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Physical Review Letters (2017). DOI: 10.1103/PhysRevLett.118.155301

The author of recent paper [1] submit the hypothesis that new dynamical effects in Bose-Einstein Condensation can be interpreted via negative masses. Despite a bibliography unusually vast for a Letter, PRL editors did not request the authors to quote the originator of negative masses, P. A. M. Dirac, because such a quotation would have brought to light the fact that negative masses violate causality in QM. In addition, PRL editors did not add comments on the interpretation of negative masses as characterizing antimatter; because such an interpretation would imply violation of the PCT theorem and other inconsistencies [2].

Since Prof. Santilli (Email: research-at-thunder-energies-dot-com) is a leading expert in the field, we contacted him and he released the following comments:

Newton's basic notion is that of "massive points," subsequently adopted by Galileo and Einstein, because necessary for compatibility with the local-differential character of the differential calculus. The representation of particles with their extended character has requested a necessary broadening of the differential calculus into the covering isodifferential calculus [3-6]. New dynamical effects not representable with Newton's laws are then evident to all.

For a truly rudimentary outline,m the first step is the isotopic (axiom-preserving) lifting of the conventional product AB of all QM quantities into the isoproduct I first presented in monographs [3] I wrote when I was at Harvard under DOE support

(1) A*B = ATB,        T > 0.

The simple realization of the isotopic element T

(2)     T = Diag (a, b, c, d),    a, b, c, d > 0

allows an immediate representation of the spheroidal share of particles with semi axes a, b, c, as well as a representation of their density d. The isotopic lifting of functional analysis, metric spaces, etc. then follows by lifting all associative products into form 91).

The next step is the generalization of ordinary numbers n, m into isonumbers with generalized unit I*, called isounit, and related rules I wrote while visiting the JINR in Dubna, Russia in 1993 [4]

(3) I* = 1/T, n' = nI*,  m' = mI*,      n'*m' = (nm)I*,   .

The generalization of numeric fields turned out to be necessary to maintain causality because isotope formulations are noncanonical at the classical level and nonunitary at the operator level (as an example the problem of causality for Dirac's negative masses is resolved by assuming a "negative"-definite new unit [2]).

The invariance of the formulation then requires the lifting of the Newton-Leibnitz differential or into the isodifferential that I first presented in the 1996 mathematical memoir [5]

(4)   d' r' = T d[rI*(r, ...)] = dr + r T dI*(r, ...)

Isomathematcs is the generalization of all branches of 20th century applied mathematics via the use of isoproduct (1) and related reformulation of functional analysis, metric spaces, geometries, Lie's theory, etc. formulated on over isofields and elaborated via the isodifferential calculus. The mathematical literature on isomathematics is nowadays quite vast. I merely quote here the six volumes by Prof. D Svetlin Georgiev [6] on the isodifferential calculus and the PhD. course going on at advanced mathematics departments .

Isomathematics was constructed to formulate a corresponding generalization of QM into into Hadronic Mechanics (HM) for the representation of particles in condition of  mutual penetration. In essence, HM is an axiom-preserving "completion" of QM according to the celebrated Einstein-Podolsky-Rosen argument, and includes isomechanics as well as the broader genomechanics for the representation of irreversibility over time, as well as the isodual mechanics for the causal description of Dirac's negative masses.The literature on HM is equally vast. I can indicate here Refs. [7-9] and vast literature quoted therein and the recent outline [10].

"In my view, the effect reported in Ref. [1] is a direct consequence of the extra term rTdI*(r, ...) in the isodifferential calculus, Eq. (4), without any violation of Newton's laws because they were conceived for exterior dynamical problems (massive points moving in vacuum), while the Bose-Einstein condensation deals with the basically different interior dynamical problems (extended particles moving within a physical medium).

Similar "negative effects" have appeared in a number of experimental verifications of HM [9,10]. The most visible evidence on the existence of novel effects at short mutual distances of particles is the synthesis of the neutron from the hydrogen atom in the core of stars studied [11,15] (see also three lectures in the link

"When using QM and its underlying differential calculus, the proton must be essentially represented as a massive point; you cannot "compress" (in Rutherford's language) the electron inside a point; and the neutron synthesis is impossible contrary to crushing evidence. In any case, the mass of the neutron is bigger than  the sum of the masses of the proton and the electron, thus implying a "mass excess" which is beyond any possible QM treatment, thus forcing the construction of the covering HM.

When the proton is represented via HM with his actual shape and density via the isotopic element T, Eq. (2), the spin-orbit coupling for the electron compressed inside the proton deviates from QM according to an effects complementary to that of Ref. [1]. In fact, the resistance experienced by the electron in its motion within the proton medium (represented with the new term in the isodifferential) turns its QM angular momentum L_e from the traditional value 0 for the ground state, to a value  "apposite" that of the electron spin S_e and equal the proton spin S_p (which value is prohibited by QM but fully  admitted by HM) according to the values illustrated in the figure below

(5)  S_p = + 1/2,   s_e = - 1/2,    L_e = - s_e = 1/2

​See the non-relativistic treatment of Ref. [11] and the relativistic representation of Ref. [12].

The above values are quite natural for interior systems because, in the event the orbital motion of the electron inside the proton is not equal to the proton spin, the electron would move against the hyperdense medium inside the proton, which is a physical impossibility. An essentially similar occurrence holds for the opposite effect of Ref. [1] without the need of retorting to causality-violating, antimatter-type, negative masses.

In summary, when the condition of compatibility with 17th century mathematics and 20th century physics is relaxed, the beautiful effect measured by M. A. Khamehchi et al.  establishes the "inapplicability (and not the "violation") of the Newton-Leibnitz differential calculus and, consequently, of QM for particles at mutual distances smaller than their size in favor of new, third millennium vistas [9] with novel applications beyond out imagination at this time." R. M. Santilli


[1]  M. A. Khamehchi et al, Negative-Mass Hydrodynamics in a Spin-Orbit–coupled Bose-Einstein Condensate, Physical Review Letters (2017). DOI: 10.1103/PhysRevLett.118.155301 

[2] R.M. Santilli, . Isodual Theory of Antimatter with Applications to Antigravity, Grand Unification and Cosmology, Springer (2006).


[3] R. M. Santilli, Foundation of Theoretical Mechanics,
Volume I (1978) and II (1982) Springer-Verlag, Heidelberg, Germany,

[4] R. M. Santilli, ``Isonumbers and Genonumbers of Dimensions 1, 2, 4, 8, their Isoduals and Pseudoduals, and "Hidden Numbers," of Dimension 3, 5, 6, 7," 
Algebras, Groups and Geometries Vol. 10, 273 (1993)

[5] R. M. Santilli, ``Nonlocal-Integral Isotopies of Differential Calculus, Mechanics and Geometries," in Isotopies of Contemporary Mathematical Structures," 
Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 7-82 (1996),

[6] S. Georgiev, Foundations of the IsoDifferential Calculus,
Volumes, I, II, III, IV,V, and VI Nova Scientific Publisher (2015 on).


[7] R. M. Santilli, "Relativistic hadronic mechanics: nonunitary, axiom-preserving completion of relativistic quantum mechanics,"  
Found. Phys. Vol. 27, 625-729 (1997),

[8] R. M. Santilli, Elements of Hadronic Mechanics,
Vol. I and Vol. II (1995) [15b], Academy of Sciences, Kiev,

[9] I. Gandzha and J. Kadeisvili , New Sciences for a New Era:
Sankata Printing Press, Nepal (2011)

[10] R. M. Santilli, "An Introduction to New Sciences for a New Era," 
Clifford Analysis, Clifford Algebras and their Applications, in press (2017)


[11] R. M. Santilli, ``Apparent consistency of Rutherford’s hypothesis on the neutron structure via the hadronic generalization of quantum mechanics - I: Nonrelativistic treatment”, ICTP communication IC/91/47 (1992)

[12] R. M. Santilli, ”Recent theoretical and experimental evidence on the synthesis of the neutron,” Communication of the JINR, Dubna, Russia, No. E4-93-252 (1993), published in the Chinese J. System Eng. and Electr. Vol. 6, 177 (1995),

[13] R. M. Santilli, ``Confirmation of Don Borghi’s experiment on the synthesis of neutrons," arXiv publication, August 15, 2006

[14] R. M. Santilli and A. Nas, ``Confirmation of the Laboratory Synthesis of Neutrons from a Hydrogen Gas," Journal of Computational Methods in Sciences and Eng, 14 (2014) 405–41

[15] R. M. Santilli, ``12 minutes Film on the Laboratory Synthesis of Neutrons from a Hydrogen Gas"

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