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Bohm's Potential, Classical/Quantum Duality and Repulsive Gravity

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Bohm's Potential, Classical/Quantum Duality and Repulsive Gravity Physics Letters B 788 (2019) 546–551 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Bohm’s potential, classical/quantum duality and repulsive gravity Carlos Castro Perelman a,b a Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA, 30314, United States of America b Ronin Institute, 127 Haddon Place, Montclair, NJ, 07043, United States of America a r t i c l e i n f o a b s t r a c t Article history: We propose the notion of a classical/quantum duality in the gravitational case (it can be extended to Received 26 September 2018 other interactions). By this one means exchanging Bohm’s quantum potential for the classical potential Accepted 7 November 2018 V Q ↔ V in the stationary quantum Hamilton–Jacobi equation (QHJE) so that V Q + V =−V 0 (ground Available online 28 November 2018 state energy). Despite that the corresponding Schrödinger equations, and their solutions differ, their Editor: M. Cveticˇ associated quantum Hamilton–Jacobi equation, and ground state energy remains the same. This is Keywords: how the classical/quantum duality is implemented. In this scenario Bohm’s quantum potential (which Bohm’s potential coincides with the attractive Newtonian potential) is now correlated to a classical repulsive gravitational Quantum mechanics potential (plus a constant). These results suggest that there might be a quantum origin to the classical (Repulsive) gravity repulsive gravitational behavior (of the accelerated expansion) of the universe which is based on this notion of classical/quantum duality. We hope that the notion of classical/quantum duality raised in this work in connection to the QHJE may cast further light into the deep interplay between gravity and quantum mechanics. © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . David Bohm showed long ago [1] that the Schrödinger equation scalar spatial curvature produced by an ensemble density of paths for the complex valued wave function (r, t) is equivalent to the associated with one, and only one particle, as shown in [2]. The h¯ 2 coupled pair of equations − constant of proportionality is 2m . It can be generalized to the rel- √ ativistic case. This geometrization process of quantum mechanics ∂ S (p)2 (∇ S)2 h¯ 2 ∇2 ρ − = + V Q + V = − √ + V (1) (not to be confused with geometric quantization) allowed to derive ∂t 2m 2m 2m ρ the Schrödinger, Klein–Gordon [2] and Dirac equations [3–5]. Most ∂ρ ∇ S recently, a related geometrization of quantum mechanics was pro- +∇·(ρ ) = 0(2) ∂t m posed [6]that describes the time evolution of particles as geodesic The first equation is the Quantum Hamilton–Jacobi Equation lines in a curved space, whose curvature is induced by the quan- tum potential. This formulation allows therefore the incorporation (QHJE) involving an external potential√ V (r), and including Bohm’s ¯ 2 ∇2 ρ of all quantum effects into the geometry of space–time, as it is the quantum potential V =−h √ (ρ is the probability density Q 2m ρ case for gravitation in the general relativity. and S is the action). The second equation is the continuity equa- The above result of Bohm can be generalized to many parti- tion. The substitution cles as well where the wavefunction depends on all of the particle iS r t h¯ coordinates (configuration space). In Bohmian mechanics the time (r,t) ≡ ρ(r,t)e ( , )/ (3) evolution of a quantum system comprised of N particles is driven into eqs. (1)–(2)yields the Schrödinger equation by the nonlocal quantum potential V Q (r1, r2, ··· , rN ; t) which is a ¯ 2 function of the entire configuration space and time. In the case of ∂(r,t) h 2 ih¯ ( ) =− ∇ (r,t)) + V (r)(r,t) (4) two particles [9], it is convenient to express all physical quantities ∂t 2m √ √ in terms of the center of mass coordinate R, and the relative radial =−h¯ 2 ∇2 Bohm’s quantum potential V Q 2m ( ρ/ ρ) was shown coordinate r given by to be proportional to the difference of the Weyl and Riemann + = m1r1 m2r2 = − E-mail address: [email protected]. R , r r1 r2 (5) m1 + m2 https://doi.org/10.1016/j.physletb.2018.11.013 0370-2693/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. C.C. Perelman / Physics Letters B 788 (2019) 546–551 547 ¯ 2 ¯ 2 2 the total mass M and reduced mass m are h 1 2 h A A − √ ∂r (r ∂r σ (r)) =− ( + ) (19) 2m σ (r) r2 2m r 4 = + = m1m2 M m1 m2, m (6) 2 2 2 m + m h¯ 1 h¯ B 1 2 − √ ∂ (R2∂ ζ(R)) =− ( ) (20) 2 R R When the temporal dependence of the action is of the form 2M ζ(R) R 2m 4 A < 0, B < 0must be negative due to the normalization condition S = S(r, R,t) = S(r, R) − Et (7) ∞ ∞ the stationary quantum Hamilton–Jacobi equation (QHJE) associ- 2 2 σ (r)4πr dr = 1, ζ(R)4π R dR = 1 (21) ated with the two particles can be decomposed in terms of the motion of the center of mass plus the motion relative to the cen- 0 0 ter of mass as follows otherwise the integrals would diverge. Despite that ζ(R) diverges √ √ = 2 2 2 2 2 2 at R 0it is normalizable. (∇r S) (∇R S) h¯ ∇ ρ h¯ ∇ ρ E = + − √r − √R + V (r) (8) Because A < 0, the quantum Bohm potential in eq. (19)leads 2m 2M 2m ρ 2M ρ to a repulsive gravitational potential plus a constant (a zero-point energy) ρ(r, R) = σ (r)ζ(R) = σ (r,θ,φ)ζ(R,θcm,φcm), R =|R|, r =|r| (9) ¯ 2 2 ¯ 2 2 − h A + A = Gm1m2 − = h A ( ) V 0, V 0 (22) S(r, R) = S1(r) + S2(R) (10) 2m r 4 r 8m To simplify matters we shall freeze the angular and temporal From eq. (22)one learns that dependence of the physical quantities and focus solely on the ra- 2 2G 2G (m1m2) dial dependence only. Hence, the functional dependence of ρ and A =− m1m2m =− ⇒ ¯ 2 ¯ 2 m + m S simplifies and reduces to the form h h 1 2 1 V = m(Gm m )2 (23) 0 2 1 2 ρ(r, R) = σ (r)ζ(R), S(r, R) = S1(r) + S2(R) (11) 2h¯ so that The quantum Bohm potential is tantamount of a repulsive gravitational potential (plus a constant), and is cancelled out 2 h¯ 1 = = √ ∂ (r2∂ (r, R)) by the attractive Newtonian gravitational potential V V (r) 2 r r ρ 2m ρ(r, R) r −(Gm1m2/r) stemming from the gravitational interaction of the ¯ 2 two particles of masses m1, m2. In doing so, the stationary QHJE h 1 2 = √ ∂r (r ∂r σ (r)) (12) (15) becomes effectively a classical-like stationary Hamilton–Jacobi 2m σ (r) r2 equation for a free particle with a shif ted energy h¯ 2 1 √ ∂ (R2∂ (r, R)) ∇ 2 2 R R ρ ( r S1(r)) 2M ρ(r, R) R E − E + V 0 = (24) 2m ¯ 2 h 1 2 = √ ∂R (R ∂R ζ(R)) (13) and from which one learns 2M ζ(R) R2 p = ∂ S (r) = 2m(E − E + V ) ⇒ S (r) = p r (25) The continuity equation in the stationary case is r r 1 0 1 r we set the constant of integration to zero in the last terms of ∇r S ∇R S ∇r (ρ ) +∇R (ρ ) = 0 (14) eq. (25). m M At this stage it is very important to emphasize that there are = Introducing a potential of the form V V (r), separating (de- key differences from our results and those of [9]. Firstly, Matone coupling) the motion of the center of mass from the motion [9]set the potential V (r) = 0, which is not the case here. One relative to the center of mass, and inserting the expressions in cannot ignore the gravitational potential generated by the presence eq. (12)–(13), leads to the following stationary QHJEs, and conti- of two masses m1, m2. Secondly, he proposed a quantum potential nuity equations of the form ∇ 2 ¯ 2 Gm1m2 ( r S1(r)) h 1 2 V =− + O(h¯ ) (26) E − E = − √ ∂r (r ∂r σ (r)) + V (r) (15) Q 2m 2m σ (r) r2 r ∇ 2 ¯ 2 which is very different from our findings in eq. (22). Our quan- ( R S2(R)) h 1 2 E = − √ ∂R (R ∂R ζ(R)) (16) tum potential generates a repulsive gravitational interaction plus 2 −2 2M 2M ζ(R) R a constant proportional to h¯ . ∇r S1(r) ∇R S2(R) The quantum potential associated to the center of mass is ∇r ( σ (r) ) = 0, ∇R (ζ(R) ) = 0 (17) m M ¯ 2 ¯ 2 2 h 1 2 h B where E is the energy associated with the center of mass motion. − √ ∂R (R ∂R ζ(R)) =− ( ) (27) 2M ζ(R) R2 2M 4 A careful inspection reveals that the following solutions (below we shall explain the physical origin of these solutions) such that 2 BR h¯ B2 Ar e (∇ S (R))2 − = 2M E ⇒ σ (r) = σoe ,(A < 0), ζ(R) = ζo ,(B < 0) (18) R 2 R2 4 2 2 with σo, ζo constants, yield the following quantum Bohm poten- h¯ B ∂R S2(R) = P R = + 2ME ⇒ S2(R) = P R R (28) tials 4 548 C.C.
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