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Non-Metrizable Time Study and Research

Winter 2005-6

Notebook of December 19, 2005 December 19:

(1) Self non-commuting time operator(2) Topologies with only left convergent limit

pointsHawking and Ellis "Large Scale Structure of Space-Time" employment of "topological induction"

Let M be an open set in Rn , with the following property:

Any finite arc in M has its endpoints in M. Then M must be unbounded. Page 1: Two time operators "before" and "after".

This is very significant in quantum theory "Before" (observation) is causal; "After" (observation) is statistical. This is the "collapse of the wave packet. Call these two operators T- and T+ . Can one make sense of a commutator [T-, T+] ? . It would seem that, under the paradigm of the collapse of the wave packet, one might have a much simpler expression | T-- T+| > h /2. The difficulty with this is that "h" has the dimension of action, which is mass (x )velocity squared x time. One can insert a "standard velocity" in

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the above relation, namely c. but what does one do with the mass? Well it's interesting, because the Wigner-Selecker clock considers this, and in fact comes up with a relation between the mass and the accuracy of a clock.

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Combining the 6 conditions for a non-metrizable time:

A] An "initial state" cannot be defined or measured. This seems to imply that its impossible to measure states altogether, since any state can be taken as an initial state. However, there are "discrete" systems in which the appearance of a state is followed by inaccessibility, followed later by the reappearance of that state.

B] Periodic systems can't be builtC] Systems in isolation can't be definedD] The 2nd Law is too strongE] Unstable Causation, alteration of the constants

of natureF] Only local time is possible, there is no global

timeCombining these, one imagines 15 universes:

[AB] No periodic systems; initial states can't be measured

[AC] There are no isolated regions, either in theory or practice, and the concept of an initial state can't be defined or measured

[AD] Strong second law. This defines a direction in time, thus ruling out periodic systems. Initial states can't be measured.

[AE] Unstable causation, initial states can't be measured. Then how does one know that causation is

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unstable? Well, this sounds like our universe, in which causation is firmly founded on the constancy of light principle, while uncertainty is founded in the value of h. c could change while h is unaffected, or vice versa, or they could both change.

[AF] No global time, no initial state measurements. Uncertainty changes as one moves about the universe!!

Examining the possibilities suggests ideas. For example, an unstable value of c or h would make it impossible to establish a time metric. Indeed one can include g in this list, since the two standard ways of establishing a time standard are the rotation of the earth and the atomic clock which depends on the release of photons. One is then led to ask if the values of c,g and h provide enough information to establish the units for all other quantities. This means that one must be able to express "space, time and matter" in units such as these 3 are all set to "1".

c = d t-1

h = md2 t-1

g = mdt-2/d-2m2 = m-1d3t-2 We can solve for "m" in h, so m =hd-2t,

and g= hd-2td3t-2 = hdt-1 = hc (!) . The units of the gravitational constant are those of action times

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velocity. Clearly what one has to do is find some standard "mass" , m which can be set equal to "1". There was a good talk by Frank Wilcjek on this subject.If m = "1", then we have d = hc/m, which gives us our unit of distance in terms of c,h and m. Time is given by t = d/c . Likewise one can get the unit for the gravitational constant, namely, the force of attraction between two particles of "unit" mass at a distance of "1 light second", etc. I won't explore it further at this point, except to pose the following question. If one should observe experimentally that the speed of light is changing, what changes would one expect to find in h, g and m, to assume that the units of time, space and matter were changing, rather than the constants of nature

One can also imagine that c , h and g all differ from one region of space to another.

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9 Categories of Temporal Translation.

Question: Is this an appropriate mathematics for each one?

(1) Translation from a past event to another closer in time to now but still past. This is a motion from "hypothetical to hypothetical". Historical causal chains. Their ultimate justification must be based on evidence found in the present T: P1 --->P2 .

(2) Translation from a "present event" to another "present". This is "pure description" without causation. Thus, a doctor monitors a patient from one day to the next, or the weather report. A census is taken every ten years T: N1 --->N2 .

(3) Translation from a "future event" to another "future event". Basically, one tabulates a "spread" of possibilities for some time t1 in the future. For each of these possibilities one has another spread, and so forth.This is the domain of graph theory, flow charts and branching diagrams T: F1 --->F2 .

(4) Translation from a present event to its past. This is the method of geology; reconstructive science T: N --->P .

(5) Translation from a present event to its future. Prediction. Statistics T: N--->F .

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(6) Translation from a future event to its past. "Based on evidence which we expect to turn up in the future, we will be able to determine when the Sphinx was constructed. T: F --->P .

(7) Translation from a future event to the present. "Future generations will look back upon us with gratitude" (Its a prediction about the behavior of a future event, which is also a prediction. Will there be any future generations? Will they bother to have opinions about us? T: F --->N .

(8) Translation from a past event to the present T: P --->N. This is basically the same as translation from a present event to the future, that is to say, a prediction.

(9) Translation from a past event to the future. "On the basis of Germany's past behavior, it is unwise that it be re-armed."T: P --->F .

Some of these may appear far-fetched, but what we're talking about is 9 possible interpretations of the deceptively elementary notion of "time translation." If f(t) is a function of time, then letg(t) = f(t-t1 ). Then if t0 is a specific value of the variable, one hasg(t0 ) = f(t0-t1 ) . Depending on the relationship of these two times to a "present time" t, one gets these 9

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different situations. The distinction is essentially epistemological, but no less scientific for all that.

Contrast this to "spatial translation" in which, rather than juggling 3 modes "Hypothesis", "Observation" and "Prediction", one need only consider "here" and "there" . Objects may be moved from here to there, from there to there , from there to here, and from here to here (stationary, absence of motion). That's only 4 categories of spatial translation.

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Notes from Hawking and Ellis "Large Scale Structure of Space-

Time"Chapter 8: Space-Time

Singularitiesdefines "b -incompleteness" (boundary

incompleteness), and"g-incompleteness" (geodesic incompleteness) A boundary -incomplete space is one in which the

singularities have been "cut out" of the space. In some sense, the metric is not singular, but it can be unbounded. In GR one is usually dealing with a curvature singularity at a point which is then deemed to be 'outside' of space time. However, there is something known as the Schmidt construction, which is a singularity that is not simply a discontinuity of the curvature. Then the families of geodesics going to this point will be 'incomplete'

In the Schmidt construction, the incomplete curves are somehow imprisoned in a compact region of space-time.

Two definitions of a singularity:(1) The metric tensor is either undefined or

undifferentiable , (which means that the curvature cannot be defined). This is the convention adopted in electrodynamics. Such points can easily be 'cut out' of

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the space, and one can treat all of space-time as non-singular.

My impression is that they find this procedure unsatisfactory and proceed to look for another way of defining a singularity.Let (M,g) be a space-time, and let the metric be positive definite. With a metric one can define a distance function (x,y) between any two points in the space time.

We will say that M is metrically complete , if all Cauchy sequences (xn , xn+1 ) converge to a point in M . It is then geodesically complete, if this is true for sequences along a geodesic.

(Geodesic completeness is a consequence of my notion of topological induction. As H&E put it, "Any geodesic can be extended to arbitrary values of its affine parameter. "

A reference for the "Geroch" construction: Geodesical completeness with an inextendible curve of bounded acceleration and finite length1968 Ann. Phys. 48, 526-40

ds be a standard Lorentz metric , g = (1,1,1,-1) Let g* = 2g , where is defined as follows:= 1 , for x between -1 and 0 , = 0 for x between

0 and 1, and = 1 for x larger than 1. Also t2 --> 0 as t -->0

Then, along the t axis (ds*)2 = (cds)2 ,

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and , that is to say, a finite length.

H&E therefore restrict their attention to situations in which all singularities derive from at least one component of the curvature which is infinite.

Penrose Theorems , 1965 . A Black Hole singularity with the following properties:

(1) The metric is Schwarzschild(2) Spherical symmetry(3) The angular momentum is "unstable and

infinite". Even a slight anomaly in the angular momentum is enough to prevent the collapse into infinite density

(4) One can still obtain an infinite collapse in the context of General Relativity

This he calls a "closed trapped surface". It is compact without boundary, a space-like 2-surface, that is to say, an "Event Horizon".

Penrose Theorem I: Space-Time (M,g) cannot be geodesically complete if

(1) for all null vectors (positive curvature condition, equivalent to a unidirectional irreversible time)

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(2) There exists a non-compact Cauchy surfaceH in M, (A well-defined 'space-like' domain one can treat as ordinary 'space')

(3) There is a closed trapped surfaceT in M Idea behind the proof:If M were null geodesically complete then the

boundary of T would be compact. Using topological induction this implies g-incompleteness. At the same time H would also be compact.

The other Penrose theorems are variations on these, and my strategy now is to find time this summer to reread the greater part of "Large Scale Structure".

aaaaaaaaaaaaaaaaaNotes from Zajonc and Greenstein : "The Quantum Challenge" (Go to page 22)

page 215. Chapter 8 on Measurement. Measurement plays an active role in Quantum Mechanics. A single photon on a photographic plate will darken a single grain. Collapse of the wave packet not implied by anything in the Schrodinger formalism. (Can we not identify the "collapse of the wave packet" with the "Now moment"? "Now" is the moment of observation . and observation 'causes' the transition from a Hermitian Operator to a Projection Operator which yields the eigenvalue. Hence, everything we are

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seeing is a collapse of all the potentials leading up to the 'isolated' events being seen. *************Pg. 23. Note on "thermal clocks". A clock which is based on averages of the random agitation of molecules may be more stable than one based on causation. Placed a heated vessel in a chamber kept at constant temperature. The rate at which the vessel loses heat to the chamber may be calibrated, time then being read as a function of temperature. When the temperature goes below a certain mark, a thermostat mechanism is triggered which reheats the vessel.

Another construction: a circular disk, rotating clockwise. The left hand side of the disk is maintained at a constant temperature. This heat propels the disk forward, so that it cools. The temperature levels of the cooled disk act on a thermostat to give the time. The cooling part of the disk rotates back into the heated section, setting up a dynamo.

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December 29: Page 227 "Quantum Challenge"Bohm: Quantum state is a set of potentialitiesShimony: Measurement actualizes potentialities

There are real physical differences between superpositions and mixtures.

" Measurement terminates the infinite regress via the projection postulate."

Rather than transcribing notes from Greenstein and Zajonc, it would be better to reread the book.

********************January 5. In "Hawking and Ellis" there is much discussion of g-incomplete spaces. These are space-times in which one finds temporal trajectories for which the affine parameter standing for time has only a finite length. This does not mean that time stops in this universe. If there is a Cauchy surface, and the finite geodesic goes to a trapped surface, the

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geodesics which don't intersect this surface can continue on indefinitely.

What happens when time "stops" at the "end point" (Black Hole). Clearly in my terms it becomes non-metrizable. The "interior" of a Black Hole therefore exhibits one or more of the basic phenomena associated with non-metrizable time:

(A) Brownian Chaos(B) An entropy spike (Hawking radiation?)(C) Violations of causality(D) Quantum chaos: Initial states can't be

measured, even defined.(E) Non-Isolation, No periodic systems. Initial

states don't reoccur(F) Non-local time. Clocks in different locations

can't be compared. ***************

Jan 27. It is well known that Newtonian Mechanics contains an inherent contradiction: the reference frame of the fixed stars. Upon among them there is absolute rest; but down here there is only relative rest.

This produces several odd conclusions. Lets say that there exists a way of determining that one is "at rest". Observer O1 on planet M1 uses this method and concludes that he is indeed "at rest".

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He calls up O2 on M2 by telephone, that is, by electromagnetic signals, who calls back to inform him that he, by his own reckoning, is at rest. O1 can now rely on this information, and use the motion of M2 as a clock !

However, no one would think of using the motion of Mars as a clock. A test to determine if one is at rest is actually a test to determine the absence of a force field in ones neighborhood. In Special Relativity this is an absolute determination, but in General Relativity, there are 3 situations:

(1) There are no forces present(2) One is accelerating(3) There is a gravitational field present.

(2) and (3) are 'equivalent'. But what about (1) ? Is this absolute or relative?

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Bell's Theorem and the EPR ParadoxCertain issues have to be avoided in the

description of any quantum experiment:2 Slit situation: One doesn't ask "which" slit did

the particle go throughDelayed Choice: One avoids the question of

alteration of the past.Certain things are not well-formed quantum

questions, and if I ever do get my ideas on this subject

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into pragmatic shape, I can couch them in such a way that they are "ill-formed" as questions.

In this treatment we use the Bohm form of the EPR experiment. A Stern-Gerlach analyzer measures spin. As we know, it can only measure one component of spin, namely its projection onto the z-axis: Sx and Sz don't commute and, since spin is a "discrete" quantity, measuring Sz eliminates the possibility of measuring its projection in any other direction. As we know, the value of Sz does allow us to determine J2 , as this commutes with any measurement in a specific direction.

Now, line up two Stern-Gerlach analyzers, arbitrarily twisted relative to each other.

There appears to be a contradiction. One can use this to get exact measurements of both Sx and Sz. Then we can get Sy. Review why this can't be done. It is relevant to the Kochen-Specker Theorem.We set up our apparatus and variables A, B with values:

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A = +1 if Alice measures spin upA = -1 if Alice measures spin downB = +1 if Bob measures spin upB = - 1 if Bob measures spin down A is pitched at angle a , B at angle b. We are

insterested in the statistics of the product AB. Note that the eigenvalues can be -1,0,+1, but that the "spins" must be either up or down. (These are the two

"spinor vectors" Basic Quantum Mechanics, as explained very well by Redhead, calculates the Expectation on the tensor product of the two apparati, as Exp (AB) = -a.b = -cos( ) , where is the angle between them. For any hidden variable theorem it is easy to show that the expectation value as calculated in this theory cannot be the quantum value above.

One sets up a situation involving 3 experiments.(1) Alice, at angle a, Bob at b, determine Exp (a,b)(2) Bob twists his apparatus to c, Alice keeps hers

at a, and they determine Exp (a,c)(3) Alice rotates her apparatus to b, Bob keeps

his fixed at c, and they determine Exp (b,c) . A simple calculation using integrals, to be

reproduced below, yields this basic equation:

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This can't be true for all values of a,b and c. For example, if a=b=c, then the quantum expectation is Eqm = -1, while

However, if we set a,b and c at 60° angles relative to each other,

then Proof of Bell's Theorem:

Since a and b are assumed independent, we can write, simply,A=A(a, ) ; B =B(b, ) , without any interaction terms. The expectation will be given by In the EPR situation A(a, ) = -B(a, ) . Therefore, we compute:

since A(a, )A(a, ) = 1, we can factor it out, to get

In other words, quantum theory contradicts a value of the expectation demanded by any kind of locality condition. Michael Redhead's treatment is better. Its somewhere in the notebooks.

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Bell's Inequalities can be tested with particles in entangled states. Source of particles with opposite spin:

this is an irreducible or indecomposable entanglement. In some sense one is not dealing with "two" particles but with a single particle in a strange geometry.

aaaaaaaaaaaaaaaVon Neumann on Projection

OperatorsTwo kinds of operators in Quantum Theory:

Hamiltonian Hermitian Operators, and Projection Operations, the U process and the R process.

A projection operator P is defined by two properties

(i) P is Hermitian(ii) P is idempotent P2 = PSuch operators can only have 1 or 0 as

eigenvalues:

A projection operator proclaims either "something is", or "something isn't"

Quantum Theory makes two kinds of calculations(1) eigenvalues of Observables, or "quantities"

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(2) probabilitiesThe probabilities for each eigenstate are the

square of the coefficients of the orthogonal expansion in terms of eigenvector functions. This follows from the calculations:

The simplest projections are those of "pure states", and can be written, in the Dirac formalism as

Let H be a Hamiltonian, a Schrodinger wave function. Then the action of H on is a "ket", which can be symbolized as H|>.Now, if we "hit" this ket with a projection operator (make an observation), the resultant can be written as

Properties of projections:If E and F are projections then EF is a projection

if and only if it is a commutation product EF = FE.E + F is a projection only if EF = 0 Now he gets onto his "U" business. If f is a

Schrodinger wave function, then its time evolution is given by

In the language of Lie Algebras, one can write the solution in the form

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The probability, or U function, evolves as

This follows from the fact that the probability is given by

(I still don't see how the probabilities are calculated. Perhaps they're determined experimentally.)

Therefore, a mixture statistical operator will be of the form:

Although the wave function evolves in time, this is a phase evolution only, and the eigenvalues and probabilities are invariants outside of time. These values replace "energy" as an invariant, which is replaced by an Expectation , which combines probabilities and eigenvalues :

where the series may have an infinite number of terms but which, we suppose, will converge!

ffffffffffffffffffffNotebook of December 25, 05Thermal Time: it is something of an anomaly that

temperature loss can be calibrated as a clock. After all. the Second Law does not say "Entropy always increases at a certain rate " , but only that entropy

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increases. Thus, I rely on the thermos bottles in which I carry about an afternoon coffee, to lose heat at a slow enough rate that I can still enjoy the coffee inside in two hours time. I also have a a strong sense of when the thermos has out for too long and the coffee is no longer drinkable.

One is reminded, perhaps, of the Hawking-Ellis g-incomplete lines. Tims just "stops" at a Black Hole, but the Hawking radiation restores it to the universe - at a predetemined rate!

Another example, of course, is found in the half life of radioactive substance. In fact one uses carbon-14 dating on the basis of this principle.

Is it possible that "time" itself is really only statistical in character? (Feynman would probably say yes.)

Note the similarity between the basic Axiom of time reckoning and of Probability. Time reckoning depends on the existence of "identity of state" of a dynamical system at two different moments. Probability depends upon the existence of "equally likely" alternative states of a given observable.

One of the important consequences of the irreversibility of time, one in fact that may be more important that irreversibility is the following: In our conception of spatial translation, it is possible to

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translate the observer also, to a new place, and back from a new place to the old one.

For temporal translation, it is impossible to translate the observer. He must wait until the translation moment arrives. One could, for example, imagine a kind of time translation in the forward direction only. Lets say that a "time machine" could project me forward into the 22nd century, but that no way exists to reverse the flow and bring me back to 2006. Would there be any logical paradoxes entailed by this? Obviously not, since the "Twins Paradox" is in some sense based upon this possibility.

aaaaaaaaaaaaaaJanuary 5, 2006:

Quantum Psychology. Observer-Observed Paradoxes are far more plentiful in psychology than they are in physics. One asks if anyone has thought seriously of developing a meta-psychology framework for psychology in analogy with the theoretical structures of Quantum Theory. Start for example, with these fundamental notions, and see if they have analogues in Psychology:

Uncertainty Principle; Wave/Particle Duality; Complementary Observables momentum/position time/ energy ; Wave Function containing all the information that one can possibly know; U process and R process; collapse of the wave packet; probabilities

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and eigenvalues; discrete energy levels; expectations; pure states; entanglements, superpositions and mixtures .

A mandatory starting point is the following: one gives up completely the dichotomies of doctor/patient ; researcher/subject. One treats the "trajectory" of the subject beforehand as unknowable, and the data gained from an observation as a set of eigenvalues, that is to say, partial information about what existed before the observation was made.

To date the best that psychology seems to have come up with is the "double bind" experiment.

aaaaaaaaaaaaaaaaaPast, Present and Future. These are translated in

epistlemologyical terms into the knowledge categories of the Unknowable, the Observable, and the Unknown. One can discuss these in their own terms without connecting them to time. This in fact is the proper approach to the "Epistlemology of Psychology"; "feeling the passage of time" doesn't represent a methodology for a theory of knowledge. The "unknowable" can also be called the "conjecturable"; the "unknown" can, in Popper's terms, be designated the "faslsifiable".

aaaaaaaaaaaaaaaaaJanuary 15 :

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"Quantum Time" versus "Relativity Time". Instead of working in a space-time with coordinates (x,y,z,t) , lets look at a 4-space with coordinates A4 = (x,y,z, a/mc) . where a is "Action" . The motivation for this is the following: If the relativistic 4-momentum is given by

one takes the time-integral of this quantity to derive:

the integral being taken from "then" to "now" .There is a difficulty of course, that a is "action divided by mass" . In some sense A4 is the "time integral" of momentum space.

Both spaces have "metrics" . One of the basic equations of Special Relativity is In fact this is a "metric" equation in Dual Minkowski space, or relativistic momentum space. If we let p4

signify the "4-momentum" of the above equation, and p3 its spatial components in the directions x, y, z , then

I'm not sure how this translates to a metric in (x,y,z, a/m) space

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Allright. In the notebook I develop a concept of generalized time, which is proper time for small arguments. So far this must be deemed unsatisfactory.

The ultimate goal is to obtain an observable which could be called the "time momentum" in analogy with the 3 components of spatial momentum. This would require differentiation by a different time coordinate, . One might, more simply, consider the equation:

Letting c = 1, the term ps is a kind of momentum observable, the derivative of the "proper time" by the "rest frame" time. Also, the metric in "momentum space" is simply the mass. Hence, and this is indeed very interesting, "mass" is in fact the value of a certain kind of Lorentzian metric.

Project: express Special Relativity in terms of this metric, treated as a "dual" form on dual-Minkowski space. Then combine Minkowski with dual-Minkowski space to create relativistic phase space.

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(2) In Special Relativity, "Space" is reduced to a single dimension, x. However, in the treatment of quantum spin it is clearly impossible to do this. What

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does this do to the combinations of relativity and quantum theory of QED and relativistic QM?

Octonions may be the answer. Combine quaternions with a new imaginary, q

For example, one might factor the expression

In order for this to work, the quantities qi, qj, qk , must be irreducible. However q2 = -1 , qi = -iq , etc. We have entered the domain of octonions.

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References Time In Quantum MechanicsSpringer 2002QC 174.13 .T56

Inconsistency, Asymmetry and Non-LocalityMathis Frisch Oxford UP 2005QC 631.3 .F 75

Brownian MotionRobert M. MazoOxford UP 2002QA 274.75. M 395

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Quantum Concepts in Space and TimePenrose and Isham Oxford 1986QC 173.96

Aristotle's "Physica"Hippocrates G. ApostlePeripatetic Press andIndiana University Press 1980Q 151 A7 A6

aaaaaaaaaaaaaaaaMarch 13, 06:

From Time in Quantum Mechanics : Preface: "...Problems .. in defining, formalizing

and measuring different time quantities in quantum theory: parametric time, tunneling times, decay times, dwell times, delay times, arrival times, or jump times ..."

The papers in this book were all presented at a workshop in Spain :

M. Beller :Presents a form of Pauli's theorem, which purports to show that a time operator cannot be defined in Quantum Theory.

Assume that there exists a self-adjoint time operator T . This would have to be non-commuting with the Hamiltonian .

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Designate an "energy eigenstate" as |E> , where E is the corresponding eigenvalue. "Measuring" by applying a unitary operator exp( -iE1T/h) produces a new eigenstate with eigenvalueE-E1 . (Don't fully understand the argument.) Since the spectrum would therefore be E is continous and unbounded , the time operator can't exist if the spectrum of H is bounded, semi-bounded or discrete ( Of course its possible that "time" could be bounded, semi-bounded or discrete) I've seen this argument elsewhere, in Schommers for example , where its more clearly presented.

The Mandelstamm-Tamm construction : They invoke the notion of an "Observable-specific time". Thus, let be some observable. The time-energy uncertainty relation can be written as

Informally speaking is the "typical or averaged time interval" required for a substantial modification of <>

fffffffffffffffAharanov-Bohm 1961 :

They conceived the idea of using a free particle's motion as a clock for measuring durations of quantum processes. Assume that a particle of momentum p

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passes a point y units from the origin. The passage time operator they conceive is written:

This operator is "maximally symmetric" which is a second-cousin to self-adjoint and, I seem to recall, has real eigenvalues, and is complementary to the

Hamiltonian Observation on page 20 "The tunneling time is one example of a quantization of a classical quantity that involves products of non-commuting variables"

fffffffffffMarch 17,2006 : It is time to study Dirac's classical text and review von Neumann. "Large Scale Structure of Space-Time" is postponed for this summer.

Paul Bush on the "Time Energy Uncertainty Relation" page 69 : The classical equation is DtDE ≥ (1/2)h

The interpretations of Bohm, Heisenberg, Pauli and Schrödinger are very different from each other. Indeed, to the present day there is no coherent agreement as to what it means. Time in Quantum Mechanics is inherently ambiguous:

(1) External time: this is the "parameter" in Schrödinger's Equation, and in the "time evolution" solution.

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(2) Intrinsic time. This reflects the dynamic behavior of systems at the quantum level. One may call it the "Event Time"Reflections on external time: Does the 'direction' of a measurement limit the accuracy of the outcomes?One might also interpret the classical equation by saying that the energy is completely indeterminate during the measurement period. This is something like the procedure in Thermodynamics, in which temperature has no meaning for a system not in equilibrium.

Intrinsic time: Every dynamic variable, or "observable" marks the passage of time. One can speak of the "characteristic time" A , of the observable A .

If A is an observable, one can define the "uncertainty" of it characteristic time as :

Thus , let our observable be the "position" of a particle Q. The "characteristic time , Q , can be taken to be the time for the bulk of the wave packet to shift forward by a distance equal to its width.

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In general, let be a Schrodinger function and P a projection operator. Define ( P) as the maximal

length of a time interval T in which Further manifestations of time in Quantum

Mechanics include: * Time delay in scattering theory

*Dwell time in tunneling* Lifetime of an unstable state They all lead to a different way of interpreting the

Time-Energy relation.aaaaaaaaaaaaaaaa

Observable Time:The time of arrival of decay products in a

detector; he "time of flight". The Pauli Theorem is stated. I need to dig up a better version of it.

aaaaaaaaaaaaNotes on the Salecker-Wigner Quantum Clock 4 Areas of contemporary research:

(1) Quantum nature of space-time on length scales larger than the Planck length

(2) Tunneling time(3) Superluminal proton propagation

( Feynman stuff, also Hawking radiation)(4) Quantum information approach

S-W Clock:

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This is a system passing through successive distinct orthogonal states at equal laboratory time intervals. There are altogether n energy eigenstates in superposition. The wave

function is therefore: For a proper Quantum Clock one would like to

have a non-demolition system, one that one can read without destroying the system.

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