Natural Science Forum / Physics / Relativity / October 2005

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 doesnt say much either, as
what else can it be if the fields dont 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 photons.  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, whats 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
whats 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, its 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 wouldnt 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.  Lets first express a line element in
terms of spherical angles

 d = d1 e1 + d2 e2 + d3 e3
   = |d|sin cos e1 +|d|sin sin e2 +|d|cos e3 (2.1)

where the spherical angles are defined as

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

The total length

)2 ] = |d|   (2.3)

is independent, hence invariant under rotation of the spherical angles
and .  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  cos  e1 +|J|sin  sin  e2 + |J|cos
e3 (2.4)

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

  =  1 e1 + 2 e2 + 3 e3
   = || sin  cos  e1 +||sin sin e2 +|

Nevertheless, since  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,

= 23 23 + 31 31 + 12 12      (2.6)

where s are unit angles on each component plane.  We then define
solid angles, 1 and 2, in terms of the plane angle components in
exact analogy to spherical angles defined in terms of line components:

1  tan-1 [312 + 232]/ 12               (2.7a)
2  tan-1 (31/ 23)                            (2.7b)

Through solid angles 1 and 2, the finite plane angle  on an
arbitrary plane can be decomposed into 3 plane components as

   = 23 23 + 31 31 + 12 12
     = | |sin 1 cos 2 23 + | |sin 1 sin 2 31 + |

The total plane angle

    = [(||sin 1 cos 2)2 + (||sin 1 sin 2)2 + (||cos
1)2 ] = | |   (2.9)

is independent of, thus invariant under arbitrary rotation of, solid
angles 1 and 2.

  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 and  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 1 and 2 (which shuffles plane angle components 23, 31
and 12) 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.relativity/browse_thread/thread/2d6d0
65d6d9f0e24/57e8e61bf9d64043#57e8e61bf9d64043
)

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:

   =    E  D  C  B  A     (3.1)

where:

A.    = exp[-i(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