From Osher Doctorow
COPYRIGHT NOTICE
Weak Heisenberg Principle
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
If there is actually a physical dimension of mass (M) on an "equal
footing" with the dimension of time (T), or in Probable Influence (PI)
several mass and time dimensions, then a "weak Heisenberg" principle or
really several different ones in different (possibly local)
circumstances make sense.
Recall that the Heisenberg Uncertainty Principle (HUP) says:
1) delta(x)delta(p) > = k
where delta(x) is the uncertainty in position (x), p is momentum, k is
a very small positive constant which is related to Planck's constant.
The Weak HUP or WHUP says:
2) xp = k_i (or ki), i in some (possibly infinite) index set
where ki > = k of (1).
Consider p = mv and now let m = f(t). Then we write (2) as:
3) xf(t)dx/dt = ki
and depending on how f(t) is chosen, position and time can vary
parabolically, exponentially, inversely or proportional to inverse
squares, etc. The cases mentioned are included in the following
possible choices of f(t):
4) f(t) = constant
5) f(t) = 1/t
6) f(t) = t
7) f(t) = t^2
Notice that if we take x as a radial position or a position along one
spatial axis, then 4-7 say that mass is respectively constant in time,
decreases with time, and (for 6 and 7) increases with time. In case
(6), dm/dt = 1, while in (7) dm/dt = 2t and Dtt(m) = 2 > 0 so mass
accelerates in time.
It turns out that if m is constant, then t is a parabolic function of
x, while if m = 1/t, then x^2 = k1t^2 + k2 for some constants k1 and k2
(k1 positive), and if m = t then exp(c) t^ki = exp((1/2)x^2) which
resembles the normal/Gaussian distribution with the power of t
replacing the probability density function. If m = t^2, then (1/2)x^2
= -ki/t + c = c - ki/t, and if c is a positive constant and ki is a
positive constant,then we get a singularity at t = 0 which might
describe a black hole.
It is obvious that x can accelerate with time (Dtt(x) > 0 where Dtt is
the second (partial or ordinary) derivative with regard to time (t)) in
some of these or similar scenarios.
Osher Doctorow
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