2 Foundations

[91.2.1] The following basic requirements define a time evolution in this chapter.

1. Semigroup
   [91.2.2] A time evolution is a pair ({Tτ(t):0≤t<∞},(Bτ,∥⋅∥)) where Tτ⁢t=T⁢t⁢τ is a semigroup of operators T⁢t:0≤t<∞ mapping the Banach space (Bτ(R),∥⋅∥) of functions fτ⁢s=f⁢s⁢τ on R to itself. [91.2.3] The argument t≥0 of Tτ⁢t represents a time duration, the argument s∈R of fτ⁢s a time instant. [91.2.4] The index τ>0 indicates the units (or scale) of time. [91.2.5] Below, τ will again be frequently suppressed to simplify the notation. [91.2.6] The elements fτ⁢s=f⁢s⁢τ∈Bτ represent observables or

   [page 92, §0]

   the state of a physical system as function of the time coordinate s∈R. [92.0.1] The semigroup conditions require

   Tτ⁢t1⁢Tτ⁢t2⁢fτ⁢t0	=Tτ⁢t1+t2⁢fτ⁢t0		(7)
   Tτ⁢0⁢fτ⁢t0	=fτ⁢t0		(8)

   for t1,t2>0, t0∈R and fτ∈Bτ. [92.0.2] The first condition may be viewed as representing the unlimited divisibility of time.

2. Continuity
   [92.0.3] The time evolution is assumed to be strongly continuous in t by demanding

   limt→0⁡T⁢t⁢f-f=0

   (9)

   for all f∈B.

3. Homogeneity
   [92.0.4] The homogeneity of the time coordinate requires commutativity with translations

   T⁢t1⁢T⁢t2⁢f⁢t0=T⁢t2⁢T⁢t1⁢f⁢t0

   (10)

   for all t2>0 and t0,t1∈R. [92.0.5] This postulate allows to shift the origin of time and it reflects the basic symmetry of time translation invariance.

4. Causality
   [92.0.6] The time evolution operator should be causal in the sense that the function g⁢t0=T⁢t⁢f⁢t0 should depend only on values of f⁢s for s<t0.

5. Coarse Graining
   [92.0.7] A time evolution operator T⁢t should be obtainable from a coarse graining procedure. [92.0.8] A precise definition of coarse graining is given in Definition 2.3 below. [92.0.9] The idea is to combine a time average 1t⁢∫s-tsf⁢t′⁢d⁢t′ in the limit t,s→∞ with a rescaling of s and t.

[92.0.10] While the first four requirements are conventional the fifth requires comment. [92.0.11] Averages over long intervals may themselves be timedependent on much longer time scales. [92.0.12] An example would be the position of an atom in a glass. [92.0.13] On short time scales the position fluctuates rapidly around a well defined average position. [92.0.14] On long time scales the structural relaxation processes in the glass can change this average position. [92.0.15] The purpose of any coarse graining procedure is to connect microscopic to macroscopic scales. [92.0.16] Of course, what is microscopic

[page 93, §0]    depends on the physical situation. [93.0.1] Any microscopic time evolution may itself be viewed as macroscopic from the perspective of an underlying more microscopic theory. [93.0.2] Therefore it seems physically necessary and natural to demand that a time evolution should generally be obtainable from a coarse graining procedure.

[93.1.1] There is a close connection and mathematical similarity between the simplest time evolution T⁢t=T⁢t and the operator M⁢t of time averaging defined as the mathematical mean

where t>0 is the length of the averaging interval. [93.1.2] Rewriting this formally as

exhibits the relation between M⁢t and T⁢t. [93.1.3] It shows also that M⁢t commutes with translations (see eq. (10)).

[93.2.1] A second even more suggestive relationship between M⁢t and T⁢t arises because both operators can be written as convolutions. [93.2.2] The operator M⁢t may be written as

where the kernel

is the characteristic function of the unit interval. [93.2.3] The Laplace convolution in the last line requires t<s. [93.2.4] The translations T⁢t on the other hand may be

[page 94, §0]    written as

where again 0<t<s is required for the Laplace convolution in the last equation. [94.0.1] The similarity between eqs. (15) and (13) suggests to view the time translations T⁢t as a degenerate form of averaging f over a single point. [94.0.2] The operators M⁢t and T⁢t are both convolution operators. [94.0.3] By Lebesgues theorem limt→0⁡M⁢t⁢f⁢s=f⁢s so that M⁢0⁢f⁢t=f⁢t in analogy with eq. (8) which holds for T⁢t. [94.0.4] However, while the translations T⁢t fulfill eq. (7) and form a convolution semigroup whose kernel is the Dirac measure at 1, the averaging operators M⁢t do not form a semigroup as will be seen below.

[94.1.1] The appearance of convolutions and convolution semigroups is not accidental. [94.1.2] Convolution operators arise quite generally from the symmetry requirement of eq. (10) above. [94.1.3] Let Lp⁢Rn denote the Lebesgue spaces of p-th power integrable functions, and let S denote the Schwartz space of test functions for tempered distributions [27]. [94.1.4] It is well established that all bounded linear operators on Lp⁢Rn commuting with translations (i.e. fulfilling eq. (10)) are of convolution type [27].

[94.2.1] For p=q=1 the tempered distributions in this theorem are finite Borel measures. [94.2.2] If the measure is bounded and positive this means that the operator B can be viewed as a weighted averaging operator. [94.2.3] In the following the case n=1 will be of interest. [94.2.4] A positive bounded measure μ on R is uniquely determined by its distribution function μ~:R→0,1 defined by

[94.2.5] The tilde will again be omitted to simplify the notation. [94.2.6] Physically a weighted average M⁢t;μ⁢f⁢s represents the measurement of a signal f⁢s using an apparatus with response characterized by μ and resolution t>0. [94.2.7] Note that the resolution (length of averaging interval) is a duration and cannot be negative.

[page 95, §1]   

[95.3.1] The averaging operator M⁢t in eq. (11) corresponds to a measure with distribution function

while the time translation T⁢t corresponds to the (Dirac) measure δ⁢x-1 concentrated at 1 with distribution function

[95.3.2] Both averages are causal, and the latter is degenerate.

[95.4.1] Repeated averaging leads to convolutions. [95.4.2] The convolution κ of two distributions μ,ν on R is defined through

[95.4.3] The Fourier transform of a distribution is defined by

where the last equation holds when the distribution admits a weight function. [95.4.4] A sequence μn⁢x of distributions is said to converge weakly to a limit μ⁢x,

[page 96, §0]    written as

if

holds for all bounded continuous functions f.

[96.1.1] The operators M⁢t and T⁢t above have positive kernels, and preserve positivity in the sense that f≥0 implies M⁢t⁢f≥0. [96.1.2] For such operators one has

[96.1.6] The preceding theorem suggests to represent those time evolutions that fulfill the requirements 1.– 4. of the last section in terms of convolution semigroups of measures.

[page 97, §1]    [97.1.1] Here δ is the Dirac measure at 0 and the limit is the weak limit. [97.1.2] The desired characterization of time evolutions now becomes

[97.2.1] Equation (29) establishes the basic convolution structure of the assertion in eq. (5). [97.2.2] It remains to investigate the requirement that T⁢t should arise from a coarse graining procedure, and to establish the nature of the kernel in eq. (5).

[97.3.1] The purpose of this section is to motivate the definition of coarse graining. [97.3.2] A first possible candidate for a coarse grained macroscopic time evolution could be obtained by simply rescaling the time in a microscopic time evolution as

where 0<t¯<∞ would be macroscopic times. [97.3.3] However, apart from special cases, the limit will in general not exist. [97.3.4] Consider for example a sinusoidal f⁢t oscillating around a constant. [97.3.5] Also, the infinite translation T∞ is not an average, and this conflicts with the requirement above, that coarse graining should be a smoothing operation.

[97.4.1] A second highly popular candidate for coarse graining is therefore the averaging operator M⁢t. [97.4.2] If the limit t→∞ exists and f⁢t is integrable in the finite interval s1,s2 then the average

is a number independent of the instant si. [97.4.3] Thus, if one wants to study the macroscopic time dependence of f¯, it is necessary to consider a scaling limit in

[page 98, §0]    which also s→∞. [98.0.1] If the scaling limit s,t→∞ is performed such that s/t=s¯ is constant, then

becomes again an averaging operator over the infinitely rescaled observable. [98.0.2] Now M⁢1 still does not qualify as a coarse grained time evolution because M⁢1⁢M⁢1≠M⁢2 as will be shown next.

[98.1.1] Consider again the operator M⁢t defined in eq. (11). [98.1.2] It follows that

and

[98.1.3] Thus twofold averaging may be written as

where

is the new kernel. [98.1.4] It follows that M2⁢t≠M⁢2⁢t, and hence the averaging operators M⁢t do not form a semigroup.

[98.2.1] Although M2⁢t≠M⁢2⁢t the iterated average is again a convolution operator with support 0,2⁢t compared to 0,t for M⁢t. [98.2.2] Similarly, M3⁢t has support 0,3⁢t. [98.2.3] This suggests to investigate the iterated average Mn⁢t⁢f⁢s in a scaling limit n,s→∞. [98.2.4] The limit n→∞ smoothes the function by enlarging the

[page 99, §0]    averaging window to 0,n⁢t, and the limit s→∞ shifts the origin to infinity. [99.0.1] The result may be viewed as a coarse grained time evolution in the sense of a time evolution on time scales "longer than infinitely long". [99.0.2] ^c (This is a footnote:) ^c The scaling limit was called "ultralong time limit" in [10] It is therefore necessary to rescale s. [99.0.3] If the rescaling factor is called σn>0 one is interested in the limit n,s→∞ with s¯=s/σn fixed, and σn→∞ with n→∞ and fixed t>0

whenever this limit exists. [99.0.4] Here s¯>1 denotes the macroscopic time.

[99.1.1] To evaluate the limit note first that eq. (11) implies

where fτ⁢t=f⁢t⁢τ denotes the rescaled observable with a rescaling factor τ. [99.1.2] The n-th iterated average may now be calculated by Laplace transformation with respect to s¯. [99.1.3] Note that

for all c∈R, where E1,2⁢x is the generalized Mittag-Leffler function defined as

for all a>0 and b∈C. [99.1.4] Using the general relation

gives with eqs. (37) and (38)

where f⁢u¯ is the Laplace transform of f⁢s¯. [99.1.5] Noting that E1,3⁢0=1/2 it becomes apparent that a limit n→∞ will exist if the rescaling factors are

[page 100, §0]    chosen as σn∼n. [100.0.1] With the choice σn=σ⁢n/2 and σ>0 one finds for the first factor

[100.0.2] Concerning the second factor assume that for each u¯ the limit

exists and defines a function f¯⁢u¯. [100.0.3] Then

and it follows that

[100.0.4] With t¯=t/σ Laplace inversion yields

[100.0.5] Using eq. (12) the result (47) may be expressed symbolically as

with t¯=t/σ. [100.0.6] This expresses the macroscopic or coarse grained time evolution T¯⁢t¯ as the scaling limit of a microscopic time evolution T⁢t. [100.0.7] Note that there is some freedom in the choice of the rescaling factors σn expressed by the prefactor σ. [100.0.8] This freedom reflects the freedom to choose the time units for the coarse grained time evolution.

[100.1.1] The coarse grained time evolution T¯⁢t¯ is again a translation. [100.1.2] The coarse grained observable f¯⁢s¯ corresponds to a microscopic average by virtue of the following result [29].

[page 101, §1]   

[101.1.2] Comparison of the last relation with eq. (44) shows that f¯⁢s¯ is a microscopic average of f⁢s. [101.1.3] While s is a microscopic time coordinate, the time coordinate s¯ of f¯ is macroscopic.

[101.2.1] The preceding considerations justify to view the time evolution T¯⁢t¯ as a coarse grained time evolution. [101.2.2] Every observation or measurement of a physical quantity f⁢s requires a minimum duration t determined by the temporal resolution of the measurement apparatus. [101.2.3] The value f⁢s at the time instant s is always an average over this minimum time interval. [101.2.4] The averaging operator M⁢t with kernel χ0,1 defined in equation (11) represents an idealized averaging apparatus that can be switched on and off instantaneously, and does not otherwise influence the measurement. [101.2.5] In practice one is usually confronted with finite startup and shutdown times and a nonideal response of the apparatus. [101.2.6] These imperfections are taken into account by using a weighted average with a weight function or kernel that differs from χ0,1. [101.2.7] The weight function reflects conditions of the measurement, as well as properties of the apparatus and its interaction with the system. [101.2.8] It is therefore of interest to consider causal averaging operators M⁢t;μ defined in eq. (17) with general weight functions. [101.2.9] A general coarse graining procedure is then obtained from iterating these weighted averages.

[102.1.1] The purpose of this section is to investigate the coarse graining procedure introduced in Definition 2.3. [102.1.2] Because the coarse graining procedure is defined as a limit it is useful to recall the following well known result for limits of distribution functions [30]. [102.1.3] For the convenience of the reader its proof is reproduced in the appendix.

[102.2.1] The basic result for coarse graining limits can now be formulated.

[104.3.1] The theorem shows that the coarse graining limit, if it exists, is again a macroscopic weighted average M⁢t;ν. [104.3.2] The condition (53) says that this macroscopic average has a kernel that is stable under convolutions, and this motivates the

[104.4.1] This nomenclature emphasizes the close relation with the limit theorems of probability theory [30, 31]. [104.4.2] The next theorem provides the explicit form for distribution functions satisfying eq. (66). [104.4.3] The proof uses Bernsteins theorem and hence requires the concept of complete monotonicity.

[page 105, §1]   

[105.2.1] Bernsteins theorem [31, p. 439] states that a function is completely monotone if and only if it is the the Laplace transform (u>0)

of a distribution μ or of a density m=d⁢μ/d⁢x.

[105.3.1] In the next theorem the explicit form of stable averaging kernels is found to be a special case of the general H-function. [105.3.2] Because the H-function will reappear in other results its general definition and properties are presented separately in Section 4.

[107.0.4] Note that hα⁢x=hα⁢s;1,0 is the standardized form used in eq. (5). [107.0.5] It remains to investigate the sequence of rescaling factors σn. [107.0.6] For these one finds