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From: Osher Doctorow on 7 Aug 2010 17:06
From Osher Doctorow

Could the 5 dimensions of classical physics, M, L, T, theta
(temperature), Q ((electric) charge) be doubled to yield the 10 of
Superstring Theory by using either Antiparticles or Multiplicative
Dimensional Inverses (the latter being for example M^(-1), L^(-1),
T^(-1) )? I will rephrase this as the following questions:

1) Could L^(-1), T^(-1), and so on reflect dimensions different from
L, T, etc.?
2) Could Antiparticles be in different dimensions from Particles?

Supposedly we ruled out (1) in the previous posts, but not quite in
the following sense. The dimension of Length, L (or of "1
dimensional component of space") has a "natural" multiplicative
inverse L^(-1) when dimensions are regarded as an Abelian
(commutative) group under multiplication, but what is the physical
justification for dividing a dimension or having a negative exponent
of a dimension? If we argue that a positive exponent of L, for
example, reflects increase in L, and a negative exponent reflects
decrease in L, then L itself (to power 1!) can be both increasing or
decreasing by definition! Why the extra exponent? One could claim
with some justification that in a physical equation with physical
variables x, y, z, z = xy, then solving for y gives y = z/x = zx^(-1),
so the same thing would work for dimensional equations, "explaining"
negative powers of fundamental dimensions. But in the equation z =
xy, not all of z, x, and y can be Fundamental Dimensions, since for
example if z were a Fundamental Dimension then it is factored into 2
other Fundamental Dimensions, contrary to definition of a "Fundamental
Dimension". So within the commutative group of Fundamental dimensions
of physics, the mystery remains as to what L^(-1) represents
physically, and so on.

One possibility is clarified by considering velocity v whose
dimensions |v| = LT^(-1) in the notation of Fundamental Dimensions in
Dimensional Analysis. What if a RATIO of two different Fundamental
Dimensions can itself have Fundamental Dimensions? An example is
Probability, which in the measurement analysis of Suppes, Tversky, and
Krantz (see the internet for citations of their work, largely in
measurement theory applicable to psychology) can be a Fundamental
Dimension in the case of "Subjective Probability". It also has ratio
or limiting ratio aspects in an "equivalent" or "almost-equivalent"
definition of Probability.

Thus, if T^(-1) is understood to generate a "ratio" Fundamental
Dimension together with some other Fundamental Dimension, or even a
new Fundamental Dimension coinciding with Probability for example,
then the 5 Fundamental Dimensions M, L, T, Q, theta and their inverses
would generate 10 Fundamental Dimensions, although one would have to
make some rules of exclusion in order to not proliferate Fundamental
Dimensions via all possible combinations.

Antiparticles are in some ways even more "amusing". What if our
ordinary "particles" are attractive while antiparticles are repulsive,
or alternatively ordinary particles are either attractive or repulsive
and antiparticles are repulsive? Then the mystery of the imbalance of
particles over antiparticles initially could be explained by an
infinite-velocity version of Inflation outward - antiparticles simply
repel to infinity when contacting particles of the same type. Note
that an attractive and repulsive object could COLLIDE under these
conditions, somewhat like "an immovable object meeting an irresistible
force" (although the analogy is not strict).

See Wikipedia's "Antiparticle," "C parity", "Quantum Number," "C-
symmetry," "Adjoint Action," "Adjoint representation of a Lie Group,"
"Charge (physics)," etc.

Osher Doctorow

From: Osher Doctorow on 7 Aug 2010 17:37
From Osher Doctorow

The probability example from the last post brings up another idea for
expanding the 5 Fundamental Dimensions of Classical Dimensional
Analysis (M, L, T, Q, theta (temperature)) to the 10 of Superstring
Theory, understanding that what we view as 3 + 1 spacetime (3 spatial
and 1 time dimensions) are really Fundamentally only 2 dimensions: L
(or L^3 for 3 dimensional space) and T.

There are 3 types or subtypes of Probability that deeply relate to
Independence vs Dependence of Random Variables:

1) Probable Causation/Influence (PI): P(A-->B) = 1 + P(AB) - P(A), or
its alternate version P ' (A-->B) = 1 + P(B) - P(A) where P(B) < =
P(A) in the second version.

2) Conditional Probability P(B|A) ("Probability of B given A") = P(AB)/
P(A) if P(A) is not 0, where / is real division and the symbol | is
merely defined by the right hand side.

3) Independent Probability/Statistics: Ones that obey P(AB) =
P(A)P(B), which is equivalent to P(B|A) = P(B) or to the equation P(A--
>B) = 1 + P(A)P(B) - P(A).

These respectively relate one-on-one to Lukaciewicz/Rational Pavelka
Multivalued Logics, Product/Goguen Multivalued Logics, and Godel
Multivalued Logic (see Pavel Hajek, "Metamathematics of Fuzzy
(Multivalued) Logics", Kluwer: Dordrecht 1998 for an excellent
exposition of the 3 Multivalued Logics, although he wasn't aware of
the Probability connection).

If we add to the above 3 the simple definition of Probability
dimensions in the sense of Suppes, Krantz, Tversky mentioned in the
previous post (for example, Subjective Probability), then we have the
9 Fundamental Dimensions:

4) M, L, T, Q, theta, P(A-->B), P(B|A), P(A)P(B) or P(AB) - P(A)P(B) =
0, P(A).

We can search for a 10th Fundamental Dimension, for example
Antiparticle. Note that (4) can be expressed as:

5) M, L, T, Q, theta, Probable Causation/Influence, Probability of B
given A, Independence or Dependence, and Probability.

Here both Probable Causation/Influence and Probability of B given A
are of main interest in dependent random variables, while P(A)P(B) is
of main interest for independent random variables. The fact that 5
dimensions are "non-probabilistic" and 4 probabilistic suggests that
the 10th Fundamental Dimension is Probabilistic - for example:

6) P(A<-->B) = P(AB) + P(A ' B ' ) (Probable Correlation).

Osher Doctorow

From: Osher Doctorow on 7 Aug 2010 17:41
From Osher Doctorow

We could also have P ' (A-->B) as the 10th Fundamental Dimension,
since:

1) P(A-->B) = 1 + P(AB) - P(A)
2) P ' (A-->B) = 1 + P(B) - P(A) for P(B) < = P(A).

Osher Doctorow
From: Osher Doctorow on 7 Aug 2010 17:49
From Osher Doctorow

P versus P ' relates to the interior vs exterior vs "overlapping
partly" set distinctions, just as the previous types of probability
relate to dependence vs independence and influence vs "given". If we
can formulate the remaining Fundamental Dimensions in terms of similar
"Deterministic" or "Almost-Deterministic" distinctions, then a 10-
dimensional Superstring Universe may make more sense in terms of
relationships deeper than merely "large vs small" or "observed vs
unobserved" or "experimentally known vs theoretically postulated".

Osher Doctorow
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