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

*Mon, 10 May 1999 09:50:56 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 298] Re: [time 252] Re: [time 251] Peter Wegner's paper"**Previous message:**Stephen P. King: "[time 296] Re: [time 295] Re: [time 291] Re: Expressiveness of interactive computing"**In reply to:**Peter Wegner: "[time 295] Re: [time 291] Re: Expressiveness of interactive computing"**Next in thread:**Stephen P. King: "[time 299] Re: [time 297] Mapping p-adic spacetime to its real counterpart"

I have been pondering more and more seriously the problem of mapping real

spacetime surface and imbedding space to their p-adic counterparts.

I proposed already earlier a possible solution of problem but it

was not satisfactory: thanks for Stephen for critical comments.

The basic problem that canonical identification mapping real coordinates

to their p-adic counterparts is not manifestly General Coordinate

Invariant concept and one should be able to identify preferred

coordinates of imbedding space, where canonical identification applies

in some form, in order to achieve GCI.

First some general comments on frames of reference question

and then a brief description of how the concept of preferred

frame appears as a purely technical concept in the formulation

of quantum TGD.

General Coordinate Invariance

The Principle of General Coordinate Invariance states that the

laws of physics cannot depend on frame of reference. A slightly different

formulation says that diffeomorphisms of spacetime do

not represent genuine physical degrees of freedom. You do not get

new physical configuration by mapping various tensor quantities describing

physical fields to their diffeomorphs. This means that GCI as a symmetry

is like gauge invariance: there are not conserved quantum numbers

associated with infinitesimal general coordinate transformations.

This is in fact leads to the basic conceptual problem of General

Relativity: one does not have any GCI definition of energy and momenta

since Noether theorem gives identically vanishing conserved diffeo

charges.

Most importantly: it does not make sense to speak about 'active

diffeomorphisms'. One can however speak of

isometries of spacetime as symmetries: in this case the action to

fields is different: one can say that fields are replaced

with general coordinate transformed counterparts but *coordinate system

is not changed*. This transformation creates genuinely new field

configuration and in case of isometries of spacetime. This new field

configuration solves the field equations.

Of course, in practice one must almost always solve field equations in

some frame of reference typically fixed to high degree by symmetry

considerations. This does not mean breaking of GCI but only finding

the coordinates in which things look simple.

For Robertson-Walker cosmology standard coordinates (t,r, theta, phi)

are special in the sense that t= constant snapshots

correspond to the orbits of Lorentz group SO(3,1)

acting as isometries of this cosmology. t= constant snapshots

are coset spaces SO(3,1)/SO(3) originally

discovered by Lobatchewski and identical

with proper time constant hyperboloids of future lightcone of

Minkowski space.

RW coordinates are *NOT UNIQUE*. For subcritical cosmology,

any Lorentz transformation generates new equally good

RW coordinates with different origin interpretable as position of

comoving observed! The cosmic time t is Lorentz invariant under Lorentz

transformations and is not changed.

Mapping problem and preferred frames

In TGD framework the problem of preferred frame (in purely technical

sense, not physically) has been one of the longstanding problems in the

attempts to understand the relationship between real and p-adic quantum

TGD (which are actually different aspects of one and same quantum TGD).

The problem boils down to the following mapping problem:

***How is real spacetime/imbedding space/configuration space of

3-surfaces/space of configuration space spinor spinor fields mapped to

its p-adic counterpart?***

Some form of canonical identification between real and p-adic imbedding

space coordinates must somehow mediate this mapping but how does it occur

precisely. Canonical identification mapping reals to p-adics

is given by

x_R = SUM_n x_np^n --> SUM x_n p^(-n)= x_p .

The problem is that canonical identification is *NOT MANIFESTLY GCI*

since it must be defined in PREFERRED COORDINATES!

For instance, if one goes to new coordinates the p-adic image of

the new spacetime surface is not identical with the original one.

GC transformations do NOT commute with canonical identification map!

How to find preferred coordinates for imbedding space?

In order to achieve GCI, one must be able to find some PREFERRED

COORDINATES for imbedding space H=M^4_+xCP_2, in which canonical

identification map is performed. If the preferred coordinates are

unique, everything is ok. If NOT, then the coordinate transformations

between preferred coordinate systems must COMMUTE with the canonical

identification map.

a) If imbedding space were not nondynamical, no preferred coordinates

would exist and p-adic quantum TGD would break GCI. This would be

the end of p-adic TGD.

b) Fortunately, imbedding space H(=M^4_+xCP_2) is NOT dynamical

but fixed by symmetry considerations and by the requirement that

configuration space geometry exists mathematically. The coordinates,

which transform *linearly under maximal subgroup* of SO(3,1)xSU(3)

(Lorentz group cross color group), form a family of preferred

coordinates. The preferred coordinates are just linear Minkowski

coordinates and complex coordinates of CP2 transforming linearly under

some subgroup U(2) of SU(3).

c) There are however QUITE TOO MANY coordinate choices

in this family parametrized by SO(3,1)xSU(3)

and SO(3,1)xSU(3) *cannot commute* with canonical

identification. One must be able to specify preferred

coordinates more uniquely. This is possible.

d) Given 3-surface Y^3 on lightcone boundary representing initial

state of a particular spacetime surface has well defined classical

momentum P^k and angular momentum vector w^k as well as

classical color charges Q_a: these charges are Noether

charges associated with the absolute minima $X^4(Y^3) of Kahler

action.

d1) One can require that the preferred Minkowski

coordinates correspond to *rest frame* of Y^3 and that

spacelike angular momentum vector w^k defines the direction

of one of the coordinate axes, say z-axs. Hence

coordinate system is specified only up to planar rotations

around the z-axis forming group SO(2).

d2) In a similar manner one can show that

the complex coordinates of CP_2 are speficified only up to a

color rotation in Cartan subgroup of U(1)xU(1) of SU(3) representing

color rotations generated by color hypercharge and color isospin.

Here very special properties of SU(3) are crucial: SU(3)

allows completely symmetric structure constants d_abc so that one

can from from the vector Q_a of classical color charges second

vector R_a = d_a^bcQ_bQ_c commuting with Q_a as an element of SU(3)

Lie-algebra. Q_a and R_a as Lie-algebra elements span the unique

Cartan Lie-algebra U(1) xU(1, which generates the allowed color rotations.

What is important is that this gives an additional item to the list of

arguments stating that CP_2 is unique choice for imbedding space.

Comment: Already at this stage one notice precise analogy with

quantum measurement theory. SO(2) belongs to and U(1)xU(1) is

the group spanned by maximal commuting set of observables associated

with isometries of H!

e) The problem is that one can specify the preferred coordinates

only up to a rotations in SO(2)xU(1)xU(1). GCI requires that these

rotations COMMUTE with canonical identification. This can be indeed

achieved by a proper definition of the canonical identification map!!

What is done is to notice that the rotations in question correspond

to *phase multiplications*, when one uses complex coordinates for CP_2

and for the plane E^2 orthogonal to momentum and spin vector w^k.

One must require that the phase exp(iphi)

of a given complex coordinate z is mapped AS SUCH such to its p-adic

counterparts: no canonical identification is involved. Geometrically

this means that products of real phase factors are mapped to products of

p-adic phase factors. The modulus |z| of z is however mapped by canonical

identification, which is continuous map.

f) This does not make sense unless phases are complex rational number

(rational numbers can be regarded as 'common' to both reals and p-adics

as far as phases are considered) and thus correspond to Pythagorean

triangle possessing rational sides

a,b,c:

a= 2rs, b= r^2-s^2, c= r^2+s^2, r and s integers.

In this case one can identify the real rational phase as such with

its p-adic counterpart. This means angle quantization.

g) Actually this applies also to the hyperbolic

phase factor exp(eta) associated with (t,z) pair of Minkowski coordinates

and in this case quantization of allowed boost velocities mathematically

equivalent with Pythagorean triangles happens so that

the group of allowed coordinate transformations extends to the

Cartan subgroup SO(1,1)xSO(2) of Lorentz group (boots in direction of

spin plus rotations orthogonal to it). Altogether this means that

only the coordinates sqrt(t^2-z^2) and rho= sqrt(x^2+y^2)

and the moduli of CP_2 complex coordinates are mapped

by canonical identification to their p-adic counterparts.

Conclusions

The conclusions are following.

a) Quantum world according to TGD has a well defined Pythagorean aspect.

Only the discrete set of Pythagorean phase angles and boost

velocities are mapped to their p-adic counterparts. By the way,

Pythagoras was a real believer: the pupil of Pythagoras, who discovered

sqrt(2) payed for his discovery with his life! Perhaps it is easier

to forgive or at least understand Pythagoras now(;-).

b) There is a deep connection with quantum measurement theory. The

phases, which are mapped as such to p-adic numbers correspond to

maximal mutually commuting set of observables formed by the isometry

charges and generating the group SO(1,1)xSO(2) xU(1)xU(1) of

the isometry group of imbedding space. Canonical identification map

commutes with the maximal mutually commuting set of observables.

c) Without the special features of SU(3) group (existence of

completely symmetric structure constants) it would not

be possible to realize GCI. Neither would this be possible

if imbedding space were dynamical as in string models.

d) The p-adic image of the spacetime surface is discrete in generic case

since only rational phases are mapped to their p-adic counterparts.

One must complete the image to a smooth surface and the phenomenona

of p-adic pseudo constants (p-adic differential equations

allow piecewise constant integration constants) and nondeterminism

of Kahler action give good hopes that p-adic spacetime surface can

satisfy the p-adic counterparts of the field equations associated with

Kahler action. Even the formal p-adic counterpats of the absolute

mininization conditions can be satisfied since they correspond to purely

algebraic conditions.

e) Similar phase preserving mapping must be applied to the basis

of configuration space spinor fields in order to achieve consistency

of canonical identification with linearity of QM and it seems that

phase preserving canonical identification provides universal solution

to the mapping problem.

Matti Pitkanen

**Next message:**Hitoshi Kitada: "[time 298] Re: [time 252] Re: [time 251] Peter Wegner's paper"**Previous message:**Stephen P. King: "[time 296] Re: [time 295] Re: [time 291] Re: Expressiveness of interactive computing"**In reply to:**Peter Wegner: "[time 295] Re: [time 291] Re: Expressiveness of interactive computing"**Next in thread:**Stephen P. King: "[time 299] Re: [time 297] Mapping p-adic spacetime to its real counterpart"

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