Download Showing-up the Extra-Dimensions of Electron

International Conference on Particle Physics
Astrophysics and Quantum Field Theory
Showing-up the ExtraDimensions of Electron
Vo Van Thuan
Vietnam-Auger Cosmic Ray Laboratory
Institute of Nuclear Science and Technology (INST)
179 Hoang Quoc Viet Street, Nghiado, Hanoi, Vietnam
E-mail: [email protected]
Singapore, 27th-29th November 2008
Contents
1. Introduction.
2. Equation of a free spinning micro-particle.
3. Equation of a free electron.
4. Semi-classical interpretation of quantum mechanics.
5. Conclusions.
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
I. Introduction (1)
•
For the early time the extra-dimensions (EDs) have been proposed by
Kaluza-1921 and Klein-1926 [1,2] in their theory 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 observation .
•
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-space is able to
obtained by reducing the ED of a massless wave equation in a higher
dimensional time-space .
•
The semi-classical approach to the equation of motion is related to the
fundamental problem of hidden parameters in quantum 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 the QM.
The experimental evidences of violation of the Bell inequality by
Freedman-1972 et al. [7] have almost closed the models of local hidden
parameters.
 However, they do not ban the non-local hidden variables .
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
I. Introduction (2)
•
1.
2.
•
Two trends:
The high dimensional superstring theory including the internal dynamical
symmetries the EDs are internal, almost abandoned to show-up.
Theories with extra-dimensions developing in geometrical dynamics following
the Kaluza-Klein formalism the EDs are of both: compacted as well as shown-up.
Two main approaches by geometrical dynamics (the 2nd trend) :
- The membrane models, for example, Randall-1999 [8]
- The so-called induced matter models - Wesson et al. [9,10].
In particular (the induced matter models):
Wesson-1992 [9] proposed a space-time-matter model regarding the proper
mass as a time extra-dimension.
Koch-2008 [11] proposed a geometrical model with an additional time
dimension for interpretation of the Klein-Gordon equation.
•
Present study (for the induced-matter approach) is aiming to meet the
physical reality based on:
-
the assumption in VVT-[12] of the space-time symmetry;
-
a simplification of the geometrical dynamics.
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
II. Equation of a free
spinning micro-particle (1)
• Defining a symmetrical Minkowski {3T,3X} time-space we introduce a
classical flat wave equation:
(1)
• 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
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 :
(2)
(3)
defines the contribution of spin (3D gyroscope)
characterizing an intrinsic transverse motion around the linear axis
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
II. Equation of a free
spinning micro-particle (2)
• 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
(1) we get an equation corresponding to the real part of the equation (1):
(4)
As
is orthogonal to
then:
Now the equation (4) reads:
(5)
(6)
•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)
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
II. Equation of a free
spinning micro-particle (3)
• Assuming: the invariant cylindrical condition in 3D-time (3) to produce the
proper mass of the material point in 4D time-space accounting a contribution of
the intrinsic spinning in 3D-space:
•Here
is the mass-scale
calibration parameter.
•The equation (6) with the condition (7) is equivalent to the system of Proca’s
quantum relativistic equations of massive vector boson (spin 1)
Thus, this leads to the equation describing micro-particle with a given spin:
(8)
•
The unification (5) of two orthogonal time axes hides the proper time (as an
independent variable) and reduces the 5D manifold into a 4D time-space :
(9)
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
III. Equation of a free electron (1)
•
•
•
Defining that an individual electron has a spin rotating freely in 3D-space,
adopting only one of the two its projections on a given space axis, as its
space-polarization. Consequently:
To get the Dirac-like equations of electron, we now apply the equation
(8) containing explicitly the spin-term a long with the proper mass to
implement the Dirac factorization.
Recalling that the wave function
of electrons consists of four
components, equivalent to a 4-spinor:
or two 2-component spinors :
which characterize
four combinations of: i/ two polarizations in 3D-time
and
ii/ two opposite orientations of spin in 3D-space.
•
The negative time polarization with
defines as an eigenstate of
positron with positive additive charges and positive energy. It means the
electric charge serves an indicator for the sign of time polarization.
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
III. Equation of a free electron (2)
•
The system of spinor equations derived from (8) now reads:
(10)
•
Where
. The spin-term (spin ½) with a fixed polarization
violates the space-parity. In the meantime, the PNC effects have been
observed by polarized (neutron or neutrino) beams only in weak
interaction [14,15]  thus, we simplify techniques by phenomenology
 which makes the spin-term at low energies effectively proportional to
the first order of Fermi constant
(why this used be neglected).
•
Recalling that our PNC experiments-1983 [15] estimated by Sushkov &
•
When we establish the differential equation of the second order (8) from
the linear equation (10), the square spin-term as an individual property
should be presented there even there is no polarization in the ensemble
of electrons.
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.
•
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.
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
III. Equation of a free electron (3)
•
In a non-relativistic situation, the square differential equation (8) turns to
a more general Schrödinger equation of a free electron which accounts
explicitly the square spin contribution:
(11)
•
Considering a 3D-space restricted condition of geodesic deviation, we get
the relation:
(12)
•
With the cylindrical symmetry, the Christoffel symbol
 which converts the last additional term in the generalized Schrödinger
equation (11) to a similar term to the so-called “quantum potential” in
the Bohm theory [5].
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
IV. A semi-classical interpretation of
quantum mechanics (1)
The imaginary part of the wave equation (1) reads:
(13)
where the signs +/- correspond to opposite polarizations in 3D space.
For individual polarized electron, the equation (13) leads to:
(14)
From (14) we get:
In according to (3) and the orthogonality of
&
From (15) we introduce a classical
continuity equation of “a single particle”:
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
(15)
and
we get:
(16)
IV. A semi-classical interpretation of
quantum mechanics (2)
Related to the physical reality of the individuality of micro-particles:
•
In according to the geometrical dynamics we are able to predict the
objective properties of an individual particle.
However,
-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 experiment
but also in the measuring equipments.
-Due to violent fluctuations in the interfering space-time structure of the unified
object-equipment system,
•
 We are able to extract the physical reality almost only statistically,
i.e. by averaging integrally over a coherent ensemble of micro-particles
(playing the role of the objects to be measured).
It is in consistency with statistical interpretation of quantum mechanics.
•
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 identical
particles carry out the same physical information as of an individual particle.
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
IV. A semi-classical interpretation of
quantum mechanics (3)
Related to the extra-dimensions:
•
Based on the space-time symmetry with a simplest mode of timespace structure  the 4D equations of quantum mechanics are
naturally derived from the 6D classical equation of a polarized wave.
•
We proved that the quantum wave function and the proper time
are the two transverse time-like extra-dimensions contributing to a full
time-space evolution of a micro-particle.
•
Being not hidden, but acting explicitly in the quantum mechanics
those EDs are not restricted in a hidden compacted volume, but
should be shown up in an acceptable time-space scale of microparticle (for electron, it is a distance of the Compton wave length).
•
Moreover, we could extract an invariant term related the micro-space
structure which defines the contribution of intrinsic angular
momentum (spin) to the rest mass of a micro-particle.
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
IV. A semi-classical interpretation of
quantum mechanics (4)
Related to the particle-wave duality:
•
On one side, the given effects of reduction (2) and (3) define
simultaneously, the localizing quantization of the 6D time-space
evolution of a micro-particle as a material point.
•
On another side, they show up a non-local 4D time-space
structure of the wave nature of the same micro-particle.
 As a result, the proposed geometrical dynamics being in
consistency with the equivalent principle of general relativity
approaches to the origin of the wave-particle duality of quantum
mechanics.
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
V. Conclusions
•
•
The formalism of the Kaluza-Klein theory has been extended to the
{3T,3X} symmetrical Minkowski time-space and then,
The Klein-Fock procedure has been applied
to reduce a 6D time-polarized wave equation to an equation of Klein-GordonFock type describing the motion of a spinning particle in 4D time-space.
 Such an equation being proved more general is able to lead to particular
equations of scalar and vector bosons as well as of electron.
•
It was found that the quantum wave function and the proper time are
two “transverse” time extra-dimensions contributing to a full space-time
realistic evolution of a micro-particle.
 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.
•
The fundamental problems of quantum mechanics (the physical reality of
an individual micro-particle, the wave-particle duality etc.) have been treated
effectively by the geometrical dynamics.
•
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 structure.
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
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Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008
Acknowledgement:
I would like to thank my colleagues in the Vietnam-Auger cosmic
ray laboratory (INST) for their cooperation.
This research has been done under financial support of
the Council of the National Program
for Research in Natural Sciences,
Ministry of Science and Technology of Vietnam (MOST).
Thank you for your attention!
Vo Van Thuan- PAQFT08, Singapore, 29 Nov. 2008