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

Physics Letters B 788 (2019) 546–551

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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 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

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 2 coupled pair of equations − h¯ 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 quantum 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 = = − E + ⇒ = The continuity equation in the stationary case is pr ∂r S1(r) 2m(E V 0) S1(r) prr (25) 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 E = − √ ∂ (R2∂ ζ(R)) (16) tum potential generates a repulsive gravitational interaction plus 2 R R − 2M 2M ζ(R) R a constant proportional to h¯ 2. ∇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. Perelman / Physics Letters B 788 (2019) 546–551 we set the constant of integration in the last term of eq. (28)to solution) obeys the Schrödinger equation (36). Since (R) is not zero. normalizable, as it is usual with the plane wave solutions one con- To finalize one needs to study the continuity equations fines the particle to a “box” of finite size; i.e. one introduces an infrared cutoff. 1 1 1 ∇ · ( (r)∇ S (r)) = ∂ ( r2 (r)∂ S (r))= 0, The normalized wavefunction solutions to the Schrödinger r σ r 1 2 r σ r 1 m m r equation corresponding to the Hydrogen atom (associated to a 1 1 1 2 Coulomb potential) in spherical coordinates are ∇R · (ζ(R)∇R S2(R)) = ∂R ( R ζ(R)∂R S2(R))= 0 (29) M M R2 The solutions to the above continuity equations, when σ (r), ζ(R) 2 (n − l − 1)! − + r = 3 e /2 l L2l 1 Y l are given by eqs. (18), will require then that the constant momenta nlm( ,θφ) ( ) n−l−1( ) m(θ, φ) nao 2n(n + l)! pr , P R in eqs. (25), (28)should be trivially zero (37) ∇ S (r) = p = 2m(E − E + V ) = 0 ⇒ E − E + V = 0 ⇒ r 1 r 0 0 where is defined by = 2r/nao, and ao is the Bohr radius m 2 2 2 2l+1 E − E = E =−V =− (Gm m ) (30) ao = h¯ /me . L − − ( ) is a generalized Laguerre polynomial of de- 0 2 1 2 n l 1 2h¯ − − l gree n l 1, and Ym(θ, φ) are the spherical harmonics. In the 2 → h¯ 2 B2 h¯ 2 B2 gravitational case one just replaces e Gm1m2 in all the expres- ∇R S2(R) = P R = + 2ME = 0 ⇒− = E (31) sions. 4 8M One may notice that the ground state wavefunction, ∼ − The ground state energy of the Hydrogen atom (ignoring rela- √n=1,l=0,m=0 exp( /2), has the same functional form as tivistic corrections) is σ (r) ∼ exp(Ar/2),√ A < 0in eq. (18). The√ expression for the quantum potential V Q ( σ (r)) based on σ (r) ∼ exp(Ar/2), A < 0, 4 2 2 me 1 e mc coincided exactly with the repulsive gravitational potential (plus E(n = 1) =− =− (m c)2 =− α2 (32) 2 e = = 2h¯ 2m hc¯ 2 a constant). The continuity equations led to pr P R 0, and fi-

nally to the value of E = E − E in eq. (34), and which has exactly we wrote the E(n = 1) in this form above to remind the reader the same expression as the ground state energy for the Hydrogen that the mean electron velocity in the ground state is c/137. By 2 atom (32), after the correspondence e ↔ Gm1m2 is made. analogy, one can deﬁne the “gravitational” ﬁne structure α from G Does this correspondence occur for the other excited states the correspondence n = 2, 3, ...?, and for other values of l, m besides l = m = 0 when 2 one includes the rotational degrees of freedom? It should occur e Gm1m2 αe = ↔ = αG (33) because the Schrödinger equation is equivalent to the coupled hc¯ hc¯ = √system of 2 differential equations (1)–(2). Namely, nlm (r, θ, φ) such that σnlm(r,θ,φ)cos(Snlm(r, θ, φ)/h¯ ) solves the Schrödinger equation 2 2 if, and only if, σnlm(r, θ, φ) and Snlm(r, θ, φ) solve the coupled sys- mc Gm1m2 mc E =−V =− ( )2 =− α2 (34) tem of 2 differential equations (1)–(2), and vice versa. 0 ¯ G 2 hc 2 The excited states will no longer correspond to static mass con-

The value of E < 0is negative as expected for a (gravitationally) figurations with respect to the center of mass, and the expression bound two-particle system. In the ground state the two particles for V Q (ρexcited) will no longer be equal to the repulsive gravita- are in static equilibrium configuration with respect to their center tional potential (up to a constant) cancelling the attractive New- of mass. tonian potential, but it will be a more complicated function. The The stationary Schrödinger equation in the spherically symmet- momenta pr will no longer be constant and there will be a non- ric case for the two-particle system can be decomposed into the trivial motion relative to the center of mass. following two Schrödinger equations. The first one is Can these results be generalized to other potentials V (r) re- flecting the interaction between two particles, besides the Newto- ¯ 2 h 1 2 nian one, or V ∼ 1/r potentials are special? Let us study the 3D − ∂r (r ∂r (r)) + V N (r)(r) = (E − E)(r) = E (r) (35) 2m r2 spherically symmetric harmonic oscillator case, assuming the interaction between the particles is governed by a harmonic oscilla- where V N is the attractive gravitational Newtonian potential √ ∼ ∼ − 2 −(Gm1m2/r). As usual, m is the reduced mass, and r =|r1 − r2| tor. The Gaussian ground state wavefunction ρ exp( λr ) is the relative separation of the two particles. The other equation yields a quantum potential is ¯ 2 ¯ 2 2 3h λ 2h λ 2 ¯ 2 V Q = − r (38) h 1 2 − ∂R (R ∂R (R)) = E(R) (36) m m 2M R2 which will cancel out the contribution of the 3D spherically sym- where M is the total mass m1 + m2, and R is the radial coordi- 1 2 2 metric harmonic oscillator potential V osc = mω r (leaving a con- nate of the center of mass. One can verify by a simple inspec- 2 √ √ stant) when (2h¯ 2λ2)/m = 1 mω2. Solving for λ = (mω/2h¯ ) gives tion that (r) = σ (r) ∼ exp(Ar/2), (A < 0), and (R) = ζ(R) ∼ 2 + = 3h¯ 2λ = 3h¯ 2 ¯ = 3 ¯ exp(BR/2)/R, (B < 0), solve the above two Schrödinger equations, V Q V osc m ( m )(mω/2h) 2 hω which is precisely the respectively. ground state energy of the 3D spherically symmetric harmonic Another physical solution√ for the center of mass motion occurs oscillator. This procedure works for other potentials since setting + = ∇ = when B = 0 ⇒ P R = 2ME, E > 0. In this case one has ζ(R) ∼ V Q (ρground) V E0, and S 0(zero momentum is a trivial 2 1 2 1/R , and the continuity equation ∂R (R ζ(R)P R ) ∼ ∂R P R = 0 solution to the continuity equation) into the QHJE leads always to MR2 = is obeyed without restricting the value √of P R to zero. One can E E0. verify again that given S2(R) = P R R = 2ME R, the wavefunc- We propose next the notion of a classical/quantum duality in ∼ 1 ¯ tion (R) R exp(iPR R/h) (a spherical analog of a plane wave the gravitational case (it can be extended to other interactions). C.C. Perelman / Physics Letters B 788 (2019) 546–551 549

By this one means exchanging V Q ↔ V in the stationary QHJE, so based on this notion of classical/quantum duality. An account of that V Q + V =−V 0 (as before) and leading to the historical developments of gravity from Newton to the repul- √ sive gravity of the vacuum energy can be found in [11]. 2 2 −2 h¯ ∇ σ (r) Gm1m2 A key remark is in order. The explicit presence of h¯ in the V Q =− ( √ ) = V N =− , 2m σ (r) r expression for V 0 (34) (and which is part of the potential V (r) in eq. (39)) is not very common in the Quantum Mechanical problems Gm1m2 V (r) =−V N − V 0 = − V 0 (39) that we are familiar with. However, as emphasized by Klauder in r his monograph [8], the principal purpose of his Enhanced Quan- We should remark that over the years dif f erent notions of classi- tization program is to describe and apply a new way to quantize cal/quantum duality than ours have appeared in the literature, see classical systems, which in turn, lead to classical enhanced Hamil- [10]for some references. One must not confuse these different no- tonians that explicitly contain nonvanishing h¯ terms. The Enhanced tions of classical/quantum duality. Quantization program of Klauder relies on the coexistence of the The differential equation to be solved now is classical and quantum world, and consequently it involves the ex- √ √ ¯ ¯ → 2 plicit presence of h, and without taking the h 0 limit. Therefore, 2 d σ (r) d σ (r) 2Gm1m2m r + 2 r − Cr (r) = 0, C = > 0 having an h¯ -dependence on V 0 is not an alien property that should 2 σ 2 dr dr h¯ be dismissed and which would disqualify V (r) as a “classical” po- (40) tential. ∞ To finalize we shall discuss another way in which repulsive The solutions must also be normalizable σ (r)4πr2dr = 1. They 0 gravity emerges. In [7]we proposed the novel equation1 which we are given in terms of modified Bessel functions of the first I1, and coined as the “Bohm–Poisson” (BP) equation (for static solutions second kind K1 √ √ ρ = ρ(r)) I (2 Cr) K (2 Cr) √ = 1 √ + 1 √ ¯ 2 ∇2 σ (r) a1 a2 (41) 2 h 2 ρ Cr Cr ∇ V Q = 4π Gmρ ⇒− ∇ ( √ ) = 4π Gmρ (43) 2m ρ The normalization condition requires to discard the I1 contribution because I1 diverges at r =∞, leaving the K1 function which van- the purpose of eq. (43)was to replace the well known (among the ishes at r =∞. In doing so, the normalization condition will fix experts) nonlinear Newton–Schrödinger equation. The fundamen- the value of the a2 coefficient in terms of C > 0. Note that C has tal quantity is no longer the wave-function (complex-valued in = ∗ an explicit h¯ -dependence, as it should, in order to have V Q = V N . general) but the real-valued probability density ρ . The logic The right hand side has no h¯ factor, so there must be a cancellation behind eq. (43)was based on the idea that the laws of physics of the h¯ factors in the left hand side. should themselves determine the distribution of matter. This is The stationary Schrödinger equation in the spherically sym- going one step further from General Relativity where a given dis- metric case involving the repulsive gravitational potential (plus a tribution of matter determines the geometry. constant) is now given by The BP equation (43)is invariant under the transformations ρ ↔−ρ, and G →−G. Thus solutions with G < 0are associated to 2 h¯ 1 Gm1m2 repulsive gravity. If, in addition to the Bohm–Poisson (BP) equation − ∂ (r2∂ (r)) + ( − V )(r) 2 r r 0 one were to add the Schrödinger equation for the complex-valued 2m r r √ ¯ wave-function ≡ ρeiS/h, one can obtain consistent solutions, = (E − E)(r) = E (r) (42) which avoids having an overdetermined system of equations, when Eq. (42)should be compared with eq. (35). They both differ how- the external potential is itself a function of ρ. The functional form ever the QHJE (15) remains the same. This is how the classi- of the potential Vcannotbe arbitrary, but it is subjected to sat- cal/quantum duality is implemented. Despite that the solutions isfy a system of equations. One equation is the BP equation (43). to eq. (42)differ from the solutions to eq. (35), the ground The second equation is the QHJE, and the third equation is the state energy solution to eq. (42)is the same as before (34) E = continuity equation. The latter two equations are equivalent to the − E =− =−mc2 2 Schrödinger equation. Therefore, one has a system of 3 equations E V√0 2 αG . One can verify this by checking that for the 3 unknowns (r), S(r), V (r). The potential itself is deter- K1(√2 Cr) ρ = a2 solves the stationary Schrödinger equation (42) Cr mined from the equations instead of being put in by hand. 2 = − E =− =−mc 2 One can also propose another system of 3 equations (for with the ground state energy E E V 0 2 αG . In this scenario Bohm’s quantum potential associated to the the 3 unknowns ρ(r), S(r), V (r)) where the first equation is = ground state V Q (σground(r)) = V N coincides with the attractive V Q (ρ) V N . The second and third equations are the usual Newtonian potential, and is now correlated with the classical QHJE and continuity equation, respectively. In principle, one could repulsive gravitational potential (plus a constant) of eq. (39). And have a family of many solutions consisting of the many triplets { } { } ···{ } vice versa, under the exchange V Q ↔ V . This 2-particle model can ρ1(r), S1(r), V 1(r) , ρ2(r), S2(r), V 2(r) , ρN (r), S N (r), V N (r) . be generalized to the N-particle case. One would have to verify There could even be an infinite number of solutions. This last set that the gravitational attraction (in the ground state configuration) of 3 equations would correlate the classical potentials V 1, V 2, ··· is compensated by the repulsive contribution of the N-particle with the quantum potential V Q = V N . Classical/Quantum duality quantum potential corresponding to the ground state probability selected one specific classical potential given by V =−V N − V 0 in density. eq. (39). The accelerated expansion of the Universe is generally assumed Extensions of the Bohm–Poisson equation to the full relativistic to be driven by a positive vacuum energy density (like a positive regime were developed in [13]. In the case of a real-valued scalar ∗ cosmological constant), or by some scalar field φ(t) (quintessence) field φ = φ , the relativistic field theory analog of the Bohm– whose potential V (φ(t)) at late stages of the universe mimics the Poisson equation in D dimensions was given by [13] behavior of the cosmological constant. Here we are proposing a very different scenario in which there might be a quantum origin to the classical repulsive gravitational behavior of the universe 1 To our knowledge eq. (43)has not appeared before. 550 C.C. Perelman / Physics Letters B 788 (2019) 546–551 √ h¯ 2 r t − 1 h¯ 2 2 φ( , ) μν ρ(dS/dx) = constant. Inserting ρ ∼ (dS/dx) 2 into Q = {S, x} ( ) 2 = 4π Gg Tμν, 4m m φ(r,t) yields the expression for Bohm’s quantum potential after some straightforward algebra [9]. 2 ≡ √1 | | μν = ∂μ( g g ∂ν), c 1 (44) Schwarzian Quantum Mechanics has recently been a very ac- |g| tive topic of research in connection to the Sachdev–Ye–Kitaev μν where the trace of the stress energy tensor T = g Tμν associated (SYK) model [18]. Another very relevant topics of current research with the scalar field appears in the right hand side. The stress en- related to the emergence of gravity are holographic quantum ergy tensor for the scalar field is defined in term of the matter complexity, entanglement entropy, information geometry, quantum √2 δSm(φ,gμν ) terms Sm in the action by Tμν =− μν . computation and information theory, black holes, Cayley graphs, g δg ··· In 4D, given the Lorentzian signature (−, +, +, +), the action , see [19] and the references therein. The close relation between in a curved background with a cosmological constant was chosen gravity and quantum mechanics has been analyzed by Susskind [20]. Our main goal, if possible, is to geometrize quantum me- to be chanics. The emergence of quantum mechanics from the fractal √ − μν 4 (R 2) g geometry of spacetime has been advanced long ago by Nottale S = d x −g − (∂μφ) (∂νφ) − V (φ) (45) 16π G 2 [12]. We hope that the notion of Classical/Quantum Duality raised in this work in connection to the QHJE may cast further light into and is associated with a canonical real scalar field φ with a poten- the deep interplay between gravity and quantum mechanics. tial V (φ). The FLRW metric is (dr)2 Acknowledgement ds2 =−(dt)2 + a2(t) + r2(d)2 , k = 1, 0;−1 (46) 1 − kr2 We are indebted to M. Bowers for assistance. − k is the spatial scalar curvature parameter with units of (length) 2. 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