Time in Physics

Note: 
1) We do not have a unique fundamental theory in physics. Instead, physics
today is based on several conceptual frameworks which are not fully consistent
with each other. Different frameworks provide different pictures of time,
and so one always has to specify the framework to which one refers. 
2) The natural language of physics is mathematics, and whenever one tries
to express physics in prose, precision and clarity are lost to some extent.

I) Classical Mechanics, classical field theory
Physical laws are given in terms of differential equations (usually
second order in time). Physical objects are particles and fields,
distributed in space and evolving with time.
A physical system is completely determined by
a) the differential equations and 
b) initial conditions (positions and velocities of particles, values of fields
   and their first derivatives with respect to time) at some time t0.
Using the differential equations and starting with the initial conditions one 
can (in priciple, i.e. with infinite computational power) determine the complete 
history of the system for all times before and after t0. 
One may therefore say that the initial time t0 implicitly "contains" all other 
times.
Furthermore physical laws are unchanged with respect to time reversal: All 
processes allowed by the laws are also allowed to happen backwards.
Cause and effect are therefore interchangeable, and the whole concept
of causality is questionable.

II) Quantum Mechanics, Quantum Field Theory
QM is conceptually very different from Classical Mechanics, 
but has Classical Mechanics as a kind of 'macroscopic limit', 
i.e. if the world is best described by QM / QFT on microscopic scales, 
it still looks 'classical' on macroscopic scales.
The role of time in QM is unclear. The Schroedinger equation
(which is the fundamental equation of QM) is a differential equation first order 
in time and therefore provides a deterministic picture of physical systems, 
described by vectors in so-called Hilbert spaces. However, the measurement
process in QM-experiments appears to be not deterministic but rather governed
by a fundamental randomness whose origin and nature are still a matter of 
intense debate.

III) Statistical Mechanics 
Statitical Mechanics is consistent with both Classical Mechanics
and QM, but it brings in a new angle: We cannot know any system at any
given time to perfect precision, our knowledge is always incomplete and tied
to mostly macroscopic observables. Each 'macrostate' (something we can observe
and describe) 'consists' of a large ensemble of microstates, and we
do not know which particular microstate is realized in the particular macrostate
we observe. It is this incompleteness of knowledge which breaks down
determinism on a practical level. 
Some macrostates are 'larger' than others, i.e. they consist of 'more'
microstates (they have a larger entropy), and therefore a transition 
from macrostate A to macrostate B may be more likely (by many orders
of magnitude) than the opposite transition from B to A. This means
that time reversal symmetry appears broken for 'macroscopic observers',
and brings causality back into play (note: it is only our incomplete
knowledge which makes causality a meaningful notion).
Another aspect of these issues is that most processes can only in one
time-direction be told in terms of macroscopic variables: A glass dropping 
to the floor and breaking into pieces makes sense to us, but the opposite
process of the pieces jumping up from the floor and combining to form
a glass does not. This doesn't mean that the latter process is impossible,
it is just very 'unlikely' in terms of macrostates, and its explanation
requires a detailed knowledge about 'all atoms in the world'.

Entropy raises many questions. Its value depends on how we combine microstates
into macrostates, and it is not entirely clear how much freedom there is 
to do this in a 'meaningful' way. Also it is unclear why the universe is so 
asymmetric in terms of  entropy, i.e. why in one time-direction (which is
necessarily the direction we consider as 'past') the entropy is so much lower 
than in the other.    

IV) Special and General Relativity
In Einstein's theories, space and time are combined into a single entity,
'spacetime', and they can to some extent be 'transformed into each other'.
In General Relativity, spacetime obtains a strucure on its own, described
in terms of topology, metric and curvature.
In spirit, Relativity is much closer to Classical Mechanics than to QM.
All classical theories with certain symmetries (in fact all classical
theories 'found' in the real world so far) can be formulated such that they
obey General Relativity. 

Since space and time are part of the same four-dimensional structure, 
a duration is equivalent to a spatial extension. A particle at rest is 
a one-dimensional entity, a line parallel to the 'time-axis'. A string
is two-dimensional, with an extension in one space- and in time-direction.
Our bodies are four-dimensional, extended a little in all three 
space-directions, and our lifetime is the extension in time-direction,
where 1 second 'IS' 300000 kilometers (the speed of light is precisely 1).
Note that we have a very 'unphysical' experience of our body, with respect
to Relativity. Instead of a four-dimesional entity embedded in a 
four-dimensional spacetime like a fossil in amber, we observe ourselves
as three-dimensional beings 'traveling through time'.