Can gravitons be effectively massive due to their zitterbewegung motion? Elias Koorambas
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Elias Koorambas. Can gravitons be effectively massive due to their zitterbewegung motion?. 2014. hal-00990723
HAL Id: hal-00990723 https://hal.archives-ouvertes.fr/hal-00990723 Preprint submitted on 14 May 2014
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E.Koorambas
8A Chatzikosta, 11521 Ampelokipi, Athens, Greece
E-mail:[email protected]
Submission Date: May 14, 2014
Abstract: Following up on an earlier, De Broglie-Bohm approach within the framework of quantum gauge theory of gravity, and based on the Schrödinger-Dirac equation for gravitons, we argue that gravitons are effectively massive due to their localized circulatory motion. This motion is analogous to the proposed zitterbewegung (ZB) motion of electrons.
PACS numbers: 04.60.-m, 11.15.-q, 03.65.Ta Keywords: Quantum Gravity, Gauge Field, Bohm Potential, Zitterbewegung Model
1. Introduction Following, D. Hestenes, the idea that the The ZB of photons is studied via the electron spin and magnetic moment are momentum vector of the electromagnetic generated by a localized circulatory motion of field. These studies show that ZB motion can the electron has been proposed independently occur only in the presence of virtual by many physicists [1]. Schrödinger’s longitudinal and scalar photons [28-30]. The zitterbewegung (ZB) model for such motion vector property of this motion is described by is especially noteworthy because it is the polarization vectors of the grounded in an analysis of solutions to the electromagnetic field [28-30]. Dirac equation [2-4]. Surely, if the ZB motion is a real physical phenomenon, then it tells us Various workers have attempted to derive something fundamental about the nature of General Relativity from a gauge-like the electron. The role ascribed to the ZB principle, involving invariance of physics motion in standard formulations of quantum under transformations of the locally (i.e. in mechanics, nonetheless, has been the tangent space at each point) acting metaphorical at best. Lorentz or Poincare group. ([5-7]). N. Wu [8−11] proposed a Quantum Gauge Theory of This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.
Can gravitons be effectively massive?
Gravity (QGTG) based on the gravitational gravitational gauge transformation, Cμ(x) gauge group (G). In Wu’s theory, the transforms as: gravitational interaction is considered as a fundamental interaction in a flat Minkowski 1 C()()()()() x C x Uˆˆ x C x U x space-time and not as space-time geometry. A (5) model of interacting massive gauge gravitons, i 1 Uˆˆ()(()), x U x and a proposed heavy gauge graviton g resulting from shell decay of Higgs bosons, whereas D transforms covariantly as: were developed recently by the author within μ D()()()()(). x D x Uˆˆ x D x U1 x (6) the framework of QGTG [12-14]. In a recent paper, we also proposed the leading order approximation, a De Broglie-Bohm approach Gravitational gauge field Cμ(x) can be within the framework of QGTG [15]. expanded in the form of linear combinations of generators of gravitational gauge group, Based on the Schrödinger-Dirac equation for gravitons, in this essay we argue that C()(), x C x Pˆ (7) gravitons are effectively massive due to a where C is the component field of the motion analogous to the ZB motion. gravitational gauge field. 2. Fundamentals of QGTG C resembles a According to N. Wu’s theory, the Although component field infinitesimal transformations of the second-rank tensor, this is not a tensor field. gravitational gauge group G can be written in The index is not an ordinary Lorentz index the form [8]: but a gauge group index. Since gravitational gauge field C has only one Lorentz index, U1, i P 0,1, 2,3, (1) it is a kind of vector field.
The strength of the gravitational gauge field where are the infinitesimal parameters of is defined by the second-order Lorenz tensor: the group, and P i/ x are the generators of the gauge group. 1 FDD ,, (8) ig It is known that these generators commute each other [8]: or:
[PP , ] 0. (2) F C()() x C x (9) igC()()()(), xC x igC xC x This property of the generators, however, does not imply that the gravitational gauge F is a vector in group space; therefore, it group is an Abelian group, since the elements of the gravitational group do not commute can be expanded in group space as: [8]: F()(). x F x Pˆ (10) [UU , ] 0. (3) 22 The explicit form of the strength of the The gravitational gauge-covariant derivative component field: FCC()() is defined by: (11) gC()() C gC C D igC() x , (4) . The strength of the gravitational gauge field where Cμ(x) is the gravitational gauge field, and g is the gravitational gauge coupling transforms covariantly under gravitational constant. C (x) is a Lorentz vector. Under gauge transformation. In analogy with μ traditional gauge field theory, the kinematical
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Can gravitons be effectively massive? term for the gravitational gauge field can be 00 g10 written as: () CC . (19) 1 0 gC() g F F . (12) 1 0 ()C 4
It can be demonstrated that this Lagrangian is Suppose that the gravitational gauge field not invariant under gravitational gauge C is very weak in vacuum, i.e. gC ≈ 0. In transformation. It transforms covariantly as follows: leading order approximation, by substituting equation (14) to equations (19), we obtain:
().Uˆ (13) 0 0 0 000 F 0. ()CC To resume the gravitational gauge symmetry (20) of the action, we introduce an essential factor in the form of: The gravitons’ equations of motion thus become: IC() gC ee1 , I(). C g C 1 (14) F 0 0. (21) The full Lagrangian is then given by: We define: e IC() , (15) 0 . FB, FE. ij ijk k 0ii (22) The action S for the gravitational gauge field is defined by: Equation (21) then takes the form:
4 S d x . (16) B 0, (23) EB 0. (24) It can be proven that this action has local t gravitational gauge symmetry [8]. According to the gauge principle, global symmetry gives Taking definitions (22) into account, we out a conserved current: derive the following equations:
E 0, (25) IC() 0 Ti e C 0 . (17) 1 ()C BE 0. (26) t
We call quantity T inertial energy- i But for their superscript α, equations (23-26) momentum tensor [8]. would be the ordinary Maxwell equations. In The Euler-Lagrange equations for C gauge conventional quantum field theory the gravitational field in vacuum is extremely fields are: weak. The gravitational wave in vacuum, . (18) therefore, is composed of four independent ()CC vector waves.
Although the gravitational gauge field is a These forms are identical with those that α occur in quantum field theory [8]. By vector field, its component fields C μ have one Lorentz index and one group index . inserting equation (15) into (18), we get: μ α Both indexes have the same behavior under Lorentz transformation – a behavior that
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Can gravitons be effectively massive? makes the gravitational gauge field to Relation (28) can be regarded as a constraint resemble a tensor field. We thus call this on the field ψα. Relation (29) can take another gravitational gauge field a pseudo-tensor form by introducing the following set of field. The spin of the gravitational gauge hermitian matrices: field, determined by its behavior under Lorentz transformation, is 2. 0 0 0 00i 00i , 0 0 0 , In conventional quantum field theory, a spin- 1 2 00i i 00 1 field is a vector field, and a vector field is a spin-1 field. In QGTG, this correspondence is 00i violated. This is because, unlike in i 00 (30) conventional gauge field theory where the 3 0 0 0 spin of a field is independent of the group . index, in QGTG the group index contributes to the spin of a field. These matrices satisfy the commutation relation: 3. Can gravitons be effectively massive due to a motion analogous to ZB motion? [,]. (31) i j ijk k In a recent paper, we proposed a De Broglie- Bohm approach to the QGTG to explain the Equation (29) can then be written as: propagation of gravitons [15]. iH , (32) Alternatively, the propagation of gravitons t α α can be described by the quantity E −iB [15], α α where E and B are the gravitational electric or, carrying off the components, and the gravitational magnetic field [17-27], i k respectively. iHij j () k ij j [15]. (33) t This description is formulated in the same way in which the Schrödinger wave function Equation (33) has the form of a Schrödinger- iS / Dirac equation. Re describes the motion of material particles in de Broglie-Bohm theory [21-22]. Following J.P. Vigierr [28], the graviton’s velocity operator in the Heisenberg picture is With this choice, the physical meaning of the defined by: graviton wave function is acquired ab initio. Indeed, there are impressive similarities dU 1 between the present formulation of QGTG ()[,]i U H (34) and de Broglie-Bohm theory [15]. dt .
We consider the complex valued quantity: Kobe [28] has shown that, for U(t), equation (34) yields the value:
1 ()E iB [15]. (27) U( t ) U (0) U (0) 2 (35) α U(0)cos t U (0)sin t Introducing ψ in the field equations of , vacuum graviton (23-26), these become: where pc / is the angular frequency of 0 , (28) the corresponding classical gravitational wave in the leading order approximation of i . (29) QGTG. t The Heisenberg equation for the displacement operator:
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Can gravitons be effectively massive?
circle of radius c / . The corresponding dx orbital momentum is and its speed on this ()[,]()i x H U t (36) dt circle of radius c / is the distance 2 (c / ) , travelled in one period divided by yields by integration: T 2/. This trembling motion of the x( t ) x (0) V (0) t graviton reduces the mass of a particle to the t (37) frequency of this motion. dtexp{ it H } U (0) 0 . The longitudinal component of the velocity The first term on the right hand side of associated with a constant energy E pc is equation (37) is the initial position of the constant. This longitudinal component is graviton. Since the graviton moves with a given by: constant longitudinal velocity operator, 22 U (0) in the direction of its constant , U|| (0) p [ p pU (0)] i pHE . (40) momentum, pˆ , the second term of equation (37) is the subsequent displacement of the Equation (40) is the corresponding graviton graviton. The third term thus yields the time operator in the direction of the momentum, dependence of the displacement Xt() due to i.e. c2 pH 1 c 2 pH E 2 corresponding to g g g , the graviton’s motion (analogous to the ZB the gravitons Dirac energy motion). After integration of equation (37) we 1 derive: 2 2 2 E pc m c (41) gg 1 . x() t x (0) U (0) U (0) sin( t ) 1 (38) Ut(0)1 cos( )
- a formula tied with a constant displacement 4. Conclusion 1 x(0) x (0) U . The last two terms of Adopting a De Broglie-Bohm approach in (38) evidently imply a spatial extension xt() Quantum Gauge Theory of Gravity QGTG, resulting from a motion analogous to the ZB and based on the Schrödinger-Dirac equation motion. for gravitons, we find that gravitons are effectively massive due to their localized The displacement operator xx(0) (0) circulatory motion which reduces the mass of the particle to the frequency of this motion. precesses about the displacement U() t t with This motion is analogous to the ZB motion of an angular frequency associated to an , electrons. amplitude c / . This suggests that the orbital angular momentum of the ZB motion corresponds to the graviton’s spin operator S, associated with an effective ‘relativistic graviton , m E// c22 c mass’ 0 . The References graviton’s spin operator S is defined by:
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Can gravitons be effectively massive?
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