# Poincare Group Not Complete Yet, Overlooked Symmetries Of Lorentz/4+1 Spacetime

*From*: "qchiang2@xxxxxxxxx" <qchiang2@xxxxxxxxx>*Date*: 5 Sep 2006 13:27:01 -0700

For more background and related topics, please refer to my website:

www.PhysicsRenaissance.com

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.) but was mistaken

as internal symmetry. This new symmetry, SO(6) or SO(10) for Lorentz

or 4+1 spacetime, is more advantageous than strings as 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 higher dimensional

symmetries of 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 waiting to be discovered.

We’ll start from some quick mathematical investigations, and then

look into the physics behind. Let’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(∂/∂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 gμν J^μ J^ν = constant (2.3)

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

I = Σ gμν Jμ Jν = 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 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 jij

J1 φ = [y(∂/∂z) - z(∂/∂y) ] exp[-i(tpt - xpx – ypy –

zpz)] = i(ypz – zpy )φ = i jyz φ (2.6a)

J2 φ = [z(∂/∂x) - x(∂/∂z) ] exp[-i(tpt - xpx – ypy –

zpz)] = i(zpx – xpz )φ = i jzx φ (2.6b)

J3 φ = [x(∂/∂y) - y(∂/∂x) ] exp[-i(tpt - xpx – ypy –

zpz)] = i(xpy – ypx )φ = i jxy φ (2.6c)

where the wave function

φ = exp[-i(pt t - px x – py y – pz z) ]

(2.7)

is the solution to the wave equation

[ ∂^2/(∂t)^2 - ∂^2/(∂x)^2 - ∂^2/(∂y)^2 - ∂^2/(∂z)^2 –

m^2 ] φ = 0 (2.8)

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

would be

W1 λ = [θzx (∂/∂θxy ) – θxy (∂/∂θzx ) ] exp[-i(jxy θxy

+ jyz θyz + jzx θzx)]

= -i(θzx jxy – θxy jzx ) λ = i Ωzx,xy λ

(2.9a)

W2 λ = [θxy (∂/∂θyz ) - θyz (∂/∂θxy ) ] exp[-i(jxy θxy +

jyz θyz + jzx θzx)]

= -i(θxy jyz – θyz jxy ) λ = i Ωxy,yz λ

(2.9b)

W3 λ = [θyz (∂/∂θzx ) – θzx (∂/∂θyz ) ] exp[-i(jxy θxy

+ jyz θyz + jzx θzx)]

= -i(θyz jzx – θzx jyz ) λ = i Ωyz,zx λ

(2.9c)

where the wave function

λ = exp[-i(jxy θxy + jyz θyz + jzx θzx)]

(2.10)

is the solution to the quantized wave equation of the Casimir

invariants (2.4)

[ ∂^2/(∂θyz)^2 + ∂^2/(∂θzx)^2 + ∂^2/(∂θxy)^2 – I^2 ]

λ = 0 (2.11)

Equation (2.11) has plane angles θij as its coordinates with angular

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

Ωij,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 true all that is

needed is the establishment of equivalency between the pl\ane 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, Fμν = Aμ

∂/∂xν - Aν ∂/∂xμ , 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 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 “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 [d2^2+ d1^2]^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 ψsin θe2 +|

α|cos ψe3 (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= α23 ξ23 + α31 ξ31 + α12 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 [α31^2 + α23^2]^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 | = [α23^2 + α31^2 + α12^2]^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 θand 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 internal symmetry when it should

actually be external. 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 spacetime and hence solid angular

momentum should contribute equally to particle symmetries. There is no

need to rush to the mysterious 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

“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.

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 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 is external and shuffles the plane components of the plane

angular momentum, thus causing parity-violation”. 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 “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 string 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.

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 - pmxm)] representing linear

(1-dimensional) momentum, including energy and mass. m 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, “... [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. 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 string 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 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 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”, 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/string 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 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 undestructible hard-cored object (e.g. string), 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 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. 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

were simply surprised at the existence of 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. 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.

References

[1] (see:

http://groups.google.com/group/sci.physics.relativity/browse_frm/thread/04fc67af618dc340?tvc=1

)

or www.PhysicsRenaissance.com

.

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