Matti Pitkanen (email@example.com)
Fri, 8 Oct 1999 20:45:08 +0300 (EET DST)
On Tue, 8 Oct 2019, Ben Goertzel wrote:
> > In a few cases... I had been studying Feynman integral myself. It
> > is hard to
> > say that it has been given a definition mathematically.
> In 2D it has been dealt with nicely using analytic continuation, but no one
> has made this work
> for real 4D space as far as I know
> Some people have dealt with the Feynman integral using some nice Hilbert
> space mathematics, but I forget
> the references
I have been editing a book about functional integrals.
What is known and obvious is that Feynmann integral cannot be defined as
a measure unlike Wiener integral. This makes the calculation of Feynmann
integral tricky. One must do analytical continuation and perturbative
approach is in practice the only possible approach.
> My inclination is to discretize everything, and then everything becomes
> automatically definable, i.e. it becomes
> a finite sum over a large number of combinations rather than a divergent
> The measure underlying the Feynman integral is not clear. Here I would like
> to introduce a notion of subjective
> simplicity, whereby e.g. the weight of a path in the measure is the a priori
> simplicity of the path. As a first
> approximation algorithmic information could be used for a simplicity
> measure. But I have never pursued this idea
> mathematically, althought it makes sense to me intuitively.
The need to get rid of Feynmann parth integrals defined
by imaginary exponent exp(iS) and replacement of
them with configuration space integrals with real exponent defining
genuine integration measure is one of the motivations of TGD
It turned out that ordinary functional integral approach based
on exp(iS_K) is horribly divergent due to the nonlinearity of Kahler
action S_K. Kahler function (absolute minimum of Kahler action) as a
functional of *3-surface* (rather than 4-surface) is however nonlocal and
the standard divergences resulting from nonlinear local interaction
terms can be avoided. There are also divergences related to
Gaussian determinants but Kahler geometry makes possible elimination
of these divergences.
> Also, if you believe the "mind over matter" results from the Princeton labs,
> these could be explained by the mind altering
> the simplicity measure underlying the Feynman integrals governing particle
> motion. But this is raw speculation
> of course!!
The sum over degenerate absolute minimum spacetime surface
associated with given 3-surface X^3 could mimick path integral.
Just a thought....
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