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Review Article Open Access Exact solutions of the -Schrödinger equation, infinite derivative and Schwarzschild atoms

Abstract Volume 1 Issue 4 - 2017 Exact solutions to the stationary spherically symmetric Newton-Schrödinger equation are proposed in terms of involving generalized Gaussians. The eigenvalues Carlos Castro Perelman are also obtained in terms of these integrals which agree with the numerical results in the Center for Theoretical Studies of Physical Systems, Clark Atlanta literature. A discussion of infinite derivative-gravity follows which allows generalizing the University, Georgia Newton-Schrödinger equation by replacing the ordinary Poisson equation with a modified Correspondence: Carlos Castro Perelman, Center for non-local Poisson equation associated with infinite-derivative gravity. We proceed to Theoretical Studies of Physical Systems, Clark Atlanta University, replace the nonlinear Newton-Schrödinger equation for a non-linear quantum-like Bohm- Atlanta, GA 30314, Georgia, Email Poisson equation involving Bohm’s quantum potential, and where the fundamental quantity is no longer the wave-function Ψ but the real-valued probability density ρ . Finally, we Received: October 07, 2017 | Published: November 10, 2017 discuss how the latter equations reflect a nonlinear feeding loop mechanism between and geometry which allows us to envisage a “Schwarzschild atom” as a spherically symmetric probability cloud of matter which curves the geometry, and in turn, the geometry back-reacts on this matter cloud perturbing its initial distribution over the space, which in turn will affect the geometry, and so forth until static equilibrium is reached.

Keywords: quantum , Newton-Schrödinger equation, infinite derivative gravity, de Broglie-Bohm ,

Introduction (such as with the proposed class of experiments put forward there), it would be sufficient to consider Newtonian gravity. This leads us The Newton-Schrödinger equation to consider what Penrose termed the Schrödinger-Newton equation. 7–17 Various arguments have been put forward from time to time to This equation has had a long history since the 1950’s. It is the name support the view that reduction is a phenomenon given to the system coupling the Schrödinger equation to the Poisson that occurs objectively, because of some gravitational influence.1–6 equation. In the case of a single particle, this coupling is effected as According to a particular argument put forward by Penrose,1 a follows: for the potential energy term in the Schrödinger equation take superposition of two quantum states, each of which would be stationary the energy determined by the Poisson equation on its own, but for which there is a significant mass displacement from a matter density proportional to the probability density obtained between the two states, ought to be unstable and reduce to one state from the wave-function. For a single particle of mass m the system consists of the following pair of partial differential equations: or the other within a certain characteristic average timescale TG . This argument is motivated by a conflict between the basic principles of The Newton-Schrödinger equation is nonlinear and nonlocal and those of . It is accordingly modification of the Schrödinger equation given by proposed that TG can be calculated in situations for which velocities  2 and gravitational potentials are small in relativistic units, so that a ∂Ψ()rt,  2    i =− ∇Ψ()rt,++V ()() rt ,,Ψ rt mVrt ( ,,)(Ψ rt ) (1) Newtonian approximation is appropriate, and TG is the reciprocal, in ∂tm2 G Planckian units, of the gravitational self-energy EG of the difference  Where V ()rt, is the external potential acting on the particle and between the mass distributions of the two states.  mVG ( r, t) is the self- gravitational potential energy arising due a There is of course a substantial literature on the problem of mass density obtained from the of the particle itself.  2 wave-function collapse and the related . See, Given the Poisson equation sourced by a mass density ρ=m Ψ() rt, for example4–6 and references therein. For a different idea about 2 2 3 ∇ΨVr(),t=4πGm () rt , (2) gravitationally-induced wave-function collapse see. It has been G 1,2 pointed out by Penrose R et al. that one can regard the basic It leads to a self-gravitational potential stationary states, into which a superposition of such states is to decay  2 into (on a timescale of order /E ), as stationary solutions of the  m Ψ()rt, 3 G V() rt,=−∫Gd r′ ( 3) Schrödinger equation where there is an additional term provided G rr− ′ by a certain gravitational potential. The appropriate gravitational Inserting equation (3) into equation (1) leads to the integro potential is the one which arises from the mass density given by the differential form of the nonlinear and expectation value of the mass distribution in the state determined by the wave-function. In the practical situations under consideration in1

Submit Manuscript | http://medcraveonline.com Phys Astron Int J. 2017;1(4):131‒137. 131 ©2017 Perelman. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your non-commercially. Copyright: Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms ©2017 Perelman 132

nonlocal Newton-Schrödinger equation

 2  2 2 ∂Ψ()rt,  2   23m Ψ()rt,  Gm i =− ∇Ψ()rt,,,+VG ( rt)Ψ() rt) −∫m d r′ Ψ( r,t (4) ∂tm2 − ′ the Newtonian form − . Because the error function Erf( r) r r r admits a series expansion around r = 0 as Relevant work pertaining pseudo differential and hyper differential operators, nonlinear partial differential equations and their 2rr35 11 r 241r 7 Erfr( )  −+ − +... ( 9) applications can be found in.18–20 ππ20 π 840 π

This equation is based on the assumption that the point-particle The potential Vr( ) is no longer singular at the origin r = 0 , but G is smeared over space such that its mass is distributed according to it behaves as its wave function. Namely, there is a mass cloud over space whose  2 24 net mass m=m∫ Ψ() rt,.d 3r′ coincides with the mass of the 2Gm r11 r Vr( )  − [1 −+ +...] (10) point-particle. The mass cloud is self-gravitating and experiences a G πσ 2σσ24 40 gravitational potential energy given by equation (3).1 The normalized Gaussian wave function 21 More recently, the authors showed that nonlinear Schrödinger 22 −r /2σ equations (NSEs) for individual particles do not follow from general  1 e Φ≡(r ) (11) relativity (GR) plus (QFT). Contrary to what is πσ3/4 3/2 commonly assumed, the NSEs are not the weak-field, non-relativistic Satisfies limit of the semi-classical Einstein equation. The wave-function in the ∞ NSEs makes sense only as that for a mean field describing a system  2 2 ∫ Φ=(r ) 4π r dr 1 (12) of N particles as N →∞ , not that of a single or finite many particles. They concluded that the origins and consequences of NSEs are very 0 different, and should be clearly demarcated from those of the semi- Let us minimize the energy functional 2 ∞∞ classical Einstein equation, the only legitimate representative of semi  222 = Φ ∇Φ2 ππ + Φ( ) Φ ( ) classical gravity, based on GR+QFT. Bearing this in mind, we proceed E ∫∫( (r) ( r))4r dr mVG [rr ]( ) 4r dr 13 2m 00 to find solutions to the NSE’s. Using this Gaussian as a trial function, and which generates the Solutions to the Newton-Schrödinger equation  Gm r regular potential at the origin V []Φ=(r ) Erf  given by G r σ Let us set the external potential to zero and look for stationary 2 -iEt/ equation (8) after solving Poisson’s equation (7). The integrals to be solutions Ψ(Φrr,t)=e ( ) such that the gravitational potential  evaluated in equation (13) are of the form becomes time independent. Ö(r ) obeys the equation ∞ 22   2  42−x 3 1 − x 2 2 Ö ∫ x−=3 x e dx −π Erf ( x) − ex 2x − 3 (14)  2 2 (r ) 3  ( ) 8 4 ( ) EÖÖ(rr)=−−∫∇ ( ) Gm d r′ Ö(r ) ( 5) 0 2m r −r′  2Which is also related to the incomplete gamma function The authors22–26 found numerical spherically-symmetric solutions 1 to equation (5). Variational forms of the stationary Newton- γπ;r= Erf (r) Schrödinger equation to find a lower bound for the 2 27 energy have been studied by several authors, see references in, and ∞ 2 2 −x 1  −  compared to numerical values in the literature. ∫ xErf( xx) e dx= 2 Erf( 2 x)− 2ex Erf ( ) (15) 4   2 0 1This may be in conflict with Born’s rule of interpreting Ψ(rt, )  After performing the definite integrals we find that as the probability density of finding a particle at the point r if one 3 π 222 Gm has abandoned the notion of point-particles. At the moment we shall E (σ ) = − (16) not be concerned with this. 4 mσ 2 π σ

If one replaces a delta function point-mass source distribution 2  3π  mδ32( rmr)= δπ( )/4 r for a normalized Gaussian mass distribution σ = (17) min 22 3 of width Gm 22 And inserting this value of σ into E (σ ) gives me−r /σ ∞ ρ(r)= ⇒=m∫ ρπ( r)4 r2 dr ( 6) 2 Gm25 Gm25 3/2 3 E = = −0.119 (18) πσ 0 min 3/2 22 3π  The solution to Poisson’s equation Which is a satisfactory value since it is above the lower energy bound 2 25 25 ∇=VrG ( ) 4πρ G( r) ( 7) 32 Gm Gm = = − ( ) 2 E 0.360 19 Is given in terms of the error function Erf( r) as follows bound 9π 222 27 25 2 r/σ 2 Derived by. and it is also close to the value −0.163Gm / Gm r Gm 1 − ( ) V( r) =−= Erf −∫ et dt ( 8) obtained numerically by.22–26 We can also verify that the value of G rrσ π −r/σ σ given by equation (17) is consistent with the virial theorem min In the asymptotic regime r→∞, Erf (∞)→1. , the potential recovers UT = 2, stating that the absolute value of the potential energy

Citation: Perelman CC. Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms.Phys Astron Int J. 2017;1(4):131‒137. DOI: 10.15406/paij.2017.01.00024 Copyright: Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms ©2017 Perelman 133

is twice the . It is also worth mentioning that a trial 3 1/2 −kr exponential ψπ(rk)= / e leads to an upper energy bound 1/2 2 2 ( )   for the ground state of27 αβ ≡≡ , ( 26) 8πGm2 2m 75 Gm25 Gm25 E ≤− =− 0.146 ( 20) And l is a physical length parameter (the gravitational analog of the 0 22 512  23 2 1 Bohr’s radius) given by l =  / Gm , such that β / l= Gm25 /  2 , 2 ( ) Which is close to the value found numerically by.22–26 However . −kr β the series expansion of e involves even and odd powers of r The energy eigenvalues are E = Ur( ) + Vr( ) .Because which is not the case of the numerical solutions provided by.22–26 l2 The spherically-symmetric solutions of the stationary Schrödinger- Ur( = 0 ) is of the form ∞× 0 which is undefined, it is more Newton equations have been solved numerically by many authors.22–26 convenient to evaluate the energy at r = ∞ where U ()∞ vanishes, as They have demonstrated numerically the existence of a discrete set follows22–26,28 of “bound-state” solutions which are everywhere finite and smooth ββ∞ 1 2 E = U ( r =∞+ ) V( r =∞= ) V0 − ∫ x S( x) dx , V = 1 (27) (and which are associated with finite energy eigenvalues), but which 222 0 lll separate ever decreasing intervals of partial solutions which diverge 0 The values of S are fine-tuned such that the ground state wave- alternately to plus or minus infinity. 0 function has no zeros, it is bounded at the origin and vanishes at infinity. The value found for the ground state energy turned out to be By demanding that Ψ and U are finite and smooth everywhere, the

5 authors showed that the n-th eigen function has n zeros and the wave- 25  Gm m 2 E =−= 0.163 − 0.163 Mc ( 21) functions are normalizable. The corresponding energy eigenvalues 0 2 planck  M planck are negative, converging monotonically to zero as n increases. Each These numerical values for the energy are very small for masses bound-state is unstable in the sense that infinite precision is required in the initial value of Sr ( = 0) = S to ensure that the solutions do not much smaller than the Planck mass. 0 diverge to infinity as r increases. For the ground state, the numerical The width (spread) σ of the Gaussian wave-function is measured value of S fell in the range given by 1.088<< S 1.090 and it led 2 25 0 0 in units of (/ Gm ), which translated into Planckian units is Gm25 3 3 1/2 to the ground state energy E = − 0.163 . (LLPS /2) ( L P) , where L is the (2Gc / ) 2 P 0 2 , and L = 2/ Gm c is the Schwarzschild radius. It was argued by  S For further details of how to obtain other energy eigen values and the authors22 that for a nucleon mass m the value of σ is vast and 24 eigen functions by choosing different values for S we refer to.23–36 is of order of10 cm . The corresponding time ∆=tE  / ∆ for the 0 The study of the Newton-Schrödinger equations in D largest (in magnitude) energy is about of 1053 seconds for the mass of other than D= 3 can be found in.28 the nucleon, and is of the order of 1 second for 1011 nucleon masses. This is perfectly satisfactory for the state vector reduction of,1 because A careful inspection of the work by22–26,28 inspired us to find the it tells us that the state of a single nucleon will not self-reduce on remarkable numerical coincidence provided by the definite a timescale of relevance to any actual particle, in agreement with ∞∞24  −+(yy )2 11 2 observation. For large collections of particles, on the other hand, the − ∫∫e y dy = −− 0.160  0.163 =  1 − x S ( x ) dx (28) 22 reduction time can become important. 00l (We setV =1). The normalization of the wave function The coupled system of differential equations (1,2) can be recast α o after integrating twice, and using dimensionless variables S, V , as22 Ψ =Sr () l2 r 2 ∞ x α 22 S( r) =−− S 1 ∫ x 1 S( x) V( x) dx ( 22) ∫ 4π r S dr = 1 (29a ) 0 2  4 l 0 r l 0 Allows to recast the right-hand side of (28) as r 1 x 2 V( r) =−− V ∫ x  1 S( x) dx ( 23) 2 0 2 r α ∞∞1 l 0 ∫∫ 4π r22 S dr − x S2 ( x ) dx (29)b 42 The value of V must be V > 0 to ensure convergence of Sr( ) 22ll00 0 0 and Vr( ) at r →∞ . Scaling arguments allow one to chooseV =1 . After relabeling the variable r for x in the above expression, 0 The numerical solutions to equations (22,23) found by,22 when V =1 equation (28) becomes 0 , are ∞∞24 2 −+ 1 4πα −− 0.160 = ∫∫e(yy )2 y dy   y − 1 y S2 ( y ) dy = − 0.163 (30) 22 4 2 l 1rS ( + 1 ) r 00 Sr( ) = S  1 −+ 0 + .... ( 24)  0 24 6ll 120 xr Which allows to identify the variables y ≡ = , and basically ll Sr22 Sr 2 4 equate the dimensionless integrals. However this does not mean that Vr( ) =−+  1 00 + .... ( 25) 624 60 the integrands are the same. The integral in the right-hand side of ll 2 1 α equation (30) is evaluated after inserting the value of 4πα /=l The wave-functions are Ψ=(r) Sr( ) , and where αβ, are 2 2 2 . The expression for Sr() corresponding to the ground state is defined by l

Citation: Perelman CC. Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms.Phys Astron Int J. 2017;1(4):131‒137. DOI: 10.15406/paij.2017.01.00024 Copyright: Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms ©2017 Perelman 134

) in the left hand side, while in the right hand side one has zeros only derived from equation (24) after inserting a value of S  1.088 o at r = 0 . If one were to equate the integrands one would arrive at (lying in the interval[1.088,1.090] ). The other “bound” states another contradiction. In the spherically symmetric case, for real- require different values of S yielding different expressions for on, valued wave-functions, and after setting y=/ rl, it gives Sr() ( Srn (=0)= Son, ) , and which have n nodes (zeros). Next we n 2 25 2 4 6 2(n+ 2) shall present the novel and most salient results of this section. To our  l 2 2 1 Gm −(yyy + + ++  y )2 (∂Ψ (yy )) 4π  e y ⇒ knowledge these findings below are new. The following remarkable 2m yn (n+ 1)22 numerical coincidences for the values of the remaining (negative) 2 4 6 2(n+ 2) 211−(yyy + + ++  y ) energy eigenvalues EEE, , ,..... have also been found. To obtain (∂Ψ ()ye)  (36) 123 yn 23 the values for the first we replace the integral in the left- (nl+ 1) 2π hand side of (28) for: Given Ψ an even, or an odd function, for n= even, odd, n ∞ 246 respectively, the left hand side is always even due to the squaring of 1 −(yyy ++ )2 n=1: − ∫ e y dy = −− 0.0302  0.0308 (31) the derivatives, as it should, since the right hand side of (36) is an even 22 0 functions.3 For this reason, if one is going to take the square root of To get the second one equation (36) we must choose n=odd ⇒ ∂Ψ = even. Hence ∞ 2468 1 −(yyyy +++)2 n=2: − ∫ e y dy = −− 0.0120  0.0125 (32) −1 2 4 6 2(n+ 2) 2 11 (yyy+ + ++  y ) 3 0 ∂Ψ(y )  e 2 , n = odd (37) yn (1n+ ) 3 The third one 2πl ∞ 2 4 6 8 10 1 −(yyyyy ++++ )2 Integrating (37) between 0 and y we arrive finally n=3: − ∫ e y dy = −− 0.00641  0.00675 (33) 2 4 0 y −1 2 4 6 2(n+ 2) The fourth one 11 (yyy+ + ++  y ) Ψ(y ) −Ψ (0)  ∫ e 2 dy , n= odd (38) nn(n+ 1) 3 ∞ 2 4 6 8 10 12 0 1 −(yyyyy ++++ + y)2 2πl n=4: − ∫ e y dy = −− 0.00399  0.00421 (34) 2 5 0 We still need to check that vanishes at , and include normalization And so forth..... The pre-factors in front of the integrals for the 23 −2 constant to enforce . ∫ |Ψ |dr =1.When n=odd, Ψ(0) = 0 n -th energy eigenvalue are (n + 1) , which are “reminiscent” because anti symmetric functions must vanish at the origin. 3This of the Balmer series for the Hydrogen atom, and the arguments of r 2 4 2(n+ 2) assumes that one can extend to extend to negative values r<0 the negative exponentials are yy+ ++ y . The authors22 ±2 ++ 22 plotted the ten energy eigenvalues as functions of n and found that , which is not unreasonable because r= xyz. Black-hole the slope of log() n versus log( E ( n )) was not −2 .22 However, the solutions can be extended to r<0 . The Schwarzschild metric solution →− →− slope is asymptotically close to −2 .35 which would be the exact value is invariant under r rm, m. for the Hydrogen atom as previously note by Bernstein et al.23–26 Here is where the contradiction arises. The integral in the right- ∞ Using these expressions in equation (28), and equations (31-34), hand side is not zero when y = . It is given by a finite number. This would Ψ∞( )=0 since Ψ (0) = 0 , for n = odd. Since the we find that energy values are remarkably close to the numerical n n results obtained in,35 and given by the numbers in the right-hand wave-functions do not vanish at ∞ these solutions are unphysical. side of equations (28, 31-34). We don’t believe this is a numerical For this reason one cannot equate the integrands in equation (35) only coincidence and could point to the fact that integrability may underlie the integrals which can be equated. the solutions of the stationary spherically symmetric Newton- To conclude this section, based on the numerical results by22–26,28 Schrödinger equations. It was emphasized by22–26,28 that one would we have arrived at the integrals. need to know the values of Sr( = 0) = S with infinite precision in n on, yy2 4 6 2(n+ 2) 22 11−(yyy + + ++  y )2 ∫∫(∂Ψyn (y )) y dy  e y dy , n =0,1,2, (39) order to ensure that Sr() does not shoot off to ±∞ at a finite value 23 n 00(nl+ 1) 2π of r . All the numerical results appearing in the right-hand side of Which furnish implicitly the stationary spherically symmetric equations (28, 31-34) are obtained by narrowing in the values of S on, wave-function solutions to the Newton-Schrödinger equation in terms within certain domains. For this reason we find the analytical results of integrals involving generalized Gaussians. The energy eigen values in the left-hand side of equations (28,31-34) very appealing. are provided by the left hand side of equations (28,31-34). Equations To conclude this section we shall recur to the virial theorem (39) must be supplemented by the normalization condition of the =− 2< > ⇒ E=< TU +− >= < T > to write the most general wave-function. The time-dependent evolution of the Schrödinger- expression for the energy after integration by parts, in the form Newton equations has been studied by many authors, in particular.36–39 The experimental tests of the validity of the nonlinear Newton- 22∞∞ *2 3 23 Schrödinger equation pose a technologically formidable challenge ETnn = − < > = ∫∫ (Ψ n (r )) ∇ Ψ n ( r ) dr = − |( ∇Ψn (r ))| dr  22mm−∞ −∞ due to the weakness of gravity and the difficulty of controlling 39 25∞ 2 4 6 2(n+ 2) quantum . 1 Gm −(yyy + + ++  y )2 − ∫ e y dy (35) 22 (1n+ )  0 Infinite derivative gravity, modified Newton-Schrödinger The wave-functions must be normalized and vanish at ±∞ . One equation and Schwarzschild atoms must emphasize that one is comparing (roughly equating) the values Since Quantum Mechanics is notoriously non-local, it is not of the integrals in (35) and not the integrands. farfetched that a theory of Quantum Gravity may require modifying It is clear that one cannot equate the integrands in equation (35). Einstein’s theory of gravity (and other local of ) to By a simple inspection one can see that there are zeros (besides r = 0 include a modified gravitational theory involving infinite derivatives,

Citation: Perelman CC. Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms.Phys Astron Int J. 2017;1(4):131‒137. DOI: 10.15406/paij.2017.01.00024 Copyright: Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms ©2017 Perelman 135

which by construction, is non-local. It turned out that the infinite derivative gravity (IDG) can resolve the problem of massive ghosts Gm2 r as well as it may avoid the singularity of the Newtonian potential at identical to − Erf (), the main difference being that r σ the origin, when one chooses the exponential of an entire function. . rr This model is also named super-renormalizable quantum gravity.40–42 Erf( )= Erf ( ) However, one does not understand fully how this non-local gravity σ 2σ  could provide a regular potential. It was argued that the cancellation Bohm’s quantum potential V =−∇ (ρρ /) has a of the singularity at the origin is an effect of an infinite amount of Q 2m geometrical description as the Weyl scalar produced by an hidden -like complex poles.42,43 ensemble density of paths associated with one, and only one particle.45,46 A regular potential at the origin of the form (8) has been studied This geometrization process of quantum mechanics allowed to by several authors44 in connection to infinite derivative gravity42,43 derive the Schrödinger, Klein-Gordon45,46 and Dirac equations.47,48 which is ghost-free and renormalizable when one chooses the Most recently, a related geometrization of quantum mechanics was exponential of an entire function in the construction of the infinite- proposed49 that describe the time evolution of particles as geodesic 42,43 derivative gravitational (IDG) SIDG For this IDG case, the lines in a curved space, whose curvature is induced by the quantum corresponding Newtonian potential generated. from the delta function potential. This formulation allows therefore the incorporation of all is non-singular at the origin. However, the authors44 explicitly showed quantum effects into the geometry of space-time, as it is the case for that the source generating this non-singular potential is given not by gravitation in the general relativity. Based on these results we propose the delta-function due to the point-like source of mass, but by the the following nonlinear quantum-like Bohm-Poisson equation for  Gaussian mass distribution. This explains clearly why the IDG with static solutions ρρ= ()r . 2 2 the exponential of an entire function yields the finite potential at the 22 ∇ ρ ∇V = 4πρ Gm ⇒− ∇ ( ) = 4πρGm (45) origin. The infinite-derivative (non-local) modified Poisson equation Q 2m ρ  42–44 associated to a point-mass particle at r = 0 is given by Such that one could replace the nonlinear Newton-Schrödinger 2 equation for the above non-linear quantum-like Bohm-Poisson σ 2 −∇ equation (45) where the fundamental quantity is no longer the 23 δ ()r (e4 ∇ ) V = 4πδ Gm ( r ) = Gm (40) Ψ 2 wave-function (complex-valued in general) but the real-valued * r probability density ρ = ΨΨ. σ Is a length scale which can be taken to be equal to the Planck- scale. Such equation (40) leads to identical solutions for the potential It has been proposed by29–31 to give up the description of physical V as the solutions for the potential in the ordinary Poisson equation states in terms of ensembles of state vectors with various probabilities, sourced by a Gaussian mass distribution,44 and which is a smeared relying instead solely on the as the description of reality. version of the delta function mass distribution. This can be shown by The time evolution of ρ is governed by the Lindblad equation.4 taking the 3D Fourier transform of equation (40) The authors31 also investigated a number of unexplored features of 2 , including an interesting geometrical structure- which 2 σ σ 2 2 k − k they called subsystem space-that they believed merits further study. Gm e 4 4 2 ⇒ e k Vk( ) = Gm Vk( ) = (41) The infinite-derivative-gravity generalization of equation (45) is k 2 2 22 2 And then performing the inverse 3D Fourier transform it yields  −∇σ 42∇ ρ Gm r −∇ (e ) ( ) = 4(πρGm 46) V( r ) = − Erf ( ) , see44 for further details. One can generalize 2m ρ G r σ the Newton-Schrödinger coupled system of equations (1,2) by The above equation is nonlinear and nonlocal. If one wishes to replacing the ordinary Poisson equation with the modified non-local introduce a temporal evolution to ρ via a Linblad-like equation, Poisson equation associated with infinite-derivative gravity (IDG) for instance, this would lead to an over determined system of  2 ρ σ 2 differential equations for (,)rt. This problem might be another −∇ 4 22 manifestation of the in Quantum Gravity. Naively (e ∇Φ ) V ( r ) = 4πρ G = 4 π Gm |( r ) | (42) 2 IDG replacing ∇ in equations (45-46) for the D’Alambertian µ µ In this case ∇∇,µ = 0,1, 2, 3 has the caveat that in QFT ρ()x no longer µ  2 has the interpretation of a probability density (it is now related to the  |Φ (r ')| V( r ) = Gm ∫ d3 r′ (43) particle number current). For the time being we shall just focus on IDG −  |'rr| static solutions ρ()r . Integrating (45) gives the integral-differential  Therefore to find (numerical) solutions of the highly nonlinear and equation for ρ()r nonlocal modified Newton-Schrödinger equation 2 2   ∇ ρ ρ()r′ 3 2 −− ( ) = Gm ∫ d r′ (47)  2    E Φ () r = − ∇Φ ()r + mV [ Φ ()] r Φ( r ) (44) 2m ρ |rr− ′ | 2m IDG

4To be more precise it is the Gorini-Kossakowski-Sudarshan- Becomes more problematic. However, solutions to the Lindblad equation infinite-derivative modified Poisson equation are not that difficult to find. For example, in the case when ρ()r is given The right hand side of (47) can be written as   by a Gaussian profile, after performing the Fourier transform GmMeff () r  ρ()r′ 3 2 − , M ( r ) = r ∫ dr′ (48) Gm r eff ′ procedure it gives V =− Erf (), which is almost r ||rr− IDG r 2σ

Citation: Perelman CC. Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms.Phys Astron Int J. 2017;1(4):131‒137. DOI: 10.15406/paij.2017.01.00024 Copyright: Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms ©2017 Perelman 136

surrounding the nucleus, and whose mass density distribution is  *  Where extreme caution must be taken because Mr() is not ρ = m ΨΨ, where Ψ()r the stationary wave-function solutions eff e the same as the enclosed mass inside a spherical region of radius to the Schrödinger equation are. In doing so we were able to obtain r encircling the origin. It is convenient to write (47) in this form the observed density after introducing an ultraviolet (48) because it reminds us of a “Schwarzschild atom” analog in the (very close to the Planck scale) and an infrared (Hubble radius) scale, spherically symmetric case, where the g metric component for and explained the origins of its repulsive gravitational . By signature (,,,)+−−− , and c =1, is 00 inspection one can see that if ρ()r is a solution to the Bohm-Poisson equation (45) with G >0, and then −ρ is a solution with −G <0. A negative gravitational constant is tantamount to repulsive gravity. For 2GMeff () r 1  gr( ) = (1−−), g = , MrMrc ( )= ( ), =1 (49) 00 rrr gr() eff eff this reason, we believe that the Bohm-Poisson equation (45), and its 00 relativistic generalization, deserves further investigations. See33,34 for many references related to the solutions (49). In the weak field limit, for slowing moving masses, one has Acknowledgments g η ++ hV=1 2 (c = 1) , and from equations (47-49) one 00 00 00 We thank M. Bowers for her assistance and to the referee for arrives at the relation between the temporal components of h and µν providing important references. the matter/probability distribution

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Citation: Perelman CC. Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms.Phys Astron Int J. 2017;1(4):131‒137. DOI: 10.15406/paij.2017.01.00024 Copyright: Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms ©2017 Perelman 137

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Citation: Perelman CC. Exact solutions of the Newton-Schrödinger equation, infinite derivative gravity and Schwarzschild atoms.Phys Astron Int J. 2017;1(4):131‒137. DOI: 10.15406/paij.2017.01.00024