Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 10 May 1999 09:50:56 +0300 (EET DST)
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
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