# [time 303] Re: [time 297] Mapping p-adic spacetime to its real counterpart

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 11 May 1999 10:25:58 +0300 (EET DST)

To say the truth, I am in awe of your knowledge of the math
involved! I
am happy that critique is useful. :) I have a few silly questions...

Matti Pitkanen wrote:
>
> 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.

Is it possible that there exist a class of pairs {M_i, I_p}, where
M represents the spacetime [hyper?]surface and I_p represents the p-adic
imbedding space, such that there is an asymptotic hierarchy of
inclusions of the {M_i, I_p} that, at the limit of +/- \inf, there is
isomorphism between M_i and I_p and a dismorphisms for any i, p > \inf.?
I hope this makes sense! ;)

[MP]
I am not sure whether you regard M_i as real or p-adic manifold and
I could not quite understand what you mean by asymptotic hierarchy of
inclusions. Certainly this kind of sequence might exists but since
I do not know about motivations for the existence of sequence I cannot
imagine any concrete example.

What I am considering is basically the problem of mapping real manifold
M_R to its p-adic counterpart M_p (in present case M is the 8-dimensional
space H where spacetimes are 4-surfaces). The idea is that this mapping
induces the mapping of submanifolds of M_R (spacetime surfaces)
to submanifolds of M_p somehow.

> 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.

The use of infinitesimal (under any circumstance!) is suspicious
to me,
since it tacitly assumes zero error (no uncertainty) observations.

[MP]
Infinitesimal is convenient physics shorthand: there is
rigorous group theory behind this all. One defines
conserved charges as variational derivatives of action with respect
to the action of various abelian subgroups generated by Lie-algebra
elements of symmetry group. For instance, angular momentum in
specific direction corresponds to the action of rotation around
the direction of angular momentum in that direction.

[Stephen]
I believe that these are *very* special cases, and need to be better
understood! One thing that infinitesimals do is that they create the
illusion that all observations use the exact same measuring 'rod'. This
may not be the case. I have been trying to explore Weyl's gauge theory
to understand how we can do physics when each observer has its own
unique measuring rod and clock, as opposed to assuming that an absolute
standard is imposed "from above"!

[MP]
Noether theorem makes the symmetry thinking of physicists mathematically
rigorous. The concept of Noether charge as completely independent
of any measurement theory: it is purely group theoretical concept:
infinitesimals are only bad linguistic habit of physicists (or who
knows!?). This group theoretical aspect becomes decisive in quantum
mechanics which to large extend reduces to a representation theory for
symmetry groups.

My personal belief that the concept of measurement comes into play only
at the level of quantum measurement theory, which is the poorly
understood part of quantum mechanics and involves
the concept of quantum jump and basically consciousness. On the other
hand, Riemannian geometry is classical theory of length and angle
measurement, and I take it as God given and go on to postulate that entire
QM with quantum jump excluded is just infinite-dimensional Riemannian
geometry. I could be wrong!!
Somehow I however believe that Riemannian geometry is something final.

I have make little quantitative progress... :(

> 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.

One thing that I have always wondered about Noeter's theorems,
which relate conservations to symmetries, is that the symmetries are
always considered using "time" as a parameter; but it is a "time" that I
would call exactly "periodic". Ben discusses a spiral/fractal time in
http://goertzel.org/ben/timepap.html that IMHO is possibly more
realistic. We must remember that the Noeter theorems are phrased in
classical thinking, and as such are ideal.

[MP]
What Noether theorem says that variation of action for a volume of
space vanishes and by equations of motion the variation reduces to
a surface integral. What happens under suitable
additional assumption is that the surface integral reduces
to contributions from two time=constant surfaces and these contributions
must cancel: this says that classical charges at these t=constant
surfaces are identical: charge is conserved. Metric signature
does not matter: only the idea that there is cylinder like region such
that charges do not flow out from the sides of the cylinder.

Since Noether theorem relies on group theory, one can generalize
the concept of Noether charge straightforwardly to quantum theory context.
Formulas are the same: now Noether charges become Hermitian operators
whose real eigenvalues correspond to quantum mechanical charges, quantum
numbers such as spin and momentum.

> 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.

One question, how would an observer *know* that their fields have
changed if their tools of measurement change also? As we consider a
transformation of fields, we must understand that the observer is *not*
independent of the transformations, as would the classical "external
observer".

[MP]
You are certainly right. I see however this question as a problem of
quantum measurement theory or basically problem of consciousness theory.
The classical theory of fields (absolute minimization of Kahler action in
TGD) as well as 'Schrodinger equation' are completely observer dependent:
there are no observers or observations in classical world nor in single
quantum mechanical time evolution/quantum history (so I believe). Quantum
histories are dead: life and observations are in quantum jumps between
challenge but a challenge to consciousness theory, which we should
construct first(;-).

With the risk of repeating myself: there is no observer in the sense
of continuous stream of consciousness residing in some corner
of spacetime or floating above the Hilbert space.
Observer exists only in the quantum jumps, moments of consciousness.
Therefore one cannot say that there is any observer subject to these
transformations: the problem disappears.

> 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.

I must confess that I do not fully understand the meaning of the
group symbols, but I am beginning to! :) Thank you Matti! ;)

> 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.

I think that Hitoshi's use of the RW metric to talk about the
expansion of an observer's space-time applies here! We must remember that
each observer, at each moment, is using a time origin unique to the
individual LS, which is the observer. Thus, when we think of modeling
the observations of co-moving observers, we are inferring from our own
time origin point.

[MP]
You can interpret different Lorentz transformation related frames
as associated with various comoving 'observers', yes. There however still
remains rotational degeneracy of the frame (rotation group SO(3)): this
is the problem from my point of view in the sequel.

[Stephen]
A local friend and I have been exploring the implications of
Lorentz
transformations, and have arrived at the conclusion that such are
restricted to the possible inferences of a single observer and can not
be assumed to well-model the actual observations of other LSs. In other
words, the Lorentz invariance of possible observations is a group that
each observer has, and there is no necessary isomorphism between the
SO(3,1) of one LS's observations and another's. All that is required for
consistency is that there is the possibility of mutual entropy in the
information that can be encoded in the SO(3,1) of each.

[MP]
This would be very nearly equivalent with the viewpoint of General
Relativity. SO(3,1) is only the group of tangent space rotations
preserving tangent space inner product and physically corresponds to
approximate Lorentz invariance of the spacetime locally. This symmetry
is however gauge symmetry and does not give rise to conserved charges such
as angular momentum and is hence problematic.

In TGD the big idea is that Lorentz invariance is actual global symmetry
of the imbedding spaceH=M^4_+xCP_2, rather than spacetime itself.
Poincare invariance is broken only by the presence of lightcone
boundaries. Immediate prediction is standard subcritical RW cosmology.

I realize that we have quite different view about observer concept. You
assume that observer is modellable mathematically at fundamental level
whereas I throw the observer out and leave only conscious observations
associated with quantum jumps replacing physical time evolution
with a new one so that also the hypothesis about single objective
reality is thrown out. With so much thrown out also many problems
disappear(;-). I hope that this what I am doing is not like solving
the problem of consciousness by saying that there is no consciousness.

[Stephen]
Thus we say that that rock is at such and such a position iff each
observer involved has information encoded is a similar enough manner. I
hope to have some more quantitative formulation of this soon! :)

> 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
>
> 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 .

Is it true that the mapping can have duplication (overlap and
underlap)? I am thinking that the equation above does not rule out the
possibility of mapping the "same" pattern in the reals onto more than

[MP]
Canonical identification from reals to p-adics is two-valued for
reals having finite pinary digits. The reason is that in this case
the pinary expansion of real is not unique (good example: 1= .9999999..
which generalizes to expansions in powers of prime p instead of 10
easily). One can however use systematically the expansion with
finite pinary digits: this is also numerically the only possibility.

I should mention also a second problem related to canonical
identification: how to map negative real numbers to their p-adic
counterparts. Just now I believe that the requirement (-x)_p = -x_p so
that negative of real number is mapped to the negative of its p-adic
image, is the correct option. It follows from the mapping of complex
numbers to their p-adic counterparts described in following by restricting
the map to real axis.

> 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).

I hate to say so, but your requirement that the imbedding space is
nondynamical is problematic. Unless we can show that it is a fixed
"portrait" of a higher dimensional dynamical space, we would not be able
to have thermodynamics on it. I may be miss understanding your

[MP]
The basic point is that dynamics for imbedding space CREATES the problems!

Only the dynamics of spacetime surfaces is needed classically.
The dynamics of surface motion makes induced spinor connection and metric
dynamical despite the fact that imbedding space metric, etc. is
nondynamical.
The classical dynamics of TGD is dynamics of shadows: for instance, the
metric of imbedding space is rigid object and induced metric is its
dynamical shadow on 3-surface moving and changing its form.

At quantum level the dynamics is at the level of configuration space
of 3-surfaces but basically reduces to the dynamica of 3-surfaces plus
that for induced spinor fields.

The existence of configuration space of 3-surfaces as a Kahler
manifold with spinor structure relies on the symmetries of the imbedding
space metric: for generic imbedding space without any symmetries
the entire construction would fail. The reason is that the symmetries
of imbedding space are 'lifted' to much larger symmetries of
configuration space and these infinite-dimensional symmetries
guarantee that Riemann connection exists (this was observed
already by Freed when he constructed the geometry of loop spaces
in his thesis). Consistency implies existence philosophy works here.

Quantum theory for dynamical imbedding space would also lead to horribly
nonrenormalizable theory: for instance, low energy limit of string models
is nonrenormalizable theory since it is defined in
10-dimensional spacetime. Also physics would come out wrong: for
instance, TWO gravitons would be predicted since the
dynamical metric of the imbedding space would also give rise to graviton
besides the graviton predicted by quantized dynamics of 3-surfaces.

> 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.

Why do we not identify coordinate choices with observers, there
are MANY of each... Is the cardinality of the class of each identical?
Each observer has its own Y^3!

[MP]
Interesting idea, I have sometimes pondered the idea that
consciousness brings in breaking of GCI by introducing coordinates.
I considered this also a possible solution of problems with GCI
but since GCI has been behind all real progress in TGD gave up
this idea.

>From TGD point of view each Y^3 (X^4(Y^3) represents entire
classical world: quantum history is superposition of these.

My idea is to fix preferred (perhaps even unique)
imbedding space coordinates. This choice depends on Y^3:
if the choice of H-coordinates is sufficiently unique, the
mapping of X^4(Y^3) to its p-adic counterpart induced by the
mapping H_R-->H_p becomes unique and one has the highly desired
GCI in the sense that possible allowed coordinate changes
commute with identification map.

> 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.

You lost me here. :( I'll re-read until it makes sense. One
question, do you think that only one group of each type "exists" in the
Platonic/ontological sense? If so, would it still make sense to think of
many "perspectives" of the One, like the many possible shadows on
Plato's cave wall?

[MP]
This is somewhat technical: the goal is to show that one can
find preferred coordinates unique modulo the phase rotations
in subgroup of SO(3,1)xSU(3), which is SO(2)xU(1)xU(1) and
corresponds to spin and color isospin and color hyper charge.
>From this GCI follows with suitable definition of identification map.

I believe that imbedding space and its isometry group is indeed unique
in Platonic sense. I would guess that the infinite-dimensional symmetry
group of configuration space isometries and corresponding veryvielbein-
group probably contain however most finite-dimensional groups as
sugroups. Note also that vielbein group which is infinite-dimensional
unitary group contais representations of all finite-dimensional
Lie groups.

> 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!

How many "mutually commutative" sets of observables could exist if
we
allow for each to be "almost" convex?

[MP]
You are probably speaking of G= SO(3,1)xSU(3). Observables correspond
group theoretically to Lie-algebra of G. Mutually commuting observables
correspond to Cartan subalgebra generating maximal abelian subgroup of G.
For G under consideration Cartan subgroup is H= SO(1,1)xSO(2) xU(1)xU(1).
Lorenz boosts in given direction and rotations around that direction
plus rotations generated by color isospin and hypercharge. Any
Cartan subalgebra gives a set of mutually commuting observables.
The space of them is the coset space G/H, H Cartan subgroup.

BTW, for SU(3) this is SU(3)/U(1)xU(1), which a mathematician Barbara
Shipman found to be related in mysterious manner to the mathematical model
of the dance of honeybee and guessed that quarks must somehow be involved!
TGD inspired explanation of the appearence of this space is on my homepage
and relies crucially on the fact that TGD predicts classical color
fields in all scales (although gluons are confined) and to to the fact
that macrosopic 3-surfaces have besides ordinary rotational degrees
of freedom also color rotational degrees of freedom: color rotating
3-surfaceshas color charges just like rotating body has angular momentum.

> 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.

Interesting!

> 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.

What happens to the remainder that is not mapped?

[MP]
That mapping is possible only for the subset of H is just what I
want since this implies that the p-adic image of spacetime surface
induced by this map is discrete in the generic case. Finite portions
of H are mapped to unions of discrete 4-dimensional surfaces. The
image of entire H is dense in H_p but does not fill H_p completely.
[The mapping is discontinuous in phase angle degrees of freedom: the
Pythagorean rays of complex plane are mapped to randomly mixed

The basic problem has been that direct canonical identification
of real spacetime surface yields continuous but non-differentiable
and hence Kahler action do not exist since derivatives
appearing in them are ill defined. The task has been to somehow
loosen the canonical identification so that image becomes discrete
and one can complete it to smooth p-adic surface by requiring
equations associated with Kahler action.

My earlier approach was to replace the direct canonical image
with its pinary cutoff, which is discrete set and hope that
surface going through these points and satisfying field equations
would exists for some maximal, or perhaps even infinite, pinary cutoff.
There earlier posting was about the possibility of infinite
pinary cutoff. The problem is
that this approach does not have any geometric appeal: it is
too numerical! And there is no connection with quantum theory.

The proposed form of real to p-adics mapping however implies
automatically that image is discrete and one could hope that
the phenomenon of p-adic pseudo constants and classical nondeterminism
of Kahler action could make it possible to find unique
p-adic spacetime surface spanned by the discrete canonical image of
real spacetime surface. Unfortunately I have no ideas about how
this could be achieved! Frustrating!!

What makes me to take this approach
seriously is the direct connection with geometry, number theory
and with quantum measurement theory and sudden consciousness
theoretic understanding of why Platon believed so firmly in
rational world(;-).

> 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(;-).

Interesting, but is it necessary and sufficient to just assume
that a single unique discrete Pythagorean phase angles *exists*, are we
not assuming an absolute basis to make this assumption? We can show that
such exists, but it is only asymptotically *knowable* and that knowledge
by one LS is not necessarily knowledge by all! I think the discussion of
the inner product problem in QGR is related... I'll try to dig something
(reference in Hitoshi's papers).

[MP]
The existence about existence of discrete Pythagorean angles reduces
to the hypothesis about existence of imbedding space H.
The angles are defined for preferred complex coordinates, which are
fixed by pure group theoretical considerations.

It is absolutely essential
that H is non-dynamical and has its symmetries: if H is dynamical or
just generic 8-dimensional Riemannian manifold,
there are no preferred coordinates and one can safely forget
on consistency implies existence philosophy.

> 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
> 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.

The non-rational remainder under mappings might be problematic...
It may connect to irreducible error in observations...

The discreteness of the image
is just what gives hopes of defining p-adic spacetime surface as
smooth surface. The rationality of cosines and boost velocities
could be translation of the belief that
all our measurements are always expressible in terms
of rational numbers. As physicists we are bound inside the confines of
rational world.

The same phase preserving mapping should work also at the level of
configuration space spinor fields but now basic QM, instead of GCI,
implies it. The nonconstant phases of basis of CH spinor fields (quantum
state basis) are mapped to their p-adic counterparts
in the same manner. This is required by the linearity of QM
and by the requirement that the action of eigen observables realized
as phase multiplication commutes with the identification map.
The phase is Pythagorean only in discrete set of CH and image is discrete.
Discreteness of the image should make it possible to complete the image
of CH spinor field to a smooth p-adic CH spinor field by requiring that
the image is eigenstate of the maximal commuting set of p-adic
observables.

> 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

Best,
MP

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