**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Wed, 6 Oct 1999 17:24:10 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 887] Re: [time 886] Unitarity finally understood!"**Previous message:**Matti Pitkanen: "[time 885] Re: [time 879] Re: [time 877] Re: Your assumption"**Next in thread:**Hitoshi Kitada: "[time 887] Re: [time 886] Unitarity finally understood!"

Dear Hitoshi,

I believe that I understand the unitarity condition now.

To streamline the notation I just denote

L_0(int)==V ,

and

L_0(free)==L_0.

|m>= |m_0> +|m_1>

is the decomposition of scattering state to free or bare state

and genuine scattering contribution (cloud of virtual particles.

S=1+T

******************************************************

1. The condition guarateing unitarity

The condition guaranteing unitarity is

V|m>=V|m_0>

The action of interaction operator to dressed

state is same as to bare state. The contribution

of virtual cloud of particles to interaction vertex

vanishes.

Vanishing coupling constant and wave function renormalization

are immediate consequences. This is just what is

expected on basis of quantum criticality.

Kahler coupling strength is fixed point of coupling constant

evolution and since it is the fundamental coupling, all

vertices are RG invariant.

p-Adic coupling constant evolution means that Kahler coupling depends

on p-adic prime labelling configuration space sectors

so that continuous coupling constant evolution

is effectively replaced by a discrete one.

2. S-matrix is trivial in real context

Usually unitarity condition in terms of scattering

matrix T (S=1+T) reads

T+T^dagger +T^daggerT=0.

Now it decomposes into two separate

conditions

T+T^dagger=0,

TT^dagger =0.

This implies that absorptive part of forward

scattering amplitude is zero and in consistency

with this total cross section is zero. This

means obviously trivial theory in real context.

3. S-matrix is nontrivial in p-adic context.

In p-adic context p-adic probability saves the day!

There is very beautiful interpretation for this.

Total p-adic probability for n-particle state

to scatter in non-forward direction vanishes.

As far as total p-adic probabilities are considered,

theory is free! In particular, single particle

states are p-adically stable!

Real probabilities obtains from p-adic probabilities by

canonical identification and normalization are however

nonvanishing so that theory is physically interesting.

Thus the theory is as close as possible to free

theory but quite not!

4. S-matrix defines p-adic cohomology theory

The conditions T=-T^dagger and T^daggerT=0 state

that iT is Hermitian nilponent operator and

defines cohomology theory! Thus Wheeler's

dream about basic physical law as equivalent

of "boundary has no boundary" is realized!

Cohomology means following.

a) States T|x> are always annihilated by T: they are exact states.

b) States |y> with the property that T|y>=0 but |y>

is not of form T|x> are closed but nonexact states and generate

cohomology group.

In TGD these states have nice interpretation: time development

operator U leaves cohomologically nontrivial states

invariant! They are fixed points of time evolution

by quantum jumps and ideal candidates for selves!!

Selves as elements of cohomology group: something which

does not come into mind spontaneously!

Note that exact states have zero norm and are not physical:

note also that p-adic Hilbert space possesses zero

norm states always and that these states are crucial

role in the construction of p-adic S-matrix.

5. p-Adic cohomologies are nontrivial and form

category closed with respect to direct sum and tensor

product

In real context Hermitian matrix cohomology

has no solutions. In p-adic context situation is

different.

Two-dimensional matrices iT of form

iT_11=a , iT12 = b= b_1+ib_2

iT21= b^* iT22= -a

such that a = sqrt(-b_1^2 -b_2^2)

are Hermitian and satisfy hermiticity and nilpotency

conditions.

Note that a is "p-adically real" number by Hermiticity:

this is possible since the square root of "negative"

p-adic number can exist as p-adically real number!

In real context this is of course impossible.

One can construct from two-dimensional iT:s and

single dimensional vanishing iT infinite number

of iT matrices by forming tensor products and

direct sums. S-matrices thus form a "category"

closed with respect to tensor product and direct

sum. Here is the fundamental category which

Stephen has been dreaming of!

I learned that in 3x3-dimensional case there exists

no solutions to the defining conditions and suspect

that all solutions are generated from two-dimensional

solutions and trivial one-dimensional solution.

Best,

MP

**Next message:**Hitoshi Kitada: "[time 887] Re: [time 886] Unitarity finally understood!"**Previous message:**Matti Pitkanen: "[time 885] Re: [time 879] Re: [time 877] Re: Your assumption"**Next in thread:**Hitoshi Kitada: "[time 887] Re: [time 886] Unitarity finally understood!"

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