The Physical Basis of The Direction of Time (The Frontiers ...

72 3 The Thermodynamical Arrow of Time
A microscopic trajectory q(t) determines all macroscopic trajectories α(t)
defined as functions of this state: α(t) := α ( p(t),q(t) ) . As discussed in
Sect. 3.1.2, the macroscopic dynamics α(t) is then in general not autonomous,
since trajectories starting from the same α(t 0 ) may evolve into different α(t 1 )
– depending on the microscopic initial state p(t 0 ),q(t 0 ). This macroscopic
indeterminism is essential for fluctuations or certain phase transitions.
The determinism of a dynamical model (such as Laplacean mechanics)
is defined by the mathematical existence of a unique mapping of appropriate
initial (or final) states onto complete trajectories. This concept of determinism
is independent of the availability of an (analytic or algorithmic) procedure for
explicitly constructing these trajectories in terms of conventional coordinates
(‘integrability’). It is therefore also independent of any practical limitation to
their computability, which forms the basis of Kolmogorov’s (1954) entropy,
and is often used in the definition of chaos (see Schuster 1984, or Hao-Bai-Lin
1987). In classical mechanics, the deterministic dynamical mapping of initial
conditions onto trajectories is a consequence of Newton’s equations under nonsingular
conditions (see Bricmont 1996 for his lucid criticism of the popular
misuse of the concept of chaos in this connection).
Trajectories could in principle be described in terms of the constants of
the motion. The latter could then be used as new coordinates or ‘co-evolving
grids’ (see Appendix B of Zurek 1989). Such constants of the motion are often
denied to exist, since they are not analytically related to conventional coordinates.
However, this does not mean that they would not exist in any absolute
sense. It was indeed one of the great lessons from the theory of relativity that
physics and spacetime geometry (‘reality’) are independent of the choice of
coordinates, while the ancient Greeks were not even able to overcome Zeno’s
paradox of Achilles and the tortoise by a transformation to more appropriate
‘coordinates’ of description. We should similarly be able to conceptually
overcome all mathematical problems in the construction of canonical transformations,
and instead rely on the assumption of a coordinate-free ‘reality’
(at least in classical mechanics).
These mathematical difficulties may nonetheless reflect the complex and
non-trivial physical relation between the Universe and its ‘observing parts’.
Observers are evidently not in any simple way related to the constants of the
motion – the reason why we feel ‘time change’. 10 Some authors have related
the problems of a universe that contains its observers (physical self-reference)
to Gödel’s undecidability theorems, which apply to logical systems that allow
formal self-reference (see Wheeler 1979). However, one cannot argue that
the existence or meaning of an observer-independent reality is excluded just
because of the observers’ limited capabilities. This insufficient argument has
even been used as an explanation of ‘quantum uncertainty’ (Popper 1950, Born
1955, Brillouin 1962, Cassirer 1977, Prigogine 1980). There is a fundamental
10 “Time goes, you say? Ah no! Alas, time stays, we go.” (Austin Dobson – discovered
in Gardner 1967.)