From Prof. David R. Finkelstein’s book Quantum Relativity pg. 21-2 and 380-2
"1.2.3 Time
As a corollary of quantum bimodality, there is a basic difference between the time concepts of quantum and classical physics.
In classical physics we can imagine that we know everything about the system at every value of the time, as though by continuous observation. The classical time parameter t is a real time, at least in imagination.
In quantum physics we recognize that we learn something about the system at just two times in every experiment, the beginning and the end. If we learn something about the system at a different time, that ends one experiment and starts another. Pictures that may seem to show how a quantum system evolves over an interval of time actually show the results of many experiments of different duration, not one continuous experiment. Where classical physics imagined that it operated in real time, quantum physics operates in sample time.
By inventing and applying the differential calculus, Newton succeeded in describing how a particle is at every instant. Newtonian motion is a continuous sequence of states of being in real time. In renouncing both states and real time, quantum physics dismantles Newton’s great construction.
Within the bimodal sample time of quantum theory there is still room for many conceptions of time. Natural languages incorporate other assumptions about time in their tense structure besides unimodality, such as the existence and absoluteness of the classes of past, present and future events at each time. Some of these assumptions are modified in classical spacetime theories and others in quantum theories. We deal in this work with two forms of dynamics, termed sequential and distributed, developed for particles and fields respectively.
A sequential dynamics assumes a single unanalyzed initial act and a like final act at two instants of time, initial and final (which may degenerate to one). It uses a Galilean, pre-Einsteinian concept of instant, and treats dynamics as a sequence of transitions between such instants, that is, as a sequence of choices. A sequential dynamics need be local since it relates one entire time-slice to another. Indeed, the historic prototype is the Newtonian of gravity, a non-local theory assuming instantaneous action at a distance.
A distributed dynamics allows interventions at all points of [a] space-time. A discrete version is a network dynamics, which assumes a network of causal connections among events, and describes how each event influences those immediately connected to it. The network itself may have a fixed structure populated by dynamical entities; or it may be a dynamical variable in its own right, like a switching network [or a gossiping hypercubic computer]. A distributed dynamics may be local. [Local system!?]
Newton assumed that time was infinitely divisible [by a single knowable universal standard or metric!?], and this is still the only working theory of time we have. This assumption led to divergent results in classical physics, and while the quantum physics improves upon these divergences, some remain. After we present the usual quantum theory, we explore a quantum network dynamics with a discrete fundamental quantum of time or chronon "\Delta t", in the search for a more consistent theory.
"12.4 Quantum Dynamics
We now consider dynamics of a more general kind than canonical. The concept of time we use is born of astronomy, the most classical of physical sciences, and is thus suspect, but is also of great pragmatic value. We define time as what clocks meter. We have not discovered yet what it is, then, that clocks meter. Similarly in hydraulics we might have defined the amount of water with a liter measure, and only later discovered what is actually metered, the number of water molecules.
In mechanics there is one independent variable, the time t, whose domain is the real axis, and as many dependent variables as we need to describe the system. Sometimes we use a discrete time whose range is the integers as a simplifying approximation.
Up to now all initial processes might have been performed with no lapse of time between them, at one common instant of time t = 0. Now we recognize explicitly that in addition to specifying the orientation of a polarizer (for example) we must in general determine when it acts upon the photon. For example, we may consider a photon traveling along (an) optical bench in a pipe filled with a clear but optically active medium such as sugar water, which rotates the polarization of a photon traveling through it. Then where we put a polarizer along the bench determines when it acts on the photon and may influence the outcome of the experiment. [The delayed choice experiment puts a twist on this statement. SPK note.]
Such experiments cause us to regard time as a numerical parameter t that must be specified in addition to an initial vector y in order to describe the initial process adequately.
If the system is extended then the experiments associated with a specified time parameter t_0 must be correspondingly non-local, since they must give maximal information about the whole system at the time t_0. This concept of experiment may bring us into conflict with general relativity, which is dominated by the assumption that the basic concepts and laws of nature are local. A deep synthesis of general relativity and quantum theory must analyze these non-local actions into local ones. We do not attempt that in this chapter, but continue to deal with global experiments described by global actions at a definite time parameter t_0.
We shall write such an initial or final action generically as
| t_0 phi> or <phi t_0|
This means that we have a separate initial vector space for each instant of time. To declare that an initial act described by a particular time t = t_0 we write a product like
| 1 |
|t’ phi > = | |
| 0 | t = t’
In principle variables too must bear such a time declaration, and are represented by matrices whose elements depend on their time parameter. In practice the time is often specified in context for external acts and variables alike.
We describe change relative to some standard of constancy. To define what it means when an initial matrix depends on time, we first state what it means to be constant. If (for example) the above column matrix stands for the vertical polarizing direction at time t_0, we must state what we shall mean by the same column matrix attached to some other time t_1. In general, we must give operational meaning to the concept of "the same action at different times t’.
To do this we may define how the frame is physically transported along light path from t’ to t".
12.4.1 Real Time and Sample Time
We have a strong illusion of continuous observation when we watch a man running or a planet. To classical intuition at least, there seems to be a position observation for every time in a certain interval of the real numbers. The time parameter of classical mechanics, therefore, seems to be a real time. We actually observe the system during its passage.
Not in quantum physics. In the three-stage experiment F Ä M Ä I we experience each photon only twice, at I and F. We still seem to have a continuum of possible times at which we might register the photon, but we choose only one of these in each experiment. To understand the continuous evolution of one photon in time, we must register many photons once in their flight, not one photon at many times. The time parameter t of quantum theory gives the time between the only two external actions on a photon. It is therefore not a real time of classical mechanics but a sample time, labeling a sample from a large ensemble of photon experiments. It is a time at which we might determine a property of a photon, but almost never do.
The continuity of quantum time means that the moment at which we sample a quantum may be changed by as small as we like between experiments, not that we watch the quantum continuously."
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Kolmogorov Complexity
From "An introduction to Kolmogorov Complexity and its Applications"
By Li Vitanyl
Pg. 140-2,
2.8 Algorithmic information Theory
One interpretation of the complexity C(x) is a the quantity of information needed for the recovery of an object x from scratch. [Here I am assuming that the author implies "scratch’ to mean "basic atomic constituents."] Similarly, the conditional complexity C(x|y) quantifies the information needed to recover x given only y. Hence the complexity is ‘absolute information’ in an object. Can we obtain similar laws for complexity based ‘absolute information theory’ as we did for the probability based information theory of Section 1.5.3?
If C(x|y) is much less than C(x), then we may interpret this as an indication that y contains a lot of information about x.
Definition 2.18 The algorithmic information about y contained in x is defined as
I_C(x : y) = C(y) – C(y|x).
If we choose the reference function f _0 in theorem 2.1 such that f _0(x, e ) = x, then
C(x|x) = 0 and I_C(x : x) = C(x).
By the additively optimality of f _0, these equations hold up to an additive constant independent of x, for any reference function f _0. In this way we can view the complexity C(x) as the algorithmic information contained in an object about itself. For applications, this definition of the quantity of information has the advantage that it refers to individual objects, and not to objects treated as elements of a set of objects with a probability distribution given on it, as treated in Section 1.5.3.
Does the new definition have the desirable properties that hold for the analogous quantities in classical information theory? We know that equality and inequality can hold only up to additive constants, according to the indeterminacy in the invariance Theorem 2.1. Intuitively, it is reasonable to require that
I_C(x : y) ³ 0,
Up to an additive fixed constant independent of x and y. Formally, this follows easily from the definition of I_C (x : y), by noting that C(y) ³ C(y|x) up to an independent additive constant.
The major point we have to address is the relation between the Kolmogorov complexity and Shannon’s entropy as defined in Section 1.5.3. Briefly, classic information theory says a random variable X distributed according to P(X =x) has entropy (complexity) H(X) = - å P(X = x)log P(X = x), where the interpretation is that H(X) bits are on the average sufficient to describe an outcome x. Algorithmic complexity says that an object x has complexity C(x) = the minimum length of a binary program for x. it is a beautiful fact that these two notions turn out to be much the same. The statement below may be called the theorem of equality between Stochastic Entropy and Expected Algorithmic Complexity.
Theorem 2.19 Let x = y_1y_2…y_n be a finite binary string with l(y_1) = … = l(y_m) = r. Let the frequency of occurrence of the binary representation of k = 1, 2, 3, …, 2^r as a y-block be denoted by p_k = d({I : y_I = k})/m. Then, up to an independent additive constant,
C(x) £ m(H + e (m)),
with H = - å p_k log p_k, the sum taken for k from 1 to 2^r, and e (m) = 2^r l (m)/m. Note that e (m) ® 0 for m ® ¥ with r fixed.
Proof. Denote 2^r by R. To reconstruct x it suffices to know the number s_k = p_k m of occurrences of k as an y_i in x, k = 1, 2,…, R, together with x’s serial number j in the ordered set of all strings satisfying these constraints. That is, we can recover x from s_1,…, s_R, j. Therefore, up to an independent fixed constant, C(x) £ 2l(s_1) + … + 2l(s_R) + l(j). By construction,
j £ æ m ö
è
s_1, …, s_R øa multinomial coefficient, page 9. Since also each s_k £ m, we find
C(x) £ 2^r+1 l(m) + æ m ö
è
s_1, …, s_R øwriting the multinomial coefficient in factorials, and using Stirling’s approximation, exercise 1.18 on page 16, to approximate j, the theorem is proved.
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