Unified Geometry:Overlooked SymmetriesOf Spacetime, UnifiedFieldTheory needs UnifiedSpacetim

INTRODUCTION

In particle interactions, the total final 4-momentum must be equal to the total initial 4-momentum in momentum space. That this must be true under an arbitrary Lorentz transformation leads ultimately to SO(3,1) Lorentz group and the angular momentum spectrum. This must also be true for total angular momentum under rotation of “angular momentum space”. Whether expressed as SO(3,1) or its isomorphisms, there are 6 planes in the Lorenz spacetime. Invariance of total angular momentum under an arbitrary rotation of the 6-d “angular momentum space” results in an SO(6) symmetry, which has always existed in mathematics, but been overlooked. A rotation in the (external) 6-d angular momentum space readily shuffles plane components and causes parity violation. Investigation of invariance in “angular momentum space” is as essential as those in linear momentum space and should be sought for before rushing into internal space and internal symmetry.

I. CRITERIA FOR ULTIMATE THEORY: SIMPLICITY, OBVIOUSNESS AND BUILDING BLOCK PROPERTIES WITHOUT SUB-CONSTITUENTS

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 currently accepted 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, protons, neutrons and electrons were able to explain the existence and properties of atoms, and quarks those of protons and neutrons, 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 being created identically as one nearby. Even the highly hoped for strings cannot answer these questions. A common “principle” (rather than a new fundamental building blocks) 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 or superstring theory, 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 that 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 anything else). 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 Lorentz, photons are created “identically anywhere in the universe”. 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 “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 space”, because the symmetries of the external (Newtonian) space and time does 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/superstring theory. In other words, what’s needed in simplifying strong/weak theory is not a change of building blocks (e.g. strings) 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 answers 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.

With this in mind, it’s not hard to see symmetry properties of Lorentz spacetime is not fully explored yet. 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 surface) and solid angular momentum should contribute equally to particle symmetries. There is no point to rush into the mysterious internal symmetry until solid angle rotation is proven to be prohibited.

II. SOLID ANGLE ROTATION

Philosophy behind
Probably because of certain incorrect understanding, solid angle rotation is taken erroneously as internal symmetry when it should actually be external. The philosophy behind is: Notice that while linear scales may be defined by propagation of light, the “equivalence” of linear scales in different dimensions are not set absolutely but defined by the circular magnetic fields running between spatial scales and the electric fields between spatial and time scales. Without such fields to define scale equivalence, light wouldn’t be measured at the same speed in different dimensions. Now, what is that field which ensures the equivalence of the 6 “plane angle scales” of a Lorentz spacetime? They also cannot be set absolutely but must be defined by physical classical fields running “from one plane to another”, i.e. in solid angles. Fields running in solid angles is conjectured to be the classical version of weak interaction. To serve this purpose, our definition of solid angle rotation is a 2-d surface rotation which leaves a finite plane angle invariant, just like a plane angle rotation leaving a line element invariant. This definition is slightly different from the one usually perceived, but the spirit remains the same.

Mathematical insight
On mathematics side, there is no reason that solid angle rotation (i.e. displacement of a 2-d surface) and solid angular momentum cannot exist. The only doubt 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, thus might be forbidden. But, this actually is not a problem because we have always overlooked the fact that “only angular momentum, but not linear momentum, is concerned in particle classifications”. On the other hand, in particle 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.

It is suspected that the so-called internal symmetry may actually be the symmetry of solid (or even higher dimensional) angle rotation “of the external spacetime”. 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 invent the internal symmetry. 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 also shuffles the plane components of the plane angular momentum, thus causing parity-violation”.

Definition Of Solid Angle

Solid angle is defined here 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. Conventionally, a solid angle is comprehended as a cone, its value as determined by the spherical surface area cut through by the cone (divided by the radius). The rotation of a solid angle can thus be thought of as shrinking or expansion of the cone.

There is however an inherent impossibility of conserving both plane angle and linear vector length under solid angle rotation. As pointed out earlier, this imperfection is reflected truthfully in observations. Thus, we define solid angle scale in such a way as to preserve only 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 dimensions). 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 x-y) plane to another (say y-z) plane and a solid angle rotation must exist between every pair of planes in the spacetime. Below shows such a rotation in terms of plane angle decomposition in a 3-space. 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 (2.1)

where the spherical angles are defined as

ψ ≡ tan-1 [d22 + d12]½/d3 (2.2a)
θ ≡ tan-1 (d2/d1) (2.2b)

The total length

|d| = [(|d|sin ψ cos θ)2 + (|d|sin ψ sin θ)2 + (|d|cos ψ)2 ]½ = |d| (2.3)

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 (2.4)

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 (2.5)

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 (2.5) as the decomposition of a plane angle into 3 2-dimensional “plane” components, rather than into 3 “vector” components. Thus, we rewrite (2.5) in terms of 3 plane components,

α= α23 ξ23 + α31 ξ31 + α12 ξ12 (2.6)

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 (2.7a)
ω2 ≡ tan-1 (α31/ α23) (2.7b)

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 (2.8)

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| (2.9)

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

Though (2.8) is similar to (2.5), their meanings are very different. Eq. (2.5) 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 (2.8) 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, (2.5) would cause an SO(4) symmetry, but (2.8) an SO(6). That they both cause the same SO(3) is only incidental in 3-dimensional space, which also hints at the two SO(3)s, one for spin and one for iso-spin.

III. STRING BEHAVIOR AND 4- AND 5-DIMENSIONAL ANGLE ROTATIONS

In Lorentz spacetime, there are 6 planes and hence a solid (3-d) angle rotation symmetry of 6-dimensional space. In the more natural 4+1 spacetime, there are 10 planes, thus that of 10-space.

(see:
http://groups.google.com/group/sci.physics...7e8e61bf9d64043
)

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 suspected that the 10 dimensions conjectured in superstring theory may actually be the “10 plane angle scales” instead of 6 curled up and 4 extended linear dimensions. In other words, the strings are circulating quantum mechanical waves confined to the 10 planes of the 4+1 spacetime. This view is more plausible than plain strings because:

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

Similarly, 4-dimensional and 5-dimensional angle rotations should also be inherent parts of the 4+1 spacetime. This means 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 (3.1)

where:

A. = exp[-iπ(p0x0 -p1x1 - p2x2 - p3x3 - pmxm)] representing linear (1-dimensional) momentum, including energy and mass. m is the extra dimension 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 (based on the “same single principle” as prescribed by Weinberg). Under this model, the external spacetime geometry, the interactions and all particles are closely related to each other as if they were each the other side of a 3-sided coin, just like Lorentz spacetime, electromagnetism and photons. Thus, it explains naturally why this, but not other, set of particles are always created and why they are created identically everywhere in the universe. In fact, only with the addition of solid angle, 4-d and 5-d angle rotations, would symmetries of Lorentz (or the 4+1) spacetime be complete.

Any agreeing or disagreeing opinions are welcome.

Qchiang

Are you aware of the following "THEORY OF EVERYTHING BEGUN FROM ABSOLUTE CONCEPT". Look it up. Your idea is worth having as an input.
jal

Hi!
qchiang2@yahoo.com...
You got to check out your thread more often.
I do not have any arguements with you.
I started a thread Inverse Square Law that, I think, said the same thing. ( In a different way)
Look it up.
I now have another thread, Entropy--Potential energy, Which should interest you.

Jal

Hi! qchiang2@yahoo.com
You said,
Nov 2 2005, 03:25 PM

I have done some work and here is what I found. See my thread at Inverse Square Law that, (I think), said the same thing.
The shape of the 3d void.

How it looks when I apply my poor drawing skills.

jal

hi Qichiang

I read you article briefly.
According to ATPS the space-time as a math model is a barrier into bridging gravitation and other three forces.
In ATPS space-time is developed into a-temporal space where time runs as a stream of material change.

see more: http://forum.physorg.com/index.php?showtopic=4321

maybe this basic approach of ATPS could give you some ideas into building up your theory

amrit

Hi!
Sorry,
I think that by looking at the image that this can be treated as a two body problem.
jal