A few things that can and have been said about "time":
Time is intrinsically a concept derived from iteration, replication, or cyclicity; it is distinctly a "counting concept" - subdivisions of time without counting are meaningless.
The ultimate quantum of "time" is the "Planck Time", about 10^(-44) sec.
There are good reasons to believe on the basis of pair production energies, space and therefore time already becomes quantized with localized energies at about 10(-13) sec. David Finkeslstein designates this as the "Heisenberg Time". an appropriate designation.
A concept of time without one also of space, is meaningless. [From Aristotle on]
The entire idea of time as perceptual, and therefore intrinsically connected to the notion of memory, of which there are many distinct neurochemical types and entire systems is being rather ignored here. More is the pity for since these, and the physical illusions that they create, are at the very foundations of our intuitive concept of time. It sometimes becomes necessary to attack concepts that are based on our rather primitive and distorted "human" perceptions
It is my purpose here to attack those very foundations, without being able to parse fully the multiple nature of the various concepts that are contained in the simple word "time". This is merely an attempt to show that "time" is definitely *not* a singular concept but that the word embraces a multiplicity of meaninings which causes the use of the word to create paradoxes which would not exist had the proper disctinctions been made in the first place. We have similar difficulties with such catchall words as "cancer", or "insanity". There is a "sense" to them which happens to remain ignorant od the complex reality that is contained within. With further investigation, knowledge can be made which can make distinctions.
My formal intuition says that the important distinctions to make are fourfold: global time of the universe, local macroscopic time, and local Heisenberg (atomic) time, local Planck time. These are physically measurable times distinct from the concept of psychologically perceived time which is absolutely a congnitive construct of direct experience. Yet we persist from Newton on in our postulates in taking this psychological time, almost subconsciously, as the model for all measures of time.
Special Relativity (SR) teaches us that time is an esentially local phenomenon, that synchronization of events is fundamentally a convention, not a physical reality, this in the context of differentiable manifolds. [Friedman 1983]
At the same time the existence of an "absolute" spacetime is also postulated.
It is basically arranged so that in the limit c→∞, SR becomes Newtonian Mechanics (NM). Where time profresses at a uniform rate through out all of space, hence becoming global.
The notion of the "uniform progression of time" is maintained as assumption of "the possible" among any set of inertial frames.
Both SR and NM fail completely to explain how this uniform progression comes about. We must recognize that the time spoken of here is a of the macroscopic variety, and neither physical theory can take its concept of "time" as anything other than some undefinable, and mysterious entity, possessing under certain circumstances in SR and under all circumstances in NM, a completely unexplained "uniformity of progression".
A simple relativization of the Schoedinger equation, does not quantize time as a countable thing, but instead creates problems of either the probability interpretation (the Klein-Gordan equation) of the energy spectrum (the Dirac equation), and of course, does not address the essential problem of the origin of the concept and assumption of "uniform progression".
With regard to the origins of this uniformity, and then also of a time mechanism associated with the energy of an oscillator as clock, there now exists a derivation of a physical local Newtonian time from the fundamental algebraic structure of FCCR.
Physical theoretical time reverts to the global Newtonian time, and even relatisitic axiomatic field theory of a multiplicity of particles reverts to the picture of an absolute spacetime, and this is supposed to be a solution to the energy spectrum problem, by introducing antiparticles. It is somewhat of a miracle of theory that this antiparticle structure seems to be right.
No existing physical theories have addressed, or explained how this apparent uniformity of the progression of physical time comes about. This is either a general failure of all physical theory, or an unknowable article of faith. I take the former view to be the case and the latter to be antiscientific.
Notes on various formal concepts of time:
n like the radial coordinate of a Robertson-Walker universe.
Expansion: n → n+1 which on the face of it appears
indeterminate. AGE of the system. (AGE of the Universe)
[A RELATIVISTICALLY EXTERNAL PROPER TIME]
At very least a local "set theoretic" relativistic
invariant. A manifestation of bean counting in a
Q-theoretical local linear phase space like FCCR.
m A counting time (m τ0) of process as invariant measure.
BEAN COUNTING is a relativstically invariant procedure.
Note: Lorentz time contraction always makes
clocks run slower
If a clock runs according to this time, it makes
a transition every τ0 and also
always makes a transition with probability 1 to the
next pointer position. The probability matrix is then
C0!(n) for "forward time" or C0(n) for a "backward time"
This is a maximum transition rate.
NB This is the opposite of the convention that we have been
using but seems somehow a better choice.
[AN INTERNAL QUANTUM COORDINATE TIME BY MATRIX ELEMENTS
OF AN OPERATOR]
A stopped clock has transition probability matrix (1/n)I(n)
and is a minimum and zero transition rate.
t(n) phase of the motion:
Derivation of a physical local Newtonian time
The time operator of [Section VII]
that exists for FCCR with finite n, but not for standard
QM. It is the cognate of a cyclic Hamilton-Jacobi
variable. A time operator can either be witin the Hilbert
space of the coordinate operator, which seems to imply at
least a noneuclidean relationship between space and time;
or, within a separate Hilbert space in a direct product
of statistically independent coordinates.
[AN INTERNAL QUANTUM COORDINATE TIME BY OPERATOR]
In the first case, it seems that the correct time operators
(forward and backward) are
(+|-)i sqrt(n)/(2π)) ln CH(n)
where CH(n) is the cyclic matrix in the eigenbasis
of the Hamiltonian.
[AN INTERNAL (INTRINSIC) QUANTUM COORDINATE TIME]
G(n) negative subspace; manifestation of the relativistic
"distinction" within the complex Hilbert space of process
that replaces the projective Hilbert space of QM states.
(<f|G(n)|f> = 0 → |f> is a light-like process.)
It should be expected that the distinction should arise
in the process Hilbert space when for operator t(n),
[Q(n), t(n)] ≠ 0. The eigenvectors of both Q(n) and t(n)
are G-null. [Refine and shorten the rambling of section XV]
[AN INTERNAL RELATIVISTIC QUANTUM COORDINATE TIME
BY STATE IS RELATED TO AN INERNAL RELATIVISTIC
QUANTUM PROPER TIME]
See ./drctprod
SEE FOLLAND P.18!
k an integer power of the stochastic matrix of
probabilities |<qk|tj>|2 that generates a markov
process by iterative mutlitiplication.
More Q-like would be a matrix iteration of transition
amplitudes and thus probabilities constructed from
<tk| Hk(n) |tj>
where |tj> is the eigenbasis of the
time operator t(n).
What is the operator and state choices that should actually
be used?
<Ek| tk(n) |Ej>
It this just a point of view?
! Note that a commutation relation between operators
is always an expression of relations between two arrays
of transition amplitudes !
Time as expected evolutionary progression from clock transition
probabilities should generally be slower than the counting time.
There should be a connection to the "Feynman kernel" which
in turn has a formal similarity to the Bergman kernel that
relates the metric on a Kaehlerian manifold to inner product
on a Hilbert space of complex analytic functions defined
on the manifold.
Feynman kernel: Cf ../paper/feyn2
NB This should be the same as "m" above.
Counting beans is a relativstically invariant procedure.
(DELTAt)
A time interval measured by macromotion:
Since there will be a nonvanishing amplitude for stasis
such an observed time interval will consist of two
kinds of time: a time of transition and a time of NO
transition (a waiting time). (DELTAt) can shrink
as a result of distributing an apparent actual increase
in the motion time and a decrease in the NO motion
time. See stuff on De Broglie Frequencies. That is,
a boost will in fact alter the t-amplitudes and presumably
then also the q-amplitudes to balance out a relativistic
invariant of their combination.
r a transition rate:
A transition within τ0 may or may not
happen. If probability of transition is p, then q = 1-p
is probability of no transition. Assume p for any of a
succession of τ0 is independent of a
p for any other and independent of any actual occurrence.
During a sequence of (n τ0), the
probability of k transitions is then
B(n k) pk (1-p)(n-k)
is distributed binomially.
The expectation value of k and the standard deviation
during the (n τ0) is given by
E(k) = np
and
σ2(n) = np(1-p)
The expected transition rate which is independent of n is then
E(k)/n = p
This, independently of the probability distribution on
transition lengths.
Calculate p or q from the transition matrix?
The transition rate should be a function
of the temperature of thre system.
α(n) = (2π/n)m
In corollary 8.11.2 it is proved that
exp( i α(n) G(n) ) = exp( i α(n) ) I(n)
and so for, say the oscillator Hamiltonian
H(n) = N(n) + (1/2)G(n)
exp( +i α(n) H(n) ) rotates Alg(n)
in the P-Q plane just like the time parameter does in CM.
α(n) acts like an infinte but discrete time parameter
that becomes continuous in the infinite n limit.
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Exercise:
Model a system that is a selfconsistent clock: one whose dynamics
proceeds according to the clock which is itself.
Email me, Bill Hammel at