Pauli's Spin Matrices |
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7th
September 2015 |
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The space-time
position (c2t2
- x2- y2-
z2), an invariant
under Lorentz transformation, is given by the
(ct, x, y, z) metric in Clifford Algebra as,![]() If
the observer has an infinite capacity we can
reduce the problem to the single measurement or
a pure two-state system. The single
measurement capacity implies relativistic limits
i.e. zero entropy ⇒
t ≈ 0*. In
that case matrix X reduces to zero-entropy
space-time matrix Xo,
![]() ![]() (We can also say that Weyl's picture SL(2,) - entropy Pauli's picture SU(2). In j-space language, Weyl's picture represents the description as measured by the macroscopic observer AKU or ObsM, and Pauli's picture represents the description as measured by the relativistic observer Obsc.) An important property to note is that Pauli Spin Matrices are their own inverse(s), which means that the definition of the origin is mandatory within the structure described by these matrices. (It is not necessarily true for a simple reflection of a structure in which case the reference origin is in the plane of the mirror, not in that of the structure itself). As we recall that the VT symmetry is required to determine the origin, hence the quanta for the measurement space is pre-defined for the structure described by these matrices. Please also note that Pauli spin matrices do not represent Majorana states. ![]()
lσx + mσy + nσz
= 0 .
-1
We plug in the
values of matrices and solve to
find,
l2 + m2
+ n2 = 0
.
-2
The
equation-2 represents an sphere in measurement
space with radius zero as measured by the
macroscopic observer ObsM. Please
note that l, m, n, are the points lying on the
measurement net. Each of them is an
integer or a ratio of integers hence rational or
"constructible"**. This is how the most
elementary geometrical symmetry i.e. the
spherical symmetry, is formed from a two-state
system.
The unit-point U will be measured as an sphere of zero radius by ObsM. Let us call it unit-point sphere SU***. This situation is rather fantastic. The quantities on LHS of equation-2 are positive, whereas RHS is equal to zero. However we must not forget Obsi, the i-space observer who has higher capacity than ObsM. Obsi will measure 0 as 0j, a finite value. Hence equation-2 can be rewritten as, l2 + m2 + n2 = 0j . -3 Above
equation represents a sphere of finite radius √0j,
a radius whose measurement is not possible for
ObsM. We should
be using the quantity i√0j
or an imaginary value for radius, but we are
not. Why? Because in
measurement space only PE1 measurable quantities
are reported, hence only 0j which
actually is the radius squared, is of
value. In other words, the
information on the internal structure of this
sphere is not available to the macroscopic
observer ObsM.
In a later blog, this argument will result in
shell-structure we are so familiar in classical
picture.
Interestingly enough, we can replace Pauli's spin matrices with the components of Quaternion matrix in equation-1. In this case also the equation-2 and the equation-3 remain unchanged. Therefore in both cases, in which the component matrices represent convex surfaces, the geometric net results in spheres of finite radii. Both these spheres represent stable structures. One just wonders, what type of current will result from such symmetry? It should be noted that the value of 0j will be different in each case. The unit-point sphere SU has a symmetry of 4π or 720o, centered around the unit point U. The 4π-symmetry is an essential feature of the systems with interacting components, or the systems whose components are not completely independent of each other. We can later build a coordinate system with independent axes with SU with 4π- symmetry, as the origin. We can visualize the symmetry of 4π as a fundamental internal symmetry of a two-state system, measured as SU by ObsM at Λ∞. One may want to think of the symmetry with respect to the integral multiple or half- integral multiple rotation of 360o, but then not all the information contained within SU would be measured. Had we considered only a classical single-state system, the 360o rotation would have been sufficient to describe the corresponding symmetry. Obviously, two 360o rotations do not represent 4π-symmetry which is the symmetry of the origin itself. In fact the information measured by ∞j × 360o rotations, will always be less than the information contained within the 4π-symmetry. In other words the resources available to a classical observer are not enough to re-define the origin and change the information structure being measured. ![]() The
unit-line Λ∞
has the property, that it is parallel to
every straight line in the anharmonic
plane. We have used this property for
drawing an important geometric physical
feature related to ObsM.
Can you figure out this feature? Think
shell theorems.
Please note
that SU will form
the initial state for the measurements
performed by ObsM,
the measurements represented by the polar Λ∞
in anharmonic coordinates. The central force
nature of the problem is quite
evident. We
have so far refrained from using
physical constants or known forces in
nature while developing this
description. One of the conditions
we have discussed, is related to the
values of scalars l, m, and
n, which have to be either integer
or rational. Consequently a
continuous spectrum of measurable
structures is prohibited in
j-space. In other words a
continuous spectrum for properties
similar to mass can not be measured in
j-space. In fact in pure j-space,
continuous spectrum corresponding to any
property related to a stable structure,
can not be measured.
__________________________________________________________ *The complete
Hamiltonian of a two-state system is
given as,
![]() We have merely eliminated
the time-dependent phase which is a
function of g0
to derive a description with the unit
point U as reference. Please note
that we have thus introduced the
relativistic description by making time
insignificant. For the unit point U
the time has no significance, as time is
the property associated with the observer
making measurement, not with the source
which is being measured. The
conventional (t, x, y, z) description is
valid only for Λ∞,
the measurement plane of the macroscopic
observer with intrinsic entropy.
** What happens if (l, m, n) are complex numbers? *** The
unit-point sphere SU
is not
the same as conventional unit sphere S
2.
![]() ![]() |
Previous Blogs: Sigma-z and I Spin Matrices Rational behind Irrational Numbers The Ubiquitous z-Axis Majorana ZFC Axioms Set Theory Nutshell-2014 Knots in j-Space Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, b-Field & Lower Mass Bound Incompleteness II The Supersymmetry The Cat in Box The Initial State and Symmetries Incompleteness I Discrete Measurement Space The Frog in Well The Einstein Theory of Relativity ![]() ![]() |
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