The Cause Of Dimensions Being Curled Up And Its Universality
qchiang
For more background and related topics, please refer to my website:
www.PhysicsRenaissance.com
Abstract
A plausible cause of curled up dimensions is proposed. The solid angle
rotation behaves like that in curled up dimensions, it actually is an
unrecognized aspect of the external spacetime. While it is not derived
from strings, it may possess string properties. There is an inherent
solid angle SO(6) associated with the Lorentz spacetime. (SO(10) with
a 4+1 spacetime.) Since these dimensions are intimately associated
with external spacetime, it answers why the same curled up dimensions
exist everywhere in the universe. This could be another revolution of
string theory, as it may establish a link between strings and the
external spacetime and escalates observations to the electroweak scale.
I. Introduction
Every semi-simple Lie algebra, say so(3), has a Casimir invariant, I =
J12 + J22 + J32 = constant. The form of this equation suggests a
brand new SO(3) symmetry. Discussion is made for the properties of
this new rotation (which may be called solid, or 3-d, angle rotation),
the reason it is overlooked and why it should account for the cause of
particle spectrum (e.g. iso-spin, strangeness, etc.). As it behaves
like strings, it could be the cause of the curled up dimensions. But
it escalates observations to electroweak scale and is economically
embedded in Lorentz (or 4+1) spacetime. It also explains parity
violation naturally. Requirements of simplicity, obviousness and
universality of the ultimate building blocks also point to these
symmetries associated with the external spacetime. Likewise, there
should be 4-d (and 5-d for 4+1 spacetime) angle rotations (without
which Poincare group is incomplete).
II. Mathematical Inevitability Of Higher Dimensional Rotation Symmetry
When internal symmetry was explored, it was believed external
symmetries were completely exhausted. What is unexpected is there may
be a series of overlooked external symmetries yet to be discovered.
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,
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 valid all that is
needed is the establishment of equivalency between the 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, 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 “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 need to rush to
the mysterious free standing internal symmetry unless solid angle
rotation is proven prohibited. The only possibility that it might be
forbidden (which is also the reason this new symmetry is overlooked) is
that solid angle rotation may not preserve the length of a vector, e.g.
linear momentum (even though it preserves a finite plane angle). But,
this actually is not a problem because we also overlooked the fact that
“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 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:
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. xm is the extra
dimension [1] and pm = mc.
B. A spinor representing plane (2-dimensional) angular momentum.
C. A solid angle spinor representing solid (3-d) angular momentum.
Solid angle rotation runs from one plane (2-brane) to another (among
the 10 planes) while preserving plane angular momentum. Symmetry of
solid angle rotation is suspected to be those of iso-spin, strangeness,
charm, etc. The interaction through solid angle rotation is believed
to be weak interaction.
D. A 4-d rotation spinor representing 4-d angular momentum. 4-d
rotation runs from one 3-plane (3-brane) to another (among the 10
3-planes) while preserving solid angular momentum. This symmetry
probably generates KL and KS, the mixtures of K0 and anti-K0 mesons.
The interactions may be the CP-violation interactions.
E. A 5-d rotation spinor representing 5-d angular momentum. 5-d
rotation runs from one 4-d plane (4-brane) to another among the 5 4-d
planes while preserving 4-d angular momentum. Fields in 5-d rotations
may be causing the strong interactions. The symmetry of 4-d angular
momentum might be the color symmetry which exists but cannot be
observed in isolation.
This shows the full symmetry property of the external 4+1 spacetime
is very rich indeed, which is enough to cover all particles (including
hadrons, leptons and photons altogether). At the same time, weak,
strong, and CP-violation interactions are but analog of
electromagnetism in solid and higher-dimensional angle rotations, while
gravitation is the interaction corresponding to the linear symmetry,
according to the 4+1 vector gravitation theory [1]. Actually, without
the addition of solid (3-d) angle, 4-d and 5-d angle rotations,
Poincare group (or the symmetries of 4+1 spacetime) would not be
complete.
VII. Simplicity, Obviousness And Universality Requirements Of Ultimate
Theory Point To Solid Angle And Higher Symmetries
In the April 10, 2000 issue of the Time magazine, one of the founders
of the standard model, Professor Steven Weinberg, prescribed the
criteria for the ultimate theory, “... [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” (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, 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 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 undestructible hard-cored object, but a (non-dissipative)
wave packet of certain 5-d, 4-d, 3-d (solid) angular momentum, plane
angular momentum and linear momentum, which is essentially the same as
a photon, except phone has only plane angular momentum and linear
momentum. This meets 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. This could be another
revolution of string theory, as it may establish a link between strings
and the external spacetime and escalates observations to the
electroweak scale.
References
[1] see: another Google discussion:
http://groups.google.com/group/sci.physics.relativity/browse_thread/thread/04fc67af618dc340/9428e30e0356d5e4#9428e30e0356d5e4
_______________________________________________________________________________
Web page of SPS: http://schwinger.harvard.edu/~sps/
Posted via: http://groups.google.com/groups?group=sci.physics.strings
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