q with all possible i , 2 =1,2,...,͑N−1͒, a necessary procedure not explicitly stated in Born’s rule. 1 ͓Kˆ ,␳␣ ␤͔ = ␤ ␳␣ ␤, ͑2͒ ប i , i , III. THE BORN RULE IN CLASSICAL MECHANICS Consider now classical mechanics. The mechanics has nu- where ␣ϵ͑␣ ,␣ ,...,␣ ͒ is the collection of eigenvalues 1 2 N merous equivalent formulations, such as Newton’s laws, the 1 ͓ ˆ ͔ ␤ϵ͑␤ ␤ ␤ ͒ associated with 2 Ki , +, and 1 , 2 ,..., N is the col- Lagrangian and Hamiltonian formulations, Hamilton-Jacobi 1 ͓ ˆ ͔ lection of eigenvalues associated with ប Ki , . theory, etc. The less familiar Hilbert-space formulation of The state of the quantum system is described by an arbi- classical mechanics used below was first established by trary density matrix ␳ in the Hilbert space under consider- Koopman ͓13͔ and subsequently appreciated by, for ex- ␳ ͓ ͔ ͓ ͔ ͓ ͔ ation, and can be expanded in terms of the basis states ␣,␤ as ample, Prigogine 14 , Zwanzig 15 , and us 17,18 in some theoretical considerations. This being the case, the above ␳ ␳ ͑ ͒ = ͚ D␣,␤ ␣,␤, 3 eigenvalue-eigenfunction structure is not unique to quantum ␣,␤ mechanics, a fact that may not be well appreciated and that is where the sum is over all eigendensities. The sum should be exploited below. understood as an integral if the spectrum is continuous. It is convenient to introduce the classical picture in the Clearly, the expansion coefficients in Eq. ͑3͒ are given by phase-space representation, although abstract Hilbert-space formulations may be used as well. Consider then the same ͓␳␳† ͔ ͑ ͒ D␣,␤ =Tr ␣,␤ . 4 case as above. Let the classical limit of the Wigner-Weyl ˆ ͑ ͒ ͑ ͒ Equations ͑3͒ and ͑4͒, i.e., the expansion of ␳ in terms of the representation of Ki be Ki p,q , where p,q are ␳ momentum- and coordinate-space variables. The complete eigendensities ␣,␤, are central to the analysis later below. Consider now the quantum probability P ͑KЈ͒ of finding set of commuting superoperators on the classical Hilbert Q ͓ ͔ ˆ Ј space is then 17 the quantum observables Ki with eigenvalues K , i i ͕ ͑ ͒ ͖ ͑ ͒ ͑ ͒ =1,2,...,N, given that the system is in state ␳. We show i Kj p,q , , Kj p,q , j =1,2, ...,N, 9 here that the Born rule is then equivalent to the statement where K ͑p,q͒ are multiplicative operators and ͕ , ͖ denotes that P ͑KЈ͒ must be proportional to the expansion coeffi- j Q the classical Poisson bracket. The simultaneous eigendensi- cient D of the given density ␳ associated with the com- KЈ,0 ties of this complete set of classical operators, denoted ␳ 1 ͓ ˆ ͔ c mon eigendistribution KЈ,0 of superoperators 2 Ki , + with ␳ ͑ ͒ ␣c,␤c p,q , satisfy eigenvalues KЈ and of superoperators 1 ͓Kˆ , ͔ with eigen- i ប i ͑ ͒␳c ͑ ͒ ␣c␳c ͑ ͒ value zero. To see this, consider first a quantum density for a Kj p,q ␣c,␤c p,q = j ␣c,␤c p,q ,

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͕ ͑ ͒ ␳c ͑ ͖͒ ␤c␳c ͑ ͒ ͑ ͒ independently obtaining Eq. ͑15͒ using this approach. To do i Kj p,q , ␣c ␤c p,q = j ␣c ␤c p,q , 10 , , so, we first expand the given density in terms of the basis ␣c ␤c ␣c ␤c R where the notation , , j, and j is introduced in parallel states KЈ,⌳. That is, with the quantum case. An arbitrary classical probability ␳ ͑ ͒ density c p,q can be expanded as ␳Ј͑ ͒ ͵ Ј ⌳ c R ͑ ͒ ͑ ͒ c K,Q = dK d DKЈ,⌳ KЈ,⌳ K,Q , 16 c c ␳ ͑p,q͒ = ͚ D c c␳ c c͑p,q͒, ͑11͒ c ␣ ,␤ ␣ ,␤ where the expansion coefficients are given by the overlap ␣c,␤c integrals where c ͵ ␳Ј͑ ͒R* ͑ ͒ ͑ ͒ D ⌳ = dK dQ K,Q ⌳ K,Q . 17 c ͵ ␳ ͑ ͓͒␳c* ͑ ͔͒ ϵ ͓␳ ␳c† ͔ KЈ, c KЈ, D␣c,␤c = dp dq c p,q ␣c,␤c p,q Tr c ␣c,␤c , Because classical probabilities in K that do not refer to Q are ͑ ͒ 12 obtained by integrating over Q, we obtain and where the sum in Eq. ͑11͒ is over all eigendensities. Consider now, within this formalism, the probability of P ͑ Ј͒ ͫ͵ ␳Ј͑ Ј ͒ͬ ͑ ͒ c K = dQ K ,Q dK. 18 ͓ ϵ͑ ͔͒ c finding K with K K1 ,K2 ,...,KN between KЈ and KЈ +dKЈ. To proceed we make a canonical transformation be- Substituting Eq. ͑16͒ into Eq. ͑18͒ and using Eq. ͑13͒ then tween representations ͑p,q͒ and ͑K,Q͒, where K are the yields ϵ͑ ͒ new momentum variables, and Q Q1 ,Q2 ,...,QN denotes P ͑ Ј͒ ͑ ␲͒N/2 c ͑ ͒ the new position variables conjugate to K. The Q can be c K = 2 DKЈ,0dK 19 obtained by regarding p as a function of q and K, defining ͑ ͒ ͐q ͑ ͒ This result is equivalent to Eq. ͑15͒, but contains an impor- the generating function S q,K = q p qЈ,K ·dqЈ, and then tant message from the perspective of mechanics in Hilbert 0 ͔ ͓ ץ ץ noting that Qi = S/ Ki 19 . In this representation the classi- ͑ ͒ ␳c space. That is, Eq. 19 indicates that, given a classical den- cal eigendensities ␣c,␤c take a rather simple form, ␳Ј sity c that describes the state, the probability of finding the ͑ ͒ ͑ ͒ observable Kj j=1,2,...,N in a regime KЈ,KЈ+dK ,is c 1 ␳ ͑p,q͒ Þ R ⌳͑K,Q͒ = ␦͑KЈ − K͒exp͑i⌳ · Q͒ ␳Ј ␣c,␤c KЈ, ͑ ␲͒N/2 proportional to the overlap between the given density c and 2 R the common eigendistribution KЈ,0 of multiplicative opera- ͑ ͒ Ј ͕ ͖ 13 tors Kj with eigenvalues Kj and of operators i Kj , with c ͓ ͔ ␣c ␤c ⌳ eigenvalue zero. This overlap is D , the expansion coeffi- with eigenvalues 20 i =Ki and i = i, for i=1,2,...,N. KЈ,0 R cient of ␳Ј in terms of R . Significantly, this connection The set of eigendistributions KЈ,⌳ are complete and or- c KЈ,0 thogonal, i.e., between classical probabilities and the overlap between the given classical-state density and a particular set of classical ͓ ͑ ͔͒ ͵ ⌳ R* ͑ Ј Ј͒R ͑ Љ Љ͒ eigendistributions Eq. 19 is the direct analog of the dK d K ,Q ⌳ K ,Q K,⌳ K, quantum-mechanical Born rule in the density matrix formal- ͓ ͑ ͔͒ ͑ ͒ ␦͑ Ј Љ͒␦͑ Ј Љ͒ ism Eq. 8 . That is, Eq. 19 is the Born rule in classical = K − K Q − Q , mechanics.

͵ R* ͑ ͒R ͑ ͒ dK dQ KЈ,⌳Ј K,Q KЉ,⌳Љ K,Q IV. QUANTUM VERSUS CLASSICAL BORN RULE IN CHAOTIC CASES = ␦͑⌳Ј − ⌳Љ͒␦͑KЈ − KЉ͒. ͑14͒ Classical-quantum correspondence of Hilbert-space struc- R ␦ The KЈ,⌳ are “improper states,” insofar as they contain tures in chaotic cases is more subtle and complicated than in functions. However, this is consistent with the fact that they integrable cases ͓17͔. Nevertheless, results above indicate are eigendistributions of classical superoperators that have that only a particular set of eigendensities are relevant in continuous spectra. If desired, a rigged Hilbert space ͓21͔ understanding the quantum versus classical Born rule. In- can be used to include these states more formally. deed, it is straightforward to show that our previous consid- ␳ ͑ ͒ Given a classical probability density c p,q that de- erations also apply to cases where there do not exist ͑N−1͒ ͑ ͒ scribes the state of the system, we can convert to the K,Q observables that commute with KN p,q . For example, con- ␳Ј͑ ͒ϵ␳ ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ representation to obtain c K,Q c p K,Q ,q K,Q . sider a chaotic spectrum case where KN p,q does not com- P ͑ ͒ The probability c KЈ of ﬁnding the observables between mute with any other smooth phase-space functions Z͑p,q͒, Ј Ј ͕ ͑ ͒ ͑ ͖͒ K and K +dK is evidently given by i.e., KN p,q ,Z p,q 0 always holds. Following Ref. ͓17͔ we consider a function ␶͑p,q͒ that satisﬁes P ͑ Ј͒ ͫ͵ ␦͑ Ј ͒␳Ј͑ ͒ͬ ͑ ͒ ͕␶͑ ͒ ͑ ͖͒ c K = dK dQ K − K c K,Q dK. 15 p,q ,KN p,q =1. Then the classical eigenfunction of ͑ ͒ the multiplicative operator KN p,q and of the operator ͕ ͑ ͒ ͖ ␭ ␰␦͓ Ј This result has an enlightening interpretation in the Hilbert i KN p,q , with eigenvalue is given by KN ͑ ͔͒ ͓ ␭␶͑ ͔͒ ␰ space formulation of classical mechanics, as can be seen by −KN p,q exp i p,q , where is a normalization con-

052109-3 P. BRUMER AND J. GONG PHYSICAL REVIEW A 73, 052109 ͑2006͒ stant. In particular, the eigenfunction with zero eigenvalue so doing we never assumed that the quantum density goes ͕ ͑ ͒ ͖ ␰␦͓ Ј ͑ ͔͒ for the operator i KN p,q , is KN −KN p,q . This smoothly, in the classical limit, to a classical density that eigenfunction defines a ͑2N−1͒-dimensional hypersurface in already has a clear probabilistic interpretation. Rather, we ͑ R phase space whereas the eigenfunction KЈ,0 in the integral have simply assumed that a system in either quantum or case defines an N-dimensional manifold͒. The overlap be- classical mechanics is described by a density operator in Hil- ␰␦͓ Ј ͑ ͔͒ bert space, and that these density operators serve the same tween this particular eigenfunction KN −KN p,q and a ␳ ͑ ͒ descriptive purpose in both mechanics. This, plus the decom- given phase-space probability density c p,q , i.e., position of the operator in terms of eigendistributions of a set ͑ Ј͒ϵ␰ ͵ ␳ ͑ ͒␦͓ Ј ͑ ͔͒ ͑ ͒ of commuting superoperators, sufficees to show that the Pc KN dp dq c p,q KN − KN p,q , 20 Born rule applies in both quantum and classical mechanics. Hence, the quantum-mechanical Born rule appears to be very ͑ Ј͒ ͑ ͒ yields the probability Pc KN dKN of finding KN p,q lying in natural in the light of quantum-classical correspondence in ͓ Ј Ј ͔ the regime KN ,KN +dKN . This is again in complete analogy how Hilbert-space structures embody measured probabilities. P ͑ Ј͒ to how the quantum probability Q KN of finding the eigen- The recognition that Born’s rule is not really a unique quan- Ј P ͑ Ј͒ value KN is determined, i.e., Q KN is given by the overlap tum element should complement, as well as impact upon, between a given quantum density ͉␺͗͘␺͉ and the eigenfunc- previous attempts to derive the Born rule ͓2,4,6,9–11͔. Fur- 1 ͓ ˆ ͔ Ј ther, it motivates numerous questions, such as, can the tion of superoperator 2 KN , + with eigenvalue KN and of 1 ͓ ˆ ͔ quantum-mechanical Born rule be derived with fewer as- superoperator ប KN , with eigenvalue zero. Such a quantum sumptions by taking advantage of the classical Born rule as a ͉ Ј͗͘ Ј͉ ͉ Ј͘ eigenfunction is simply given by KN KN , where KN is the limit? how can one reconcile derivations involving purely ˆ Ј eigenfunction of KN with eigenvalue KN. These analyses quantum language with the existence of a classical Born make it clear that even for chaotic cases, the Born rule for- rule?, etc. These, and related issues, are the subject of future mulated in terms of eigendensities in the associated ͑classical work. or quantum͒ Hilbert space exists in both quantum and classical mechanics. ACKNOWLEDGMENTS

V. CONCLUDING REMARKS This work was supported by a grant from the National Science and Engineering Research Council of Canada. We In summary, with the quantum-mechanical Born rule for- thank Professor. R. Kapral and Professor S. Whittington, mulated in the Hilbert space of density matrix, we have dem- University of Toronto, for comments on an earlier version of onstrated an analogous Born rule in classical mechanics. In this manuscript.

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