Re: Black Hole Bouncing vs Evaporation vs Control
From: OsherD (mdoctorow_at_comcast.net)
Date: 03/17/05
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Date: 17 Mar 2005 15:02:04 -0800
>>From Osher Doctorow
Let's look (from last time) at:
1) E = M^a L^b T^c (theta)^d (charge)^e
where a, b, c, d, e may change in different "phases" but remain usually
constant within a particular phase.
We know that one of the big arguments between the David Bohm and
Copenhagen and other theoretical physics schools concerns the question
of whether certain variables or observables are selected as more
important or more "central" than others (aside from the dimensions
mass, length, time, temperature, (electric) charge, which are
respectively M, L, T, theta, charge in (1), although the idea of
phase-related exponents may be susceptible to this criticism.
It might be argued that (1) selects energy as more important than other
variables, especially if it is postulated that (1) holds in different
scenarios with phase-related exponents. However, there is a
"superselection rule" which is somewhat "natural". Consider the
definition of decision in statistical decision theory (in graduate
statistics). A decision is an action based on various inputs or
stimuli such as information/knowledge/sampling. Although statistical
decision theory limits actions to statistical actions such as
estimating and hypothesis testing, it can easily be generalized to any
actions. In this. sense, an action can be regarded as the same as a
behavior, which is energy, whether potential or kinetic. So let's
write:
2) D = E/I
where D is decision, E is energy (generalizing action), I is input or
stimuli. Note that I'm not talking about Hamiltonian "least action"
here, whether or not it turns out to be related.
So what stimuli or inputs should we use for I? Well, the dimensions
are certainly "superselected" in that they are regarded as the most
fundamental irreducible components of physics: mass (M), length (L),
time (T), temperature (Theta), (electric) Charge (Q). With the
provision about phases indicated by variable exponents from phase to
phase, we can put:
3) I = M^a L^b T^c (theta)^d Q^e
We then obtain:
4) D = E/(M^a L^b T^c (theta)^d Q^e
and multiplying both sides by the right-hand-side denominator assuming
that it is not 0 yields:
5) E = DM^a L^b T^c (theta)^d Q^e
So we essentially have (1) except for D (decision). However, in
statistics, randomized decision rules are key in statistical decision
theory, and nonrandomized ones can be regarded as special cases of
randomized ones, so D inserts one extra dimension: Probability (P),
with an exponent f. To avoid a flood of objections, I'll say a few
words about Probability as a dimension after writing the new equation:
6) E = M^a L^b T^c (theta)^d Q^e P^f
It could be argued that D involves dimensions besides Probability, but
if they're one or more of M, L, T, theta, or Q, they can be
incorporated into the other factors with new exponents, so (6) pretty
much sums up the situation.
It is typical to regard Probability as dimensionless because of the
representation in the frequentist approach of Probability as a limit of
relative frequences as the sample size increases (to infinity if the
population is infinite, otherwise to the finite population). Here a
relative frequency is the frequency of the event in question, call it
A, to the total frequency of all events, call it the universe U.
However, the frequentist approach to probability is not the only
approach, and besides that the question of whether a limit of a ratio
of similar things is dimensionless is much more subtle than most people
realize. For example, a mass m can be regarded as a numerical limit of
ratios simply by approximating an irrational-valued mass by a sequence
of rational numbers divided by 1, which is a limit of ratios. Less
trivial examples are easy to construct. Yet mass is not dimensionless.
What determines whether a quantity is dimensional or not is not
whether it is the limit of a ratio of quantities having the same
dimension, or even whether it is a number or not, but whether it
involves a fundamental irreducible quantity or thing (called a
dimension). Probability qualifies as a dimension on all counts.
Osher Doctorow
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