Necessary Multiple Concepts of Time "Time is defined in such a way as to make dynamics look simple"
-- A. Einstein
FCCR Table of Contents

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)
		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.

	        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.
		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.

	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]
                See ./drctprod

	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.

		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

                     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.



     Model a system that is a selfconsistent clock: one whose dynamics
     proceeds according to the clock which is itself.

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Created: August 1997
Last Updated: August 5, 2000
Last Updated: June 8, 2004