ad nqatmter,adacrigt tnadCopenhagen standard encompasses to system according the and of theory, actually quantum is measurement in of hand, process a we these whether pursue equations) deterministically Hamilton’s will eq th system Hamilton’s the solving classical ( by that numbers configurations pressuposed In faces past of all immediately set retrodict formulations. and a one conn future by disparate obstacles the described very grasp main completely have is to the theories attempts of two many One the been have mechanics. there classical years the Over Introduction 1 xsec fmte eoe nxrcbylne otepoeso m of process the to linked inextricably trajecto becomes defined process well matter measurement a of measu the of existence actual by notion The disturbed the is Futhermore, system measurement. wavefunction. wavefunct the a the that of that in sense outcome namely the the in for deterministic predictions not is description This arXiv:quant-ph/0110062v1Keywords: PACS: 10 Oct 2001 ∗ † [email protected] [email protected] omfruain ial,w ics oeapcso h dy the of aspects some discuss we Finally, a with w formulation. associated by-product, Bohm probabilities a and As values function. expectation Wigner the Thi the of quasi-distributions. formalism of the method with the through space phase 36.a 36.b 03.65.Ge 03.65.Db; 03.65.Ca; eso htted rgi-omitrrtto a eeasil be can interpretation Broglie-Bohm de the that show We aslItrrtto n unu hs Space Phase Quantum and Interpretation Causal eBoleBh nepeain us-itiuin,Wge quant Wigner quasi-distributions, interpretation, Broglie-Bohm De nvriaeLsooad uaiae Tecnologias e Humanidades Lus´ofona de Universidade v ap rne 7,14-2 iba Portugal Lisboa, 1749-024 376, Grande, Campo Av. all h nomto bu h tt ftesse hti cesbeto accessible is that system the of state the about information the eatmnod Matem´atica de Departamento oa uoPrata Jo˜ao Nuno uoCsaDias Costa Nuno coe 6 2018 26, October Abstract ~x, ~p 1 tagvntm,fo hc n a rdc all predict can one which from time, given a at ) eea bevbei h eBroglie- de the in observable general ∗ † ehdetbihsaconnection a establishes method s yi pie n hsteobjective the thus and spoiled is ry namics. banterlsfrevaluating for rules the obtain e ldfie rjcois(ouin of (solutions trajectories defined ll nepeain h wavefunction the interpretation, ain fmto.Mrvr tis it Morover, motion. of uations mlmne nquantum in implemented y easurement. eetpoesi problematic, is process rement efre rnt nteother the On not. or performed eutn nteclas fthe of collapse the in resulting cinbtenqatmand quantum between ection o nyprisprobabilistic permits only ion oytesaeo h system the of state the eory eie ntefc that fact the in resides mmechanics um us. Possibly, the most celebrated attempts to formulate classical and quantum mechanics in terms of a common language are the de Broglie-Bohm [1]-[3] interpretation and the Wigner (or, more generally, the quasi-distribution) formulation [4]-[17] of quantum mechanics. The purpose of this letter is to establish a relation between the two formulations. We will show that the de Broglie- Bohm theory can be implemented through a quasi-distribution in quantum phase space. In doing so, we shall obtain a systematic (albeit mathematically involved) method for computing the local expectation value of a generic operator Aˆ in a particular state |ψ>. As a by-product, we will write a previously missing formula for evaluating the probabilties associated with a generic observable. Finally, we discuss the Heisenberg picture of the de Broglie-Bohm formulation.
2 Quasi-distributions in quantum phase space
Let ψ(~x, t) be the wave function representing the state of a one-particle system at any given time t. The wave function is a solution of the Schr¨odinger equation:
∂ψ h¯2 ih¯ (~x, t)= − ∇~ 2 ψ(~x, t)+ V (~x)ψ(~x, t), (1) ∂t 2m x where V (~x) is the potential and ~x = (x, y, z). The momentum ~pˆ is canonically conjugate to ~xˆ: [ˆxi, pˆj]= ihδ¯ ij, i, j =1, 2, 3. According to Cohen’s classification [5], the f-quasidistribution function for the pure state ψ(~x, t) is given by:
1 1 1 ~ ′ F f ~x, ~p t dξ~ d~η d~x′ ψ∗ ~x′ h~η t ψ ~x′ h~η t f ξ,~ ~η t eiξ·(~x −~x)e−i~η·~p. ( ; )= 6 − ¯ ; + ¯ ; ( ; ) (2) (2π) Z Z Z 2 2 The average value of a generic operator Aˆ(~x,ˆ ˆ~p, t) can be evaluated according to:
f < ψ|Aˆ(~x,ˆ ˆ~p, t)|ψ>= d~x d~p A˜f (~x, ~p, t)F (~x, ~p, t), (3) Z Z where A˜f (~x, ~p, t) is a c-function known as the ”f-symbol” associated with the operator Aˆ:
3 h¯ ~ iξ~·~xˆ+i~η·ˆ~p −1 ~ −iξ~·~x−i~η·~p Vf Aˆ(~x,ˆ ˆ~p) = A˜f (~x, ~p, t) ≡ dξ d~ηT r Aˆ(~x,ˆ ˆ~p)e f (ξ, ~η, t) e . (4) 2π ! Z Z n o ∗ The application Vf : Aˆ(H) →A(T M) is called the ”f-map” [4] and, at each time t, can be regarded as an isomorphism between the quantum algebra of operators, Aˆ(H), acting on the Hilbert space H, and a ”classical” algebra of c-functions, A(T ∗M) in the phase space T ∗M. Consequently, the map Vf allows for a formulation of quantum mechanics in terms of ”classical-like” objects. ~ The most celebrated example corresponds to fW (ξ, ~η, t) = 1, in which case (4) yields the Weyl- symbol [6]:
3 h¯ ~ iξ~·~xˆ+i~η·ˆ~p −iξ~·~x−i~η·~p VW Aˆ(~x,ˆ ˆ~p) = A˜W (~x, ~p) ≡ dξ d~ηT r Aˆ(~x,ˆ ~pˆ)e e , (5) 2π ! Z Z n o
2 where VW is the Weyl map. The corresponding quasi-distribution function (2) is proportional to the Weyl-symbol of the quantum density matrixρ ˆ = |ψ ><ψ|. It is named the Wigner function [7]: 1 1 1 1 F W ~x, ~p, t V ρ d~ye−i~y·~pψ∗ ~x h~y, t ψ ~x h~y, t . ( ) ≡ 3 W (ˆ)= 3 − ¯ + ¯ (6) (2πh¯) (2π) Z 2 2 It is easy to check that it is a real function. The Weyl-symbol of the operator product and of the quantum commutator are known as the ∗-product and the Moyal bracket [8], respectively:
ih¯ Jˆ VW Aˆ · Bˆ = A˜W ∗W B˜W = A˜W (~x, ~p)e 2 B˜W (~x, ~p), (7) ¯h VW A,ˆ Bˆ = A˜W , B˜W =2iA˜W (~x, ~p) sin Jˆ B˜W (~x, ~p), M 2 h i h i ← → ← → ← → where Jˆ is the ”Poisson” operator: Jˆ ≡ 3 ∂ ∂ − ∂ ∂ , the derivatives ∂ and ∂ acting on i=1 ∂qi ∂pi ∂pi ∂qi A˜W and B˜W , respectively. From eq.(1) itP is possible to obtain the differential equation governing the dynamics of F W :
W 2 ∂F 1 W ~p = H˜W , F , H˜W (~x, ~p)= + V (~x), (8) ∂t ih¯ M 2m h i where H˜W (~x, ~p) is the Weyl symbol of the quantum Hamiltonian. Eq.(8) can be regarded as a deformation of the classical Liouville equation. A remarkable theorem by Baker [9] states that provided a real function F W satisfies (8) and some normalization requirement, then there exists a complex function ψ which (i) is a solution of the Schr¨odinger equation (1) and (ii) is related to F W according to eq.(6). Consequently, the Wigner function encompasses the same information about the state of the system as does the wave function. Indeed, it is possible to evaluate all the mean values (using eq.(3)) and also all probabilities according to the following procedure. Consider a complete set of commuting observables Aˆ1, ··· , Aˆn, and let |a1, ··· , an > be a general eigenstate, satisfying: Aˆi|a1, ··· , an >= ai|a1, ··· , an >, (i = 1, ··· , n). The projector |a1, ··· , an >< a1, ··· , an| also satisfies the previous eigenvalue equation:
Aˆi|a1, ··· , an >< a1, ··· , an| = ai|a1, ··· , an >< a1, ··· , an|, i =1, ··· , n. (9)
1 The probability that a measurement of the operator Aˆi yield the eigenvalue ai is given by :
2 da1 ··· dai−1 dai+1 ··· dan | < ψ|a1, ··· , an > | = Tr {ρˆ|ai >< ai|} , (10) Z Z Z Z where |ai >< ai| is also a projector, which reads:
|ai >< ai| ≡ da1 ··· dai−1 dai+1 ··· dan |a1, ··· , an >< a1, ··· , an|. (11) Z Z Z Z
Notice that |ai >< ai| is also a solution of the eigenvalue equation (9). 1If the spectrum is discrete, the integrals are replaced by sums.
3 In the formalism of the Wigner function, one introduces the so called stargenfunctions [14]-[17]:
W W δ∗ A˜1W − a1 ∗W ···∗W δ∗ A˜nW − an = VW (|a1, ··· , an >< a1, ··· , an|) (12) ˜ W ˜ 1 +∞ ik(AiW −ai) δ∗ AiW − ai = VW (|ai >< ai|)= 2π −∞ dk e∗W , R A(~x,~p) ∞ 1 n∗W ˜ ˆ ∗ where AiW = VW Ai (i =1, ··· , n). The ∗-exponential is defined by e W = n=0 n! (A(~x, ~p)) , n∗W n−1∗W 0∗W where A = A∗W A and A = 1. These stargenfunctions are solutionsP of the stargenvalue equations:
W W W W A˜iW ∗W δ∗ A˜1W − a1 ∗W ···∗W δ∗ A˜nW − an = aiδ∗ A˜1W − a1 ∗W ···∗W δ∗ A˜nW − an n o W W A˜iW ∗W δ∗ A˜iW − ai = aiδ∗ A˜iW − ai , (13) which are the Wigner-Weyl counterparts of the eigenvalue equations (9). Moreover, the probability (10) can be evaluated according to the following suggestive formula2:
W W P1 (Ai(t)= ai)= d~x d~pF (~x, ~p, t)δ∗ (A˜iW − ai). (14) Z Z ˆ ˆ ˆ W ′ ′ W ′ ′ In particular, for A = ~x, ~p, it can be shown that: δ∗ (xi −xi)= δ(xi −xi) and δ∗ (pi −pi)= δ(pi −pi) (i =1, 2, 3), and we thus get the classical-like results:
′ W 3 ′ 2 P1(~x = ~x ) = d~x d~pF (~x, ~p)δ (~x − ~x )= |ψ(~x)| , R R (15) ′ W 3 ′ 2 P1(~p = ~p ) = d~x d~pF (~x, ~p)δ (~p − ~p )= |φ(~p)| , R R where φ(~p) is the Fourier transform of ψ(~x). Notice the remarkable similarities between the function W δ∗ and the ordinary δ-function. Not only does it satisfy eqs.(13,14) and:
d~x d~p A˜ (~x, ~p)δW (A˜ − a) W ∗ W = a, (16) W R R d~x d~pδ∗ (A˜W − a)
R R W it is also anh ¯-deformation of the δ-function: δ∗ (A˜W − a)= δ(A˜W − a)+ O(¯h). It therefore works as a δ-function in quantum *-phase space. For these reasons it is called the *-delta function. From eqs.(3,15) one could be tempted to interpret the Wigner function as a probability distri- bution in phase space. However, this interpretation is immediately spoiled, if one realizes that it can take on negative values. It is a well known fact, that some undesired property will hinder a full classical interpretation for any quasi-distribution. This is usually interpreted as a manifestation of Heisenberg’s uncertainty principle. Finally, the Wigner-Weyl formalism can be regarded as a basis for other quasi-distributions (i.e. with f(ξ,~ ~η, t) =6 1). It can be checked (cf.(2,4)) that the f-symbol of a certain observable and the 2 Here Ai denotes the observable Aˆi irrespective of the particular representation chosen, i.e. irrespective of the particular function f ξ,~ ~η, t .
4 corresponding quasi-distribution can be obtained from the Weyl symbol and the Wigner function, according to the followig prescription [4, 5]:
−1 A˜f (~x, ~p, t)= f −i∇~ x, −i∇~ p, t A˜W (~x, ~p), (17) f W F (~x, ~p, t)= f i∇~ x, i∇~ p, t F (~x, ~p, t), where ∇~ = ∂ , ∂ , ∂ . This permits us to write the counterpart of eqs.(7), (12), (13) and (14) p ∂px ∂py ∂pz for different choices of function f(ξ,~ ~η, t). Starting with equation (7), we have:
˜ ˜ −1 ~ ~ ˜ ˜ −1 ~ A ~ B ~ A ~ B Af ∗f Bf = f −i∇x, −i∇p, t AW ∗W BW = f −i∇x − i∇x , −i∇p − i∇p , t × (18) ~ A ~ A ~ B ~ B i¯h ~ A ~ B ~ A ~ B ˜ ˜ ×f −i∇x , −i∇p , t f −i∇x , −i∇p , t exp 2 ∇x · ∇p − ∇p · ∇x Af Bf , h i where the superscripts A and B mean that the derivatives act on A˜f and B˜f respectively. Using this definition of the star-product ∗f , we can further define the f-stargenfunction: ∞ 1 + ˜ δf A˜ a dk eik(Af −a), ∗ ( f − )= ∗f (19) 2π Z−∞ which can easily be shown to be the f-symbol of |a>< a| (cf.(4,11,12)). Furthermore, from (13), (14) and (17) it follows that:
f f A˜f ∗f δ∗ (A˜f − a)= aδ∗ (A˜f − a) (20) f f P1 (A(t)= a)= d~x d~pF (~x, ~p, t)δ∗ A˜f (t) − a . R R 3 Causal interpretation
In the de Broglie-Bohm interpretation [1]-[3] of quantum mechanics, the wave function is written in the form: i ψ(~x, t)= R(~x, t) exp S(~x, t) , (21) h¯ where R(~x, t) and S(~x, t) are some real functions. Substituting this expression in eq.(1) we obtain the dynamics of R and S:
∂R 1 ~ 2 ~ ~ ∂t = − 2m R∇xS +2∇xR · ∇xS , h i (22) 2 2 ∂S 1 ~ ¯h 1 ~ 2 ∂t = − 2m ∇xS − V (~x)+ 2m R ∇xR. 2 2 The distribution P2(~x) ≡|ψ(~x)| = R (~x) corresponds to the probability of finding the particle at ~x. In terms of P2, we can rewrite eq.(22) as:
∂P2 ~ P2 ~ ∂t + ∇x · m ∇xS =0, (23) 2 2 ∂S 1 ~ ¯h2 1 ~ 2 1 1 ~ + ∇xS + V (~x) − ∇ P − 2 ∇xP =0. ∂t 2m 4m P2 x 2 2 P 2 2 5 P2 ~ The first equation is a statement of probability conservation, with associated flux m ∇xS. The second equation is interpreted as a Hamilton-Jacobi equation. The solution S(~x, t) corresponds to a set of trajectories, known as Bohmian trajectories, stemming from some original position ~x with probability P2(~x), for a particle under the influence of a classical potential V (~x) and a quantum potential: 2 2 h¯ 1 ~ 2 h¯ 1 ~ 2 1 1 ~ 2 Q(~x, t) ≡ − ∇xR = − ∇xP2 − 2 ∇xP2 . (24) 2m R 4m "P2 2 P # 2 Morover, the momentum ~p is subject to the constraint:
~p = ∇~ xS, (25) holding at all times (including t = 0). We can thus conclude that, for a pure state, the quantum phase space distribution is given by [1]:
B 3 ~ F (~x, ~p; t) ≡ P2(~x, t)δ ~p − ∇xS(~x, t) . (26) The dynamics of F B is obtained from eqs.(22,23): ∂F B (~x, ~p; t)= H(~x, ~p)+ Q(~x; t), F B(~x, ~p; t) , (27) ∂t P n o B where H is given by eq.(8) and {, }P is the Poisson bracket. F is positive defined, real and has the marginal distribution:
2 B P2(~x; t)= R (~x; t)= d~pF (~x, ~p; t), (28) Z which coincides with P1(~x; t) (eq.(15)). Integration over the positions yields the momentum prob- ability distribution P2(~p, t). However, this distribution is different from the one in eq.(15). In the de Broglie-Bohm formulation it corresponds to the true momentum of the particle [1]. In order to distinguish the two, we use the subscripts 1 and 2. Mean values of observables are evaluated according to the formula:
B < A>ˆ = d~x d~pAB(~x, t)F (~x, ~p, t), (29) Z Z where the ”local expectation value” AB is given by [1]: ψ∗(~x, t)(Aψˆ )(~x, t) AB(~x, t)= Re . (30) |ψ(~x, t)|2 ! With this definition, we get for instance:
~pB(~x, t)= ∇~ xS (Momentum) 2 (∇~ xS) H (~x, t)= + Q(~x, t)+ V (~x) (Energy) B 2m (31) L~ B(~x, t)= ~x × ∇~ xS (Angular momentum)
2 h2 2 L~ 2 (~x, t)= ~x × ∇~ S − ¯ ~x × ∇~ R B x R x 6 It is important to stress that these objects are not f-symbols of the corresponding quantum mechanical operators. In fact, and contrary to ordinary f-symbols, these objects live in configuration space. A second remark, this time regarding the dynamics, is in order. The classical looking equation (27) does not yield the complete time evolution. This is because if we want F B to preserve its form with time (and this is crucial for the causal interpretation), we have to evolve the local expectation values as well. We shall come back to this later.
4 De Broglie-Bohm formulation as a quasi-distribution
4.1 Quasi-distribution and Bohm-symbols In this section we show that eq.(26) can be obtained from eq.(2) for a particular choice of the function f: d~u|ψ(~u; t)|2 exp i~η · ∇~ S(~u; t)+ iξ~ · ~u ~ u fB(ξ, ~η; t)= . (32) R ∗ ¯h ¯h iξ~·~v d~vψ ~v − 2 ~η; t ψ ~v + 2 ~η; t e This is the most important formula ofR this work. Substituting in eq.(2) , we obtain:
~ ~ ~ B 1 ~ −iξ·~x−i~η·~p 2 i~η·∇uS(~u,t)+iξ·~u F (~x, ~p; t)= (2π)6 dξ d~η d~ue |ψ(~u, t)| e = R R R (33) ~ 1 −i~η·~p 2 i~η·∇uS(~u,t) 3 2 3 ~ = (2π)3 d~η d~ue |ψ(~u, t)| e δ (~u − ~x)= |ψ(~x, t)| δ ~p − ∇xS(~x, t) , R R and we do indeed recover eq.(26). Moreover the Bohm-symbol of an operator Aˆ(~x,ˆ ˆ~p) can be obtained from the Weyl-symbol according to eq.(17):
˜ ˆ ˆ ˆ −1 ~ ~ ˜ AB(~x, ~p, t)= VB A(~x, ~p) = fB −i∇x, −i∇p, t AW (~x, ~p). (34) If we expand eq.(32) in powers of ~η, we get:
2 h¯ ~ ~ f −1(ξ,~ ~η; t)=1+ < eiξ·~x >−1< (∂ ∂ ln R) eiξ·~x > η η + ··· (35) B 4 i j i j where we used the notation: < g(~x) >= R2(~x)g(~x)d~x for any function g(~x). From the previous ˆ ˆ ˆ ˜ ˜ equation and eq.(34), we conclude that ifR A is independent of ~p or linear in ~p, then AB = AW . Consequently:
VB V (~xˆ) = V (~x) VB (ˆpi) = pi (36) V Lˆ = ǫ x ∗ p = ǫ x p + i¯h δ =(~x × ~p) B i j,k ijk j k j,k ijk j k 2 jk i Let us go back to the remark madeP at the end of theP previous section. The Bohm-symbols are phase space quantities and must therefore differ from the local expectation values in equations (30,31), which live in configuration space. This notwithstanding both sets will yield identical predictions, as we shall see.
7 For operators Bˆ(ˆx, pˆ) quadratic in the momentum, the Bohm-symbol becomes more involved. 2 ˆ ˆ ~pˆ ˆ ˆ As a concrete example, consider the kinetic energy, T (~p) = 2m . Since T is independent of ~x, we obtain from (34,35):
2 2 2 2 2 ∇~ 2 ˆ ~p ¯h ∂ ∂ ~p ~p ¯h xR 1 ~ 2 VB T = − < (∂i∂j ln R) > = − < − 2 (∇xR) >= 2m 4 ∂pi ∂pj 2m 2m 4m R R (37) 2 2 ~ 2 2 ~p ¯h ∇xR ~p = 2m − 2m < R >= 2m + < Q(~x; t) > . The mean total energy is then: ˆ ˆ ˆ ~p 2 2 2 3 ~ < H(~x, ~p) >= d~x d~p 2m − d~uR (~u, t)Q(~u, t)+ V (~x) R (~x, t)δ ~p − ∇xS(~x, t) = n o R R R (38) ∇~ 2 2 ( xS(~x,t)) = d~xR (~x, t) m + V (~x)+ Q(~x, t) , ( 2 ) R 2 2 ~ˆ ~ˆ 2 3 2 3 as expected. Now let us consider L . Since VW (L )=(~x × ~p) − 2 h¯ , we get : 2 2 ˆ 2 3 2 ¯h ~x·∇~ x −1 ~x·∇~ x ∂ ∂ 2 VB(L~ )=(~x × ~p) − h¯ − < e > < (∂i∂j ln R) e > (~x × ~p) = 2 4 ∂pi ∂pj
2 3 2 ¯h2 ∂ =(~x × ~p) − h¯ − < ∂i∂j ln R> + [< xk∂i∂j ln R> − < xk >< ∂i∂j ln R>] + 2 2 ∂xk n 1 + 2 [< xkxl∂i∂j ln R> −2 < xk >< xl∂i∂j ln R> +
∂ ∂ 2 +2 < xk >< xl >< ∂i∂j ln R> − < xkxl >< ∂i∂j ln R>] (δij~x − xixj)= ∂xk ∂xl o 2 3 2 2 ~ 2 1 2 2 =(~x × ~p) − 2 h¯ +¯h < ∇ ln R> − 2 (~x − < ~x >)+(xi− < xi >) < xi > + h i 2 1 +¯h < ∂i∂j ln R> 2 (xixj− < xixj >) − (xj− < xj >) < xi > h i 2 ~ 2 ¯h2 ¯h2 2 ~ 2 −h¯ < xi∇ ln R> − < xj∂i∂j ln R> × (xi− < xi >)+ 2 < xixj∂i∂j ln R> − 2 < ~x ∇ ln R > . ˆ2 The previous expression is the Bohm symbol associated with L~ , which is strikingly different from ˆ 2 its local expectation value (31). We now show that the two predictions for the mean value of L~ coincide. 2 2 ~ˆ ~ˆ 2 3 ~ ~ 2 3 2 ¯h2 < L >= d~x d~pVB(L )R (~x)δ ~p − ∇xS =< (~x × ∇xS) > − 2 h¯ + 2 < xixj∂i∂j ln R> R R ¯h2 ~ 2 2 ~ 2 3 2 ¯h2 ¯h2 2 ~ 2 − 2 < ~x 2∇ ln R>= d~xR (~x) (~x × ∇xS) − 2 h¯ + 2 xixj∂i∂j ln R − 2 ~x ∇ ln R . n o 3 R2 ¯h2 ~ Taking into account that 2 h¯ = 2 ∇x · ~x, we get after a few integrations by parts: 2 2 ~ˆ 2 ~ 2 2 1 ~ 2 1 ∂ R 2 2 1 ~ 2 < L >= d~xR (~x) (~x × ∇xS) + 2¯h (~x · ∇x)R +¯h xixj − h¯ ~x ∇xR . (39) Z ( R R ∂xi∂xj R ) 3 ∂ i Sum over repeated indices is understood and ∂ = ∂xi .
8 2 2 1 ~ The latter three terms can be rewritten as −h¯ R ~x × ∇x R, and we thus recover eq.(31). 4.2 Stargenfunctions and probabilities In order to compute of the probability functionals associated with projectors of the form |a>
′ B 3 ′ 2 ′ P1(~x , t)= d~x d~pF (~x, ~p, t)δ (~x − ~x )= R (~x , t). (41) Z Z ˆ This means that P1(~x, t) is indeed equal to P2(~x, t) (28). However, the ∗-genvalue equation for ~p is more involved: ˆ~p|φ >< φ| = ~p′|φ >< φ| =⇒ |φ >< φ| = |~p′ >< ~p′|. Upon application of the B ′ B Bohm-map, we obtain: ~p ∗B Θ (~x, ~p, t)= ~p Θ (~x, ~p, t), with solution: B ′ ′ B ′ −1 ~ 3 ′ Θ (~x, ~p, t)= VB (|~p >< ~p |)= δ∗ (~p − ~p , t)= fB 0, −i∇p, t δ (~p − ~p ), (42) which is not identical to δ3(~p − ~p′). Therefore the marginal probability distribution for ~p is given by: ′ B B ′ P1 (~p , t)= d~x d~pF (~x, ~p, t)δ∗ (~p − ~p , t). (43) Z Z ′ ′ This expression coincides with P1(~p , t) in eq.(15), but not with P2(~p , t). For a generic operator Aˆ and an eigenstate |a>, we get from (19,34): B ˜ −1 ~ ~ W ˜ δ∗ (AB − a, t) ≡ VB (|a> 4.3 Dynamics and the Heisenberg picture From eqs.(32,34) one realizes that, in the Schr¨odinger picture, where the observables remain unal- tered and the wave function evolves in time, the Bohm-symbol will also evolve. This is clear from (30,31) as well. We conclude that, contrary to what happens in the Wigner-Weyl case, the time- evolution of the distribution function (eq.(27)) is not sufficient to pin down the complete dynamics. On the other hand, the Heisenberg picture [16] seems to be more adjusted to the de Broglie-Bohm B case. If the wave function does not evolve, then F and fB do not evolve either and the complete dynamics is encapsulated in the time evolution of the Bohm-symbols, according to [4, 5]: ∂A˜B 1 −1 (~x, ~p; t)= f −i∇~ x, −i∇~ p A˜W (~x, ~p; t), H˜W (~x, ~p) = ∂t i¯h B M h i 2 ~ A ~ A ~ H ~ H −1 ~ A ~ H ~ A ~ H = ¯h fB −i∇x , −i∇p fB −i∇x , −i∇p fB −i∇x − i∇x , −i∇p − i∇p × (45) ¯h A H A H × sin ∇~ ∇~ − ∇~ ∇~ A˜BH˜B = A˜B(~x, ~p, t), H˜B(~x, ~p) . 2 x p p x B h i h i 9 5 Example The previous results will be illustrated with the example of a coherent state. We shall adopt the Heisenberg picture, where the wavefunction is time independent: 1 (x − x )2 i ψ x 0 p x . ( )= 1 exp − 2 + 0 (46) [2π(∆x)2] 4 " 4(∆x) h¯ # Since S(x)= p0x, a trivial calculation, yields (cf.(26,32)): h¯2η2 fB(ξ, η)= fB(η) = exp , (47) " 8(∆x)2 # and: 1 (x − x )2 F B x, p δ p p 0 . ( )= 1 ( − 0) exp − 2 (48) [2π(∆x)2] 2 " 2(∆x) # From (41) it is easy to obtain: 1 (x − x )2 x 0 R2 x , p δ p p . P1( )= 1 exp − 2 = ( ) P2( )= ( − 0) (49) [2π(∆x)2] 2 " 2(∆x) # 2 2 − h¯ η B ′ −1 8(∆x)2 Let us now compute δ∗ (p − p ) from eq.(42). Notice that fB (η)= e . 2 2 2 2 h¯ ∂ h¯ ∂ ′ B ′ 8(∆x)2 ∂p2 ′ 1 8(∆x)2 ∂p2 ik(p−p ) δ∗ (p − p )= e δ(p − p )= 2π dk e e = R (50) h¯2k2 ′ 1 − +ik(p−p ) 2 2 2 1 8(∆x)2 2(∆x) (∆x) ′ 2 = 2π dk e = π¯h2 exp −2 ¯h2 (p − p ) . h i h i Let us verify that this isR indeed a solution of the star-genvalue equation (cf.(18)): −1 B ′ B ′ ∂fB ∂ ∂ B ′ p ∗B δ∗ (p − p )= pδ∗ (p − p ) − i ∂η 0, −i ∂p fB 0, −i ∂p δ∗ (p − p )= B ′ ¯h2 ∂ B ′ = pδ∗ (p − p )+ 4(∆x)2 ∂p δ∗ (p − p )= (51) B ′ ¯h2 (∆x)2 ′ B ′ ′ B ′ = pδ∗ (p − p )+ 4(∆x)2 −4 ¯h2 (p − p ) δ∗ (p − p )= p δ∗ (p − p ). h i B ′ And so, δ∗ (p − p ) is indeed the solution of the stargenvalue equation, as expected. Substituting (50) in eq.(43), yields: 1 2 2 2 ′ B B ′ 2(∆x) (∆x) ′ 2 P1(p )= dx dp F (x, p)δ∗ (p − p )= 2 exp −2 2 (p − p0) , (52) Z Z " πh¯ # " h¯ # which coincides exactly with |φ(p′)|2, where φ(p′) is the Fourier transform of (46). Let us now compute the time evolution (45) of a generic observable A˜B(x, p) induced by a Hamiltonian with ˜ p2 Weyl symbol HW (x, p)= 2m + V (x): ∂ 1 h¯2 ∂2 p2 A˜ x, p t A˜ x, p t , V x . B( ; )= exp 2 2 W ( ; ) + ( ) (53) ∂t ih¯ 8(∆x) ∂p ! " 2m #M 10 First of all notice that: 2 1 −1 ∂ ∂ p 1 −1 ∂ ∂ ∂A˜W f −i , −i A˜W (x, p; t), = f −i , −i p = i¯h B ∂x ∂p 2m M m B ∂x ∂p ∂x h i (54) − 1 2 2 p −1 ∂ ∂ ∂A˜W i ∂fB ∂ ∂ ∂A˜W p ∂A˜B ¯h ∂ A˜B = m fB −i ∂x , −i ∂p ∂x − m ∂η −i ∂x , −i ∂p ∂x = m ∂x + 4m(∆x)2 ∂x∂p . Similarly: ∂ ∂ 1 −1 ˜ 1 ˜ fB −i , −i AW (x, p; t),V (x) = AB(x, p; t),V (x) , (55) ih¯ ∂x ∂p ! M ih¯ M h i h i where we used the fact that fB is independent of ξ. Consequently: ˜ 2 2 ˜ ∂ 2 ∂AB p h¯ ∂ AB 1 < Aˆ(t) >= dx dp R (x)δ(p − p0) + + A˜B,V (x) . (56) ∂t ∂p m 4m(∆x)2 ∂x∂p ih¯ M ! Z Z h i 2 ∂R ∂R 2 (x−x0) Since ∂x =2R ∂x = R (x) (∆x)2 , we get after an integration by parts: ∂ h¯2(x − x ) ∂A˜ 1 p2 < Aˆ t > dx dp R2 x δ p p 0 B A˜ , V x . ( ) = ( ) ( − 0) 4 + B + ( ) (57) ∂t Z Z 4m(∆x) ∂p ih¯ " 2m #M ! 2 2 2 2 ¯h (x−x0) ∂ ¯h ¯h (x−x0) ∂Q Now notice that: − 4m(∆x)4 = ∂x 4m(∆x)2 − 8m(∆x)4 = ∂x (x), where Q(x) is the quantum potential associated with (46). Finally, weh get: i 2 ∂ B p ih¯ < Aˆ(t) >= dx dp F (x, p) A˜B(x, p; t), + V (x)+ Q(x) . (58) ∂t Z Z " 2m #M Notice that this formula is valid both for the observable Aˆ as well as for the stargenfunction associ- B ated with the projector |a> 6 Conclusions We have established a connection between the de Broglie-Bohm interpretation and the quasi- distribution formulation of quantum mechanics. For the theorist familiar with the latter formu- lation, the causal interpretation corresponds (at least formally) to just another quasi-distribution. Consequently, the whole machinery for calculating average values, probability functionals and the dynamics can be trivially applied. Our aim was not to construct a proper causal interpretation in non-commutative quantum phase space. That has been the purpose of some attempts in the past, [18]-[20]. However, to our result that 11 the de Brolie-Bohm interpretation can be implemented via a quasi-distribution there is a converse result. Namely, that any quasi-distribution can be cast in a form almost identical to the causal form (26), except that the underlying variables ~x and ~p no longer commute. The Wigner function can then be regarded as ah ¯-deformation of the Bohm function (26). This will be the subject of a future work [21]. Acknowledgments We would like to thank Jo˜ao Marto for useful suggestions and for reading the manuscript. This work was partially supported by the grants ESO/PRO/1258/98 and CERN/P/Fis/15190/1999. References [1] P.R. Holland, The Quantum Theory of Motion, (1993) Cambridge University Press. [2] L. de Broglie, C.R. Acad.Sci.Paris 183 (1926) 24; Nature 118 (1926) 441; C.R. Acad.Sci.Paris 184 (1927) 273; C.R. Acad.Sci.Paris 185 (1927) 380; J. de Phys. 8 (1927) 225. [3] D. Bohm, Phys. Rev. 85 (1952) 166; Phys. Rev. 85 (1952) 180. [4] H.W. Lee, Phys. Rep. 259 (1995) 147. [5] L. Cohen, J. Math. Phys. 7 (1966) 781. [6] H.Weyl, Z. Phys. 46 (1927) 1. [7] E.Wigner, Phys. Rev. 40 (1932) 749. [8] J.Moyal, Proc. Camb. Phil. Soc. 45 (1949) 99. [9] G.Baker, Phys. Rev. 109 (1958) 2198. [10] R.F.O’Connell, E.P.Wigner, Phys. Lett. (1981) 121. [11] P.Carruthers, F.Zachariasen, Rev. Mod. Phys. 55 (1983) 24. [12] N.Balazs, B.Jennings, Phys. Rep. 104 (1984) 347. [13] N.C.Dias, J.N.Prata, e-print: quant-ph/0003005 (2000). [14] T.Curtright, D.Fairlie, C.Zachos, Phys. Rev. D58 (1998) 025002. [15] D.B.Fairlie, hep-th/9806198. [16] N.C.Dias, J.N.Prata, e-print: quant-ph/0110035 (2001). [17] F. Bayen et al., Annals of Physics 111 (1978) 61; Annals of Physics 110 (1978) 111. 12 [18] H.W. Lee, M.O. Scully, Found. Phys. 13 (1983) 61. [19] P.R. Holland et al., Phys. Rev. A 33 (1986) 4380. [20] G.G. de Polavieja, Phys. Lett. A 220 (1996) 303. [21] N.C.Dias, J.N.Prata, in preparation. 13