International Conference on Particle Physics Astrophysics and Quantum Field Theory Showing-up the Extra- Dimensions of Electron Vo Van Thuan Vietnam-Auger

International Conference on Particle PhysicsInternational Conference on Particle Physics Astrophysics and Quantum Field Theory Astrophysics and Quantum Field Theory

Showing-up the Extra-Showing-up the Extra-Dimensions of ElectronDimensions of Electron

Vo Van ThuanVo Van Thuan

Vietnam-Auger Cosmic Ray LaboratoryVietnam-Auger Cosmic Ray LaboratoryInstitute of Nuclear Science and Technology (INST)Institute of Nuclear Science and Technology (INST)

179 Hoang Quoc Viet Street, Nghiado, Hanoi, Vietnam179 Hoang Quoc Viet Street, Nghiado, Hanoi, VietnamE-mail: E-mail: [email protected]@vaec.gov.vn

Singapore, 27Singapore, 27thth-29-29thth November 2008 November 2008

mailto:[email protected]

ContentsContents

1.1. Introduction.Introduction.

2.2. Equation of a free spinning micro-particle.Equation of a free spinning micro-particle.

3.3. Equation of a free electron.Equation of a free electron.

4.4. Semi-classical interpretation of quantum mechanics.Semi-classical interpretation of quantum mechanics.

5.5. Conclusions.Conclusions.

Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008

I. Introduction (1)I. Introduction (1)• For the early time the extra-dimensions (EDs) have been For the early time the extra-dimensions (EDs) have been

proposed by Kaluza-1921 and Klein-1926 [1,2] in their theory proposed by Kaluza-1921 and Klein-1926 [1,2] in their theory where the 5th space axis being adopted as a realistic one, but where the 5th space axis being adopted as a realistic one, but is to be compacted in a tiny cylinder then should be hidden from is to be compacted in a tiny cylinder then should be hidden from observation observation . .

• Klein and Fock-1926 [2,3] were the first to show a technique Klein and Fock-1926 [2,3] were the first to show a technique that the equation of motion of a massive particle in 4D time-that the equation of motion of a massive particle in 4D time-space is able to obtained by reducing the ED of a massless space is able to obtained by reducing the ED of a massless wave equation in a higher dimensional time-spacewave equation in a higher dimensional time-space . .

• The semi-classical approach to the equation of motion is related The semi-classical approach to the equation of motion is related to the fundamental problem of hidden parameters in quantum to the fundamental problem of hidden parameters in quantum mechanics set up by de Broglie-1927, Bohm-1952 et al. [4,5]mechanics set up by de Broglie-1927, Bohm-1952 et al. [4,5] . .

• Bell-1964 [6] shown the way to test the hidden parameters vs Bell-1964 [6] shown the way to test the hidden parameters vs the QM. The experimental evidences of violation of the Bell the QM. The experimental evidences of violation of the Bell inequality by Freedman-1972 et al. [7] have almost closed the inequality by Freedman-1972 et al. [7] have almost closed the models of local hidden parameters.models of local hidden parameters.

However, they do not ban the non-local hidden variablesHowever, they do not ban the non-local hidden variables ..


I. Introduction (2)I. Introduction (2)• Two trends:Two trends:1.1. The high dimensional superstring theory including the internal The high dimensional superstring theory including the internal

dynamical symmetriesdynamical symmetries the EDs are internal, almost abandoned to show- the EDs are internal, almost abandoned to show-up.up.

2.2. Theories with extra-dimensions developing in geometrical Theories with extra-dimensions developing in geometrical dynamics following the Kaluza-Klein formalismdynamics following the Kaluza-Klein formalism the EDs are of both: the EDs are of both: compacted as well as shown-up.compacted as well as shown-up.

• Two main approaches by geometrical dynamics (the 2Two main approaches by geometrical dynamics (the 2ndnd trend) : trend) : - The membrane models,- The membrane models, for example, Randall-1999for example, Randall-1999 [8] [8] - The so-called induced matter models - - The so-called induced matter models - Wesson et al.Wesson et al. [9,10]. [9,10]. In particular (In particular (the induced matter modelsthe induced matter models):):Wesson-1992 [9]Wesson-1992 [9] proposed a space-time-matter model regarding the proposed a space-time-matter model regarding the proper mass as a time extra-dimensionproper mass as a time extra-dimension. . Koch-2008 [11]Koch-2008 [11] proposed a geometrical model with an additional time proposed a geometrical model with an additional time dimension for interpretation of the Klein-Gordon equationdimension for interpretation of the Klein-Gordon equation..

• Present study (Present study (for the induced-matter approachfor the induced-matter approach) is aiming to meet ) is aiming to meet the physical reality based on:the physical reality based on:- - the assumption in VVT-[12] of the space-time symmetrythe assumption in VVT-[12] of the space-time symmetry;;

- - a simplification of the geometrical dynamicsa simplification of the geometrical dynamics.. Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008

II. II. EEquation of a free quation of a free spinning micro-particle spinning micro-particle (1)(1)

• Defining a symmetrical Minkowski {3T,3X} time-space we introduce a Defining a symmetrical Minkowski {3T,3X} time-space we introduce a classical flat wave equation:classical flat wave equation:

• Following Zeldovich-1968 [13] we assume, qualitatively, that the fluctuations of a Following Zeldovich-1968 [13] we assume, qualitatively, that the fluctuations of a physical vacuum as a global effect of the averaged cosmological constant would be physical vacuum as a global effect of the averaged cosmological constant would be a mechanism for rolling the transverse time axes.a mechanism for rolling the transverse time axes.

• Defining the wave (1) polarized circularly along the axis thus its evolution keeping from the past to the future is restricted by a cylindrical condition .

The polar coordinates are used for the 3D-time and the wave (1) now reads :

(1)(1)

(2)(2)

(3) (3)

defines the contribution of spin (3D gyroscope) defines the contribution of spin (3D gyroscope) characterizing an intrinsic transverse motion around the linear axischaracterizing an intrinsic transverse motion around the linear axis



• The equation (2) performs a separation of variables and reduces the 6D time-space The equation (2) performs a separation of variables and reduces the 6D time-space into 5D-manifold which can be reminiscent of the Kaluza-Klein theory. Putting (2) in into 5D-manifold which can be reminiscent of the Kaluza-Klein theory. Putting (2) in (1) we get an equation corresponding to the real part of the equation (1):(1) we get an equation corresponding to the real part of the equation (1):

(4)(4)

(5)(5)

(6)(6)

As is orthogonal to then:

Now the equation (4) reads:

•Equivalent to the traditional Lorentz condition an orthogonal condition is proposed here to compensate contribution of the longitudinal terms in the parentheses of (6) as:

(7)(7)



• The unification (5) of two orthogonal time axes hides the proper time (The unification (5) of two orthogonal time axes hides the proper time (as an independent as an independent variablevariable) and reduces the 5D manifold into a 4D time-space ) and reduces the 5D manifold into a 4D time-space ::

• Assuming: the invariant cylindrical condition in 3D-time (3) to produce the Assuming: the invariant cylindrical condition in 3D-time (3) to produce the proper mass of the material point in 4D time-spaceproper mass of the material point in 4D time-space accounting a contribution of the intrinsic spinning in 3D-space:

(8)(8)

•Here is the mass-scale calibration parameter.

(9)(9)

•The equation (6) with the condition (7) is equivalent to the system of Proca’s The equation (6) with the condition (7) is equivalent to the system of Proca’s quantum relativistic equations of massive vector boson (spin 1) quantum relativistic equations of massive vector boson (spin 1) Thus, this leads to the equation describing micro-particle with a given spin:Thus, this leads to the equation describing micro-particle with a given spin:


III. III. Equation of a free electron (1)Equation of a free electron (1)

• Defining that an individual electron has a spin rotating freely in Defining that an individual electron has a spin rotating freely in 3D-space, adopting only one of the two its projections on a given 3D-space, adopting only one of the two its projections on a given space axis, as its space-polarization. space axis, as its space-polarization. ConsequentlyConsequently::

• To get the Dirac-like equations of electron, we now apply the To get the Dirac-like equations of electron, we now apply the equation (8) containing explicitly the spin-term a long with the equation (8) containing explicitly the spin-term a long with the proper mass proper mass to implement the Dirac factorizationto implement the Dirac factorization..

• Recalling that the wave function of electrons consists of four Recalling that the wave function of electrons consists of four components, equivalent to a 4-spinor: components, equivalent to a 4-spinor:

or two 2-component spinors : which characterize or two 2-component spinors : which characterize

four combinations of: i/ two polarizations in 3D-time four combinations of: i/ two polarizations in 3D-time and and ii/ two opposite orientations of spin in 3D- ii/ two opposite orientations of spin in 3D-

space. space.

• The negative time polarization with defines as an eigenstate The negative time polarization with defines as an eigenstate of positron with positive additive charges and positive energy. of positron with positive additive charges and positive energy. It It means the electric charge serves an indicator for the sign of time means the electric charge serves an indicator for the sign of time polarizationpolarization. .



• The system of spinor equations derived from (8) now reads:The system of spinor equations derived from (8) now reads: (10)(10)

• Where . The spin-term (spin ½) with a fixed polarization Where . The spin-term (spin ½) with a fixed polarization violates the space-parity. In the meantime, the PNC effects have been violates the space-parity. In the meantime, the PNC effects have been observed by polarized (neutron or neutrino) beams only in weak observed by polarized (neutron or neutrino) beams only in weak interaction [14,15] interaction [14,15] thus, we simplify techniques by phenomenology thus, we simplify techniques by phenomenology which makes the spin-term at low energies effectively proportional which makes the spin-term at low energies effectively proportional to the first order of Fermi constant (to the first order of Fermi constant (why this used be why this used be neglectedneglected). ).

• Recalling that Recalling that our PNC experiments-1983 [15] estimated by Sushkov & our PNC experiments-1983 [15] estimated by Sushkov & Flambaum-1982 [16] proved clearly that the PNC contribution without any Flambaum-1982 [16] proved clearly that the PNC contribution without any special enhancement is of a factor of ~10E-7 less than the nuclear force. special enhancement is of a factor of ~10E-7 less than the nuclear force.

• When we establish the differential equation of the second order (8) When we establish the differential equation of the second order (8) from the linear equation (10), the square spin-term as an individual from the linear equation (10), the square spin-term as an individual property should be presented there even there is no polarization in property should be presented there even there is no polarization in the ensemble of electrons. the ensemble of electrons.

• As less as the square spin-term is of the order of As less as the square spin-term is of the order of its contribution in (8) used hardly to separate from the total rest mass. its contribution in (8) used hardly to separate from the total rest mass.



• In a non-relativistic situation, the square differential equation In a non-relativistic situation, the square differential equation (8) turns to a more general Schrödinger equation of a free (8) turns to a more general Schrödinger equation of a free electron which accounts explicitly the square spin contribution:electron which accounts explicitly the square spin contribution:

(11)(11)

• Considering a 3D-space restricted condition of geodesic Considering a 3D-space restricted condition of geodesic deviation, we get the relation:deviation, we get the relation:

(12)(12)

• With the cylindrical symmetry, the Christoffel symbol With the cylindrical symmetry, the Christoffel symbol

which converts the last additional term in the generalized which converts the last additional term in the generalized Schrödinger equation (11) to a similar term to the so-called Schrödinger equation (11) to a similar term to the so-called “quantum potential” in the Bohm theory [5].“quantum potential” in the Bohm theory [5].


IV. A semi-classical interpretation IV. A semi-classical interpretation of quantum mechanics (1)of quantum mechanics (1)

The imaginary part of the wave equation (1) reads:The imaginary part of the wave equation (1) reads:

(13)(13)

where the signs +/- correspond to opposite polarizations in 3D space.where the signs +/- correspond to opposite polarizations in 3D space.For individual polarized electron, the equation (13) leads to:For individual polarized electron, the equation (13) leads to:

(14)(14)

From (14) we get:From (14) we get: (15)(15)

In according to (3) and the orthogonality of and we get: In according to (3) and the orthogonality of and we get: &&

From (15) we introduce a classical From (15) we introduce a classical continuity equation of “a single particle”:continuity equation of “a single particle”:

(16)(16)Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008

IV. A semi-classical interpretation of IV. A semi-classical interpretation of quantum mechanics (2)quantum mechanics (2)

Related to the physical reality of the individuality of micro-particles:Related to the physical reality of the individuality of micro-particles:

• In according to the geometrical dynamics we are able to In according to the geometrical dynamics we are able to predict the objective properties of an individual particle. predict the objective properties of an individual particle. However, However, -Due to the non-locality of the time-like extra-dimension variables which is, in -Due to the non-locality of the time-like extra-dimension variables which is, in principle, varying continuously not only in the micro-objects of a given principle, varying continuously not only in the micro-objects of a given experiment but also in the measuring equipments. experiment but also in the measuring equipments.

-Due to violent fluctuations in the interfering space-time structure of the unified -Due to violent fluctuations in the interfering space-time structure of the unified object-equipment system, object-equipment system,

• We are able to extract the physical reality almost only We are able to extract the physical reality almost only statistically, i.e. by averaging integrally over a coherent ensemble of statistically, i.e. by averaging integrally over a coherent ensemble of micro-particles (micro-particles (playing the role of the objects to be measuredplaying the role of the objects to be measured). ).

It is in consistency with statistical interpretation of quantum It is in consistency with statistical interpretation of quantum mechanics. mechanics.

• The situation would be better when we can measure an isolated particle or can The situation would be better when we can measure an isolated particle or can create some coherent effects in macroscopic scale, where an ensemble of create some coherent effects in macroscopic scale, where an ensemble of identical particles carry out the same physical information as of an individual identical particles carry out the same physical information as of an individual particle. particle.



Related to the extra-dimensions:Related to the extra-dimensions:

• Based on the space-time symmetry with a simplest mode of Based on the space-time symmetry with a simplest mode of time-space structure time-space structure the 4D equations of quantum mechanics the 4D equations of quantum mechanics are naturally derived from the 6D classical equation of a are naturally derived from the 6D classical equation of a polarized wave. polarized wave.

• We proved that the quantum wave function and the proper We proved that the quantum wave function and the proper time time are the two transverse time-like extra-dimensions contributing to are the two transverse time-like extra-dimensions contributing to a full time-space evolution of a micro-particle. a full time-space evolution of a micro-particle.

• Being not hidden, but acting explicitly in the quantum mechanics Being not hidden, but acting explicitly in the quantum mechanics those EDs are not restricted in a hidden compacted volume, but those EDs are not restricted in a hidden compacted volume, but should be shown up in an acceptable time-space scale of micro-should be shown up in an acceptable time-space scale of micro-particle (particle (for electron, it is a distance of the Compton wave lengthfor electron, it is a distance of the Compton wave length). ).

• Moreover, we could extract an invariant term related the micro-Moreover, we could extract an invariant term related the micro-space structure which defines the contribution of intrinsic space structure which defines the contribution of intrinsic angular momentum (spin) to the rest mass of a micro-particle. angular momentum (spin) to the rest mass of a micro-particle.



Related to the particle-wave duality:Related to the particle-wave duality:

• On one side, the given effects of reduction (2) and (3) On one side, the given effects of reduction (2) and (3) define simultaneously, the localizing quantization of the define simultaneously, the localizing quantization of the 6D time-space evolution of a micro-particle as a material 6D time-space evolution of a micro-particle as a material point. point.

• On another side, they show up a non-local 4D time-space On another side, they show up a non-local 4D time-space structure of the wave nature of the same micro-particle. structure of the wave nature of the same micro-particle.

As a result, the proposed geometrical dynamics As a result, the proposed geometrical dynamics being in being in consistency with the equivalent principle of general relativityconsistency with the equivalent principle of general relativity approaches to the origin of the wave-particle duality of approaches to the origin of the wave-particle duality of quantum mechanics.quantum mechanics.


V. V. ConclusionsConclusions

• The formalism of the Kaluza-Klein theory has been extended to the The formalism of the Kaluza-Klein theory has been extended to the {3T,3X} symmetrical Minkowski time-space {3T,3X} symmetrical Minkowski time-space and thenand then,,

• The Klein-Fock procedure has been applied The Klein-Fock procedure has been applied to reduce a 6D time-polarized wave equationto reduce a 6D time-polarized wave equation to an equation of Klein- to an equation of Klein-Gordon-Fock type describing the motion of a spinning particle in 4D Gordon-Fock type describing the motion of a spinning particle in 4D time-space. time-space. Such an equation being proved more general is able to lead to particular Such an equation being proved more general is able to lead to particular equations of scalar and vector bosons as well as of electron.equations of scalar and vector bosons as well as of electron.

• It was found that the quantum wave function and the proper time It was found that the quantum wave function and the proper time are two “transverse” time extra-dimensions contributing to a full are two “transverse” time extra-dimensions contributing to a full space-time realistic evolution of a micro-particle. space-time realistic evolution of a micro-particle. They are not restricted in a hidden compacted volume, but should be shown They are not restricted in a hidden compacted volume, but should be shown up in an acceptable time-space scale of a nominal micro-particle.up in an acceptable time-space scale of a nominal micro-particle.

• The fundamental problems of quantum mechanics (The fundamental problems of quantum mechanics (the physical reality the physical reality of of an individual micro-particle, the wave-particle dualityan individual micro-particle, the wave-particle duality etc.) have been etc.) have been treated effectively by the geometrical dynamics. treated effectively by the geometrical dynamics.

• In particular, it was shown that the “quantum potential” of a micro-particle, In particular, it was shown that the “quantum potential” of a micro-particle, such as of an electron, originates from spinning of its non-local space-time such as of an electron, originates from spinning of its non-local space-time structure.structure.


ReferencesReferences

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Thank you for your attention!Thank you for your attention!

AcknowledgementAcknowledgement: :

I would like to thank my colleagues in the Vietnam-Auger I would like to thank my colleagues in the Vietnam-Auger cosmic ray laboratory (INST) for their cooperation. cosmic ray laboratory (INST) for their cooperation.

This research has been done under financial support of This research has been done under financial support of the Council of the National Program the Council of the National Program

for Research in Natural Sciences, for Research in Natural Sciences, Ministry of Science and Technology of Vietnam (MOST).Ministry of Science and Technology of Vietnam (MOST).


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International Conference on Particle Physics Astrophysics and Quantum Field Theory Showing-up the Extra- Dimensions of Electron Vo Van Thuan Vietnam-Auger