International Journal of Sciences

Research Article Volume 5 – May 2016 (05)

The CPT-RICCI Scalar Symmetry in Quantum Electro- 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 .

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 . 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 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  | | 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) qq  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  2pn   m c 1  2  c   mc  (9) VV    2 2 2qu 2 2  2 qu m c1 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 ) where the quantum potential reads

2      | | V   , (11) qu m | | and , by using (9), that

Vqu( n ) E p  m q1   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   qi   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 S  Vqu   1 2 2 2 (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 ct   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 ct   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 8G 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 8G2 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. and by the author himself [14,22], it is noteworthy to observe that, due to the biunique relation between the quantum hydrodynamic equations (2.a-2.b) and the quantum equation (1), equations (23-24) are equivalent to the Klein-Gordon one 2 2 1  m c    g g      2  , (32) ; g 

(where the semicolon stands for the 4-D covariant derivative) that through the QEITD (25) couples to the quantum gravity equation (1) independently by the hydrodynamic approach .

2.2 The Overall Cosmological Constant and the Classical Limit http://www.ijSciences.com Volume 5 – May 2016 (05) 41

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

If we re-write the QEITD as

1 2 2  1 S  m| | c Vqu  T T  CL   1   g   (33) m c2t  mc 2       where

1 1 S  m| |2 c 2 T CL  2  u  u  , (34) m c t   is the classical limit of the QEITD (22.a) [22] , equation (28) reads

1V  8G R g R  (1  1 qu )  g  T   2   4  CL (35) 2 mc  c

Vqu that in the classical limit (when it holds Vqu  0 as well as 1  1 )) leads to the equation of the mc 2 general relativity without the cosmological term that is canceled by an opposite contribution of quantum origin.

3. Gravitational Quantum Equations for Scalar Charged Particles When we consider charged particles, we have to consider the electromagnetic (EM) interaction. This can be done by T introducing in the quantum gravitational equation (1) the energy-impulse tensor for the EM field  f and for the  T T  g  matter-field interaction m f m  f (derived in the following section) leading to [26]

1 8G R g  R     T   T  . (36) 2 c4  f m  f  coupled with the Klein-Gordon equation

1 A  g m2 c 2 D  g D         2 (37) g  g 

e (where DA   ) and with the Maxwell one  i 

  F ;  4 J (38)

T coupled to (36) through the energy-impulse tensor density  f that reads [27]

1   1   T   F F  F F g   (41) f 4  4 

 where J(*DD*)     , where [26] 2im  

http://www.ijSciences.com Volume 5 – May 2016 (05) 42

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

F  A ;  A;     A   A , (39) and where

  A  ( , - A ) (40) c i is the potential 4-vector,

3.1 The Qeitd for Scalar Charged Particles As done in section 2, we derive the mixed QEITD from which we can obtain the covariant one. The euclidean quantum hydrodynamic equations of motion for charged particles (from which the mixed QEITD can be derived) corresponding to the KGE

m2 c 2 DD    (42)   2 can be obtained by applying the minimal coupling correspondence

S E   p     eA  ( , pi ) (43) q c

(where   is the mechanical momentum) to the equations (2.a) so that the H-J hydrodynamic equation reads [21]

  S  S    | |  ( q,t )  eA  ( q,t )  eA   m2c2   2       (44)  q  q  | | united to the conservation equation

J   0 (45) q

where the 4-current J  reads

 e   e  J(c,j) i  (*     A          A*)    2im i   i    2e  ( *        *)  A   *  (46) 2im i  | |2 |  | 2  p  eA     m   m  where

| |2 S      e 2   . (47) mc  t 

Moreover, analogously to (8,12), from (44) it follows that

http://www.ijSciences.com Volume 5 – May 2016 (05) 43

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

1 S   n  e  q n 2    c t  (48.a) E e n q  p  eA c2 n  and by (A.5) (see appendix A), that

1 S     e  q 2    c t  (49) E e q  p  eA c2    that leads to

1 dS S  S1  S e       L   qi   p q  2  2   p  ( p  eA ) dt t  qi c  t c  (50) 1    p J that, as a function of  and A , reads

1        ln[ ]   ln[ ]  ln[ ] i 2  * 2ie  *  * 2ie  L   c     (  A ) (51) 2  t   q  q       from which, by using (19,50) with the help of (16,48), it follows that

1 1 S   2         T  ||pqpq       2   e     pJpJ       m f mc t   (52) 1  eA    2 2 4 S      Vqu  eA   | | m c  e  1       t  m2 c 2 mc 2 m 2 c 2         and finally that

2 1    e| |     T   m  J  J  J  J    A  J  A  J     m  1  V   qu e     u 1  2  A u mc  mc   (53) m| |2 c 2 1  S        e     2mc 2 t 1   V    qu e     1  1  2  A u    mc  mc      

Moreover, by using (3,43) we can express the QEITD as a function of the wave function as

http://www.ijSciences.com Volume 5 – May 2016 (05) 44

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

1   S   T  | |2 c 2   e p  eA p  p  eA p   m f              t         ln[ ]   ln[ ]   * 2ie  *   A        q  q 2 1    (54) 2 c    2ie     | |  ln[ ]      2i  t  *          ln[ ]   ln[ ]    * 2ie  *     A          q  q        

4. Half Spin Charged Particles For half spin charged particles, the QGEs are given by the gravitational one (36) coupled both to the Maxwell T equation (38) (through the EM energy impulse density tensor  f ) where[22]

  J    , (55)

  0         , (56)  0   where   ( 0 , i ) are the 4-D extended Pauli matrices [22], where

  ,  (57)

  *,  * , (58) and to the

   ie   i    A   mc   0 (59)     

 T T  g  (through the energy impulse density tensor (to be defined)) m f m  f . 4.1 The Quantum Energy Impulse Density Tensor and the Cosmological Constant for Half Spin Particles   As shown by Bialiniki et al. [22] the components  and  of the bispinor     in the   Dirac equation (56), that in the Schrödinger-like form reads

   0 i    0 2   i t    c      eA   mc  e   , (60)     i    are not independent given that

i       . (61) mc 

http://www.ijSciences.com Volume 5 – May 2016 (05) 45

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

Thence, as shown by Guvenis [21], by (56 or 60) for the  it holds the equation

2 2  e   e   m c ec    A   A       i Bi (62)  i  i   2  where B    A and where it has been used the Lorentz gauge  A  0 . By using the standard hydrodynamic notation [22] we can express the  as

1 S    1    |  | exp i    (63) 2     2  where

 1  exp[i / 2] cos / 2      (64)   2   exp[i / 2] sin / 2  and where  and  are the angles in spherical coordinates of the spin versor

 ni  (sin cos,sin sin,cos )   i . (65) and the associated 4-vector

2  n  (| | ,ni )    (66)

Moreover, the current  e   e  J ( c ,  ji )  (  *    A          A    *)) 2im i   i    2e  (**)A*            2im i  (67) | |2     S  cos     eA   m 2  | |2 |  | 2  p  eA     m   m  where, in agreement with the result in ref.[22]

S   p    cos , (68) q  2 q  is still conserved, since

 *  i Bi  i Bi *  0 , (69) and leads to the conservation equation,

http://www.ijSciences.com Volume 5 – May 2016 (05) 46

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

   S  | |2   eA   0      (70) q   q  that by the hydrodynamic identity (43) leads to

 2  J  | |    m q   m  0 , (71) q q q with

| |2  S         cos  e  . (72) mc 2  t 2 t 

By equating the real and imaginary part of equation (60), the conservation equation (71) is obtained together with the quantum hydrodynamic Hamilton-Jacobi motion equation [21]

 V  B   S( q,t )  S( q,t )   1 2 2 qu j  i i  j   eA   eA     m c 1   (73)     q  1  2 2   q       mc mc  where the quantum potential V 2     qu1      | |   1   1    1  V          ln    ln  ln  (74) qu V  m  | |             qu 2    2   2   2  contains the contribution from the spin distributions 1,  2 , where the hydrodynamic spin vector i reads  1  1 0 e  1         (75) i  1  i     0  2  mc  2  and where the “mechanical moment” (see (A.5) in appendix A) reads

  E e  cos  S   2 t q    cos   eA  p  eA . (76) c2  q2  q    

Moreover, by using (68,73), it follows that

 S  S   ( q,t )  eA  ( q,t )  eA             q  q   2        (77)  E  e  cos  V  2 t   qu j i Bi  j      p  eA 2  m2c 2 1    2 i i   2 2   c   mc mc      that summing over the index j=1,2 leads to

http://www.ijSciences.com Volume 5 – May 2016 (05) 47

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

SS    ( q,t )eA ( q,t )  eA          q q  2     E e  cos    (78) 2 t 2 Vqu  B    p  eA   m2 c 2 1   i i  2 i i    2 2  c mc mc     

V  V  B   B qu1 qu 2  i i 1  i i 2 (where V  i Bi  ) and, hence, for eigenstates it follows that qu 2 2 (9)

  V  B  E e  cos    m  c2 1 qu  i i  n 2 2  (79) 2 t mc mc  that gives (for matter or antimatter eigenstates (see appendix A))

  Vqu iB i p  m q 1    eA  , (80) Moreover, by n mc2 mc 2     using (76) it follows that 1 S    cos  e   L  p q    t2  t p   2   c  (81)   1    p J from which by using (5), and the identity

  1  2 S  ln , (82) 4i  1 *  2 * the QEITD, as a function of the wave function, reads

1 S    T | |2 c 2  e   cos  p p   eA   p p   eA     m f        t2  t    1 1 S        (83)  2  e  cos     p J  p  J    mc t2  t   2 1   e| |      m  J J  J  J    A  J  A  J     m 

http://www.ijSciences.com Volume 5 – May 2016 (05) 48

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

1 4ic 2     4ie    2  1 2  T m f | |  ln    2i cos    t  1 *  2 *  t           ln 1 2   ln 1 2    1 *  2 * 4ie   1 *  2 *   A  q    q      (84)               ln 1 2   ln 1 2    *  * 4ie  *  *    1 2  A  1 2           q  q         

Finally, by using (81), the CC reads

1 S  0  e 8GG 8    SS     | |2 L  |  | 2 t 0 0  eA  40 4 2     (85) c c c   q q    that for classical matter or antimatter states reads

8G mc2  B    | |2  1 i i  eA q   c4 mc 2     (86) 8G 2    4 | | E0  e   eA q  c where the lower minus refers to antimatter.

5. Symmetry Of The Half Spin Charged Particles QGE The symmetry characteristic of the QGEs (36,59, 84-85) can be straightforwardly derived by observing that the Dirac equation (56) is invariant under the CPT transformation. In fact, the charge, time and parity inversion transformations lead to the substitutions

C  e  e P  z  z (87) T  t  t and hence to

  CPT      P (88)   CPT  A  A P (89)

CPT  Bi  Bk Pki (90)

CPT  i   k Pki (91)

http://www.ijSciences.com Volume 5 – May 2016 (05) 49

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

1 0 0 0    1 0 0   0 1 0 0   where P    and P  0 1 0 , that, due to the Hermitian property of  B , applied  0 0 1 0  ki   i i   0 0 1 0 0 0 1 to (62) leads to

 e  e   m2c 2 ec   CPT   A    A      B        2 i i    i  i       (92)  e  e   m2c 2 ec      A   A CPT       B CPT         2 i i   i  i      that, being equal to the Dirac equation for the complex conjugated wave function, leads to

CPT    * (93) from where we can see that the CPT leads a particle in its and vice versa. Moreover, given that P( )    , on the basis of the above identities we can apply the CPT on the QEITD to obtain

1 4ic2    *  * 4ie   CPT T   T  (*) | |2  ln 1 2    2icos    m f   m f     t  1  2  t          ln 1 2   ln 1 2    1 *  2 * 4ie   1 *  2 *   A  q    q      (94)      T    m f         ln 1 2   ln 1 2    *  * 4ie  *  *    1 2  A  1 2         q    q          

 Moreover, by using the time-inversion identity T  q  q  and given that , according to (A.8) it holds

Vqu  SBn   ( n )  i i  lim0  e  cos    lim   0 1   t2  t  mc2 mc 2   

SB0   i i    e  1  2  t  mc  under CPT, it follows that

http://www.ijSciences.com Volume 5 – May 2016 (05) 50

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

  iBS i    0   CPT1    CPT   e   mc2  t      

Sn     lim0 CPT   e  cos    t2  t   , Sn    lim0   e  cos   t2  t  V  qu ( n ) BB   lim 1  i i    1  i i  0 mc2 mc 2  mc 2     that mc2  B  CPT L  CPT  1 i i  eA q   0  2   mc  (95) mc2  B    1 i i eA q    L  2   0 mc  and hence that

8GG2  8  2 CPT  CPT | | L0    |  | L 0   . (96) c4  c 4   8G Thence, if we consider the trace T      R , where R is the trace of the Ricci curvature tensor, it c 4 follows that

4 4 c   c   CPT R   CPT T    T    R (97) 8G   8G   given that the trace of the (antisymmetric) electromagnetic QEITD is null. From the above result, we can infer that the CPT transformation leads to the change of the sign of the space-time scalar curvature (i.e., R  R ) so that QGEs are invariant under the overall CPTR inversion transformation.

6. General Comment the wave-function, the QGEs are independent by the On one side the quantum equation defines the hydrodynamic formalism. evolution of the particle wave function and the associated spatial mass density distribution. On the The CPTR symmetry of the QGEs embodies the CPT other side, the gravity equation defines how the 4-D symmetry of quantum mechanics in a wider one that curvature is generated by the mass distribution and requires that matter and antimatter bend the space in its movement through the associated tensor of opposite way. energy-impulse density. This property implies interesting consequences to the The hydrodynamic approach allows to obtain the quantum-gravitational dynamics. energy-impulse tensor once the wave function of the particle is defined leading to a complete and well First of all, it leads to have a repulsive matter- defined system of differential equations of evolution. antimatter Galilean gravitational field [20,31]. Even if this output is the subject of discordant opinions The biunique correspondence between the standard [15], if confirmed it may bring to the elegant solution quantum mechanics and the hydrodynamic of the problem of the zero-point of quantum energy representation [22,30] warrants that once the tensor density of vacuum, an enormous amount of energy of energy-impulse density is written as a function of that in the gravitational approach would lead to a http://www.ijSciences.com Volume 5 – May 2016 (05) 51

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

very high curvature of the space without matter. 8) Hollands, S., Wald, R., M., „Quantum field in curved space time“, arXiv 1401.2026 [gr-qc] 2014. 9) Wang A., Wands, D., Maartens, R., ““, arXiv 0909.5167 If the antimatter brings a negative curvature (respect [hep-th] 2010. to that of matter) and since the vacuum is represented 10) Witte, E., “Anti De Sitter space and holography”, Adv. by a sea of virtual particles and antiparticles [29] (in Theor. Math. Phys., 2, 253-91, 1998. equal number) the total curvature will be 11) Corda C., “Bohr-like model for black holes” Class. Quantum Grav. 32, 195007 (2015) spontaneously null . 12) Zee, A. (2010). in a Nutshell (2nd Moreover, eq. (35-36) shows the presence of the ed.). Princeton University Press. p. 172. cosmological constant that, due to the quantum 13) F. Finster, J. Kleiner, Causal Systems as a Candidate potential derivatives, is different from zero just in the for a Unified Physical Theory, arXiv:1502.03587v3 [mat-ph] 2015. places where the mass is localized (in quasi-punctual 14) P. Chiarelli, The quantum lowest limit to the black hole mass particles) so that the spatial mean can lead to the right derived by the quantization of Einstein equation, Class. order of magnitude of the cosmologically observed Quantum Grav. ArXiv: 1504.07102 [quant-ph] (2015) values [23]. 15) Cabbolet, M., J., T., F., Elementary Process Theory:a formal axiomatic system with a potential application as a foundational framework for physics underlying gravitational Finally, the matter-antimatter repulsion can lead to repulsion of matter and antimatter, Annalen der Physik, their phase separation generating cosmological 522(10), 699-738 (2010). domains (or even universes) where matter (or 16) Morrison, P., Approximated Nature of Physiocal Symmatries, Am. J. Phys., 26 358-368 (1958). antimatter) prevail [32] allowing the solution of the 17) Schiff, L.,I., Sign of the gravitational mass of positron, Phys. enigma of the abundance of matter. Rev. Let., 1 254-255 (1958) 18) Good, M., L., K20 and the , Phys. Rev. 7. Conclusions 121, 311-313 (1961). 19) Chardin, G. And Rax, J.,M., CP violation: A matter of In this work the quantum gravitational equations are (anti)gravity, Phys. Lett. B 282, 256-262 (1992). obtained for particles interacting by means of the 20) P. Chiarelli, The antimatter gravitational field, ArXiv: electromagnetic force and owning half unit of spin. 15064.08183 [quant-ph] (2015). This is achieved by defining the energy-impulse 21) H. Guvenis, Hydrodynamische Formulierung der relativischen Quantumchanik, The Gen. Sci.. J., 2014. tensor density through the hydrodynamic quantum 22) I. Bialyniki-Birula, M., Cieplak, J., Kaminski, “Theory of formalism. The electro-QGEs show to be invariant Quanta”, Oxford University press, Ny, (1992) 87-111. under the CPT inversion associated to the change of 23) P. Chiarelli, “Theoretical derivation of the cosmological sign of the trace of the Ricci tensor of curvature. constant in the framework of the hydrodynamic model of quantum gravity: the solution of the quantum vacuum catastrophe?” accepted for publication on Galaxies, (2015) References 24) Madelung, E., Quantum Theory in Hydrodynamical Form, Z. Phys. 40, 322-6 (1926). 1) M. Aguilar et al. First Result from the Alpha Magnetic 25) Jánossy, L.: Zum hydrodynamischen Modell der Spectrometer on the International Space Station: Precision Quantenmechanik. Z. Phys. 169, 79 (1962).8. Measurement of the Positron Fraction in Primary Cosmic 26) Landau, L.D. and Lifsits E.M., Course of Theoretical Rays of 0.5–350 GeV, Phys. Rev. Lett. 110, 141102 – Physics, vol.2 p. 309-335, Italian edition, Mir Mosca, Editori Published 3 April 2013 Riuniti 1976. 2) Sahni, V., The cosmological constant problem and 27) Ibid [26] p. 363. , Class. Quant. Grav. 19, 3435, 2002. 28) Dirac, P.,A.,M., A theory of electrons and protons, Pro. A. 3) Patmanabhan, T., Cosmological constant: the weight of the Roy. So. A. A126, 360-5, 1930. vacuum, Phys. Rep. 380, 235, 2003. 4) Copeland, E., Sami, M., Tsujikawa, S., Dynamics of Dark 29) Feynman, R.,P., space-time approach to quantum energy, Int. J. Mod. Phys. D 15, 1753, 2006. electrodynamics, Phys. Rev. B76, 769-89, 1949. 5) Hartle, J., B. and Hawking, S.,W., “The wave function of the 30) Tsekov, R., Bohmian Mechanics Versus Madelung Quantum universe”, Phys. Rev. D 28, 2960, 1983. Hydrodynamics, arXiv:0904.0723v8 [quantum-ph] (2015). 6) Susskind, L., “ and the principle of black hole 31) Villata, M., CPT symmetry and antimatter gravity in general complementarity”, Phys. Rev. Lett. 71 (15): 2367-8, 1993. relativity, Europhysics Lett. , 94,2001. 7) Nicolai H., “Quantum gravity: the view from particle 32) P. Chiarelli, On the cosmological matter abundance versus physics”,arXiv:1301.5481 [gr-qc] 2013. the quantum matter-antimatter symmetry, HSFM J.9 (7) 2015.

http://www.ijSciences.com Volume 5 – May 2016 (05) 52

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

Appendix A

S S0 In this section we calculate and its limit for a generic superposition of states with positive (i.e., matter) t t and negative (antimatter) energy eigenvalues. To this end we use the hydrodynamic expression of the wave-function

iS iSn |  |exp[ ]   |  |exp[ ] (A.1) n n n  where

iSn |  | exp[ ] (A.2) n n  are the eigenfunctions (that for sake of simplicity we suppose with discrete eigenvalues). For systems with time independent Hamiltonian we can write

iS iSn |  | exp[ ]   |  |exp[ ] (A.3) n n n  and, hence,

  iSn iS n  S ln[ ]  ln[n |  n |exp[ ]]  ln[   n |  n |exp[  ]]  . (A.4) 2i * 2 i n n   By using (A.3) we obtain both

iS | |i  S  exp[n ] n n |  |  n nq   q  n p     iE t |  |exp[n ] n n n 

iSn | n |i  S n   nexp[ ]  |  n |  n q   q      cos iE t  |  |exp[ n ] 2 q (A.5.a) n n n  iS S |  |exp[n ] n n n n  q      cos iS  |  |exp[n ] 2 q n n n 

http://www.ijSciences.com Volume 5 – May 2016 (05) 53

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

iSn     n|  n |exp[ ] p n  cos    n  2 q   p   cos  iS  |  |exp[n ] 2 q n n n 

iSn     n|  n |exp[ ]  n eA   cos    n  2 q      cos iS  |  | exp[n ] 2 q n n n  (A.5.b)    E e  cos  iS n  |  |exp[n ] 2 t n n n 2   c     q  eA iS    | |exp[n ] n n n     E e  cos   2 t  2 q  eA  c   

  E e  cos  S   p  eA 2 t q    cos   eA (A.6)   c2  q2  q   and

iSn  n|  n | E n exp[ ]  S n   E    (A.7) t iSn  n|  n |exp[ ]  n  

Vqu  2 ( n ) iB i   E  m c1     e   cos  . (A.8) n mc2 mc 2  2 t  

     VVquB iS t qu  B iS t a | |1( n ) i i  exp[ n ]  b |   |  1  ( n )  i i  exp[  n ]  n nmc2 mc 2   n n  mc 2 mc 2    n   n    S m c2  iS t  iS t  a | |exp[n ] b |  |exp[  n ]  t n n  n n  n n      e  cos   2 t  (A.9) Moreover, given that for the classical limit

http://www.ijSciences.com Volume 5 – May 2016 (05) 54

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

2 iB i  lim0 En   m c1 2   e   n (A.11) mc  it follows that iS t iS t  exp[0 ]  exp[  0 ] n  n   S SB      lim0   m c2 1 i i n n  e   0 2  iS t iS t  (A.12) t  t mc  exp[0 ]  exp[  0 ] n  n    n n   and, hence, for pure matter or antimatter states, that

SB0 2  i i   E0   m c1 2   e  (A.13) t mc  that for a spinless uncharged particle reads

S0 2   m c (A.14) t

Appendix B broaden the solutions so that not all momenta Analysis of the Quantization Condition and solutions of the hydrodynamic equations can be Determination of the Quantum Eigenstates in the solutions of the Schrödinger problem. Quantum Hydrodynamic Description As well described in ref.[12], the state of a particle in If we look at the mathematical manageability of the QHEs is defined by the real functions quantum hydrodynamic equations of quantum | |2  n p  S mechanics (2.a-2.b) no one would consider them. (q, t) and ( q,t ). Nevertheless, the QHEs attract much attention by The restriction of the solutions of the QHEs to those researchers. The motivation resides in the formal ones of the standard quantum problem comes from analogy with the classical mechanics that is additional conditions that must be imposed in order appropriate to study those phenomena connecting the to obtain the quantization of the action. quantum behavior and the classical one. The integrability of the action gradient, in order to In order to establish the hydrodynamic analogy, the have the scalar action function S, is warranted if the gradient of action has to be considered as the probability fluid is irrotational, that being momentum of the particle. When we do that, we

q q   S( q,t )   dl S   dl p (B.1) q0 q0 is warranted by the condition

  p  0 (B.2) so that it holds http://www.ijSciences.com Volume 5 – May 2016 (05) 55

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

 c   dl  m q  0 (B.3)

Moreover, since the action is contained in the exponential argument of the wave function, all the multiples of

2 , with

q S  S  2n  S  dl  p  2n n  0, 1, 2, 3, ... n ( q,t ) 0( q,t ) 0( q0 ,t )  (B.4) q0 are accepted.

Quantum Eigenstates Below, we will show how the problem of finding the quantum eigenstates can be carried out in the hydrodynamic description. Since the method does not change either in classic approach or in the relativistic one, we give here an example in the simple classical case of a classical harmonic oscillator. In the hydrodynamic description, the eigenstates are identified by their property of stationarity that is given by the “equilibrium” condition

 p  0 (B.5.a)

(that happens when the force generated by the quantum potential exactly counterbalances that one stemming from the Hamiltonian potential) with the initial “stationary” condition

 q  0 . (B.5.b) .

 The initial condition (B.5.b) united to the equilibrium condition leads to the stationarity q  0 along all times and, therefore, by (B.5.a) the eigenstates are irrotational. Since the quantum potential changes itself with the state of the system, more than one stationary state (each one with V its own qu n ) is possible and more than one quantized eigenvalues of the energy may exist. p 2 For a time independent Hamiltonian H   V , whose hydrodynamic energy reads 2m ( q ) p 2  [31] E  V V , with eigenstates  n ( q )(for which it holds p  m q  0 ) it follows that 2m ( q ) qu

t t p  p S  dt( V V )  (V  V ) dt  E ( t  t ) (B.6) n  2m ( q ) qu n ( q ) qu n  n 0 t0 t0

V  V ( ) where qu n qu n , and that

V  E V (B.7) qu n n ( q ) where (B.7) is the differential equation, that in the quantum hydrodynamic description, allows to derive to the eigenstates. http://www.ijSciences.com Volume 5 – May 2016 (05) 56

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

m 2 For instance, for a harmonic oscillator (i.e., V  q 2 ) (B.7) reads ( q ) 2 2 2 2  1 m q V  ( )| |    | | E  . (B.8) qu 2m n n n 2

If for (B.8) we search a solution of type

2 | |  A exp  aq (B.9) (q, t) n(q)  ,

m A  H m we obtain that a  and n(q) n( q) (where Hn(x) represents the n-th Hermite polynomial). 2 2 Therefore, the generic n-th eigenstate reads

i  m 2   iE t    | | exp [ S ]  H m exp  q exp n  n( q ) (q, t) (q,t) n( q)   , (B.10)  2  2    

From (B.10) it follows that the quantum potential of the n-th eigenstate reads

2 n  V  ( ) | |    | | qu 2m q q   m   2   Hn1  2( n 1)Hn2   m 2  1   q  n    (B.12) 2  H  2   n       m 2 1   q 2  ( n  ) 2 2 where it has been used the recurrence formula of the Hermite polynomials

m H  qH  2nH , (B.13) n1  n n1 that by (B.7) leads to 1 E  V V  ( n  ) (B.14) n qu n ( q ) 2

The same result comes by the calculation of the eigenvalues that read

http://www.ijSciences.com Volume 5 – May 2016 (05) 57

The CPT-RICCI Scalar Curvature Symmetry in Quantum Electro-Gravity

 E  | H |   * H op dq n n n  (q, t) (q, t) -  2 n  | | H  V dq   (q, t) qu  -  2    2 m m n  n  q  ( q  q )2  V dq  (q, t)  2 2 qu  (B.15) -    2  1 2 m n   n  S  ( q  q )2  V dq  (q, t) 2m ( q ) 2 qu -    m 2 m2 1  1  n  ( q  q )2  ( q  q )2  ( n  )dq  ( n  )  (q, t) 2 2 2 2 -  

2 2 op   n   *  where H   V( q ) and where (q, t) (q, t) (q, t). Moreover, by using (B.6, B.14-B.15) for 2m q 2 eigenstates it follows that

 1 p  ( H  V )  (( n  ) )  0 , (B.16) qu 2

 S( q,t ) q   0 , (B.17) m

Confirming the stationary equilibrium condition of the eigenstates.

Finally, it must be noted that since all the quantum In the relativistic case, the hydrodynamic solutions states are given by the generic linear superposition of   are determined by the eigenstates  n ,  n the eigenstates (owing the irrotational momentum derived by the irrotational stationary equilibrium  field m q  0 ) it follows that all quantum states are condition applied to the momentum fields of matter and antimatter of equation (23), respectively . irrotational. Moreover, since the Schrödinger description is complete, do not exist others quantum irrotational states in the hydrodynamic description.

http://www.ijSciences.com Volume 5 – May 2016 (05) 58