NeuroQuantology | March 2014 | Volume 12 | Issue 1 | Page 35-39 35 Glinka LA., Do global virtual axionic gravitons exist?
Do Global Virtual Axionic Gravitons Exist?
Lukasz Andrzej Glinka ABSTRACT Recently, the author has proposed the global one-dimensionality conjecture which, throughout making use of the method of the primary canonical quantization of General Relativity according to the Dirac-Arnowitt-Deser-Misner Hamiltonian formulation, formally avoids the functional differential nature which is the basic obstacle of quantum geometrodynamics, a model of quantum gravity based on the Wheeler-DeWitt equation and resutling from quantization of the Einstein field equations. The emergent quantum gravity model is straightforwardly integrable in terms of the Riemann-Lebesgue integral, has an unambiguous physical interpretation within quantum theory, and introduces a consistent concept of global graviton. In this article, the axionic nature of the global gravitons, that is the hypothetical particles responsible for gravitation in the global one-dimensional quantum gravity model, is shortly described and, making use of fundamental methods of quantum field theory, the virtual states of such gravitons are generated. Key Words: global one-dimensionality conjecture; quantum gravity; quantum geometrodynamics; graviton; axion DOI Number: 10.14704/nq.2014.12.1.716 NeuroQuantology 2014; 1: 35-39
1. Introduction1 results in replacement of the functional Recently the Wheeler-DeWitt equation was evolution parameter which is the metric of considered in the framework of the global one- space by a variable which is the metric dimensionality conjecture (Glinka, 2012; 2010). determinant. The ramifications of this change Recall that this conjecture formally reduces the are highly non-trivial. Namely, the resulting functional differential nature of the Wheeler- model of quantum gravity significantly differs DeWitt quantum geometrodynamics from the Wheeler-DeWitt equation, because has throughout the primary canonical quantization a manifestly well-defined mathematical of General Relativity in the Dirac-Arnowitt- consistency defined in terms of classical Deser-Misner Hamiltonian formulation integrals and, therefore, is straightforwardly (Glinka, 2012; 2010). integrable. Moreover, the model has both a Let us sketch the state of affairs. The clear and unambiguous physical interpretation reduction follows from a simple and well- within the formalism given by quantum field known Jacobi's formula connecting the theory and, consequently, establishes a new variations of a metric and its determinant, and concept of graviton. This “global graviton” arises from a certain specific global optimization of the Wheeler-DeWitt quantum Corresponding author: Lukasz Andrzej Glinka geometrodynamics and, for this reason, its Address: Hattorfer Platz 8, 36269 Philippsthal (Werra), Deutschland physical existence is much more fundamental Phone: +49-15143497665 e-mail [email protected] than the quanta of gravitation defined by the Relevant conflicts of interest/financial disclosures: The authors standard quantum geometrodynamics based on declare that the research was conducted in the absence of any the Wheeler-DeWitt equation, as well as more commercial or financial relationships that could be construed as a potential conflict of interest. likely than the gravitons understood in terms of Received: December 2, 2013; Revised: December 9, 2013; the Yang-Mills theories, which constitute the Accepted: February 12, 2014. leading reasoning line of modern particle eISSN 1303-5150 www.neuroquantology.com
NeuroQuantology | March 2014 | Volume 12 | Issue 1 | Page 35-39 36 Glinka LA., Do global virtual axionic gravitons exist? physics (Cf. relevant references discussed in
Glinka 2012; 2010). 2 i NNNN i j In this short article, the axionic nature g = . (3) N h of the global graviton is consistently described i ij throughout the global one-dimensional version of the quantum geometrodynamics and, making use of certain fundamental methods of The model (2) makes sense if and only if for the quantum field theory, the existence of the non- dimension of space D is different from 0 and 2. trivial states of such global gravitons, which we Let us notice that, in general, axions can call the virtual states of global gravitons, are be understood as “axial” particles, that is the deduced. particle-like solutions to an arbitrary one- dimensional quantum evolution equation. The 2. Global Quantum Gravity single dimension is then considered as a global Let us focus our attention on the global one- affine parameter of the quantum evolution dimensional quantum gravity, recently which plays the role of either time variable or investigated in (Glinka 2012; 2010), in the single space dimension. Interestingly, such version appropriate for any Lorentzian higher- property gives automatically a supersymmetric dimensional space-time which is strictly nature to axions. The quantum theory (1) is analogous to the case of 3+1 dimensional space- manifestly a one-dimensional quantum time. Making use of certain standard classical in mechanics, where the evolution parameter is conventions, one can show that in such a the determinant h of a metric of space. situation the quantum gravity is described by According to the investigations (Glinka 2010; the one-dimensional Klein-Gordon equation: 2012), this parameter is the global dimension of the model, and, consequently, the gravitons understood as the particle-like solutions to the 2 evolution equation (1), have axionic nature. The 2 2 = 0, (1) one-dimensional nature gives naturally h inherited supersymmetry in such a model of quantum gravity, what means that the global gravitons are new kind of supersymmetric 2 where is the following parameter axionic particles. In the context of quantum field theory, ()D which is the mathematical basis of particle 2 R 2 2 1 = 2 2 , (2) physics, the model of quantum gravity (1) is the 2 D ( D 2) h Klein-Gordon equation. Its solutions have the physical sense of particles if and only if the gravitational frequency is constant, that is ()D where R is the Ricci scalar curvature of the = const . In such a situation, a geometry of the D-dimensional space embedded in the D+1 D -dimensional space is determined through space-time, is the cosmological constant, the following Ricci scalar curvature = T n n is the energy density of Matter fields created by the double projection of the (D )R= 2 2 2 2 D 2 ( D 2) 2 h , (4) stress-energy tensor of Matter fields T onto the unit normal vector n , in which the corrections due to the frequency effects can be treated as small in the light of the 8G 2 86 2 = fact that 10 N . In such a situation, the c4 particle solutions to the quantum gravity (1) can be constructed by making use of the Fourier transforms and the secondary quantization in is the Einstein constant, and h= det hij is the the Fock space. determinant of a metric of space hij defined by the D+1 decomposition of a space-time metric eISSN 1303-5150 www.neuroquantology.com
NeuroQuantology | March 2014 | Volume 12 | Issue 1 | Page 35-39 37 Glinka LA., Do global virtual axionic gravitons exist? 3. Axionic Graviton and, therefore, the Heisenberg-type equations The particle-like solution of the quantum theory of motion hold (1) can be constructed straightforwardly in terms of the annihilation and creation operators † and which constitute the appropriate =i , (14) Fock space h † =i† , (15) † h (h ), ( h ) = 1, (5)
††(h ), ( h ) = 0, (6) and the initial data satisfy the relations (h ), ( h ) = 0, (7) † 0, 0 = 1, (16) or, in other words, by the secondary †† , = 0, (17) quantization of the wave function 0 0 ()()h † h , = 0. (18) (h ) = . (8) 0 0 2 Consequently, the graviton which is the axion This is the global one-dimensional quantum has the form gravity in the case of constant , which is the 1 i()() h h0 i h h 0 † quantum field theory, that is the quantum (h ) = e0 e 0 , (19) 2 theory of the global graviton. It is easy to see that in the case of a constant value of the quantum theory (1) has and, for this reason, the relations hold the unique general solution in the form of a † (h ) = ( h ), (20) pattern quantum superposition i()() h h i h h sin (h h ) (h ) = e0 A e 0 B , (9) (h ), † ( h ) = i , (21)
(h ), † ( h ) = 0. (22) where A and B are the constants of integration, and h0 is a certain initial value of the global dimension. Writing One sees also that the conjugate momentum field
()h i()() h h i h h A =0 , (10) 0 0 † (h ) = = i e0 e 0 (23) 2 h 2 † B =0 , (11) =i ( h ) † ( h ) , (24) 2 2
† has the following properties where 0 and 0 are the initial values of the creation and annihilation operators, one obtains the dynamical Fock basis which spans ( (h ))† = ( h ), (25) the Hilbert space in the model of quantum gravity (h ), ( h ) = i cos ( h h ), (26)
† (h ), ( h ) = i sin ( h h ). (27) i() h h0 (h ) = e 0 , (12)
††(h ) = ei() h h0 , (13) 0 eISSN 1303-5150 www.neuroquantology.com
NeuroQuantology | March 2014 | Volume 12 | Issue 1 | Page 35-39 38 Glinka LA., Do global virtual axionic gravitons exist? 0 4. Virtual States i() h h b (h ) = i e 0 . (37) The equation (21) can be used for generation of 3,4 the quadratic equations In this manner, this is clear that a ()h and 1,2 b ()h are independently existing quantum (ei()() h h0 )2 2 ( h ), † ( h ) e i h h 0 1 = 0, (28) 3,4 0 states of the same graviton. Moreover, one can i()() h h2 † i h h easily see that 0 0 (e ) 2 ( h ), ( h0 ) e 1= 0, (29) a a † sin (h h ) 1(h ),( 1 ( h )) = i , (38) which can be solved immediately sin (h h ) a(h ),( a ( h ))† = i , (39) 2 2 2 ei() h h0 = ( h ), † ( h ) 1 2 ( h ), † ( h ) , b(h ),( b ( h ))† = 0, (40) 0 0 3 3 (30) b b † 4(h ),( 4 ( h )) = 0, (41)
and, consequently, the only state a ()h agrees 2 1 ei() h h0 = (),() h † h 1 2 (),(), h † h 0 0 with the commutation relation (21), while the a b b (31) remaining states 2 ()h , 3 ()h , 4 ()h do not satisfy this relation. This suggests that the a b b quantum states 2 ()h , 3 ()h , 4 ()h of the and, consequently, the graviton solution (23) a possesses four nonequivalent quantum states graviton are virtual, while the state 1 ()h is which by making use of the definitions physical.
5. Summary 01 † The global one-dimensional quantum gravity =0 0 = (h 0 ), (32) 2 (Glinka 2012; 2010), in itself is a consistent quantum theory of gravitons defined as the 0 † axionic particles carrying gravitation in a global =i0 0 = ( h 0 ), (33) 2 way. On the background of modern models of quantum gravity, such like the attempts offered by gauge field theories, loop quantum gravity, can be written in the following form string theory, and other much more advanced mathematical physics for presently technologically available experimental a 0 † 1,2(h ) = i ( h ), ( h 0 ) verification (Cf. discussion in Glinka 2012; 2 (34) 2010), the existence of global gravitons is a 1 2 (h ), † ( h ) , 0 0 unique potentially verifiable physical situation. In this paper, constructive application of
the fundamental methods of quantum field b 0 † theory, such celebrated in particle physics, 3,4(h ) = i [ ( h ), ( h 0 ) straightworwardly demonstrated that the 2 (35) 1 2 † unique hypothetical features of the global 1 (h ), ( h0 ) ]. gravitons create an inherent and coherent scenario for existence of their virtual quantum states. The virtual gravitons could be Now, by taking into account the formula (21), responsible for the specific particle-like one can present these states in the following gravitational interactions rather than form measurable inertial effects analogous to the 0 astronomical phenomena which give the a 0 1,2(h ) = cos ( h h 0 ) sin ( h h 0 ), (36) support to validity of the Mach principle on the macroscopic scales. eISSN 1303-5150 www.neuroquantology.com
NeuroQuantology | March 2014 | Volume 12 | Issue 1 | Page 35-39 39 Glinka LA., Do global virtual axionic gravitons exist? However, this question needs further detection, and description of the microscopic clarification due to both the phenomenology gravitational waves, possibly at high or even and experiments of the joined best effort of ultra-high energetic regions. observational astronomy, high energy physics, particle astrophysics, and space science. References Glinka LA. Aethereal Multiverse: A New Unifying Nevertheless, looking from the present-day Theoretical Approach to Cosmology, Particle Physics, theoretical point of view, the model reasoning and Quantum Gravity. Cambridge International presented in this paper allows to make use of Science Publishing 2012. the hypothetically existing virtual axionic Glinka LA. Global One-Dimensionality Conjecture within Quantum General Relativity. Grav Cosmol 2010; 16(1): particle-like global gravitons in order to search, 7-15.
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