Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 15 Apr 1999 09:56:38 +0300 (EET DST)
Is classical TGD exactly solvable?
The field equations of classical TGD are horribly nonlinear and the
hopes of writing exact solutions for them seem to be rather meager.
The intensive discussions a couple weeks ago in Time however stimulated me
to ponder whether *p-adic conformal invariance could provide a possible
general solution to field equations associated with Kahler action*. Also
the earlier criticism of Stephen King relating to the problems related
to General Coordinate Invariance served as a stimulus.
What is so magic in p-adic conformal invariance is that it makes sense
for arbitrary spacetime dimensions and that the unique algebraic extension
allowing square roots of ordinary p-adics is 4-dimensional for p>2! Thus
one could regard 4-dimensional p-adic spacetime as a local number field
and p-adicity would 'explain' spacetime dimension.
The 4-dimensional extension is unique: it is extension allowed
square roof of any ordinary p-adic number but of course not square root
of arbitary number in extension.
Z= x+iy +sqrt(p)(u+iv)
is the form of extension for p modulo 4 =3 allowing sqrt(-1)=i.
********************
I glue a piece of text from 'Mathematical Building Blocks'
in 'TGD and p-adic numbers'
(http://www.physics.helsinki.fi/~matpitka/padtgd.html)
to clarify the physical and mathematical background.
******
Background
One of the original motivation of the p-adic approach was based on the
following argument. Since two-dimensional critical systems are described
by conformal field theories, also quantum TGD, which describes
quantum critical Universe (in several senses of the word), must be
described by conformal field theory, which is however 4-dimensional. In
real context 4-dimensional conformal invariance is rather tiny symmetry
as compared to its 2-dimensional counterpart. In p-adic context however
situation is completely different since p-adic numbers allow algebraic
extensions of arbitrarily high dimension. The fact that for p>2 the
algebraic extensions of p-adic numbers allowing square root of ordinary
p-adic numbers are 4-dimensional, makes the idea even more attractive.
It was also found that p-adically analytic maps ('analytic' is understood
in the following as a synonym for 'holomorphic') maps from M^4_+ to CP_2
define a very general class of extremals of the p-adic Kaehler action
in the approximation that induced metric is flat. This enhanced the hopes
about the description of quantum TGD or at least, of its QFT limit, as
a p-adic conformal field theory in p-adic spacetime. Somewhat
disappointingly, it turned out that spin glass analogy seems to be a more
natural manner to understand the emergence of the p-adic topology.
Furthermore, the direct p-adic image of the real surface in the canonical
identification is not even p-adically differentiable so that this very
attractive idea had to be given up.
One could however wonder whether this extremely beautiful
generalization of conformal invariance suggested
by physical considerations could not be realized
in the basic structure of the theory in some delicate
manner. This hope is also encouraged
by the following argument. Quantum TGD is characterized
by generalized conformal invariance made possible by the magic conformal
properties of the light cone boundary. 'Ontogeny repeats phylogeny'
metaphor states that the general features of the quantum TGD defined
at the configuration space level are realized also at the level
of spacetime at p-adic QFT limit. Could it be that
the extension of the conformal invariance
from light cone boundary to p-adic conformal
invariance at the level of spacetime surface and imbedding space
is possible after all? Or putting it more precisely:
*Could it be possible that the most general solutions of the p-adic field
equations defined by Kaehler action having 4-dimensional CP_2 projection
corresponds to p-adically holomorphic maps from spacetime surface
to imbedding space in suitably chosen spacetime- and imbedding
space coordinates and subject to classical Virasoro conditions?*
The argument for exact solvability
It might be that the dream about exact solvability could be realized!
The core of the argument goes as follows.
1. Basic hypothesis
Field equations reduce to *two* separate terms O^1 and O^2 coming from
the variations of the Kaehler action with respect to *induced metric* and
*induced Kaehler form* (nonlinear Maxwell field) respectively. For all
known solutions of field equations these terms vanish separately.
Hypothesis: this occurs for all absolute minima and the vanishing of O^1
follows from p-adic analyticity (holomorphy) of imbedding map whereas
the vasishing of O^2 can be interpreted as generalized Virasoro conditions
so that the general solution of classical field equation in TGD has same
formal structure as in string models.
2. p-Adic imbedding space M^4_+xCP_2 allows generalized Hermitian metric
p-Adic conformal invariance requires that the metric of the imbedding
space and Kaehler form of CP_2 have no diagonal components in generalized
complex coordinates H^1,H^2 of H (H^i are two 4-coordinates of H analogous
to ordinary 2-coordinate z). That is: imbedding space metric is Hermitian
in generalized sense. This is strong requirement since it is not all
clear that Minkowski metric could be interpreted as a generalized
Hermitian metric. This turns out to be the case! Same is true for CP2
metric.
The crux of the matter is that sqrt(p) appears in extension: this
makes possible Minkowski signature. The quintessence of the argument
becomes clear by studying two-dimensional algebraic extension based on
sqrt(p). In two dimensional case one has z= x+sqrt(p)y and
z_c= x-sqrt(p)y. The metric ds^2= dzdz_c reads as ds^2= dx^2-pdy^2 and
indeed has Minkowski signature.
3. Induced metric and Kaehler form and energy momentum tensor are
Hermitian for analytic imbeddings.
For p-adically analytic imbeddings (H^1,H^2)= (f^1(Z),f^2(Z)), where
Z is complex four-coordinate of spacetime surface, the induced metric and
Kaehler form for analytic imbeddings are also *nondiagonal* in complex
coordinates. Also energy momentum tensor is *non-diagonal*. This is of
utmost importance.
4. Energy momentum tensor and second fundamental tensor have no common
components!
*Second fundamental form* consisting of covariant derivatives
of the gradients of imbedding space coordinates is however *diagonal* in
complex coordinates for p-adically analytic imbedding maps.
Consequence: the metric induced term O^1 in the field equations vanishes
identically since it involves contraction of second fundamental form
with energy momentum tensor which have no common components!
This phenomenon is completely analogous to what happens for Laplace
equation in plane in complex coordinates: Laplace equation reads
g^(zbarz)partial_Zpartial_barz Phi=0
and is identically satisfied by analytic or antianalytic functions. Also
string model field equations and minimal surface equations for soap films
are solved exactly by the same ansatz. Therefore TGD:eish spacetime is be
very much like 4-dimensional soap film spanned by a frame on lightcone
boundary.
5. Vanishing of the term O^1 is equivalent with classical Virasoro
conditions
The second term in the field equations involves the contraction of the
vacuum current associated with induced Kaehler field with
certain quantity involving Kahler form. The vanishing of this term
*selects one or possibly several analytic imbeddings* from all possible
analytic imbeddings.
For all known solutions this term vanishes because Kaehler current
*either vanishes or is lightlike*. The vanishing
of this term is equivalent with generalized Virasoro conditions selecting
one (or possibly several) analytic imbedding maps satisfying.
The point is that the variation of Kaehler action under infinitesimal
conformal transformation of imbedding space comes from this term and is
by definition proportional to the classical charge associated with the
infinitesimemal generator of the variation. Vanishing of these classical
charges is nothing but classical Virasoro conditions.
Similar conditions are encountered in string model: in string
models Virasoro conditions state that the induce metric has the
form g_zbarz in complex coordinates chosen.
Consequences
What is especially important is that p-adic solution ansatz seems to
make sense also in the real context!! The number field property of
the 4-dimensional algebraic extension of p-adics is not needed at all.
This would mean following:
a) p-Adic and real solutions are in one-one correspondence
and it the mapping of real theory to p-adic theory would be possible
in extremely elegant manner.
b) p-Adic length scale would manifest itself directly in properties of
the real spacetime surfaces since powers of p would appear in the
expansions of p-adically analytic functions.
c) A complete solution of both real and p-adic field equations
would be possible. An open question is whether p-adic analyticity
guarantees automatically absolute minimum property. This might be the
case!
d) If the solution ansatz really works, TGD would belong with classical
string model and Euclidian Yang Mills theories to the respected company of
exactly solvable nonlinear field theories. Even more, in TGD classical
theory is exact part of quantum theory rather than only an approximation.
Best,
Matti Pitkanen
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