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Download Full International Journal of Sciences Research Article Volume 5 – May 2016 (05) The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity Piero Chiarelli1 1National Council of Research of Italy, Area of Pisa, 56124 Pisa, Moruzzi 1, Italy Interdepartmental Center “E.Piaggio” University of Pisa Abstract: In this work the quantum electro-gravitational equations are derived for half spin charged particles with the help of the hydrodynamic quantum formalism. The equations show the reversing of the trace of the Ricci curvature in passing from matter to antimatter and that the CPT transformation, associated to the reversing of the trace of the Ricci curvature, leads to a more general symmetry in quantum gravity. PACS: 04.60.-m Keywords: Quantum Gravity, Quantum Kaluza Klein Model, CPT Symmetry in Quantum Gravity Introduction kinetics of mass density at the Planck scale such as One of the intriguing problems that the theoretical forbidding the formation of a black hole with a mass physics is facing is the integration of the general smaller than that one of the Planck [14]. relativity with the quantum mechanics. The Einstein gravitation has a fully classical ambit, the quantum Another measurable output of the theory is the mechanics mainly concerns the small atomic or sub- detailed description of the gravitational field of atomic scale, and the fundamental interactions. antimatter. Many and discordant are the hypotheses on the gravitational features of the antimatter [15-19]. Many are the unexplained aspects of the matter on The hydrodynamic model shows that the Ricci tensor cosmological scale [1]. Even if the general relativity associated to an antimatter distribution has a negative has opened some understanding about the sign respect to that one of the same distribution of cosmological dynamics [2-4] the complete matter. This fact is due to the negative value of the explanation of generation of matter and its energy function for the antimatter states [20]. This distribution in the universe needs the integration of fact is of paramount importance in making the CPT the cosmological physics with the quantum one. To symmetry compatible with the matter-antimatter this end the quantum gravity (QG) represents the goal repulsive behavior [20]. of the theoretical research [5-11]. Nevertheless, difficulties arise when one attempts to apply, to the The objective of this work is to generalize the force of gravity, the standard recepy of quantum field quantum gravitational equations (QGEs)[14] to theories [12-13]. charged particles with half spin and to show that the CPT symmetry of euclidean quantum mechanics is Recently, the author has shown that by using the the particular case of a more general one that quantum hydrodynamic formalism is possible to comprehends the inversion of the curvature of space- achieve a non contradictory coupling of quantum time. equations with the gravitational one via the derivation The work is carried out by utilizing the of the impulse energy tensor [14]. The result can be hydrodynamic representation of quantum mechanics easily translated into the standard quantum formalism where the problem is solved as a function of two real giving rise to equations that are independent by the variables, | | and S , that lead to the standard hydrodynamic approach and that have clear meaning and appear well defined [14]. iS complex wave function | | exp[ ] [21-25]. A first outcome of the model shows that quantum The paper is organized as follows: in the first section effects play an important role on the gravitational the hydrodynamic QGEs are briefly resumed. Then This article is published under the terms of the Creative Commons Attribution License 4.0 Author(s) retain the copyright of this article. Publication rights with Alkhaer Publications. Published at: http://www.ijsciences.com/pub/issue/2016-05/ DOI: 10.18483/ijSci.1012; Online ISSN: 2305-3925; Print ISSN: 2410-4477 Piero Chiarelli (Correspondence) [email protected] +39-050-315-2359 The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity they are generalized to half spin charged particles. In this section we will use the euclidean hydrodynamic Finally, the CPT symmetry is analyzed respect to the representation [21] of Klein-Gordon equation (KGE) to QGEs. derive the mixed energy-impulse tensor density. 2. The Impulse-Energy Tensor of Quantum States The hydrodynamic form of the Klein-Gordon equation Derived Via the Hydrodynamic Quantum (for scalar uncharged particles) Equations 2 2 m c , (1) 2 is given by the Hamilton-Jacobi (H-J) type equation S( q,t ) S( q,t ) 2 | | 2 2 g m c 0 (2.a) q q | | coupled to the current conservation one [21] S J | |2 m 0 (2.b) q q q where S ln[ ] (3) 2i * and where i * J c,J i ( * ) (4) 2m q q is the 4-current. Moreover, being the 4-impulse in the hydrodynamic analogy E S p ( , pi ) (5) c q it follows that the 4-current reads 2 p J ( c , J ) | | (6) i m where | |2 S (7) mc 2 t Moreover, by using (5), equation (2.a) leads to 2 SSE 2 2 2 Vqu p p 2 p m c 1 2 (8) qq c mc 2 where p pi pi is the modulus of the spatial momentum. Generally speaking, the hydrodynamic function EE ( t ) , but for eigenstates, for which it holds E En const , it follows that http://www.ijSciences.com Volume 5 – May 2016 (05) 37 The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity 2 En 2 2 2 Vqu 2pn m c 1 2 c mc (9) 2 2 2VVqu 2 2 2 qu m c1 2 m q 1 2 mc mc from where it follows that (see appendix A) Vqu E m c2 1 (10) n mc2 (where the minus sign stands for antiparticles) where the quantum potential reads 2 | | V , (11) qu m | | and , by using (9), that Vqu( n ) E p m q1 n q . (12) n mc2 c 2 Thence, the Lagrangian form of the quantum hydrodynamic equation of motion (2.a) reads [14] L p , (13.a) q L p (13.b) q where dS S S L qi p q (14.a) dt t qi that, for eigenstates, reads 2 mc Vqu L() 1 (14.b) mc2 where the minus accounts for antiparticles. p The motion equation can be obtained by deriving ( q,q ) from (13.a) and then inserting it into (13.b). In the quantum case, the equation of motion is also coupled, through Vqu , to the mass distribution | | of the conservation equation (2.b). For 0 it follows that Vqu 0 and the classical equations of motion are recovered. Thence, the quantum hydrodynamic motion equation for eigenstates (omitting the subscript n) reads http://www.ijSciences.com Volume 5 – May 2016 (05) 38 The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity dp d V L mcu 1 qu ds ds 2 c mc q (15) Vqu mc 1 q mc2 where u q , (16) c that leads to Vqu du d Vqu Vqu T mc 1 mcu 1 mc 1 (17) 2 ds ds 2 2 c mc mc q mc q where, for eigenstates, the quantum energy-impulse tensor (QEIT) T reads[14] 2 L mc Vqu T q L 1 u u . (18) 2 q mc leading to the quantum impulse energy tensor density (QIETD) [14] L 2 L 2 T q L | | q L | | T (19) q q where L | |2 L is the (quantum hydrodynamic) Lagrangian density and L is the quantum hydrodynamic Lagrangian function.. It must be noted that the hydrodynamic solutions solution of equation (17) allows to solve the quantum given by (17) represent an ensemble wider than the problem. quantum one since not all the field solutions p Since for a generic matter-antimatter superposition of warrant the existence of the integral action function states the energy is neither positive nor negative S so that the irrotational condition has to be definite, the Lagrangian function as well as the imposed (see references [14,22]). QEITD must be re-write in a more general form. To this end we observe that by using (8) it follows that Equation (17) (following the method described in ref. (see (A.6) in appendix A) [14] ) can be used to find the eigenstates of matter wave n (by considering the upper positive sign in 1 S p q , (20) 2 (18)) and the antimatter eigenstates n by using c t the lower minus sign in (18). Furthermore, being the eigenstates irrotational (see example in appendix B) and that and hence, all their linear superpositions, the http://www.ijSciences.com Volume 5 – May 2016 (05) 39 The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity 1 1 dS S S S | |2 S 2 L qi p q c p p 2 p J dt t qi t mc t 1 1 2S 2 2 Vqu (21) p J c m c 1 2 t mc 1 ln[ ] ln[ ] ln[ ] i *** c2 2 t q q i and that 1 2 2 S T | | c p p p p t 1 1 S (22.a) 2 J p J p mc t 1 m| |2 c 2 1 S V u u 1 qu 2 2 m ct mc 1 T m J J J J 1 2 2 ln[ ] (22.b) 2 2 2im c * m | | c t 2 ln[ ] ln[ ] ** Vqu +1 2 2mc q q mc 2.1 Non-Euclidean Generalization Since any mass distribution leads to a non-flat space, equation (2.b, 15, 22) must be expressed in a non- euclidean space and read, respectively, 1 S g g | | 0 (23) g q q du 1 g u u ds 2 q (24) d Vqu Vqu u ln 1 ln 1 2 2 ds mc q mc http://www.ijSciences.com Volume 5 – May 2016 (05) 40 The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity 1 S p p Vqu T T g | |2 m 2 c 4 1 g 2 2 2 t m c mc 1 (25.a) m | |2 c 2 1 S V u u 1 qu g 2 2 m ct mc 1 2 2 ln[ ] 2 2 2im c * T m| | c t (26) 2 ln[ ] ln[ ] ** Vqu +1 2 g 2mc q q mc where 2 1 Vqu g | | , (27) m | | g 1 where | g | J 2 , where J is jacobian of the transformation of the Galilean co-ordinates to non- g ac ac euclidean ones and where g is the metric tensor defined by the quantum gravitational equation [14] 1 8G R g R g T (28) 2 c4 where the constant , that warrants the principle of minimum action and the correct Einstein classical limit [14], reads 2 2 1 8G2 8 G m| | c 1 S0 4| | L0 4 2 (29) c c m c t where, for scalar uncharged particles (see appendix A) 1 2 2 4 S0 mc L0 m c (30) t where the lower minus refers to antimatter and where (A.4)) S lim ln[ ] (31) 0 0 2i * As shown by Bialiniki-Birula et al.
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