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Abstract

A plausible cause of curled up dimensions is proposed. The solid angle

rotation behaves like that in curled up dimensions, it actually is an

unrecognized aspect of the external spacetime. While it is not derived

from strings, it may possess string properties. There is an inherent

solid angle SO(6) associated with the Lorentz spacetime. (SO(10) with

a 4+1 spacetime.) Since these dimensions are intimately associated

with external spacetime, it answers why the same curled up dimensions

exist everywhere in the universe. This could be another revolution of

string theory, as it may establish a link between strings and the

external spacetime and escalates observations to the electroweak scale.

I. Introduction

Every semi-simple Lie algebra, say so(3), has a Casimir invariant, I =

J12 + J22 + J32 = constant. The form of this equation suggests a

brand new SO(3) symmetry. Discussion is made for the properties of

this new rotation (which may be called solid, or 3-d, angle rotation),

the reason it is overlooked and why it should account for the cause of

particle spectrum (e.g. iso-spin, strangeness, etc.). As it behaves

like strings, it could be the cause of the curled up dimensions. But

it escalates observations to electroweak scale and is economically

embedded in Lorentz (or 4+1) spacetime. It also explains parity

violation naturally. Requirements of simplicity, obviousness and

universality of the ultimate building blocks also point to these

symmetries associated with the external spacetime. Likewise, there

should be 4-d (and 5-d for 4+1 spacetime) angle rotations (without

which Poincare group is incomplete).

II. Mathematical Inevitability Of Higher Dimensional Rotation Symmetry

When internal symmetry was explored, it was believed external

symmetries were completely exhausted. What is unexpected is there may

be a series of overlooked external symmetries yet to be discovered.

Weâ€™ll start from some quick mathematical investigations, then look

into the physics behind. Letâ€™s start from a 3-space where a length

is invariant under any rotation,

L^2 = x1^2 + x2^2 + x3^2 (2.1)

An SO(3) symmetry results with infinitesimal rotation operators

J1 = x2(âˆ‚/âˆ‚x3) â€“ x3(âˆ‚/âˆ‚x2)

(2.2a)

J2 = x3(âˆ‚/âˆ‚x1) â€“ x1(âˆ‚/âˆ‚x3)

(2.2b)

J3 = x1(âˆ‚/âˆ‚x2) - x2(âˆ‚/âˆ‚x1)

(2.2c)

As well known every semi-simple Lie algebra has a Casimir invariant

I = Î£ gÎ¼Î½ J^Î¼ J^Î½ = constant (2.3)

The Casimir invariant for the so(3) of 3-space is its angular momentum

I = Î£ gÎ¼Î½ JÎ¼ JÎ½ = J1^2 + J2^2 + J3^2 = J^2= constant (2.4)

Equation (2.4) has the same form as equation (2.1) and hence should

generate a new SO(3) symmetry. It is the main topic of this paper to

examine the properties of this rotation (which may be called solid

angle rotation, as to be explained later), the reason itâ€™s overlooked

and why it should account for the cause of particle spectrum, e.g.

iso-spin, strangeness, etc.

Upon realization of the new SO(3) symmetry, the infinitesimal rotation

operators would be written as

W1 = J2(âˆ‚/âˆ‚J3) â€“ J3(âˆ‚/âˆ‚J2) (2.5a)

W2 = J3(âˆ‚/âˆ‚J1) â€“ J1(âˆ‚/âˆ‚J3) (2.5b)

W3 = J1(âˆ‚/âˆ‚J2) â€“ J2(âˆ‚/âˆ‚J1) (2.5c)

The eigenvalues of the infinitesimal operators J are the angular

momenta jij

J1 Ï† = [y(âˆ‚/âˆ‚z) - z(âˆ‚/âˆ‚y) ] exp[-i(tpt - xpx â€“ ypy â€“

zpz)] = i(ypz â€“ zpy )Ï† = i jyz Ï† (2.6a)

J2 Ï† = [z(âˆ‚/âˆ‚x) - x(âˆ‚/âˆ‚z) ] exp[-i(tpt - xpx â€“ ypy â€“

zpz)] = i(zpx â€“ xpz )Ï† = i jzx Ï† (2.6b)

J3 Ï† = [x(âˆ‚/âˆ‚y) - y(âˆ‚/âˆ‚x) ] exp[-i(tpt - xpx â€“ ypy â€“

zpz)] = i(xpy â€“ ypx )Ï† = i jxy Ï† (2.6c)

where the wave function

Ï† = exp[-i(pt t - px x â€“ py y â€“ pz z) ]

(2.7)

is the solution to the wave equation

[ âˆ‚^2/(âˆ‚t)^2 - âˆ‚^2/(âˆ‚x)^2 - âˆ‚^2/(âˆ‚y)^2 - âˆ‚^2/(âˆ‚z)^2 â€“

m^2 ] Ï† = 0 (2.8)

In the same manner, the eigenvalues of the infinitesimal operators W

would be

W1 Î» = [Î¸zx (âˆ‚/âˆ‚Î¸xy ) â€“ Î¸xy (âˆ‚/âˆ‚Î¸zx ) ] exp[-i(jxy Î¸xy

+ jyz Î¸yz + jzx Î¸zx)]

= -i(Î¸zx jxy â€“ Î¸xy jzx ) Î» = i Î©zx,xy Î»

(2.9a)

W2 Î» = [Î¸xy (âˆ‚/âˆ‚Î¸yz ) - Î¸yz (âˆ‚/âˆ‚Î¸xy ) ] exp[-i(jxy Î¸xy +

jyz Î¸yz + jzx Î¸zx)]

= -i(Î¸xy jyz â€“ Î¸yz jxy ) Î» = i Î©xy,yz Î»

(2.9b)

W3 Î» = [Î¸yz (âˆ‚/âˆ‚Î¸zx ) â€“ Î¸zx (âˆ‚/âˆ‚Î¸yz ) ] exp[-i(jxy Î¸xy

+ jyz Î¸yz + jzx Î¸zx)]

= -i(Î¸yz jzx â€“ Î¸zx jyz ) Î» = i Î©yz,zx Î»

(2.9c)

where the wave function

Î» = exp[-i(jxy Î¸xy + jyz Î¸yz + jzx Î¸zx)]

(2.10)

is the solution to the quantized wave equation of the Casimir

invariants (2.4)

[ âˆ‚^2/(âˆ‚Î¸yz)^2 + âˆ‚^2/(âˆ‚Î¸zx)^2 + âˆ‚^2/(âˆ‚Î¸xy)^2 â€“ I^2 ]

Î» = 0 (2.11)

Equation (2.11) has plane angles Î¸ij as its coordinates with angular

momenta jij [in (2.10)] as its corresponding momenta. The eigenvalues

Î©ij,jk [in (2.9)] are solid angular momenta and the rotations Wi [in

(2.5)] solid angle rotation. In other words, these equations treat

plane angle scales as linear scales. For these to be valid all that is

needed is the establishment of equivalency between the plane angle

scales so that a rotation (or reshuffling of the 3 Jâ€™s in eq. (2.4))

would not alter the total value of the Casimir invariant I.

III. Physics Defining The Equivalency Between Plane Angle Scales

Notice when one writes down the length invariant (2.1), what is not

explicitly spelled out is that the linear scales x1 , x2 and x3 cannot

be arbitrarily defined, but should be the spatial components of Lorentz

scales (or something properly defined). An arbitrarily defined scales

cannot ensure equivalency between the three linear scales x1 , x2 and

x3 , thus a linear rod cannot be measured invariant after a rotation,

light wonâ€™t be measured at the same speed in different directions and

the SO(3) group cannot form. In other words, the validity of eq. (2.1)

and the associated SO(3) is not unconditional but is based on the

unsaid condition that the 3 linear scales be defined by the real

physics of electromagnetism.

For the same reason, the validity of the Casimir invariant, eq. (2.4),

also is not unconditional but is based on an unsaid condition.

Obviously, (2.4) cannot be valid for any arbitrarily defined plane

angle scales. Then what is that condition? Or, what is the physical

interaction based on which equivalency of plane angle scales for J1 ,

J2 , J3 are defined? The interaction must exist in the form of (2.5),

i.e. rotating between planes (like magnetic fields, FÎ¼Î½ = AÎ¼

âˆ‚/âˆ‚xÎ½ - AÎ½ âˆ‚/âˆ‚xÎ¼ , rotating between lines), to make the 3

plane angle scales equivalent. We shall call this kind of rotation

solid angle rotation (to be explained later). Note that solid angle

rotation is not limited to semi-simple Lie algebras but should exist

between any pair of planes which have equivalent plane angle scales.

While that interaction is not identified, we know it exists because we

know (2.4) is valid and equivalency of plane angle scales exists, and

consequently the new SO(3) symmetry also exists in Nature. It is

conjectured that this required interaction is just the classical

version of weak interaction and the new SO(3) symmetry is related to

iso-spin.

IV. Solid Angle Rotation And Its Definition Through Plane Angle

Decomposition

The reason we name the rotation between planes solid angle rotation is

because conventional concept of solid angle is like a cone; its

rotation is the shrinkage (or expansion) of the cone from a plane to a

needle then back to the â€œsameâ€ plane. Though it does not rotate to

a different plane, it is a rotation from plane to plane. Thatâ€™s why

we borrow the term solid angle rotation for the rotation between

planes. However, one is free to call it 3-d rotation, or anything

he/she likes. We shall call it solid angle rotation in this paper.

There is an inherent impossibility of conserving both a finite plane

angle arc and a linear vector length under solid angle rotation. It

will be shown that this imperfection is reflected truthfully in

observations. We shall define solid angle scale in such a way as to

preserve only the plane angle arc in order to allow consistent

comparison of plane angle scales on different planes (just like plane

angle rotation preserving the length of a vector allows comparison of

linear scales on different axes). Such kind of rotation does not, and

is not intended to, preserve vector lengths. Nor is it intended to be

represented and visualized in â€œCartesian coordinatesâ€. The

rotation can be thought of as a cone that does not shrink/expand but

remains always as a plane-cone rotating from one (say x1-x2) plane to

another (say x2-x3) plane and a solid angle rotation must exist between

every pair of planes in the spacetime.

Below defines solid angle by means of plane angle decomposition (into

plane components). Such definition allows its rotation to leave

invariant a plane angle arc (and hence angular momentum) in exact

analogy to plane angle rotation leaving invariant the length of a

vector. Letâ€™s first express a line element in terms of spherical

angles

d = d1 e1 + d2 e2 + d3 e3

= |d|sinÏˆ cosÎ¸ e1 +|d|sinÏˆ sinÎ¸ e2 +|d|cosÏˆe3 (4.1)

where the spherical angles are defined as

Ïˆ â‰¡ tan-1 [d2^2+ d1^2]^Â½/d3 (4.2a)

Î¸ â‰¡ tan-1 (d2/d1) (4.2b)

The total length

|d| = [(|d|sin Ïˆï€ cos Î¸ï€ )^2 + (|d|sin Ïˆï€ sin Î¸ï€ )^2 +

(|d|cos Ïˆï€ )^2 ]^Â½ = |d| (4.3)

is independent, hence invariant under rotation of the spherical angles

Î¸ï€ and Ïˆ. SO(3) symmetry arises naturally from this invariance.

In the same way, by treating angular momentum as a 3-vector, we can

decompose an angular momentum into 3 components

J = |J|sin Ïˆï€ cos Î¸ï€ e1 +|J|sin Ïˆï€ sin Î¸ï€ e2 + |J|cos

Ïˆï€ e3 (4.4)

Obviously, if this decomposition can be done to angular momentum, it

can also be done to any finite plane angle Î±,

Î± = Î±1 e1 + Î±2 e2 + Î±3 e3

= |Î±| sin Ïˆï€ cos Î¸ï€ e1 +|Î±|sin Ïˆï€ sin Î¸ï€ e2 +|

Î±ï€ |cos Ïˆï€ e3 (4.5)

Nevertheless, since Î± is actually not a 1-dimensional vector but an

angle on a 2-dimensional plane, we would like to treat it exactly as an

angle and consider (4.5) as the decomposition of a plane angle into 3

2-dimensional â€œplaneâ€ components, rather than into 3 â€œvectorâ€

components. Thus, we rewrite (4.5) in terms of 3 plane components,

Î±ï€ = Î±ï€ 23 Î¾23 + Î±ï€ 31 Î¾31 + Î±ï€ 12 Î¾12 (4.6)

where Î¾â€™s are unit angles on each component plane. We then define

solid angles, Ï‰1 and Ï‰2, in terms of the plane angle components in

exact analogy to spherical angles defined in terms of line components:

Ï‰1 â‰¡ tan^-1 [Î±31^2 + Î±23^2]^Â½/ Î±12 (4.7a)

Ï‰2 â‰¡ tan^-1 (Î±31/ Î±23) (4.7b)

Through solid angles Ï‰1 and Ï‰2, the finite plane angle Î± on an

arbitrary plane can be decomposed into 3 plane components as

Î± = Î±23 Î¾23 + Î±31 Î¾31 + Î±12 Î¾12

= | Î±ï€ |sin Ï‰1 cos Ï‰2 Î¾23 + | Î±ï€ |sin Ï‰1 sin Ï‰2 Î¾31 + |

Î±ï€ |cos Ï‰1 Î¾ 12 (4.8)

The total plane angle

| Î± | = [Î±23^2 + Î±31^2 + Î±12^2]^Â½

= [(|Î±|sin Ï‰1 cos Ï‰2)^2 + (|Î±|sin Ï‰1 sin Ï‰2)^2 + (|Î±|cos

Ï‰1)^2 ]^Â½ = | Î±ï€ | (4.9)

is independent of, thus invariant under arbitrary rotation of, solid

angles Ï‰1 and Ï‰2.

Though (4.8) is similar to (4.5), their meanings are quite different.

Eq. (4.5) is the decomposition of a vector into 3 â€œlinearâ€

components and rotation of plane angles Î¸ï€ and Ïˆï€ preserves the

length of the â€œvectorâ€. But (4.8) is the decomposition of a plane

angle into 3 2-d â€œplane angle componentsâ€ and rotation of solid

angles Ï‰1 and Ï‰2 (which shuffles plane angle components Î±23,Î±31

and Î±12) preserves the â€œtotal plane angleâ€. If they were for a

4-dimensional space, (4.5) would cause an SO(4) symmetry, but (4.8) an

SO(6). That they both cause the same SO(3) is incidental in

3-dimensional space, which also hints at the two SO(3)s, one for spin

and one for iso-spin. For Lorentz spacetime, there should be an SO(6)

solid angle rotation symmetry (or its isomorphism) and for 4+1

spacetime [1] an SO(10) (or its isomorphism).

Thus, an extended polar coordinates should work like this: a point in a

3-space can be identified by first identifying the plane on which it

resides in terms of solid angles Ï‰1 and Ï‰2 , then the line on the

plane by plane angle Î¸ and lastly its position r on the line.

V. Solid Angle Rotation As Cause Of Particle Spectrum

Probably because of certain incorrect understanding, solid angle

rotation was taken erroneously as â€œfree standingâ€ internal symmetry

totally unrelated to external spacetime, when it should actually be

external (or at least closely related to external spacetime).

Currently, only symmetries under linear displacement (displacement of a

0-d point) and plane angle rotation (displacement of a 1-d line) are

recognized. I.e. only linear and angular momenta are recognized.

However, a little sense of mathematics would dictate that solid angle

rotation (or, displacement of a 2-d plane) is also an inherent part of

the external spacetime and hence solid angular momentum should

contribute equally to particle symmetries. There is no need to rush to

the mysterious free standing internal symmetry unless solid angle

rotation is proven prohibited. The only possibility that it might be

forbidden (which is also the reason this new symmetry is overlooked) is

that solid angle rotation may not preserve the length of a vector, e.g.

linear momentum (even though it preserves a finite plane angle). But,

this actually is not a problem because we also overlooked the fact that

â€œonly angular momentum, but not linear momentum, is concerned in

particle classificationsâ€. On the other hand, in particle (weak)

interactions where linear momentum must be conserved, solid angular

momentum (i.e. the suspected iso-spin, strangeness, etc.) rightly fails

to conserve. This shows observations agree exactly with mathematical

imperfection.

The fact that solid angle rotation leaves total plane angular momentum

invariant may have misled us to conclude that particle spectrum is

â€œindependentâ€ of external spacetime and hence invented the (free

standing) internal symmetry (which may be caused by the curled up

dimensions under the string model). But not only the origin of the

internal space is mysterious, it also cannot explain P-, C- and

CP-violations. The virtue of solid angle rotation is that, â€œwhile it

preserves total plane angular momentum it is external and shuffles the

plane components of the plane angular momentum, thus causing

parity-violationâ€. Another utterly important virtue of the rotation

being external will be exemplified later.

VI. String Behavior, Extended Polar Coordinates And 4- And 5-d Angle

Rotations

In Lorentz spacetime, there are 6 planes and hence a solid (3-d)

angle rotation symmetry of 6-dimensional space. In the 4+1 spacetime

which is more reasonable and natural [1], there are 10 planes, thus

that of 10-space.

Since what on each plane is â€œnot a pointâ€ but a â€œcirculatingâ€

quantized wave of certain angular momentum, it would behave like a

string. It is therefore conjectured that the curled up dimensions of

string theory may actually be the plane angle scales of the solid angle

rotation. In other words, the strings are circulating quantum

mechanical waves confined to the 6 or 10 planes of the Lorentz or 4+1

spacetime. This view is more plausible than plain strings because:

1. It escalates the 10 dimensions of strings to observable

â€œelectroweakâ€ scales.

2. It is highly economical as the 10 dimensions are embedded in a 4+1

spacetime.

3. It reduces the complexity of strings drastically.

Extended polar coordinates

Just as plane (2-surface) can be decomposed into (or represented by)

plane components, arbitrary 3-surface can be represented as a summation

of 3-surface components and arbitrary 4-surface summation of 4-surface

components. Thus, as if it were an extended polar coordinates, a point

in a 4+1 spacetime can be identified by: first identifying the

4-surface the point is on (in terms of its 4-surface components), then

the 3-surface in the 4-surface, then the plane (2-surface) in the

3-surface, then the line (1-surface) on the plane (2-surface), lastly

the position of the point on the line. From symmetryâ€™s point of

view, this would be the more proper way to identify a point than

through Cartesian coordinates. Thus, particle spectrum is but a

representation of the full symmetries of the â€œexternalâ€ 4+1

spacetime, in the same way photon is to the Lorentz spacetime. Here is

a similarity to the M-theory. The complete wave function of a particle

would be of form:

Î¨ï€ = âˆ‘ E Ã— D Ã— C Ã— B Ã— A (6.1)

where:

A. = exp[-iÏ€(p0x0 -p1x1 - p2x2 - p3x3 - pmxm)] representing linear

(1-dimensional) momentum, including energy and mass. xm is the extra

dimension [1] and pm = mc.

B. A spinor representing plane (2-dimensional) angular momentum.

C. A solid angle spinor representing solid (3-d) angular momentum.

Solid angle rotation runs from one plane (2-brane) to another (among

the 10 planes) while preserving plane angular momentum. Symmetry of

solid angle rotation is suspected to be those of iso-spin, strangeness,

charm, etc. The interaction through solid angle rotation is believed

to be weak interaction.

D. A 4-d rotation spinor representing 4-d angular momentum. 4-d

rotation runs from one 3-plane (3-brane) to another (among the 10

3-planes) while preserving solid angular momentum. This symmetry

probably generates KL and KS, the mixtures of K0 and anti-K0 mesons.

The interactions may be the CP-violation interactions.

E. A 5-d rotation spinor representing 5-d angular momentum. 5-d

rotation runs from one 4-d plane (4-brane) to another among the 5 4-d

planes while preserving 4-d angular momentum. Fields in 5-d rotations

may be causing the strong interactions. The symmetry of 4-d angular

momentum might be the color symmetry which exists but cannot be

observed in isolation.

This shows the full symmetry property of the external 4+1 spacetime

is very rich indeed, which is enough to cover all particles (including

hadrons, leptons and photons altogether). At the same time, weak,

strong, and CP-violation interactions are but analog of

electromagnetism in solid and higher-dimensional angle rotations, while

gravitation is the interaction corresponding to the linear symmetry,

according to the 4+1 vector gravitation theory [1]. Actually, without

the addition of solid (3-d) angle, 4-d and 5-d angle rotations,

Poincare group (or the symmetries of 4+1 spacetime) would not be

complete.

VII. Simplicity, Obviousness And Universality Requirements Of Ultimate

Theory Point To Solid Angle And Higher Symmetries

In the April 10, 2000 issue of the Time magazine, one of the founders

of the standard model, Professor Steven Weinberg, prescribed the

criteria for the ultimate theory, â€œ... [it] has to be simple - not

necessarily a few short equations, but equations that are based on a

simple physical principle ... it has to give us the feeling that it

could scarcely be different from what it is... More and more is being

explained by fewer and fewer fundamental principles... no further

simplification would be possible.â€ Unfortunately, the current

standard model is not as simple and obvious as desired. (I.e. the real

ultimate theory seems yet to be discovered.) Equally important, the

ultimate theory should answer the ultimate questions below:

1. Why the ultimate building blocks behave the way they do, not by

lower level constituents, but by â€œitselfâ€.

2. Why it is this but not other set of building blocks which is chosen

as the ultimate building blocks of Nature.

3. What ensures the same building blocks be created identically

everywhere in the universe.

In the past, atoms were able to explain the existence and properties

of molecules, and protons, neutrons and electrons the existence and

properties of atoms, but none were able to explain their own existence

and properties. Neither could they explain why they are created

identically universally, e.g. an electron one billion light years away

be created identically as one nearby. A common â€œprincipleâ€(rather

than a new fundamental building blocks) which rules â€œthroughout the

universe simultaneouslyâ€ must exist to ensure all building blocks be

created identically at such a distance.

Electromagnetism as a model

Unlike the standard model, electromagnetism has reached such a simple

and obvious level as prescribed by Weinberg, and its quanta, photon,

answers all the ultimate questions perfectly. (It appears obviousness

and simplicity go hand in hand with the 3 ultimate questions). Observe

there are 2 Maxwell equations when expressed in 3+1 Lorentz spacetime.

The first is essentially equivalent to a definition of electric and

magnetic fields. The only real equation of motion is the second which

simply demands conservation of the fields defined by the first equation

(i.e. it doesnâ€™t say much either, as what else can it be if the

fields donâ€™t conserve?) It is really â€œsimple and obviousâ€ (i.e.

â€œcan scarcely be different from what it isâ€). Photon emerges from

quantization of electromagnetic field, which on the other hand serves

to define the Lorentz spacetime. Photons, electromagnetism and Lorentz

spacetime are intimately tied to each other as if they were other sides

of the same 3-sided coin. Symmetries of photon is just symmetries of

the external spacetime. â€œNo other choice would be possibleâ€, as no

symmetry properties of Lorentz spacetime is not represented in photon.

It exists by itself â€œwithout lower level constituentsâ€. Andas long

as the local spacetime is Lorentzian, photons are created

â€œidentically anywhere in the universeâ€. Not surprisingly, the

first half of 20th century witnessed a flourishing era for physics as

culminated by the extremely accurate verification of quantum

electrodynamics (QED).

It makes sense to emphasize that electromagnetism being simple and

obvious is â€œnotâ€ because we have chosen the right quanta, photon,

but because we have chosen the right (Lorentz) spacetime. Imagine if

Lorentz spacetime were not discovered, electromagnetism would appear as

mysterious as strong and weak forces. Even photon would be complex and

considered as associated with â€œinternal symmetryâ€, as the

symmetries of the external (Newtonian) space and time do not match that

of photonâ€™s. But as soon as Lorentz spacetime is used, the theory

changes immediately from mysterious and complex to obvious and simple.

Similar dramatic change also happened when Ptolemy planetary model was

changed to Copernican. Complexity and mysteriousness mixed with

certain plausibility are typical symptoms of physics expressed in

â€œwrongâ€ spacetime, which seem to be shared by the standard model.

In other words, whatâ€™s needed in simplifying strong/weak theory is

not a change of building blocks but a refinement of spacetime.

Mimicking electromagnetism

In this respect, it is insightful to point out that Lorentz

spacetime is defined by nothing but electromagnetism itself. Yet, the

only thing standard model did not mimic electromagnetism is that strong

and weak interactions are not expressed in an (external) spacetime

geometry defined by the interactions themselves. All contemporary

theories are constructed to fit the already-defined Lorentz spacetime

(i.e. to fit straightly the data measured under Lorentz scales), while

whatâ€™s needed may actually be a â€œspacetime geometry that isdefined

to fitâ€ the interactions, just like Lorentz spacetime was defined to

fit electromagnetism.

If such a spacetime can be found, then complexity and mysteriousness

may turn into simplicity and obviousness, while particles, interactions

and the (external) geometry would form an intimately related 3-sided

coin like photons, electromagnetism and Lorentz spacetime.

Consequently, symmetries of all particles would coincide with that of

the â€œexternalâ€ spacetime and hence answer all the 3 ultimate

questions in the same way photon does. Actually, it seems that an

(external) spacetime defined by strong/weak interactions is the

â€œonlyâ€ answer to the 3 ultimate questions, because the onlything

that exists â€œthroughout the universe simultaneouslyâ€ seems to be

the external spacetime itself, and it appears there is no way â€œa

priori building blocksâ€ is able to answer its own properties without

referring to one more level of sub-constituents. As explained earlier,

assigning interactions in solid angle and higher dimensional angle

rotations (of the external spacetime) to strong/weak interactions

readily fits all the above prescriptions perfectly.

Under this view, the reality of the ultimate building blocks of Nature

is not any undestructible hard-cored object, but a (non-dissipative)

wave packet of certain 5-d, 4-d, 3-d (solid) angular momentum, plane

angular momentum and linear momentum, which is essentially the same as

a photon, except phone has only plane angular momentum and linear

momentum. This meets Weinbergâ€™s criteria of all being based on a

simple physical principle as well. Under this view, particle (a wave

packet) is more like an illusion than a real object, as it is but the

envelop of the superposition of mono waves.

VIII. Conclusion and Discussion

As inherent parts of spacetime geometry, a complete Poincare group

should include the symmetries from linear displacement, plane angle,

solid angle and 4-d rotations (and 5-d rotations for 4+1 spacetime).

Associated with each of them may be gravitation, electromagnetic, weak,

CP-violation and strong interactions, respectively. The reason we

never thought about solid angle rotation and beyond is because we

always â€œassumedâ€ the equivalence between plane angle scalesand

hence the need for solid angle rotation to ensure their equivalence

disappeared. This works fine with electromagnetism (and gravitation)

because EM concerns only with the equivalence between linear scales

(which requires plane angle rotation), but not that between plane angle

scales (which requires solid angle rotation). When weak force came up,

we were simply surprised at the existence ofthe spectrum of numerous

particles without a bit clue that they could also originate from the

symmetries of the (external) spacetime just like the 2 photon states.

This shows an â€œassumptionâ€ in the definition of spacetime geometry

may lock the door to a new geometric aspect. Until these fundamental

geometric aspects are exhausted and excluded, there seems to be no

reason to rush to other exotic ideas, especially in the form of a

deeper layer of un-ending building blocks. This could be another

revolution of string theory, as it may establish a link between strings

and the external spacetime and escalates observations to the

electroweak scale.

References

[1] see: another Google discussion:

[url]http://groups.google.com/group/sci.physics.relativity/browse_thread/thread/04fc67af618dc340/9428e30e0356d5e4#9428e30e0356d5e4[/url]

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Web page of SPS: [PLAIN]http://schwinger.harvard.edu/~sps/ [Broken]

Posted via: http://groups.google.com/groups?group=sci.physics.strings

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# The Cause Of Dimensions Being Curled Up And Its Universality

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