Matti Pitkanen (matpitka@pcu.helsinki.fi)
Wed, 6 Oct 1999 17:24:10 +0300 (EET DST)
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
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