Stephen P. King (stephenk1@home.com)
Mon, 10 May 1999 12:06:55 -0400
Dear Matti,
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! ;)
> 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. 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"!
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.
> 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".
> 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.
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.
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
> 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 .
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
one p-adic set.
> 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
thinking... I'll read and think about this some more...
> 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!
> 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?
> 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?
> 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?
> 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
up about this. It is mentioned in Conceptual Problems of Quantum Gravity
(reference in Hitoshi's papers).
> 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.
The non-rational remainder under mappings might be problematic... It
may connect to irreducible error in observations...
> 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
Onward to the Unknown!
Stephen
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