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

*Thu, 15 Apr 1999 09:56:38 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 231] Re:"**Previous message:**Hitoshi Kitada: "[time 229] Direction of time or Free will"

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

**Next message:**Hitoshi Kitada: "[time 231] Re:"**Previous message:**Hitoshi Kitada: "[time 229] Direction of time or Free will"

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