CCoherent Distributions and Quantization
Marius GrigorescuThis work presents a selective review of results concerning the mathematical interfacebetween the classical and quantum aspects encountered in problems such as the nuclearmean-field dynamics or quantum Brownian motion. It is shown that the main differ-ence between classical and quantum behavior arises from the coherence properties of thephase-space distributions known as ”action waves” and Wigner functions. The quantumwave functions appear as elementary degrees of freedom for the phase space granularity. Based on the research proposal 338620ERC-2013-ADG ”Functional Coherent Distributions on Gran-ular Phase-Space and Quantization”, November 22 (2012). a r X i v : . [ qu a n t - ph ] J a n Introduction
In classical statistical mechanics, the entropy of a many-body system is defined withrespect to a partition of the 2 n -dimensional one-particle phase-space in elementary cells,but the size of these cells is not specified. The occurrence of h = 6 . × − J · s in thePlanck distribution for thermal radiation was considered as evidence for such a granularstructure, with cells (”quantum states”) of volume h n . The old quantum mechanicsmaintained this view, as for the integrable systems in action-angle coordinates a cellstructure was introduced by the set of stationary orbits (invariant tori) selected by theBohr-Sommerfeld quantization conditions. In quantum mechanics however, the physicalstates of microparticles are described by rays in an abstract Hilbert space, and its mainelements (wave-functions and operators) are expressed in terms of only n coordinates.This formalism, though quite successful in atomic physics, is not complete, because itrequires additional rules for ”quantization” (e.g. algebraic (Dirac), geometric [1, 2, 3, 4])and interpretation (e.g. probabilistic interpretation of the scalar product). These rulesprovide an interface between the physical observables defined at classical level and theabstract Hilbert space in some important situations of interest (e.g. the measurementprocess), but are not enough general to join classical and quantum mechanics into asingle theory. One of the difficulties in formulating such a theory is that while thermalfluctuations in classical systems can be explained by the action of random forces, theorigin of the quantum fluctuations is unknown. The conceptual gap between classicaland quantum mechanics is illustrated for instance by the early (1927) Bohr-Einsteindebate on the intrinsic statistical character of quantum mechanics, the paradox of the”Schr¨odinger’s cat experiment”, or the Zeno paradox at the continuous measurementprocess [5]. There was also a constant effort to bridge this gap by non-standard theo-ries such as the thermodynamics of the isolated particle of L. de Broglie, the Bohmianinterpretation of quantum mechanics, or the hypothesis of spontaneous wave functioncollapse [6].Considering quantum mechanics as fundamental, in the case of a mixed system con-taining classical and quantum components one can start with a quantum description ofthe whole system, and then perform partial tracing over the variables known as classical.Along this line, the classical dynamics of a nuclear collective model can be derived asconstrained quantum dynamics if in the quantum time-dependent variational principle(TDVP) for the Schr¨odinger Lagrangian L ψ = (cid:104) ψ | i (cid:126) ∂ t − ˆ H | ψ (cid:105) , ψ belongs to a suitablemanifold of trial wave functions (e.g. coherent states) [7, 8]. A possible mechanismto generate such a collective phase-space as a symplectic submanifold of the quantumHilbert space is the spontaneous symmetry breaking [9].Classical degrees of freedom in nuclear, atomic or molecular systems can be intro-duced using the Born-Oppenheimer approximation. However, attempts to formulatea theory of genuine mixed classical-quantum dynamics appear in 1994, in the form ofcoupled Hamilton-Schr¨odinger [10] or Hamilton-Heisenberg [11] equations, derived froma mixed TDVP. The mixed Lagrangian contains a classical part L cl ( x, ˙ x ), the quantumpart L ψ , and an interaction term L int ( x, ψ ) depending both on the coordinates x ( t ) of2he classical component and the quantum wave function ψ ( q, t ).In the case of a quantum particle coupled to a thermal bath of classical harmonicoscillators by a bilinear interaction the Hamilton-Schr¨odinger equations can be reducedto a non-linear Schr¨odinger-Langevin equation, containing noise and non-linear frictionterms (nonlinearity due to the backreaction of the classical environment) [10]. Furtherapplications present two variants of this approach:- for a many-body quantum system, described within some mean-field approximation, ψ is constrained to a symplectic submanifold of the quantum Hilbert space. The exampleof a superfluid quantum many-fermion system in a thermal environment was consideredin [12, 13].- for the study of decoherence L ψ can be expressed in terms of the density matrix,such that instead of the Schr¨odinger-Langevin equation one obtains a quantum Liouville-Langevin equation [14]. Applications to atomic transition rates in thermal radiationfield and decoherence time for a two-level system with Ohmic dissipation are presentedin [14]. Numerical integration shows that dissipation produced by the non-linear frictionterm alone (at zero temperature) resembles the spontaneous decay obtained when theclassical environment is quantized. The realistic situation of dissipative atomic tunnelingin an asymmetric quartic (double well) potential at finite temperature is presented in[15]. Analytically it was shown that if the nonlinear friction term is neglected, then bytaking the ensemble average, the results are equivalent to the integration of a quantumFokker-Planck equation for the density matrix [15].The extensive literature on these subjects also includes a detailed analytical study ofthe environmental decoherence during macroscopic quantum tunneling in a cubic poten-tial using a quantum Kramers equation for the reduced Wigner function of the tunnelingparticle [16], or a variational principle describing a classical statistical ensemble on theconfiguration space interacting with a quantum system, applied to couple quantum mat-ter fields and classical metric [17].The variants [10, 12, 14] are summarized in [18], using for the Lagrangian L tot amore compact form, with the term L ψ alone, but the trial wave functions ψ expressedas a product between quantum, quasiclassical (e.g. mean-field) and classical (actionphase-factor) components. The quantum transport equation derived in [15] is also im-proved by the non-linear friction term to get a Fokker-Planck equation, and integratednumerically for the two-level system. The results show the advantage of the non-linearLiouville-Langevin equation, because the presumed non-linear friction term in the quan-tum Fokker-Planck equation does not ensure thermalization.Attempting to find a random force which could simulate quantum fluctuations, Ihave arrived at the unexpected result that there is no such force, but instead that acertain ”granularity” is required [19], like in the old quantum mechanics. Thus, whilethe random force destroys the classical coherence expressed by the Hamilton-Jacobiequation, discretization at a certain scale may induce the ”quantum” type of coherence.The next section reminds the framework of statistical mechanics in which the ”clas-sical” coherent distributions are defined, Section 3 outlines the transition to Wignerfunctions, while the Fokker-Planck equation is discussed in Section 4.3 The Liouville equation
Let ( M µ , ω µ ) be the phase space of a classical elementary system µ (molecule) with n degrees of freedom [20], and ( M Γ , ω Γ ) the 2 nN - dimensional phase space of the ensembleΓ (gas) consisting of N identical elementary subsystems, M Γ = M × M × ... × M N , ω Γ = N (cid:88) µ =1 ω µ . (1)In particular, the state of a system composed of N identical point-like particles is de-scribed on the 6 N dimensional manifold M Γ ≡ T ∗ R N by a representative (”phase”)point m of coordinates ( Q, P ). To obtain a statistical description of the ensemble eachmanifold M µ is divided in K infinitesimal cells { b j ⊂ M, b j (cid:54) = ∅ ; b i ∩ b j | i (cid:54) = j = ∅ , i, j ∈ I b } , I b = { , K } , M µ = ∪ j ∈ I b b j , (2)of volume Ω jµ = (cid:90) ¯ b j Ω µ , Ω µ = | ω µ | / d qd p . (3)Therefore, we also obtain a partition of the manifold M Γ in N B = K N cells B j of volumeΩ j Γ , j = 1 , N B . Denoting by w j the probability to find the representative point m ∈ M Γ at the time t in the cell B j , the ratio P j = w j / Ω j Γ defines the distribution function ofthe probability density P , normalized by (cid:90) M Γ Ω Γ P = 1 , Ω Γ ≡ dQdP . It is important to remark that to address the issues of continuity and unicity of P itmight be necessary to consider instead of a partition (2) an indexed system of open sets { U i , i ∈ I } covering M µ and a system of q -cochains [21], q = 0 ,
1, associating to each setof q + 1 indices i , ..., i q from I a function P q ( i , ..., i q ) ∈ R on U i ∩ U i ... ∩ U i q .As the Hamiltonian flow F t on M Γ preserves the volume element Ω Γ , the probabilitydensity behaves as a perfect fluid described by the continuity (Liouville) equation ∂ t P + L H P = 0 , (4)where L H P ≡ −{ H , P} is the Lie derivative defined by the Poisson bracket and H is thetotal Hamiltonian, including interaction terms.The dimensionality of P depends on the dimension of M Γ , 2 nN . Because in general M Γ is not a metric space it is convenient to introduce a fundamental unit h for ω µ , suchthat h Nn is the fundamental unit for Ω Γ . The ratio γ j = Ω j Γ / h Nn is the weight of thecell B j , while ¯ P = h Nn P is dimensionless, normalized by (cid:90) M Γ Ω Γ h Nn ¯ P = 1 . (5)4he expectation value of a many-body observable A ∈ F ( M Γ ) (smooth function on M Γ ),defined by < A > = (cid:90) M Γ Ω Γ h Nn ¯ P A , evolves in time according to d < A >dt = < { A , H } > . (6)The expectation value of − k B ln ¯ P , where k B is the Boltzmann constant, defines theentropy S = − k B (cid:90) M Γ Ω Γ h Nn ¯ P ln ¯ P . (7)The one-particle probability density ρ (or ¯ ρ = h n ρ ) on the phase-space M µ is related tothe density P on M Γ by the projection given by integration over N − M µ , ρ ( q , p ) = (cid:90) d q ...d q N d p ...d p N P ( q , q , ..., q N , p , p ... p N ) . (8)This is well defined because the particles are identical, and although the permutationsof coordinate indices 1 , , ...N → { i , i , ..., i N } yield different phase points, P remainsinvariant. For instance, if P is a symmetric functional P ( Q, P ) = 1 N ! (cid:88) { i ...i N } ρ ( q i , p i ) ρ ( q i , p i ) ...ρ N ( q i N , p i N )of L ≤ N distinct distribution functions ρ λ , λ = 1 , L on M µ , then ρ ( q , p ) = 1 N L (cid:88) λ =1 N λ ρ λ ( q , p ) , (9)where N λ is the number of particles assigned to ρ λ .The ensemble of identical particles can also be described using the Boltzmann repre-sentation of ”occupation numbers” in the µ -space. Thus, on M µ = T ∗ R , each particleis represented by a point of coordinates ( q , p ) i , i = 1 , N . Let (2) be a partition of M µ and N j the average number of particles localized in the cell b j . The ratio f j = N j / Ω jµ defines the distribution function of the particle density f = N ρ on M µ , normalized by (cid:90) M µ Ω µ f = N . (10)If there are no interactions between particles, f satisfies the one-particle Liouville equa-tion ∂ t f + L H f = 0 (11)5here H ∈ F ( M µ ) is the one-particle Hamiltonian and L H f ≡ −{ H, f } . If M µ = T ∗ R ,then L H = ( ∇ p H ) · ∇ − ( ∇ H ) · ∇ p , (12)where ∇ p ≡ (cid:126)∂ p , ∇ ≡ (cid:126)∂ q , and for H ( q , p ) = p m + V ( q ) (13)(11) becomes ∂ f ∂t + p m · ∇ f − ∇ V · ∇ p f = 0 . (14) To solve (14) it is convenient to use the Fourier transform ˜ f ( q , k , t ) in momentum,˜ f ( q , k , t ) ≡ (cid:90) d p e i k · p f ( q , p , t ) , (15)which is a density on the configuration space R related to the particle ( n ) or current( j ) densities by n ( q , t ) ≡ (cid:90) d p f ( q , p , t ) = ˜ f ( q , , t ) , (16) j ( q , t ) ≡ (cid:90) d p p m f ( q , p , t ) = − i m ∇ k ˜ f ( q , , t ) . (17)Thus, if f ( q , p , t ) is a solution of (14), then its Fourier transform ˜ f ( q , k , t ) will satisfy ∂ t ˜ f − i m ∇ k · ∇ ˜ f + i k · ( ∇ V )˜ f = 0 . (18)An important class of solutions for the one-particle Liouville equation (14) is representedby the ”action distributions” f ( q , p , t ) = n ( q , t ) δ ( p − ∇ S ( q , t )) . (19)These are coherent functionals in the sense that remain all the time a product between n ( q , t ) and δ ( p − ∇ S ( q , t )). The two real functions of coordinates and time, n ( q , t ) and S ( q , t ) are related by the Hamiltonian flow because for˜ f ( q , k , t ) = n ( q , t )e i k ·∇ S ( q ,t ) (20)(18) reduces to the system of equations ∂ t n = −∇ j (21) n ∇ [ ∂ t S + ( ∇ S ) m + V ] = 0 (22)6here j ≡ n ∇ S/m is the current density (17). Thus, presuming the existence of a ”mo-mentum potential” S ( q , t ) we get both the continuity and Hamilton-Jacobi equations.In general the solutions of (22) are multi-valued, and f is a sum f = (cid:88) i n i δ ( p − ∇ S i )over different branches. The solutions ˜ f ≡ n [ S ] corresponding to the same function S satisfy the superposition principle, ( n + n ) [ S ] = n [ S ]1 + n [ S ]2 , and can be called ”actionwaves”.If n ( q ) is a solution of the system (21), (22), then − n ( q ) is also a solution. To obtainonly positive solutions it is convenient to search n = ˜ f | k =0 of the form n = | ψ | , where ψ can be a complex function. When ψ = √ n exp(i S/σ ), with σ a dimensional constant,then ˆ ω ≡ (cid:90) d q ( d n ∧ dS ) = − i σ (cid:90) d q ( dψ ∗ ∧ dψ ) (23)is the symplectic form induced by the complex structure of the Hilbert space H = { ψ ∈L ( R ) } . The constant σ and h from (5) have both dimensionality of action, and in thequantum theory σ = h / π ≡ (cid:126) , where h is the Planck constant.It is important to remark that while the singularity of f is necessary for coherence,it yields infinite entropy, and therefore is not realistic. To obtain finite entropy we canreplace for instance the delta function δ ( p − ∇ S ) by a Gaussian g ( p ) = 1( πb ) / e − ( p −∇ S ) /b , where b is a finite constant, but in general f = n g is not coherent. However, in the par-ticular case of the harmonic oscillator ( V ( q ) = mω q /
2) we can find coherent solutionsof the form ρ G ( q , p ) = g X ( q ) g Y ( p ), g X ( q ) = 1( πa ) / e − ( q − X ) /a , g Y ( p ) = 1( πb ) / e − ( p − Y ) /b , if b/a = m ω and X , Y are time-dependent variables which satisfy the classical equa-tions of motion, ˙ X = Y /m , ˙ Y = − mω X . These are solutions of constant entropy(classical, S ( ρ G ) = − k B < ln ¯ ρ G > = 3 k B (1 − ln 2), and quantum, [22]), which can alsobe written in the form ρ G ( q , p ) = 1(2 π ) (cid:90) d k e − i k · p ψ G ( q + σ k ψ ∗ G ( q − σ k , σ = √ ab (24)where ψ G ( q ) = (cid:112) g X ( q )e i q · Y /σ . If σ = (cid:126) then ψ G are (up to a phase factor) thenonstationary solutions of the TDSE known as Glauber coherent states. An applicationof such states to describe ”preformed” alpha particles in heavy nuclei, with relevance forthe Geiger-Nuttall law [23], was presented in [24]. At astronomic scale, we may presumethat for a suitable constant σ similar considerations might explain the ”preformation”of planets along the orbits described by the Titius-Bode law. Gaussian distributions usually describe fluctuations around a mean. Discretization, coherence and quantization
The partial derivative k · ∇ S ( q , t ) in (20) is the limit of k(cid:96) [ S ( q + (cid:96) k k , t ) − S ( q − (cid:96) k k , t )] , (25)with k = | k | (cid:54) = 0, when (cid:96) →
0. However, if k → (cid:96) can bearbitrarily small, but finite. It seems though that for microparticles there is a physicallimit (cid:96) , and (cid:96) → (cid:96) >
0. The existence of an elementary length (cid:96) >
0, proposedby W. Heisenberg ( (cid:96) ∼ − m) and M. Planck ( (cid:96) ∼ − m), was developed inthe framework of general relativity theory, by the model of crystalline lattice of thephysical space-time [25], or in string theory [26]. Independently of these considerations,the assumption of a finite limit (cid:96) , expected for each massive particle near its Comptonwavelength ( (cid:96) ∼ /m ), was used in [19, 27, 28] to justify the transition from a classicalcoherent distribution (20) to a ”quantum” distribution of the form (24). Let us presumethat in the action wave (20) we approximate the phase k · ∇ S ( q , t ) by (25) and theamplitude n ( q ) by √ n + n − , n ± = n ( q ± (cid:96) k k ) . Thus, the space derivative ∂ q i S ( q , t ) is replaced by the finite differences expression withrespect to a minimum length (cid:96) i = (cid:96)k i /k , while the integration on k is limited by the sizeof the domain in which n ( q , t ) (cid:54) = 0. In terms of the new parameter σ = (cid:96)/k , if k (cid:54) = 0then ˜ f ( q , k , t ) = lim σ → ˜ f ψ ( q , k , t ) (26)where ˜ f ψ ( q , k , t ) ≡ ψ ∗ ( q − σ k , t ) ψ ( q + σ k , t ) (27)with ψ = √ n exp(i S/σ ). Moreover, in the limit k → S ( q ± σ k , t ) = S ( q , t ) ± σ k · ∇ S ( q , t ) + σ k · ∇ ) S ( q , t ) ± ... and if the terms containing ( σ k ) m , m ≥
3, are neglected, then k · ∇ S ( q , t ) ≈ σ [ S ( q + σ k , t ) − S ( q − σ k , t )]for any dimensional constant σ >
0. Therefore, within a suitable domain for k , we mayconsider σ from (26),(27) as a finite constant, related eventually to the size of the cells b j used in the partition (2). If σ = (cid:126) then f ψ obtained inverting (15), f ψ ( q , p , t ) = 1(2 π ) (cid:90) d k e − i k · p ˜ f ψ ( q , k , t ) (28) This means ln n ≈ (ln n + + ln n − ) /
2, which indicates that such an n ( q ) is up to the factor N a ”pure”1-particle probability density rather than an ensemble average.
8s the Wigner transform [29, 30] of the complex ”wave function” ψ = √ n exp(i S/σ ).Some properties of this functional are summarized below:- f ψ is not positive definite, and in general it cannot represent particle density. Howeverit is integrable, and the normalization condition (5) takes the form (cid:90) d qd p f ψ ( q , p , t ) = (cid:90) d q | ψ ( q , t ) | ≡ (cid:104) ψ | ψ (cid:105) = N , (29)where N can be seen as the number of particles (or identical 1-particle systems) used todefine the probability density ρ = n /N .- The ”overlap” integral between two distributions f ψ , f ψ , over the phase-space is [27]( f ψ , f ψ ) ≡ (cid:90) d qd p f ψ f ψ = N h < ¯ f ψ > | ρ ψ = |(cid:104) ψ | ψ (cid:105)| (2 πσ ) (30)where (cid:104) ψ | ψ (cid:105) ≡ (cid:90) d q ψ ∗ ( q , t ) ψ ( q , t ) (31)is the scalar product between ψ and ψ as elements of the quantum Hilbert space H .Thus, the overlap (30) is positive and directly related to the statistical interpretationof the scalar product in quantum mechanics, suggesting again the choice σ = (cid:126) . Inparticular, the overlap between f ψ and the Gaussian (24) is positive.- The expectation value of a classical observable A such as q , p , p , L = q × p , < A > f ψ = (cid:90) d qd p f ψ A = (cid:90) d q ψ ∗ ( q , t ) ˆ Aψ ( q , t ) ≡ (cid:104) ψ | ˆ A | ψ (cid:105) = (cid:104) ˆ A (cid:105) ψ is just a matrix element of the usual operator ˆ A on H associated to the observable A :ˆ q = q , ˆ p = − i σ ∇ , ˆ p = − σ ∆, ˆ L = − i σ q × ∇ . For the Hamiltonian (13) < H > f ψ = (cid:104) ˆ H (cid:105) ψ , ˆ H = − σ ∆ / m + V , and the energy density becomes w q = (cid:90) d p f ψ H = σ m [ |∇ ψ | −
14 div( ψ ∗ ∇ ψ + ψ ∇ ψ ∗ )] + V ψ ∗ ψ . (32)- A symplectic diffeomorphism Φ of ( M, ω ) which acts on f ψ ∈ F ( M ) by Φ ∗ f ψ = f ψ (cid:48) yields, according to (30), a unitary transformation ˆ U Φ of the state vectors ψ ∈ H of theform ψ (cid:48) = ˆ U − ψ , such that Φ ∗ f ψ = f ˆ U − ψ . (33)In particular, when Φ is the action of a Lie group G , the infinitesimal transformationstake the form ˆ U (cid:15) = 1 + i (cid:15) ˆ J , where ˆ J are Hermitian operators associated to the elementsof the Lie algebra g of G . Thus, the main difference between the classical and quantumoutcome of dynamical symmetries (e.g. at spontaneous symmetry breaking) is due totheir realization within spaces having different coherence properties ( f and f ψ ).In general, a functional f [ n ,S ] of n and S , will be called coherent with respect tothe classical Liouville equation if during time-evolution it remains the same functional,9lthough n and S may change. According to [19], if the potential in H is a constant,linear, or quadratic polynomial of q , then f ψ is an exact solution of the Liouville equation ∂ t f ψ = { H, f ψ } if ψ is an exact solution ofi σ∂ t ψ = ˆ Hψ , ˆ H = − σ m ∆ + V , (34)formally identical to the time dependent Schr¨odinger equation (TDSE). Thus, f is co-herent for any Hamiltonian, but f ψ is coherent only for polynomial potentials of degree atmost 2. This restriction was derived before using different arguments both in algebraicand geometric quantization [1, 28].Beside the stability of the functional form, another aspect of the ”coherence” prop-erty is that for f and f ψ the two functions n and S play also the role of canonicallyconjugate variables. This aspect is particularly important for real waves quanta suchas photons or phonons [31], and it can be shown [19] that the equations of motion forthese variables can be derived from a variational principle related to infinite-dimensionalHamiltonian systems of the form i X W ˆ ω = dW , (35)where ˆ ω = d ˆ θ , ˆ θ = (cid:90) d q n dS , X W = (cid:90) d q ( ∂ t n ∂∂ n + ∂ t S ∂∂S ) , and W = (cid:82) d q w is the classical ( w ≡ w cl = n H ( ∇ S, q )) or quantum, ( w ≡ w q , (32) )energy functional of n and S .For an integrable distribution f ∈ L ( M ) on the symplectic manifold M = T ∗ Q ,the ”coordinates” ( n , S ) presume a foliation of M by Lagrangian submanifolds Λ S ⊂ M generated by S ∈ C ( W ) , W ⊂ Q , and the projection π : L ( M ) (cid:55)→ L ( Q ) , π ( f ) = n defined by integration on Λ S . If S is the solution of the Hamilton-Jacobi equation, thenthe asymptotic solution of the Schr¨odinger equation (34) in the WKB approximation ψ ∼ exp(i S/σ ) is related to the subspace of polarized sections r ∈ Γ L ( M, Λ S ) autoparallelon Λ S ( ∇ X r = 0 , ∀ X ∈ T Λ S ) in a complex Hermitian line-bundle with connection( L, ∇ ) over M [28, 3, 2]. Thus, the subspace of the coherent functionals f ψ defined bythe Wigner transform arises by a peculiar lift of Γ L ( M, Λ S ) to Γ L ( M ).Although the relationship between ψ and f ψ is nonlinear, and f ψ + ψ (cid:54) = f ψ + f ψ , wenote that if ψ , ψ are solutions of TDSE, and f ψ , f ψ satisfy the Liouville equation,then f ψ + ψ is also a solution of the Liouville equation . Denoting by ˆ P ψ ≡ | ψ (cid:105)(cid:104) ψ | the The spectacular case in which the interference terms between ψ and ψ may produce the resonanttransfer of a heavy particle across a macroscopic distance is discussed in [32] using estimates based on afinite triple-well potential. ψ , (cid:104) ψ | ψ (cid:105) = 1, the distribution f ψ (28) takes the form f ψ ( q , p ) = W ( ˆ P ψ ) ≡ π ) (cid:90) d k e − i k · p (cid:104) q | ˆ U k/ ˆ P ψ ˆ U k/ | q (cid:105) (36)with ˆ U k/ = e i k · ˆ p / . Thus, between f ψ and ˆ P ψ there exists a linear relationship by thetransform W . Moreover, if C denotes a complete set of states, ( (cid:80) ψ ∈C ˆ P ψ = ˆ1), then (cid:88) ψ ∈C f ψ ( q , p ) = 1(2 πσ ) . (37)In terms of group actions, the configuration space Q = R is homogeneous space for theLie group G = R of the space translations, T q Q (cid:39) T e G ≡ g , the momentum space T ∗ q Q is parameterized by p ∈ g ∗ , and k , the Fourier dual to p , enters in (36) as a parameteron G .The eigenfunctions ψ λ of ˆ H , ˆ Hψ λ = E λ ψ λ (38)are stationary solutions of (34), ψ λ ( t ) = e − iE λ t/σ ψ λ (0), and correspond to distributions f ψ λ independent of time, of energy E = < H > | f ψλ = (cid:90) d qd p f ψ λ H = (cid:104) ψ λ | ˆ H | ψ λ (cid:105) = E λ . (39)It is important to remark that this equality, which is used in many stationary variationalcalculations, holds for any Hamiltonian of the form (13) [33]. The problem of relativistic Wigner functions and Schr¨odinger equation for massive par-ticles was studied in [27] within the extended phase-space M e = T ∗ R presented in [34].Thus, the energy ( E ) and time ( t ≡ q /c ) become conjugate variables, evolving withrespect to a true parameter u , called universal time. A particular class of coherent solu-tions for the relativistic Liouville equation (RLE) consists of the ”action distributions” f e ( q e , p e , u ) = n e ( q e , u ) δ ( p − ∂ S ) δ ( p − ∇ S ) , (40)where n e is the probability density of localization in space-time. Considering ∂ u S = m c , in the case of a free particle we get the continuity equation m ∂ u n e = ∂ ( n e ∂ S ) − ∇ · ( n e ∇ S ) , (41)and ( ∂ S ) − ( ∇ S ) = m c . (42)For a density n e ( q e , u ) = δ ( q − cu ) n ( q , u ), localized in time, (41) reduces in the nonrel-ativistic limit to the usual continuity equation δ ( t − u )[ m ∂ u n + ∇ · ( n ∇ S )] = 0. The11 igure 1. m c / Γ from experimental data ( ∗ ) and the interpolation functions 2 . C/ Γ (solid)for 32 light unflavored meson resonances ( ω, η, φ, π, ρ, a, b, f ) with Γ ≥ .
43 MeV (A) and 48baryon resonances ( N, ∆ , Λ , Σ) with Γ ≥ . nonrelativistic identification of u as time may appear when t is a quasiclassical variable[18], described by a Gaussian wave-packet such that (cid:104) t (cid:105) = u . The width of this wave-packet sets a lower limit for the classical time-intervals, and a fundamental space-length (cid:96) . Evidence for the existence of such an elementary time-interval δt = (cid:96) /c = (cid:126) /m c was found in the particle data [27]. Thus, the ratio m c / Γ = τ L /δt between the mass(in MeV) and decay width (Γ), calculated using the experimental data for meson andbaryon resonances is well interpolated by functions of the form 2 . C/ Γ, where C is1222 MeV for mesons and 1487 MeV for baryons (Figure 1, Appendix 1), indicating thatthe lifetime τ L = (cid:126) / Γ is limited below by 2 δt .A quantum distribution˜ f e Ψ ( q e , k e , u ) ≡ Ψ( q µ + σk µ , u )Ψ ∗ ( q µ − σk µ , u ) , (43)is a ”static” solution ( ∂ u ˜ f e Ψ = 0) of the RLE if − σ (cid:3) Ψ = m c Ψ, (cid:3) ≡ ∂ − ∇ . When σ = (cid:126) this represents the Klein-Gordon equation.The extended phase-space is also the suitable framework to describe the electromag-netic field [36]. In vacuum the electric and magnetic fields E and B appear as coefficientsof two dual 2-forms ω f , ω ∗ f on the space-time manifold R , ω f = − B · d S + E · dq ∧ d q , ω ∗ f = E · d S + B · dq ∧ d q , (44)where dS = dq ∧ dq , dS = − dq ∧ dq , dS = dq ∧ dq . In the presence of the field thecanonical symplectic form ω e on T ∗ R for a relativistic massive particle which carriesthe electric charge q e and the magnetic charge q m becomes [36] ω e = ω e + q e c ω f + q m c ω ∗ f , (45)to account for the Lorentz forces F B = q e v × B /c and F E = − q m v × E /c . By specificintegrality conditions these two forms provide electric or magnetic charge quantization,while the exterior derivatives in vacuum dω f = dω ∗ f = 0 yield the wave equation (cid:3) E = Some of these old data do not appear anymore in [35]. B = 0. Such an equation has as coherent solutions any vector function of τ = t ± n · q /c ,where n is the unit vector along the propagation direction. For instance, E can be aharmonic or a Gaussian function of τ , as we may have a plane wave or a localized pulse.However, τ is not Lorentz-invariant, and instead it is convenient to consider coherentfunctionals of a Lorentz-invariant phase function ϕ ( q , q ) ∼ τ .The photon, as a relativistic particle of vanishing rest mass and energy (cid:15) = c | p | ,associated with the (real) electromagnetic waves, can be introduced considering theenergy-density continuity equation ∂ t w f = − div Y , (46)and the eikonal equation ( ∇ ϕ ) = ( ∂ ϕ ) , (47)which are similar to (41) and (42) with m = 0. Here w f = ( E + B ) / Y = c E × B is the Poynting vector. Because photons are freeparticles there are no ”zero-point energy” terms in the Planck distribution, (accuratelyretrieved in the 2.7 K cosmic microwave background spectrum [37]), or in the vacuumenergy density [38].The case of particles in states of negative energy ( E <
0) is peculiar because accordingto [34, 27], in such states the Lorentz group SO (1 ,
3) is replaced by SO (4), locallyisomorphic to SU (2) × SU (2). This means that for E < M [39], ds = (1 − γ G Mrc ) dq − (1 − γ G Mrc ) − dr − r ( dθ + sin θdϕ ) ,γ G = 6 . · − Nm /kg , shows no clear distinction between space and time coordinatesat r < R g = 2 γ G M/c , when formally the gravitational (binding) self-energy approaches − M c . Beside this similarity, we may speculate that the effect of the inertial parameter(mass) on the metric described at macroscopic scale by the general relativity may turnat atomic scale into an effect on the constant (cid:96) . At finite temperature ( T ) the thermal noise affects the statistical ensemble of the ”pure”action distributions (19), which evolve towards the classical equilibrium density f e , f e = N h e − βH Z µ , Z µ = (cid:90) M µ Ω µ h e − βH ,β = 1 /k B T , according to the Fokker-Planck equation ∂ t f + 1 m p · ∇ f − ∇ V · ∇ p f = γ ∇ p · ( p m + ∇ p β ) f , (48)13here γ denotes the friction coefficient. By the Fourier transform in momentum (48)becomes ∂ t ˜ f − i m ∇ k · ∇ ˜ f + k · (i ∇ V + γm ∇ k )˜ f = − γk B T k ˜ f . (49)A function ˜ f of the quantum form ˜ f ψ ( q , k ) = ψ ( a ) ψ ( b ) ∗ , (Appendix 2), with a = q + (cid:126) k / b = q − (cid:126) k /
2, can be written as a matrix element ˜ f ab = (cid:104) a | ψ (cid:105)(cid:104) ψ | b (cid:105) of the operator | ψ (cid:105)(cid:104) ψ | between the eigenstates | a (cid:105) , | b (cid:105) of the position operator ˆ q . With the notationˆ f ab = ˜ f ab / h we also get ∂ t ˆ f ab = ( ∂ t ˆ f ) ab , k = 1 σ ( a − b ) , k ˆ f ab = 1 σ [ˆ q , ˆ f ] ab , k ˆ f ab = 1 σ [ˆ q · , [ q , ˆ f ]] ab , ∇ k = σ ∇ a − ∇ b ) , ∇ k ˆ f ab = σ {∇ , ˆ f } (cid:48) ab , ∇ k · ∇ ˆ f ab = σ , ˆ f ] ab , where { , } (cid:48) denotes the anticommutator. At small kσ k · ∇ V ˆ f ab ≈ ( V a − V b )ˆ f ab = [ V, ˆ f ] ab , and as indicated in [19], if σ = (cid:126) , ˆ f /N = ˆ ρ = ˆ¯ ρ/ h , T r ˆ¯ ρ = 1, then for a single microscopicparticle (49) takes the form of the quantum Fokker-Planck equation for the usual densityoperator ˆ¯ ρ , i (cid:126) ∂ t ˆ¯ ρ = [ ˆ H, ˆ¯ ρ ] + γ m [ q , ·{ ˆ p , ˆ¯ ρ } (cid:48) ] − i γk B T (cid:126) [ˆ q , · [ˆ q , ˆ¯ ρ ]] . (50)This equation is similar to the one proposed in [40], and it takes the form considered in[18] if ˆ p from { ˆ p , ˆ¯ ρ } (cid:48) is replaced by (cid:104) ˆ p (cid:105) . Though, none of them has a satisfactory form,apparently due to the dissipative term. In classical mechanics, the effect of dissipationis not only energy loss, but also decrease in the phase-space volume. Thus, density fluc-tuations near the volume of the elementary cell may change the dissipation mechanism.In fact, when ˆ H = ˆ p / m the quantum equilibrium distributionsˆ¯ f ± = 1e β ˆ H − α ± , T r ˆ¯ f ± = N , (51)can be obtained as stationary solutions of the nonlinear equationi (cid:126) ∂ t ˆ¯ f = [ ˆ H, ˆ¯ f ] + γ m [ q , ·{ ˆ p , ˆ¯ f (1 ∓ ˆ¯ f ) } (cid:48) ] − i γk B T (cid:126) [ˆ q , · [ˆ q , ˆ¯ f ]] , (52)in which ∓ γ { ˆ p , ˆ¯ f } (cid:48) / m could be assigned to a density-dependent correction term to thefriction force [33]. However, before thermalization, while the thermal noise decreases thecoherence domain, one can expect a transition from complex ( ψ ) ”probability waves” toreal ( n ) density waves [19]. 14 Conclusions
The Fourier transform in momentum ˜ f of the distribution function on the classical phasespace is a density on the configuration space Q , such that coherent solutions of the Li-ouville equation, expressed as functionals of only two functions on Q , n and S , can befound. The action waves f (19) are localized in momentum and evolve according tothe Hamilton-Jacobi equation at infinite entropy. Coherent distributions (24) of finiteentropy can be found in the particular case of the harmonic oscillator. These are co-herent not only as Gaussian distributions on the phase-space, but also as the Wignertransform of the Glauber states, coherent for TDSE. In general, a functional f takesthe form of the Wigner function f ψ (28) by space discretization. Although f ψ is notpositive definite, it has a positive overlap with (24), and in this sense can be consideredas particle (or probability) density. During time evolution f ψ , with ψ an exact solutionof TDSE, remains coherent only for polynomial potentials of degree at most 2. Thismeans for instance that in a Coulomb potential either the classical Liouville equation,or TDSE should contain correction terms, which can be calculated and compared toother corrections (e.g. relativistic, QED [41]), or experimental data (e.g. the transitiontime in single atoms [42]). However, the equality (39) < H > | f ψ = (cid:104) ˆ H (cid:105) ψ , which can beused in time-independent variational calculations for the dominant part of the (quasi)stationary equilibrium distributions, holds for any potential [33].Relativistic Wigner functions can be defined similarly, by space-time discretizationof the action distributions on the extended phase space. Though, the presumed depen-dence of the minimum interval of time (the ”present”) on the inertial parameter, orthe problem of negative mass, indicate that the suitable framework for discussion is thegeneral relativity.At finite temperature the Fourier transform (49) of the classical Fokker-Planck equa-tion takes the form of the quantum transport equation (50) simply by considering thedensity ˜ f ψ ( q , k ) as a matrix element. However, to obtain the quantum equilibrium dis-tributions (51) as stationary solutions an additional, density-dependent correction to thedissipative term, is necessary.The results summarized above indicate that the functional coherent distributions onthe classical phase-space may provide the missing link between classical mechanics andquantum phenomenology. The ”action waves” (19) and the Wigner functions (28) aretwo examples of coherent distributions f [ n ,S ] related to the classical and quantum behav-ior, respectively, but the space of such solutions, its relationship to ”granularity”, andthe various aspects of ”decoherence” remain so far unexplored.15 Appendix 1: Particle data tables
Table 1.
Comparison between the experimental value of the mass ( M ) and the estimate2 . M, Γ from [35]).Resonance Γ (MeV) J P C M (MeV) 2 . f (1285) 24.2 ± . ++ ± . ± . η (1295) 55 ± − + ± ± . f (1500) 109 ± ++ ± ± . π (1800) 208 ±
12 0 − + ± ± . Table 2.
Comparison between the experimental value of the mass ( M ) and the estimate2 . M, Γ from [35]).Resonance Γ (MeV) J P M (MeV) 2 . ± − ± ± . − − − Let (2) be a partition of the µ -space M = T ∗ R and χ i : M → R the characteristicfunction of the cell b i . If all cells have the same volume Ω kµ = Ω , (the granularity), thenthe elements of the set C b = { χ i : M → R / χ i | b k = δ ik , i, k ∈ I b } have the properties:1 . (cid:90) M Ω µ χ i = Ω , . ( χ i , χ j ) = (cid:90) M Ω µ χ i χ j = Ω δ ij , . (cid:88) i ∈ I b χ i ( m ) = 1 , ∀ m ∈ M . (53)The partition is presumed adapted to a real polarization of (
M, ω ) such that the cellscan be separated by Lagrangian submanifolds Λ Q , Λ P , and χ i ( q , p ) is separable in thecoordinates ( q , p ) as χ i ( q , p ) = χ Qi ( q ) χ Pi ( p ), ∀ i ∈ I b . The cells are considered identical:cubic, simply connected, although in principle we may consider also quasi-degeneratepartitions with elongated, string-like cells, or with flat cells, almost plane.An observable f ∈ F ( M ) is called macroscopic with respect to C b if it can beaccurately approximated by f b = (cid:80) k ∈ I b f k χ k with f k = ( f, χ k ) / Ω . If ρ is a macroscopicprobability distribution and A a macroscopic observable then Ω ρ k is the probability oflocalization in the cell b k and < A > ρ = ( A, ρ ) = (cid:88) k ∈ I b ( A, χ k ) ρ k = Ω (cid:88) k ∈ I b A k ρ k (54)is the expectation value of A .If we change C b to C b (cid:48) = { χ (cid:48) k , k ∈ I b (cid:48) } by global translations, rotations or varia-tions in the shape of the cells at constant volume then formally χ (cid:48) i = (cid:80) k ∈ I b P bik χ k with16 bik = ( χ (cid:48) i , χ k ) / Ω ≥
0, but the matrix P b may be singular because χ (cid:48) i are not macro-scopic observables. The set { P bik , k ∈ I b } does not specify b (cid:48) i completely, but it canbe interpreted as its probability distribution over the partition { b k , k ∈ I b } , related to { b (cid:48) i ∩ b k , k ∈ I b } .Let ˜ f , ˜ f be the Fourier transforms in momentum of f , f ∈ F ( M ). Then( f , f ) = (cid:90) M d qd p f ( q , p ) f ( q , p ) = 1(2 π ) (cid:90) d qd k ˜ f ( q , k ) ˜ f ( q , − k ) . (55)Introducing new variables ( a , b ): a = q + (cid:126) k / b = q − (cid:126) k /
2, and the notationˆ f ab = ˜ f ( q , k ) / h , ˆ f ∗ ab = ˆ f ba , (55) becomes( f , f ) = h (cid:90) d ad b ˆ f ab ˆ f ba ≡ h T r ( ˆ f ˆ f ) . (56)In particular, when A ∈ F ( M ) is 1, q i or p i we get ˆ1 ab = δ ( a − b ), (ˆ q i ) ab = a i δ ( a − b ),(ˆ p i ) ab = − (i (cid:126) / ∂ a i − ∂ b i ) δ ( a − b ).A function χ ψ ∈ F ( M ) is called associated to a ”quantum cell” q ψ (topological sub-space of R ) if ( ˆ χ ψ ) ab is separable in the variables a and b such that ( ˆ χ ψ ) ab = ψ ( a ) ψ ( b ) ∗ (or ˆ χ ψ = | ψ (cid:105)(cid:104) ψ | ≡ ˆ¯ ρ ψ ) with ψ ∈ L ( R ), (cid:104) ψ | ψ (cid:105) = 1.Let E = { ψ n ∈ H / (cid:104) ψ n | ψ n (cid:48) (cid:105) = δ nn (cid:48) , n, n (cid:48) ∈ I q } be a countable orthonormal basis in H = L ( R ). Then the elements of the set C q = { χ ψ n ∈ F ( M ) , ψ n ∈ E} have theproperties:1 . ( χ ψ n ,
1) = h , . ( χ ψ n , χ ψ n (cid:48) ) = h δ nn (cid:48) , . (cid:88) n ∈ I q χ ψ n ( m ) = 1 , ∀ m ∈ M , (57)similar to (53) with Ω = h , while ρ ψ = χ ψ / h is the Wigner transform of ψ . Areference set is C ˆ H = { χ ψ n ∈ F ( M ) / ˆ Hψ n = E n ψ n , (cid:104) ψ n | ψ n (cid:48) (cid:105) = δ nn (cid:48) , n, n (cid:48) ∈ I q } ,defined by the eigenfunctions (38) of the 1-particle Hamiltonian ˆ H . With respect tothis set, a macroscopic (thermal) probability distribution on M has the form ρ q = (cid:80) n ∈ I q w n ρ ψ n , with w n ≥ (cid:80) n ∈ I q w n = 1, h ρ ψ n = χ ψ n ∈ C ˆ H , or ˆ ρ q = ˆ¯ ρ q / h whereˆ¯ ρ q = (cid:80) n ∈ I q w n | ψ n (cid:105)(cid:104) ψ n | is the density operator.For an arbitrary function ψ ∈ H , (cid:104) ψ | ψ (cid:105) = 1, the coefficient P qψψ n = ( χ ψ , χ ψ n ) / h = |(cid:104) ψ | ψ n (cid:105)| is interpreted as non-thermal probability distribution related to the expansion ψ = (cid:80) n ∈ I q (cid:104) ψ n | ψ (cid:105) ψ n . Because in principle supp ( χ ψ ) covers all phase space M thesubsets where χ ψ < (cid:104) ψ | ψ (cid:48) (cid:105) = 0 then ( χ ψ , χ ψ (cid:48) ) = 0. Toillustrate this situation in the case M = T ∗ R let us consider the orthogonal states ψ + and ψ − , ψ ± = η ± ( ψ G,d ± ψ G, − d ), where ψ G, ± d ( x ) = ( c/π ) / e − c ( x ∓ d ) / are Gaussianwave packets localized at x = ± d , c is a constant ( c = mω/ (cid:126) for the harmonic oscillatorground state), and η ± = (2 ± − cd ) − / are normalization factors. The correspondingWigner functions are ρ ± = η ± ( ρ d + ρ − d ± ρ i ) where ρ ± d ( x, p ) = 1 π (cid:126) e − c ( x ∓ d ) − p /c (cid:126) , ρ i ( x, p ) = 2 π (cid:126) e − cx − p /c (cid:126) cos(2 dp/ (cid:126) ) , (58)17uch that if d > ρ + , ρ − ) = 0. The negative values of ρ ± indicate that for thelocalized distribution ∆ q p ( q, p ) = δ ( q − q ) δ ( p − p ), the operator ˆ∆ q p ,( ˆ∆ q p ) ab = 2h ( ˆΠ q ) ab e i( a − b ) p / (cid:126) , ˆΠ q ψ ( x ) = ψ (2 q − x ) , (59)is not positive. For instance, if ( q , p ) = (0 ,
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