4. Final reflections and open issues
Considering the general question placed by Barbour (1994):
is time a basic concept? The result of the analysis of
arguments and of theoretical discourse developed in the previous two sections, as well as the mathematical results obtained
in the previous section, leads to an answer to the above question, with another question: what time?
Indeed, the result of the work developed above shows that
temporality and the notion of time go beyond the more restrictive chronometrizable notion
of time, that is internally definable with respect to a space-time geometry.
The arguments of Barbour (1994), of Kauffman and Smolin (1997),
and Smolin (2001), along with the Wheeler-DeWitt equation show how a chronometrizable physical clock time may indeed not be
a basic concept .
However, whenever we consider a quantum computation, when we address a timeindependent ket , or the relations between different physical observables’
eigenbasis, we find a basic temporality that is definable with respect to the configuration of the relational
nexus of a system of relations between objects.
In physics, as we saw in the previous section, if we accept the fundamental role of a configuration space, we find this temporality present,
even in the absence of any kind of fundamental clock time.
Therefore, we are led to a conceptual need of defining a relation time, as a fundamental (ordinal) time, which is
the time of the order of the objects’ positions in the relation, and that ultimately proceeds from the connection of
two individuations that are separated, but linked by the relation, and, thus, are temporally connected in the temporality that is
the order of terms in the relation.
For relational structures such that, given any two objects X and Y , N(X&Y ) is either N(X
Y ), N(X Y ) or N(X↔Y ), it is possible to obtain a numeric scale for the objects that reflects the relation time, by introducing the structure of prevalences
in relational nexus (O,%N ), defined as:
X
%N Y
if, and only if, N(X&Y )=N(XY ) ∨N(X&Y )=N(X↔Y )
X
sY if, and only if, N(X&Y )=N(X↔Y )
for any
X, Y ∈O.
This order expresses the
relation time that results from the peculiar order of terms in the relational nexus N(X&Y ). Now, if we were to combine the objects with probabilities, and introduce von
Neumann and Morgenstern’s (1953, [1990]) axiomatic for expected utility, a chronometric could, then, be assigned to
the above structures, reflecting the order %N , which would result in an axiomatic for expected time. Of course, this is a special case, but it
serves to show how a chronometric time may emerge from a purely relational temporal background.
In this sense, one may be inclined to agree with the position that a
chronometric time may not be fundamental, and may emerge
from a more fundamental temporal structure that is purely ordinal. The above result is, at least, sufficient to show that
an ordinal temporality, that is a time that emerges within a relational
structure, can be more fundamental from a physical point of view,
playing a foundational role in an emergence of a space-time chronometrizable physical time.
Commentary by Tony B.
The above paper is imo a brilliant exercise in addressing
the question of time. Carlos Pedro is a professional mathematician (or possesses the information and nous to act and write
as such) and Maria Odete is similarly well versed in questions of philosophy, scientific history and metaphysics.
This paper is rather technical and 'hard to read' for
the underinformed, including myself.
However the crux of the matter strikes at the core of
my own decade long efforts to describe the underpinning critera in refards the workings of the universe through a pertinent
cosmology.
So I shall try to simplify a bit and also add some details
from my own meanderings for Carlos and Maria to perhaps consider as 'missing details and/or initial conditions' in their opus
on time.
We know, that Galilean Relativity and time utilised the
Newtonian 'absoluteness', which was then displaced by Einstein's Special Relativity (SR), which semingly 'did away' with any
absolute reference frames.
History then shows, how SR became extended to General
Relativity (GR) in redefining the acceleration of inertia as a curvature in spacetime itself and leading to an alternative
geometrical description for the phenomenon of gravitation. This spacetime is 4D and Minkowski 'flat' in SR and so became 'curved'
in the metric mathematical representations expressing such concepts as 'distance between two points' and the 'time taken to
travel this distance'.
Time is however still considered to be chronological
and measured by synchronised clocks in GR, which now must hower take the effect of masses upon time measurements into account,
as the masses curve spacetime and so also 'bend' time.
Enter quantum mechanics and we have Schroedinger's Equation
in both a time-independent formalism, which describes 'bound systems', say a 'quantum particle in a box', upon which say an
electron 'bound' to an atom, can be modelled. This time-independent formalism then describes 'Standing Waves' and
boundary conditions (nodes) as say the linear dimensions of the 'box' or the atom. This equation is a simple outgrowth of
the Bohr Atom, with momentum- and position coordinates replaced by quantum operators.
The time-dependent form of Schroedinger's Equation describes
a 'free particle' however and now the time-coordinate is to first order and the position coordinates is to second order and
subsequently not consistent with SR.
Then we have the Wheeler-de Witt Equation and the Wavefunction
of the Universe after Hartle and Hawking; both of which carry a status of time-independence (see the Goncalves-Madeira
paper for details).
So pondering a 'free particle' in the universe would become simplified, should this 'freedom' be
only a local measurement. This means that should the universe as a whole be rendered 'boxed'; then the time-independent formalisms
would suffice to describe the dynamical eigenstates of the 'particles/wavelets' in a simplified cosmology.
Goncalves-Madeira emphasise a principle of order to precede the notion of chronological time, requiring
some cyclic mechanism for measurement. They term it before/input followed by after/output. They also emphasise the arbitrariness
of any such time-interval as any two ordinal number coordinates, provided the ordinals are ordered.
This is also, what I have found in my considerations in Quantum relativity (QR). I have found a
possible universal wavefunction to follow as a first approximation to say the Hawking-Hartle approach from the following Differential
Equation:
dB/dT + aB(n) = 0; a being the Electromagnetic Finestructure as the probability of light-matter interaction
(~1/137).
This has a solution: B(n) = Bo.exp[-a.T(n)]; Bo=2e/hA from QR boundary conditions
defining:
T(n)=n(n+1) as the Feynman Path-Summation of particular histories under the pentagonal supersymmetry given in the
identity:
XY=X+Y=-1=i^2=exp[ip] and lim [n->X]{T(n)}=1
This allows the Normalisation of the [Y]^2 wavefunction to sum to unity in B(n)=(2e/hA).exp[-a.n(n+1)] with Functional Riemann Bound FRB=-1/2, centred on the interval
[Y,...-1,...-X,...-1/2,...(X-1),...0,...X].
Interval (-n,Y,-1] sets F-Space; interval [-1,0] sets M-Space with uncertainty interval [-X,(X-1)] and interval
[0,X,n) sets the C-Space, encompassing OmniSpace. One can label F-Space as 12D 'Father'-Space; M-Space as 11D 'Mother=Mirror=Manifold'-Space
and C-Space as 10D 'Child-Space' in a somewhat semiotic manner of poetry.
n<0 is imaginary as real reflection of real n>0 of the C-Space, metrically defined at the coordinate n=0
mapping n=nps, which is the instanton tps=fss=1/fps and using
modular string-duality T, with inversion properties for displacements relating 'winded low frequency (ss)' and 'vibrating
high frequency (ps)' energy-momentum eigenstates.
Cycletime n is defined in GR as dimensionless TauTime in curvature radius Rc=c.dt/dt for the pathlength of x=ct and become dn/dt=Ho, n=Hot in QR, with Ho the nodal HubbleConstant
defined in c=HoRmax=lps.fps.
But this now allows us to explicitely define a NOW-Time as the instanton: nps=lps/Rmax=Minimum Metric Displacement/Maximum Metric=Ho.tps. This definition renders the mathematical Null-Coordinate
n=0 as 'quantum smeared' or 'quantum foamed' or 'quantum looped' in a precise metric limit, which QR terms the Weyl-Geodesic
or Kerr Torus Ring Singularity.
The chronological time so is quantised in multiples of HoRmax=lps.fps.3.33..x10^-31 seconds in QR and for a minimises lightpath of x=c.t.ps=lps.=10^-22 meters
as a looped perimeter or wavelength.
The Feynman Path so sums both negative and positive integers as: -n......-3...-2...-1...0...1...2...3......n
=T(n) in absolute value to double the infinities as the entropy reversal of lightpath x=c.t=(-c)(-t).
lim [n->X]{T(n)}=1 now can be defined to collect the 'Infinities'
of Cantor, say the Cardinality Aleph Null for the limit of T(n), as n approaches the infinite count of
the integers. (I am no expert mathematician so this statement is heuristic and not rigorous).
Cantor Cardinality Aleph-Null is thus Unitised in Aleph-All, counting infinities as if they were integers of the
Feynman Path. This allows the Feynman interpretation of Quantum Mechanics as alternative to the formulations of Schroedinger
(fermions 1/2spin) and Klein-Gordon (bosons) as timeindependent and timedependent (free particle form inconsistent with SR
in Schroedinger in 1st order t & 2nd order x), formulations respectively.
In the above, the before/input interval of Goncalves/Madeira is mathematically described by (-Infinity,..,Y,..-1)
and the after/output interval by (0,..,X,..,+Infinity).
The Unitary Interval (-1,0) then describes a binary quantum realm, which can be said to comprise an abstract
'surface mapping' connecting the input from a lower dimension to an output in a higher dimension (by a count of 2).
The above so describes the universal wavefunction as a gaussian distribution of spacetimes, centred on an
uncertainty interval, described by the M-Space (or Witten-Membrane-Space). Chronological time depends on the activation of
any such spacetime, which follows the postulates in SR for nonaccelerating frames of reference and the premises of GR, where
the acceleration becomes an effect of the curvature of spacetime of the 'universe in a box'.
All 'activated' spacetimes must activate at the X-coordinate in C-Space in a 'self-relative' eigenstate
and as defined in a one-to-one correspondence to the M-Space, which both reflects and refracts the information after mapping
(as some sort of Memory) from its 'Moebian' surface back to C-Space and 'out' to F-Space.
I have discussed more details elsewhere, but wish to close min recommending the Gonsalves-Madeira paper for
the readers and contributors, which are intyerested and can follow the formalisms in that paper.
As a final point however, let me say, that the principle of the input/output as described in the Gonsalves-Madeira
paper finds its beautiful origin (imo) in the following algorithmic statement and which is the basis for all of the above.
The principalities became defined in the first subroutine of the generating master algorithm, which could be stated in
generalised form in the following manner to
DEFINE THE (ABSTRACT) CONFIGURATION SPACE in the form of AWARENESS-TRIPLETS represented by:
{OLDSTATE, EXPERIENCE, NEWSTATE} and selfiterative in the following algorithmic statement:
GENERATE NEWSTATE IN ADDING THE SECOND NEWEST OLDSTATE AS EXPERIENCE
TO THE NEWEST NEWSTATE AS THE NEWEST OLDSTATE.
...BEGIN (0,0,0)-DEFINE SELFSTATE=(1,0,1)-REDEFINE BINARY EIGENSTATE
=(01,01,01)=(0+1,1,1+0)=(1,1,10)=(1,1,2)-(2,1,3)-(3,2,5)-...CONTINUE
INITIALISE: N=0; LIMIT=[REDEFINITION OF EIGENSTATE]={GOOGOLPLEX E}
-FOR (N=N+1 TO LIMIT) GOTO
-SUBROUTINE (DEFINE PARAMETERS FOR EIGENSTATE FOR M=M+1) GOTO
-SUBROUTINE (ALGO[M]) GOTO
-SUBROUTINE (...) GOTO
-SUBROUTINE (...) GOTO
-SUBROUTINE (SEARCH FOR LIMIT=[!])-STOP GOTO
-SUBROUTINE (!- GOTO BEGIN -REDEFINE SELFSTATE IN [!]) GOTO
-CONTINUE FOR O=M FOR LIMIT=[EIGENCODE IN !]={GOOGOLPLEX E} GOTO
-REPEAT FOR SUBROUTINE (O- [O]=[!], [!]=[!+1], [?]=[!]) GOTO
-REPEAT FOR LIMIT GOTO BEGIN...
Tony B.
Carlos Pedro Gonsalves replied to the above heading:
Thanks Pedro for your constructive reply and observations. Please allow me to intersperse your
reply.
Hi Tony,
Many thanks for your kind words, and for the perspective on the
paper, you've given us a lot to think
about.
I'm not as well versed as you in string theory and M theory, I'm
more familiar with loop quantum gravity,
because it was a good place
to start, in a first approach for connections with a quantum
computational framework for
quantum space-time. Nonetheless, M
theory is on my future agenda.
TB>I am no expert on string theory; however I had developed a similar framework in the 1980's
where there exist five subclasses of elementary entities, which are called strings nowadays. Then after reading Brian Greene's
book "The Elegant Universe" in 2001, it suddenly dawned on me, that my five classes could be identified with the string classes
with one major difference.
The Planck string of class I (both open and closed) had the Planck energy, but the other four
classes were transformation of that primary. As you know, the string theorists consider all classes to act on the Planck scale.
The first transformation (class IIB) was 'my' magnetic GUT-monopole of precisely 2.7x10^16 GeV, which was 'selfdual' and became
attached to a 'brane' from its closed eigenstate. The monopole transformed to the heterotic class (HO32) in a bifurcation
which allowed the fermionic eigenstates to be 'born' from the bosonic substrate (of 26 dimensions of 10 manifested and 16
chirally absorbed). I had called this class as the X-L Boson of energy just a little below the monopole class. The beauty
of this is, that the X-L Boson string could drop a lot in energy in a physically metric cosmology and become a K-L Boson,
which then would split into a quark-lepton duality in a base neutron and a base muon, which would then charge bifurcate into
a fermionic proton with a fermionic muon say.
What is called strongweak string duality now relates the monopole class to the IIA class
of Witten's 11th M-dimension and a class, which I have (correctly imo) identified as the instigator of the Cosmic ray spectra,
which are upper limited by the monopole class. I had called this string classs the ECosmic Boson of Cosmic ray energy (called
the knee).
The last class is the strongweak duality between the heterotic classes in the HE64. This one
is the boundary condition between the string epoch of inflation and the thermodynamic universe of Planck's Black Body Radiator
and the metrication of General Relativity. Its energy becomes pervasive in Quantum Relativity and is sometimes quoted in works
of Ng and Van Dam and similars, which attempt to approach the Planck-Length transformation in magnification. It is for example
the displacement in the gravitational wave detectors in sensitivity to 10^-22 meters.
All of my work is based on this assumption. Spacetime will become wormholed at this wavelength
and a concentrated particle energy of 0.002 Joules or so 1.24x10^7 GeV as the Weyl-String (say).
QR will stand or fall when this energy realm can be examined by future experiments.
Smolin's QLG fits in nicely into my framework (of boundaries and initial conditions and not a
fully fledged physical theory imo btw), as QR considers all of spacetime to be granulated in the Weyl-String or (heterotic
class 8x8 however magnified as Planck-string); as one should be able to model the adjacency of the 'Weyl-quanta' or perhaps
Weyl-qubits as a nonphysical 'connector spacetime' akin a distribution of vertices.
I am in no way informed enough so, to rigorously establish this mathematically. I hope however,
that more adept people like yourself, can someday use my boundary/initial conditions to formulate the 'New Theory' incorporating
Quantum Relativity.
In a first reading, as far as I am able to understand your proposal,
you have a very interesting approach
to quantum cosmology and to
physics, not only by making the Feynman path integral a fundamental
formulation, but,
also, in the relational structures with which it
works, as well as by introducing a quantization of chronological
time,
which seems to me that should be a natural choice for
quantization within quantum gravity.
Loop quantum gravity,
in its bottom-up construction, may have
problems in addressing this issue, unless the links between the
different
quantum spin networks, in a quantum causal history, are
assumed to occur with quantized units of time, which would mean
that
one should add extra quantum numbers, representing temporal
quantization of network connections, labeling the
links in the
quantum causal histories connecting two spin networks. I am not
aware of research being done in this
direction, except, perhaps,
some possible integration with Lucien Hardy's quantum gravity
computer. Nonetheless, the
traditional quantum causal histories are
not exactly fit for this, since a temporal quantization would
eventually
have to introduce a fuzzy causality structure to the
causal histories, as discussed in our paper.
TB>I see no problem here, as time in QR is indeed quantized by the Weyl-qubits. The timeinstanton
is 1/fps with shorter duration limited by the Planck-Time and an interval, which describes
the inflationary dynamics.
The Weyl-qubits are toroidal volumars or 3-branes in 10 dimensions, which carry the same coefficient
as the macroquantised Hubble-Volumar as a 3dimensional surface. A granular chronological time would so consist of NOW-intervals
given by the Weyl-qubit.
The fuzzy causality could relate to the Uncertainty Principle, where Planck's Constant is finestructured
in the Weyl parameters as h=Eps/fps=wavelength/e*c (for e*=2Rec^2
Re the classical electron radius in string units).
>>In the above, the before/input
interval of Goncalves/Madeira is
>>mathematically described by (-Infinity,. .,Y,..-1) and the
>>after/output
interval by (0,..,X,..,+ Infinity) .
>>The Unitary Interval (-1,0) then describes a binary quantum realm,
>>which
can be said to comprise an abstract 'surface mapping'
>>connecting the input from a lower dimension to an output
in a
>>higher dimension (by a count of 2).
This is a very interesting example of a fundamental relation where
the two intervals (-Infinity,. .,Y,..-1), (0,..,X,..,+ Infinity) are
the "objects" connected by a unitary interval
(the "relation").
The relational structures, addressed in the paper, are very general
mathematical structures with
a naturally emergent ordinal
temporality, that arises from the order of terms in the relation. In
this way, they are
more general than the mathematical categories,
from category theory, which correspond to substructures within a
relational
structure where the relations are morphisms satisfying a
number of conditions.
The temporality emergent from the
relational structures is a
temporality found deep within the realm of the pure mathematics,
which, in itself, is very
interesting, but, for physics, the most
important step is to find fundamental relations, that should be the
object
of formalization, in the form of a relational nexus or a
subset of a relational nexus. We addressed some of these in the
paper, in the form of the unitary connection between two bases,
which brings a temporal connection to the Hilbert
space geometry
producing a processual expression of a vector in a Hilbert space.
TB>Here you are much more informed than myself. I have only a BSc. done almost 30 years ago
and I have never studied the abstract mathematics of quantum mechanics formally. However I can intuit what you mean. The Hilbert
space has a metric basis and is so relevant for the 3-geometries. Time extends this to a 4-geometry in Minkowski flatness
applicable to SR. Now I'd say that any dynamical interpretation of momentum-position coordinates so defines the c-invariance
as a partitioned movement through time and space. Only a relative 'standing still' in metric space, will allow
maximum dynamics through time at lightspeed c.
This to me, implies, that the most fitting cosmology should introduce a dynamics of duality -
using the higher dimensions, which are albeit colocal with the familiar hypersphere dimensions as a limit of the 4-sphere(dV/dR=2Pi^2R^3
for V=Pi^2R^4/2).
Using this, the metric Hilbert space will act in a dimensionality one lower than its 'envelope',
which then is akin your 'procession' say in an asymptotic approach of the 'inner' universe towards its 'outer' boundary.
But the 'envelope' will not be restricted by the metricated inertia and so will be purely electromagnetic
without requiring accelerating inertia for its generation (it will require instead acceleration of the e* magnetocharges of
the string epoch though).
In a convergence with your approach, we indeed show that a vector,
independent from a chronological time,
can be expressed by a sum
over histories, which brings the issue of the centrality of the path
integral formulation
to the study on the foundations of physics.
Nonetheless, the issue of identifying the fundamental relations,
connected
to fundamental symmetries and, eventually, to also,
fundamental symmetry breaking processes, seem to be an important
step
in finding a way to formalize quantum cosmology with a sensible
and fruitful approach.
TB>I have done a lot of work to examine the symmetry breaking between the string classes Pedro.
I have posted this, but noone seems able to make any sense of it. I am almost inclined to say, that this process underpins
the entire cosmology. If you wish, and if it seems appropriate for your own endeavours, I shall engage in a detailed
discussion on this with you.
So far as I am able to understand your proposal, it seems sensible
and fruitful.
In the relation defined by the interval (-1,0) you
seem to identify a fundamental relation, and introduce a fundamental
relational connection that is able to produce a cosmological
formulation where Witten's Membrane-Space becomes the
mean, median
and mode of the set of potential physical geometrical structures,
which is an important result, from
the point of view of M theory.
TB>Indeed, the M-Space is the mean and in discussion with a logician, who believes that the
'void' has a fractal negative dimensionality of -0.62; a similar scenario emerges in the 'crossing of boundaries'.
I feel that many approaches could converge, should the works proceed in cooperations. The logiocian's
name is HansDieter Franke and he posts on patternville group (see my posts on crossing the boundaries) if you choose).
The
emergence of chronometric time from a more fundamental ordinal
time does also seem to be present within your theory, which
goes in
line with the discussion in our paper's conclusion.
TB>Yes, after I perused your paper I could see the convergences and that is why I decided
to comment upon it. I am very pleased that you replied.
Thus, although the mathematical formalism of the relational
structures applies to different physical formulations of quantum
cosmology, there is a great number of points of application
of the
basic results from our paper in how you're approaching physics and
cosmology. Especially taking into account
the way in which
chronological time emerges from a more fundamental ordinal structure.
All the best,
C.
Pedro
Same to you. It was a pleasure to converse with you.
Tony B.
