Download The Symmetries of the DFSD Space

The Symmetries of the DFSD Space
Eliahou Tousson
5455 Sylmar Avenue #1501, Sherman Oaks, California 91401, USA
Email: [email protected]
December 2003
Abstract
The nature and the symmetries of the Electrodiscrete interaction’s DFSD space [1] is
further evaluated. Conservation laws and empirical laws of nature, as well as uncertainty
rules, are discussed as the consequences of the various symmetries (or invariances). Nonelementary particles (or systems of particles) and photons are also discussed.
The DFSD Space Model [1] Basic Assumptions (See Figures 1-3)
1. There is a Basic-Generator, like a universal metronome with period TB. Everything
in the universe is synchronized with the Basic-Generator (BG), which can be understood
as a common correlation which has been carried over since the beginning of time (the Big
Bang).
2. Space is four-spatial-dimensional, composed of the familiar 3D space plus one more
spatial dimension (which we do not perceive directly) which is curved (in a fifth
dimension) and closed on itself like a ring (for every "point" of "our" 3D space).
The radius of the ring at any "point" of the "regular" 3D space is determined by the
energy (mass) at that "point". The radius of the ring is inversely proportional to the
energy. The closed-fourth-spatial dimension manifests as our perception of time.
3. Space is discrete, composed of space-segments the size of DB which constitute its
basic elements.
4. The speed of light, c, is defined as c = DB/TB.
5. The 4D building blocks of the Discrete-Four-Spatial Dimensional (DFSD) space,
can be each “turned on” (excited) to the Basic-Energy EB for the duration of TB.
6. Planck’s constant, h, is defined as h/2 = EBTB, where h/2 is the smallest and the
basic unit of action.
7. An elementary particle of any permissible mass, moving at any permissible velocity,
can be described by a single type of entity, the Basic-Entity, that is moving at the speed of
light in the DFSD space. The Basic-Entity is described by the Basic-Energy EB. The
Basic-Entity is formed by the excitation of the 4D building blocks of the DFSD space.
The Basic-Entity propagates by the sequential excitation of the building blocks, thus
propagates around the closed-fourth-spatial dimension at the speed of light.
A particle at rest is described by a closed “ring” of the closed-fourth-spatial dimension.
The particle’s “ring” is composed of N segments, the size of DB each, identifying the
particle’s rest energy (E0=EB/N). When the same particle is moving (in the “regular” 3D
1
space), it is described by a spiraled-string composed of the same number of segments, N,
between any two consequent intersections with the “regular” 3D space. The particle’s
Basic-Entity propagates (spirals around) in a direction such that the curvature remains
constant for any 3D velocity (R0=cT0/2π=NDB/2π - It always completes one turn relative
to the closed-fourth-spatial dimension in N “steps”).
The description of a matter particle in the DFSD space is as follows:
(cT0) = (cT) + r0,
(cT0)2 = (cT)2 + r02,
where a boldface letter denotes a vector and a boldface parenthesis also denotes a single
vector as shown in figures 1-3.
8. The energy of a particle is defined as E = EBTB/T (the rest energy is E0=EBTB/T0).
The particle’s energy is inversely proportional to the radius of the particle’s spiral, or the
radius of the closed-fourth-spatial dimension at the particle’s location.
9. The speed of a particle is defined as v = r0/T0.
10. The electromagnetic-interaction-charge of a particle is a function of EB, and its sign
is determined by the handedness of the particle’s spiral [1].
Note that assumptions 7-10 above give the basic description of a matter particle in the
DFSD space. See reference [1] for more details.
About the Nature of the DFSD Space
As discussed before [1], space and time are the outcome of the Electrodiscrete
interaction. This is how we perceive the interaction. Thus, the DFSD space [1] cannot
really be considered as composed of pre-existing building blocks “waiting” to be excited.
These building blocks are exist only when and where an excitation is taking place.
Excitations propagate in a sequential manner, in such a way that the subsequent one is
induced by the one that is preceding it, and is in “average” time of TB and “average”
distance of DB away from it, in the direction that the particle’s ring or spiraled string [1]
is “formed”.
From our 3D point of view, a particle is a pulse of energy EB that appears for a duration
of TB over a space segment of DB [1]. It appears repeatedly with a time period of T and at
a 3D space distance of r0 in the direction of the particle’s velocity [1]. Then, a particle
appears in our 3D view as a single pulse that propagates at the particle’s velocity by
appearing for a duration of TB, disappearing and reappearing after a time duration of T at
a distance of r0 in the direction of the particle’s velocity.
Only when we look in space (3D) by time, meaning, looking over the “surface” ∆r⋅∆t,
we will be able to “see” a series of pulses, that can be treated as a periodic function
(superposition of harmonic oscillations with frequencies lying in a finite band). This is
implying the existence of another kind of symmetry, in the “surface” ∆r⋅∆t, in addition to
the symmetries in our 3D space, in time and in the 4D of the DFSD space [1].
Those symmetries of the DFSD space and the laws of physics that they imply are the
main subject discussed in this article.
2
Consequences of the Discreteness of the DFSD Space
According to the Electrodiscrete theory [1], a CFSD (continuous four-spatialdimensional) space would have resulted in a universe with no energy (or mass) as we
know it. The mechanism that gives us energy (or mass), as defined in our 3D space and
our time, is described in details in reference [1]. This mechanism is a direct consequence
of the discreteness of the DFSD space. In a CFSD space there would be no energy (no
inertial mass), no momentum and no inertial forces as we know them. There would be no
gravity either.
On the other hand, nature is beautifully described in the DFSD space that is described
above and in reference [1]. Let us consider the followings:
1. In the Electrodiscrete theory’s DFSD space, the particle’s energy is derived directly
from the discreteness [1]:
E = EBTB/T = (h/2)/T.
(1)
2. The momentum of a particle can be derived as follows:
m = E/c2,
(2)
where the inertial mass, m, is defined this way (in the Electrodiscrete theory). Only its
use in empirical laws has given it a distinct character. The momentum is then [1]:
P = mv = Ev/c2 = (h/2)/(Tc2/v) = (h/2)/(c2TT0)/r0) ≡ (h/2)/LPc,
(3)
where v is the velocity of the particle, and it is in the direction of r0.
3. The inertial force description for a particle is as follows:
F = ma = (E/c2)(∆v/T) = (E/c2)(∆r0/T0T) = (h/2)/(T(c2T0T/∆r0))
(4)
≡ (h/2)/(TLFc) = ((h/2)/T)⋅∆(1/LPc) = (1/T) )⋅∆((h/2)/LPc) = ∆P/T,
where a is the acceleration of the particle, and it is in the direction of ∆r0 (as a vector
subtraction of the two r0 involved). The T here is the latter one.
4. The strength of the electromagnetic field is defined by the magnitude of the electric
charge. It is conveniently defined in terms of the dimensionless charge strength:
ke2/(ch/2π) ≅ 1/137.
(5)
Correcting this charge strength as required by the discreteness of the DFSD space reveals
a new charge which includes electromagnetism and gravity (Newton’s gravity) as a
unified picture [1]:
(ke2)NEW ≅ (ke2)(1 - (1/2)(TB/T0)2).
(6)
As for the force (the electrostatic force between two electrons):
F = ∆P/∆t = (h/2)/(∆r∆t),
(7)
ke2/r2 = (h/2)(a1r)-1(a2r/c)-1,
(8)
(ke2)(a1)(a2/c) = h/2,
(9)
where the dimensionless coefficient a1 describes ∆r in terms of the distance r in the
electron’s field and the dimensionless coefficient a2 describes ∆t in terms of the time r/c
which is the time it takes the field to develop or change at the distance r in the electron’s
field. By comparison with equation 5, we have: a1a2 ≅ 430.5 (or 137π). However, these
coefficients can be calculated as follows:
The electrostatic force between two electrons at a distance r apart is (see equation 4):
ke2/r2 = ma = (h/2)/(TLFc).
(10)
Let us choose r = cT, use equations 9 and 10, and compare to equation 4. Then we get:
3
(ke2/c2T2)(a2T)(a1cT) = h/2,
(11)
(cT)
(12)
F(cT)⋅T⋅LFc = F(cT)⋅T⋅(137πcT) = h/2,
where (by comparison to equation 4):
cT0/∆r0(cT) = 137π,
(13)
and the fine-structure constant, then, is:
(14)
α = π∆r0(cT)/(cT0) = ∆r0(cT)/(2R0).
Note that equation 14 reveals the physical meaning of the fine-structure constant.
The Various Symmetries of the DFSD Space
Every symmetry leaves something unchanged. For each continuous symmetry there is a
corresponding conservation law (Noether’s theorem). The classical conservation laws the conservation of momentum, angular momentum and energy - are related to
invariances in our 3D space and our time description of nature.
The consequences of a continuous space (our 3D) and a continuous time dimension (as
related to the closed-fourth-spatial dimension [1] of the DFSD space) are listed in table 1
below. These laws of nature, which are the consequences of continuous space and time,
are only approximations that are valid for large distances, rotation angles and time
periods in comparison to the discreteness level of the DFSD space (as appropriate to our
3D point of view and time sensation).
The corrections to the continuous symmetries driven laws, required by the discrete
reality described by the DFSD space, appear as additional laws of nature. There are two
levels of discreteness to be considered. With each of which are associated different laws
of nature. The “fine” discreteness level is at the basic structure of the DFSD space (DB
and TB). The outcome of it is the basic characteristic behavior of nature that we
experience. The “coarse” discreteness level is related to our limited point of view (3D
space and time sensation) of the DFSD space reality. The outcome of it are the
uncertainty rules.
The consequences of the “fine” and the “coarse” discrete symmetries are listed in table
1 below.
However, there are other symmetries which are associated with the Electrodiscrete
interaction’s DFSD space. Not all of them are listed in table 1. Some are described
below:
1. The invariance (in the DFSD space) of the particle’s ring or spiraled string variable,
cT0 [1]. For the same kind of particles (same rest mass), the particle’s ring or spiraled
string (see figures 1-3) is invariant in its length and invariant in its curvature [1]. This
invariance is a consequence of the structure of the DFSD space as it is representing a
matter particle:
(15)
(cT0) = (cT) + r0,
2
2
2
(16)
(cT0) = (cT) + r0 ,
where a boldface letter denotes a vector and a boldface parenthesis also denotes a single
vector as shown in figures 1-3, and where the length (cT0) is an invariant which is
associated with the rest mass of the particle. This invariance leads to the Lorenz
invariance of nature, as described in reference [1]. It is listed too in table 1 below.
4
2. The existence of four different spirals (indicated by the vector-charge of the
Electrodiscrete interaction [1]) leads to the four Electrodiscrete charges. See reference
[1] for the details.
Inertia Versus Gravity
According to the Electrodiscrete theory, the kinetic energy of a matter particle is a
consequence of the projection of the particle's string on the closed-fourth-spatial
dimension (which is the size of the closed-fourth-spatial dimension at the particle’s 3D
location) [1]. This kinetic energy contributes to the particle’s inertial mass, m. However,
the Electrodiscrete theory does not offer a mechanism for this kinetic energy to have
gravity too. Furthermore, the correction to the electromagnetic force, that satisfies the
discreteness of nature, is calculated [1] to be exactly gravity, with m0 (the rest mass) as
the gravitational mass. A simple conclusion is that the kinetic energy of a matter particle
does not have gravity. This suggests a modified equivalence principle.
As for the gravitational energy itself, the Electrodiscrete theory suggests that this energy
appears in the form of the rest masses of the matter particles. The rest masses of matter
particles were zero at the big-bang. The rest masses (and the associated gravitational
energy) kept growing ever after (as inflation took place) [1].
Note: One way of understanding inertia is possibly by considering the “conservation”
of the “global” center of mass of the universe (in the DFSD space) which is ensured by
Newton’s laws of motion and the classical conservation laws. These laws ensure the
conservation of the local systems’ centers of mass which in turn ensure the
“conservation” of the center of mass of the universe (which is at the “location” of the
Big-Bang).
Non-Elementary particles and Systems of Particles
The energy of a non-elementary particle or a system of particles is the sum of the
energies of its elementary particles members. For each one of the n elementary particles
members that make up the system the energy is (see equation 1):
Ei = (h/2)/Ti,
(17)
thus, the energy of the system is:
(18)
ΣE = E1 + E2 + E3 + ⋅⋅⋅ + En = (h/2)(1/T1 + 1/T2 + 1/T3 + ⋅⋅⋅ + 1/Tn).
Let us define TΣ, such that:
(19)
1/TΣ = (1/T1 + 1/T2 + 1/T3 + ⋅⋅⋅ + 1/Tn),
then, we get:
(20)
ΣE = (h/2)/TΣ,
or:
(21)
(ΣE)TΣ = h/2.
At zero temperature or when all the members of the system have no kinetic energy, we
get:
(22)
ΣE = nE0 = n(h/2)/T0,
5
or:
(23)
(ΣE)(T0/n) = h/2.
When all the members of the system have the same kinetic energy, we get:
(24)
ΣE = nE = n(h/2)/T,
or:
(ΣE)(T/n) = h/2.
(25)
This is practically the case when all the members of the system have the same rest mass,
m0, and have low velocities in comparison with the speed of light, vi<< c.
Note that above zero temperature and when the members of the system do not have the
same velocities, TΣ is “smeared” with a time spread bandwidth as large as the difference
between T0 and the smallest T in the system. Thus, the time spread bandwidth is a
function of the highest kinetic energy of the members of the system.
As for the momentum of the system, a somewhat similar treatment gives the
followings:
Pi = (h/2)/(Tic2/vi),
(26)
for an elementary particle (see equation 3). Then, for the system we have:
(27)
ΣP = (h/2)(1/c2)(v1/T1 + v2/T2 + ⋅⋅⋅ + vn/Tn),
where boldface letters denote vectors. Let us define:
(28)
LPcΣ ≡ c2/(v1/T1 + v2/T2 + ⋅⋅⋅ + vn/Tn),
then, we get:
(ΣP)⋅⋅LPcΣ = h/2.
(29)
For a rigid body (made up of n members and move at the speed of v) we can write:
(30)
(ΣP)⋅LPcΣ = (nPv)⋅(LPcv/n) = Pv⋅LPcv = h/2,
where Pv is each member’s component of the momentum in the direction of v, and LPcv is
a function of v and an average T (see equation 3).
Photons
The description of a photon in the DFSD space model is discussed in reference [1] and
is possibly as illustrated in figure 4, showing:
(31)
(cT0) = (cT) + (r0),
for the first half cycle, and:
(32)
(cT0) = -(cT) + (r0),
for the second half cycle. A boldface letter denotes a vector and a boldface parenthesis
also denotes a single vector as shown in figure 4. The “average” values over one
complete cycle are:
(33)
(cT)Average = 0, (cT0)Average = r0,
as the basic-entity propagates back and forth around the closed-fourth-spatial dimension.
For the (cT0) vector, this “average” means that the average of the vector components in
the direction of r is r0 while the average of the vector components in the direction of (cT)
is zero. In addition, for each one of the halves of the cycle:
(34)
(cT0)2 = (cT)2 + r02.
The energy of a photon is (see reference [1] and figure 4):
6
EP = EBW = EBTB/T = (h/2)/(TP/2) = h/TP,
(35)
where T ≡(cT)/c and where TP is the time-period of the wave. The appearance (in our
regular 3D space) of the basic-energy, in the case of a photon, occurs every half cycle
because of the vibratory nature of the photon. The objectivity factor W = TB/T, is
“responsible” for the particle-like nature of the photon [1]. The energy of the photon is a
consequence of the “switching-on” (appearance in our regular 3D space) of the basicenergy. Its zero-charge is a consequence of the alternating, every half cycle, types of the
spiral (right-handed and left-handed) as the direction of motion of the basic-energy
relative to the closed-fourth-spatial dimension alternates [1].
The spin of the photon, as described in the DFSD space model, is zero in the r direction
because, as described in reference [1], at the speed of light, the spin has no component in
the r direction in contrast to any other direction in the regular 3D space. Hence, a spin
direction can be defined only as forward or backward relative to the direction of motion
and hence, for the alternating spirals of the photon it adds up to zero. However, the spins
of the two components (right-handed and left-handed spirals) of the photon are in the
plane perpendicular to r (in the regular 3D space) and the spin of the photon is interpreted
as spin 1 - as the superposition of the spins of the two components of the photon.
The speed of the photon, as seen in our regular 3D space, is derived as follows (see
figure 4):
(36)
vP = r0/T0 = (cT0)Average/T0 = c,
where T0 ≡(cT0)/c. One way to understand the derivation of equation 36 is through our
perception of time, as associated with the closed-fourth-spatial dimension’s “clocks” [1].
Over one complete cycle, the T “clock” average to zero as (cT)Average = 0. Then, the
photon’s self-clock appears not to be advancing at all. This, in turn, is perceived by us as
a movement at the speed of light, c, for the photon, through the Lorenz invariance of
nature [1]:
T = T0(1-vP2/c2)1/2 = 0 ⇒ vP = c.
(37)
The non-relative nature of the speed of light may be understood through this derivation.
More about the photon (see figure 4). The energy of the photon is (see equation 35
above):
EP = h/TP,
(38)
where the photon’s time period is:
(39)
TP = 2T,
and its wavelength is:
(40)
λP = cTP = 2cT.
But also (see figure 4):
(41)
λP = 2r0,
then, for the photon:
cT = r0,
(42)
and also, by using equation 34:
cT0 = 21/2cT = 21/2r0.
(43)
Note that equation 34 works for both halves of the cycle separately but not for the cycle
as a whole, then, the same is true for equations 43-45. Thus, we get for the photon:
7
cT0 = (21/2/2)λP,
(44)
and:
T0 = (21/2/2)TP.
(45)
The direction of the vector λP is along r, because the path of the basic-entity is along
(cT0) which, over a complete cycle, has non zero component along r only.
Notes
1. The field of force of two interacting particles may possibly be the result of the
“twist” of the DFSD space between and around the interacting particles, which is induced
by the interacting particles. This field is a property of the interaction. It is an “interaction
field” of the two particles (or more), where we may call one of them the source particle
and the other one the test particle. This “twist” of the DFSD space between the
interacting particles may be considered as the mediator of the force between the particles
- the virtual photons if we may.
This “twist” of the DFSD space, which is induced by the interacting particles, is like
“ripples” in space which result in a “wavy” field of force that has wave-like properties
and so can interfere with itself. The wave-like properties of the field of force are
described above, to some extent, and also listed in table 1 (third row from the bottom).
The speed of propagation of the “ripples” (virtual photons) is the speed of light, c, to
account for the speed at which the field of force of the particles (electron-like or positronlike) can be developed or changed.
2. For two interacting particles (electron-like and/or positron-like), there is a mutual
“cT” which is a relative feature of the two interacting particles - this is how one is
“viewed” from the other’s system of reference.
3. In the DFSD space description of nature, the position of the basic-entity around the
closed-fourth-spatial dimension may be considered as the phase of the particle. Then, the
phase velocity of a particle in the DFSD space is the speed of light, c, for any permissible
particle velocity v [1]. The ratio of phase velocity to particle velocity is c/v.
Therefore, for a different description of a particle, where it is composed of partial waves
traveling at the speed of light, their phase velocity must be c2/v in order for the ratio of
phase velocity to particle (group) velocity to still hold: (c2/v)/c = c/v.
Particle, Wave and the Field of Force
Table 1 can tell us even more. The third column from the left displays a particle-like
behavior while the fourth column from the left displays a wave-like behavior.
Furthermore, the field of force could be associated with the second row from the bottom.
As we can see through the symmetries of the DFSD space, the Electrodiscrete theory
description of a particle, its wave behavior and its field of force, reveals a unified picture
which improves our understanding and demonstrates once again the beauty and the
elegance of the Electrodiscrete theory.
8
Table 1 - A Summary
The dimension(s) or
“surface” or variable
of the DFSD space
(associated with
the invariance)
Closed-fourthspatial dimension
(Timedisplacement)
Consequences of
continuous
symmetries
Consequences of
“fine”-discrete
symmetries
Consequences of
“coarse”-discrete
symmetries
Conservation of
energy
Existence of inertial
mass (m = E/c2)
E⋅T = h/2
Our regular 3D:
1. Translation
2. Rotation
Conservation of:
momentum
angular momentum
Existence of:
P = m⋅v
JS = rS × m⋅v
P⋅LPc = h/2
S⋅(2π) = h/2
3D × Closed-4thD
“surface”
Newton’s third law
of motion
4D of the DFSD
Conservation of
electrical charge
Existence of inertial
forces (Newton’s
first and second
laws of motion)
F = m⋅a
Existence of
gravitational mass
(m0 = E0/c2) and
force [1]
The variable (cT0)
of the particle’s ring
or spiraled string in
the DFSD space 3
1
Lorenz
invariance [1]
Existence of
maximum speed for
a matter particle [1]
vmax< c
F⋅T⋅LFc = h/2
(ke2)(π/αc) = h/2
2
Not
applicable
Table 1: A list of the various laws of physics that are associated with the various
symmetries of the DFSD space of the Electrodiscrete interaction. See reference [1] for
more details. Vectors are denoted by boldface letters.
1
Conservation of the Electrodiscrete charge in general [1].
2
The speed of light, c, in this equation should be replaced by the speed [1]:
vmax = c(1(TB/T0)2)1/2,
as required by the discreteness of the DFSD space, to obtain the corrected charge which
includes electromagnetism and gravity (Newton’s gravity) as a unified picture [1].
α is the fine-structure constant.
3
An invariant variable in its length and in its curvature.
9
References
[1] E. Tousson, A New Description of Nature - The Way to Unification,
http://members2.1stnetusa.com/~etousson (1996).
Figure Captions
Figure 1. A description of the closed-fourth-spatial dimension, for: (a) A particle at
rest in our regular 3D space; (b) The same particle moving in the r direction in our
regular 3D space, relative to the system of reference of figure 1(a); (c) The same particle
approaching the speed of light.
Figure 2. A vector description of one unfolded turn of the closed-fourth-spatial
dimension of figure 1(b), describing a moving particle.
Figure 3. A description of the closed-fourth-spatial dimension, for: (a) A particle at
rest in our regular 3D space; (b) The same particle moving in the r direction in our
regular 3D space, relative to the system of reference of figure 3(a).
Figure 4. A vector description of one unfolded turn of the closed-fourth-spatial
dimension, describing a photon.
10
Figure 1
(cT)=(cT0)
(cT)
(cT0)
r0 =0
0
(a)
r
r0
(b)
(cT)→
→(cTB)
0
r
r0→(cT0)
(c)
Figure 2
(cT0)
(cT)
0
r
r0
11
Figure 3
(cT)=(cT0)
(cT)
R0
R
Q
r0 =0
0
(a)
(cT0)
r
r0
(b)
Figure 4
(cT0)
(cT)
(cT0)
r
r0
2r0
-(cT)
12