--- In
undernetphysics@yahoogroups.com,
"video_ranger"
wrote:
> It's true that the transition probability is a
sum over real
numbers
> each of which depends on one pair of paths,
since if {w_i} is any
> finite sequence of N unimodular complex
numbers:
>
> F^2 = |w_1+...+ w_n|^2 = N + Sum over (i,j): w_i w_j* =
N + Sum
over
> all pairs (i,j): 2 Re (w_i w_j)
mmmmmmm.
interesting.
> The terms of the sum aren't independent since there're
N(N-1)/2 of
> them and the original sequence contains only N real numbers
worth
of
> information.
makes sense.
> On the
other hand, although F^2 is clearly a function only of the N-
1
> phase
angles theta_i between consecutive pairs w_i and w_(i+1) I
> don't believe
- if this is what David was suggesting - that it's
> possible to write F^2
as a sum of N-1 real numbers each of which is
> the function of only one
of those N-1 phase angles.
I think that may be what I was suggesting in
PI-2: basically S' = the
phase angle between w_i and w_(i+1). But I've sorta
convinced myself
that my derivation only applies to situations where S'' =
S''' =
S'''' = ... = 0, and that a more general solution needs to
include
the terms S'', S''', S'''', ... (This is basically a
mathematical
guess based on the same physical reasoning that got me to come
up
with PI-2 in the first place.)
That would be
> equivalent to
saying that the partial derivative of F^2 with
respect
> to the phase
angle between w_i and w_(i+1) doesn't depend on any of
> the other w_i's
which is not true mathematically.
I don't know how exactly you reasoned
out that statement, but it
seems right to me anyway and I won't argue. In
fact it gives me a
thought about analytical functions .. see
below.
> Paths normally can only be meaningfully indexed by an
infinite
> dimensional (or infinite number of) variables and I'm not
sure
> there's any rigorous definition of convergence.
Here's a
mathematical question for you. Consider an analytic function
f(x) where x is
any real number. If I provide you with the
following
numbers:
f(x_0)
f'(x_0)
f''(x_0)
f'''(x_0)
...
ie
I tell you all the derivatives at x_0, then is this sufficient
information to
calculate f(x) for all x? If so, then I have a
mathematical reason to belive
that my above guess should work.
IOW: Index the paths by the variable
lambda = l (which as you say
above may turn out to be infinite dimensional).
S(l) is the action of
the l-th path. Consider path l_0, and suppose we are
provided
with:
S(l_0)
S'(l_0)
S''(l_0)
S'''(l_0)
S''''(l_0)
...
[S'
= ds/dl. For the sake of notational simplicity I'm assuming for
the time
being that l is a scalar.]
That means we have enough information to
calculate S(l) for all l;
IOW each individual path "knows" the action of
every other path, even
paths that are far away, based solely on information
that is locally
available to it (ie based only on the characteristics of the
paths
that are in its immediate vicinity). Thus, each individual
path
"knows" (has enough information to calculate) the total
amplitude = sum of
all the individual amplitudes. So whatever
algorithm [1] we use to assign
nonnegative, real-valued probability
measures to each path, there should be a
way to calculate this
quantity given only values that are evaluated at each
path (S, S',
S'', S''', ...). All that's left is to figure out the
"algorithm", ie
figure out the equation [2] to calculate m from S, S', S'',
...
Of course, this is all assuming analyticity ...
DS
[1]
See my previous post, where I give an "existence proof" that it
is possible
to come up with an algorithm to assign a nonnegative,
real probability
measure m to each path, such that the sum of m over
all paths = |(sum of
phase over all paths)| squared = probability.
[2] right now I'm guessing
-- and this is just my first guess, not
churned out -- that the equation is
similar to what I had in PI-2.
Except that instead of:
m = (1/2) ^
n(S')
we should have:
m = (1/2) ^ [n_1(S') + n_2(S'') + n_3(S''') + ...
]
where the n's are (perhaps) what I calculated in
PI-2.
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