CC BY
19
Bulletin of PFUR Series Mathematics. Information Sciences. Physics. No 4. 2009. Pp. 79-82
UDC 535.42:681.786.23
Correlations in Quantum Mechanics as Origin of Modified Newtonian Dynamics
V. V. Asadov *, O. V. Kechkin t
* Neur OK-III, Center for Informational Technologies-104 Vorob'jovy Gory, Moscow, 119899, Russia t Institute of Nuclear Physics, M.V. Lomonosov Moscow State University Vorob'jovy Gory, Moscow, 119899, Russia
We present derivation of modified Newtonian force from quantum mechanics with Coulomb potential. We show, that MOND-like theory arises in classical limit of quantum dynamics if nontrivial correlations of canonical variables are taken into account.
Key words and phrases: quantum and classical correlations, effective force, MOND.
1. Introduction
Modified Newtonian Dynamic (MOND) was proposed by Milgrom in 1981 [1,2]. The most successful relativistic version of MOND, known as 'TeVeS', was introduced by Bekenstein in [3-5]. Although MOND provides impressive experimentally verified predictions [6,7], there is still lack of fundamental origin of this theory. In this paper we show that nontrivial correlations leads to MOND-like modification of Newtonian force at classical limit of the standard quantum theory. These correlations include dispersions of coordinates, i.e. we propose the nonlocal origin for MOND. Similar approach for relativistic scenario will be considered in forthcoming publication.
2. Classical Theory with Nontrivial Correlations
Let us consider non-relativistic quantum system with one-component wave function ^ = ^(t, xk), where xk (k = 1, 2, 3) — coordinates of the three-dimensional space. We use exponential parametrization ^ = exp(iS/h), where the complex phase S is represented in terms of the real functions S1 and S2 as
i h
s = Si + 2 S2. (1)
In [8] it was shown, that the matrix quantities A1 ki = S1,XkXl and A2 ki = S2,XkXl define correlations between canonical variables of the theory in the classical limit of h ^ 0. There it is established, that all these correlations vanish if A21 ^ 0, where A2 is the matrix with the coefficients A2,ki.
Now let us study the system of Kepler type with (for example) planet and Sun, with the masses m and M, respectively (where m ^ M). Then its Hamiltonian reads:
2
pi GMm
k
V = V ^ - ^^ , (2)
2m r
where pt = ihdXk, and G — Newton's gravity constant. In the classical limit one obtains the following modified Hamilton's equations [8]:
d^fc Pk iA-ur dPk GMmxk . n r /„N
-JT =--(A1 )kl r,xi, -r— =--2---)kl r,xi, (3)
dt m dt r2 r
Received 4th June, 2009.
Authors are really grateful to prof. B. Ishkhanov, Yu. Rybakov, G. Costa and M. Matone for fruitful discussions while this investigation was carried out.
where the function r is defined as
r = -ASi, (4)
m
and A = J2i(^i)2 is the Laplace operator. We use these dynamical equations to calculate the physical force according the Second Newton's law:
d2xk d f dxk\ Fk = m^2T = ^ ) ; (5)
the result reads:
i
GMmxk d[(A^i)kl ASi,Xl ] 1 ,_i) (6)
rk =--2----72---)kiAbi,Xl. (6)
r2 r di m
Here the first term coincides with the conventional gravitational force, whereas the second and third ones maintain its dispersion corrections. By introduction of the normalized function
£ = * (7)
m
one modifies the correctional terms to make them proportional to the inertial mass m. Thus, the full force is proportional to the inertial mass — this fact proves the equivalence principle in this theory.
Actually, using of the function £ makes the Hamilton-Jacoby equation Si, t+E = 0 free of the mass parameter m:
+ 1E - ™ = 0, (8)
2 k r
where we mean the classical limit of the theory [8]. Thus, any solution £(t, xk) of this equation does not depend on m. Then, the function S2 satisfies the relation
S2,t + E £,*fc ^ =AE, (9)
k
which is also independent on the inertial mass. From this it follows that the whole theory under consideration yields the equivalence principle.
3. Modified Newtonian force
To study the total force (6) in the theory, let us represent it in the following form:
Fk = Fo k - m$k. (10)
Here F0k = —GmMxk/r3, whereas the normalized correlation term reads:
$fc = (£A_2^ ki A£, xi + ek, (11)
where
efc = -d di
(12)
and £ means the matrix with the components £,xkxi. Taking into account, that the total ¿-derivative is given by the relation
d = +dXn Q = Q + di di Xn
£,xn — [A-2 M A£,as dXn, (13)
V / ns J
XL
Correlations in Quantum Mechanics as Origin of Modified Newtonian Dynamics
81
and performing the calculations, one obtains the following result:
e=-1MklA (E2- L+
+ +
(VE + EV) ki (V) (V) (AEXki Xk2 E,xk s2,xki Xk2
1 V / lk2
E,xn - (V) ae
V / ns
—1\ A V ( /1—1\ ( A — 1
ae,xi+
x
ki ae,xikki ik2 S2,xki xk2 1
In the infinitesimal correlation regime, where A2 ^ 0, for the dispersion term one obtains:
1 2
+ (a^Ê + 2EA21) ki AE, „ + E,fa1) ki AE, . (14)
Note, that Eq. (14) does not include nonlinear correlation terms.
Now let us study one concrete solution of the dynamical system under consideration, which corresponds to the spherically-symmetric potential E. Thus, let us consider the anzats with E = E(t, r), where r means the radial coordinate (i.e., the distance between planet and Sun). In the classical case with separated variables,
*fc--1 (V) kl A(E^) +
nt
E = - y + f (r), (15)
where a = const, one deals with
/< = }"2. da)
For the isotropic matrix A^1 (which describes the correlations between the Cartesian coordinates Xk of the planet, see [8])
{A~21)ki = (17)
where e ^ 0 in the infinitesimal limit under our study, one obtains:
$k = 2e (E, XkXl AE, Xl — E, XkXlXnE, XlXn). (18)
Note, that Eq. (17) means localization of the planet in the sphere of the ^ \fe radius with the center in its classical position. Using the identity
E ,XkXi = V^k xi + £5ki, (19)
where £ = f'/r and ^ = £'/r, and also the relation
E, Xkxixn = —XkXlXn + i] (xkSin + xn5ki + xrfnk), (20)
it is possible to calculate the final form of the correlation potential and for the total force as well. This term (and the force) is radial, i.e. $k = $ Xk/r, where the defining magnitude $ reads:
3GM\ 2
£6 +—)
a + a +
$ 2GM . (21)
r
Note, that e > 0 for the well-localized position of the planet (see [8]), whereas the constant a is of the arbitrary sign. It is seen, that $ ~ r-3 for r ^ so F ~ r-2 for the large distances. Then, the small distances case (r ^ 0, for the case of a > 0) corresponds to F ~ r-4. This type of force behavior is actually natural to MOND, see [1,2].
4. Conclusion
In this article we have derived corrections to the standard gravitational (or Coulomb) force, which can be used for the MOND approach verification. These are given by Eq. (21), which follows from the standard quantum mechanics in the classical limit with nontrivial correlations of the canonical variables. This result can be generalized to the temperature-dependent case, which can be interested in the cosmo-logical framework. In [8] it was shown, that the corresponding consideration needs in introducing of the complex parameter of evolution to the theory.
References
1. Milgrom M. Solutions for the Modified Newtonian Dynamics Field Equation // Astrophys. J. — 1986. — Vol. 302. — Pp. 617-625.
2. Milgrom M. Do modified Newtonian Dynamics Follow from the Cold Dark Matter Paradigm? // Astrophys. J. — 2002. — Vol. 571. — Pp. L81-L83.
3. Bekenstein J. Invited Talk at 28th Johns Hopkins Workshop on Current Problems in Particle Theory: Hyperspace, Superspace, Theory Space and Outer Space // PoS JHW2004. — 2005. — P. 012.
4. Bekenstein J. Relativistic Gravitation Theory for the MOND Paradigm // Phys. Rev. — 2004. — Vol. D70. — P. 083509.
5. Large Scale Structure in Bekenstein's Theory of Relativistic Modified Newtonian Dynamics / C. Skordis, D. Mota, P. Ferreira, C. Boehm // Phys. Rev. Lett. — 2006. — Vol. 96. — P. 011301.
6. Famaey B., Binney J. Modified Newtonian Dynamics in the Milky Way // Mon. Not. Roy. Astron. Soc. — 2005. — Vol. 363. — Pp. 603-608.
7. Sanchez-Salcedo F., Reyes-Iturbide J., Hernandez X. An Extensive Study of Dynamical Friction in Dwarf Galaxies: the Role of Stars, Dark Matter, Halo Profiles and Mond // Mon. Not. Roy. Astron. Soc. — 2006. — Vol. 370. — Pp. 1829-1840.
8. Asadov V. V., Kechkin O. V. // Moscow University Physics Bulletin. — 2008. — Vol. 2. — P. 105.
УДК 535.42:681.786.23
Корреляции в квантовой механике как основа модифицированной ньютоновской динамики
В. В. Асадов *, О. В. Кечкин t
* НейрОК-Ш, Технопарк МГУ-104 Россия, 119899, Москва, Воробьёвы Горы
1 НИИЯФ МГУ Россия, 119899, Москва, Воробьёвы Горы
Даётся вывод модифицированной ньютоновской силы из модифицированной квантовой механики с кулоновским потенциалом. Показывается, что теория типа МОНД появляется в классическом пределе квантовой динамики при учёте нетривиальных корреляций канонических переменных теории.
Ключевые слова: квантовые и классические корреляции, эффективная сила, МОНД.
