String Behavior From Inherent Solid Angle Rotation Imbedded in Lorentz/4+1 Spacetime at Electroweak Scale

For more background and related topics, please refer to my website: www.PhysicsRenaissance.com

Abstract

2-dimensional space (a plane) has 2 ways to generate symmetries, namely, linear displacement and plane angle rotation (disregarding reflections).  Common sense dictates there should be n ways to generate symmetries in an n-dimensional spacetime.  Investigation is made of overlooked symmetries (e.g. that of solid angle rotations, etc.) of Lorentz/4+1 spacetime (thus, completing the Poincare group).  What’s exciting is it reveals the properties of strings at electroweak scale.  There is an inherent solid angle SO(6) associated with Lorentz spacetime (SO(10) with a 4+1 spacetime.)  What’s more: While solid angle rotation behaves like that in curled up dimensions, it actually is an unrecognized aspect of the external spacetime, hence answers why the same dimensions are curled up everywhere in the universe.  It establishes a link between strings and the external spacetime. 

I. Introduction

One dimensional space (a line) has one way to generate symmetry, namely, linear displacement (disregarding reflections).  2-dimensional space (a plane) has 2 ways to generate symmetries, namely, linear displacement and plane angle rotation.  It's easy to see there should be n ways to generate symmetries in an n-dimensional spacetime.  Nevertheless, the current Poincare group for the 4-dimensional Lorentz spacetime is concerned only with linear displacement and plane angle rotation.  Where are the other 2 ways of symmetry generation?  Obviously, we are missing some important symmetries of Lorentz spacetime (namely, one generated by displacement of one whole plane and another by displacement of one whole 3-d space in the 4-d spacetime) which is very likely the sought-for cause of particle spectrum, e.g. iso-spin, etc. 

Let’s explore the missing symmetries using 3-d space as an example.  Looking at the form of the Casimir invariant of plane angle rotation group, I =  J₁² + J₂² + J₃² =  constant, we see immediately that a brand new SO(3) exists on top of the existing SO(3).  Discussion is made of the properties of this new rotation (which may be called solid angle, 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.). 

It is interesting to note that some string properties may be revealed in solid angle rotation.  Hence, if we take solid angle rotation as strings, certain mysterious questions may be answered, e.g. 1. What is the connection between the small size of strings and the external Lorentz spacetime at observable scales, 2. What causes dimensions being curled up.  3. What ensures the same strings to be created (and the same dimensions curled up) everywhere in the universe.  Additionally, it explains parity violation naturally.  Requirements of simplicity, obviousness and universality of the ultimate building blocks also point to these symmetries closely tied up with the external spacetime.  Likewise, there should be inherent 4-d (and 5-d for 4+1 spacetime) angle rotations, without which Poincare group would not be complete. 

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. 

In a 1-dimensional space (a line), there is one way to make displacement (a line displacement, which moves a point) and look for symmetry.  In a 2-dimensional space, there are 2 ways to make displacement (a line displacement and a rotation, which moves a whole line) and look for symmetries.  In a 3-dimensional space, there should be 3 ways to make displacement (a line displacement, a rotation and solid angle rotation, which moves a whole plane) and look for symmetries.  In the 3+1 Lorents space, there should be 4 ways to make displacement and look for symmetries.  Obviously, 2 more symmetries in the Lorents spacetime is not uncovered yet and the Poincare group is not complete.  Isn’t it weird that we spend billions of dollars on far-fetching concepts, such as strings, while overlooking the close-by symmetries which are more natural.  We’ll start from some quick mathematical investigations, then look into the physics behind.  Let’s start from a 3-space where a length is invariant under any rotation,

L² = x₁² + x₂² + x₃²                                       (2.1)

An SO(3) symmetry results with infinitesimal rotation operators

J₁  = x₂(∂/∂x₃) – x₃(∂/∂x₂)                              (2.2a)

J₂  = x₃(∂/∂x₁) – x₁(∂/∂x₃)                              (2.2b)

J₃  = x₁(∂/∂x₂) - x₂(∂/∂x₁)                               (2.2c)

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

I =  Σ  g_μν J^μ J^ν = constant                                (2.3)

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

I =  Σ  g_μν J^μ J^ν = J₁² + J₂² + J₃² = J² =  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 it’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(∂/∂J₃) – J₃(∂/∂J₂)                          (2.5a)

W₂  = J₃(∂/∂J₁) – J₁(∂/∂J₃)                          (2.5b)

W₃  = J₁(∂/∂J₂) – J₂(∂/∂J₁)                          (2.5c)

The eigenvalues of the infinitesimal operators J are the angular momenta j_ij

J₁  φ = [y(∂/∂z) - z(∂/∂y) ] exp[-i(tp_t - xp_x – yp_y – zp_z)] = i(yp_z – zp_y )φ = i j_yz φ (2.6a)

J₂  φ = [z(∂/∂x) - x(∂/∂z) ] exp[-i(tp_t - xp_x – yp_y – zp_z)] = i(zp_x – xp_z )φ = i j_zx φ (2.6b)

J₃  φ = [x(∂/∂y) - y(∂/∂x) ] exp[-i(tp_t - xp_x – yp_y – zp_z)] = i(xp_y – yp_x )φ = i j_xy φ (2.6c)

where the wave function

 φ = exp[-i(p_t t - p_x x – p_y y – p_z z) ]                                                      (2.7)

is the solution to the wave equation

[ ∂²/(∂t)² - ∂²/(∂x)² - ∂²/(∂y)² - ∂²/(∂z)² - m² ] φ = 0   (2.8)

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

W₁  λ = [θ_zx (∂/∂θ_xy ) – θ_xy (∂/∂θ_zx ) ] exp[-i(j_xy θ_xy + j_yz θ_yz + j_zx θ_zx)]

         = -i(θ_zx j_xy – θ_xy j_zx  ) λ = i Ω_zx,xy λ                                                       (2.9a)

W₂  λ = [θ_xy (∂/∂θ_yz ) - θ_yz (∂/∂θ_xy ) ] exp[-i(j_xy θ_xy + j_yz θ_yz + j_zx θ_zx)]

         = -i(θ_xy j_yz – θ_yz j_xy  ) λ = i Ω_xy,yz λ                                                       (2.9b)

W₃  λ = [θ_yz (∂/∂θ_zx ) – θ_zx (∂/∂θ_yz ) ] exp[-i(j_xy θ_xy + j_yz θ_yz + j_zx θ_zx)]

         = -i(θ_yz j_zx – θ_zx j_yz  ) λ = i Ω_yz,zx λ                                                       (2.9c)

where the wave function

 λ = exp[-i(j_xy θ_xy + j_yz θ_yz + j_zx θ_zx)]                                                           (2.10)

is the solution to the quantized wave equation of the Casimir invariants (2.4)

[ ∂²/(∂θ_yz)² + ∂²/(∂θ_zx)² + ∂²/(∂θ_xy)² - I² ] λ  = 0           (2.11)

Equation (2.11) has plane angles θ_ij as its coordinates with angular momenta j_ij [in (2.10)] as its corresponding momenta.  The eigenvalues Ω_ij,jk [in (2.9)] are solid angular momenta and the rotations W_i [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 3 plane angle scales so that a rotation (or reshuffling of the 3 J’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 won’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, F_μν = A_μ ∂/∂x_ν - A_ν ∂/∂x_μ , 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 same) plane.  Though it does not rotate to a different plane, it is a rotation from plane to plane.  That’s why we borrow the term solid angle rotation for the rotation from one plane to another (or same) plane.  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.  The purpose of solid angle rotation is to establish equivalency between plane angle scales, the effect is preservation of any plane angle arc under this (solid angle) rotation. 

  However, 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 disregard linear vector length and 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 “Cartesian coordinates”.  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.  Let’s first express a line element in terms of spherical angles

  d = d₁ e₁ + d₂ e₂ + d₃ e₃      

    = |d|sinψ cosθ e₁ +|d|sinψ sinθ e₂ +|d|cosψ e₃ (4.1)

where the spherical angles are defined as

    ψ ≡ tan^-1 [d₂² + d₁²]^½/d₃      (4.2a)

    θ ≡ tan^-1 (d₂/d₁)             (4.2b)

The total length

|d| = [(|d|sin ψ  cos θ )² + (|d|sin ψ  sin θ )² + (|d|cos ψ )² ]^½ = |d|   (4.3)

is independent, hence invariant under rotation of the spherical angles θ  and ψ.  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 ψ  cos θ  e₁ +|J|sin ψ  sin θ  e₂ + |J|cos ψ  e₃ (4.4)

Obviously, if this decomposition can be done to angular momentum, it can also be done to any finite plane angle α,

  α =  α₁ e₁ + α₂ e₂ + α₃ e₃      

    = |α| sin ψ  cos θ  e₁ +|α|sin ψ sin θ e₂ +| α |cos ψ e₃ (4.5)

Nevertheless, since α 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 “plane” components, rather than into 3 “vector” components.  Thus, we rewrite (4.5) in terms of 3 plane components,

α = α ₂₃ ξ₂₃ + α ₃₁ ξ₃₁ + α ₁₂ ξ₁₂      (4.6)

where ξ’s are unit angles on each component plane.  We then define solid angles, ω₁ and ω₂, in terms of the plane angle components in exact analogy to spherical angles defined in terms of line components:

ω₁ ≡ tan^-1 [α₃₁² + α₂₃²]^½/ α₁₂               (4.7a)

ω₂ ≡ tan^-1 (α₃₁/ α₂₃)                             (4.7b)

Through solid angles ω₁ and ω₂, the finite plane angle α on an arbitrary plane can be decomposed into 3 plane components as

   α = α₂₃ ξ₂₃ + α₃₁ ξ₃₁ + α₁₂ ξ₁₂

      = | α |sin ω₁ cos ω₂ ξ₂₃ + | α |sin ω₁ sin ω₂ ξ₃₁ + | α |cos ω₁ ξ ₁₂        (4.8)

The total plane angle

| α | = [α₂₃² + α₃₁² + α₁₂²]^½

     = [(|α|sin ω₁ cos ω₂)² + (|α|sin ω₁ sin ω₂)² + (|α|cos ω₁)² ]^½ = | α |   (4.9)

is independent of, thus invariant under arbitrary rotation of, solid angles ω₁ and ω₂. 

Though (4.8) is similar to (4.5), their meanings are quite different.  Eq. (4.5) is the decomposition of a vector into 3 “linear” components and rotation of plane angles θ and ψ  preserves the length of the “vector”.  But (4.8) is the decomposition of a plane angle into 3 2-d “plane angle components” and rotation of solid angles ω₁ and ω₂ (which shuffles plane angle components α₂₃, α₃₁ and α₁₂) preserves the “total plane angle”.  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 ω₁ and ω₂ , then the line on the plane by plane angle θ 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 “free standing” 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 reason 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 may be 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 “only angular momentum, but not linear momentum, is concerned in particle classifications”.  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 “independent” 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, “while it preserves total plane angular momentum, it is external and shuffles the plane components of the plane angular momentum, thus causing parity-violation”.  Another utterly important virtue (its universality) 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 “not a point” but a “circulating” 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: 

It escalates the 10 dimensions of strings to observable “electroweak” scales. 

It explains why the same strings are created (and the same dimensions are curled up) everywhere in the universe. 

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

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 symmetry’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 “external” 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: 

    Ψ =   ∑ E × D × C × B × A              (6.1)

where: 

A.    = exp[-iπ(p₀x⁰ -p₁x¹ - p₂x² - p₃x³ - p_mx^m)] representing linear (1-dimensional) momentum, including energy and mass.  x^m is the extra dimension [1] and p_m = 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 K_L and K_S, 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, “... [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.”  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 “itself”. 

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 “principle” which rules “throughout the universe simultaneously” 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 doesn’t say much either, as what else can it be if the fields don’t conserve?)  It is really “simple and obvious” (i.e. “can scarcely be different from what it is”).  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.  “No other choice would be possible”, as no symmetry properties of Lorentz spacetime is not represented in photon.  It exists by itself “without lower level constituents”. And as long as the local spacetime is Lorentzian, photons are created “identically anywhere in the universe”.  Not surprisingly, the first half of 20^th 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 “not” 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 “internal symmetry”, as the symmetries of the external (Newtonian) space and time do not match that of photon’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 “wrong” spacetime, which seem to be shared by the standard model.  In other words, what’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 what’s needed may actually be a “spacetime geometry that is defined to fit” 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 “external” 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 “only” answer to the 3 ultimate questions, because the only thing that exists “throughout the universe simultaneously” seems to be the external spacetime itself, and it appears there is no way “a priori building blocks” 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 un-destructible 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 photon has only plane angular momentum and linear momentum.  This meets Weinberg’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.  This would be the most natural origin of weak and other interactions, just like electromagnetism.  They are unified in concept as all being rotations.  The difference is only that they are rotations of different dimensions. 

If strings are not connected with external spacetime, it would simply be another layer in the series of un-ending sub-constituents of matter, which would not be able to answer its own properties without one more layer of sub-constituent, neither could it explain its universality.  If it is connected with the external spacetime like solid angle symmetry, it is not matter any more, but a geometric feature providing required symmetry, hence can explain its own property without sub-constituent and can explain its universality.  This could be another revolution of string theory, as it may establish a link between strings and the external spacetime and escalates observations to electroweak scale. 

The reason we never thought about solid angle rotation and beyond is because we always “assumed” the equivalence between plane angle scales and 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 simply invented internal symmetry because we were surprised at the 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 “assumption” in the definition of spacetime geometry may lock the door to a new geometric aspect. 

References

 [1] see: another Google discussion: 

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

or

- K.C.Chiang: A Unified Gravitational And Quantum-Mechanical Space-Time Structure Through A Unified Origin Of Inertial And Gravitational 

Masses And A Discussion Of The Foundations Of Special Relativity, Il Nuovo Cimento Vol. 68B, N.2, p.322 (1982) 

and 

-  K. C. Chiang: A Critical Problem, The Artificial Curvature In Current Assumption Of Parametrization In General Relativity, in Proceedings of 

the XXTH International Conference On Differential Geometric Methods in Theoretical Physics, p.727, World Scientific, 1992. 

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