Some comments:
My earlier post was rather glib. Consider in general a clock as any causal sequence of events (each tick causing the next tick). One can then compare and calibrate clocks and observe "good" clocks as those for which the dynamics of physical systems has the simpler form (within the constraint of being equally useful for prediction) when the systems are parametrized by their clicks.
Consider this in the sense of
general covariance where we allow arbitrary re-parametrization of time in our dynamic equations. (possibly but not necessarily constrained by monotonicity e.g. preserving time ordering).
Prathyush said:
First is in special relativity we have world lines evolving in a space time continuum with the Lorenz transformation the space time metric.
It helps to understand here that the space-time continuum itself is a continuous array of
events (including null events e.g. "nothing happened here-now"). As time is used here it is inextricably mixed with our notions of causal ordering of events and hence of our notions of
causality and
events themselves.
But note that here, time (and also in the relativistic case space as well) is a
parametric quantity which is to say something we construct to index the causal relationship between events as opposed to being representative of any physical observable of existent objects. It is the events which we observe and thus which have direct physical meaning (clicks of particle detectors and collisions of galaxies or simply the fact that nothing extra-ordinary has happened). I think that you can't isolate time in this setting. Speak only of space-time structure as it models the transitive causal relationship between events:
A affects B
B affects A
A and B are causally isolated from each other.
Mathematically such a set of relationships (with "affects" being transitive A->B->C => A->C) forms a lattice structure. This plus topology gives us the metric space-time structure since knowing the boundaries of the future light-cones for (events which may be affected by) each event defines the metric on a topological space.
The second is in quantum mechanics there seems to be a world clock that ticks at a constant rate. The phase of each energy eigenstate evolves as
(e^iEt/h)
In quantum field theory also time is a unitary evolution. However the Lagrangian is taken to be Lorentz invariant.
The "world clock" in this case is the laboratory clock and we must invoke some form of covariance to express how e.g. different laboratory frames will see the same events. I'm not sure what you mean by "seems to tick at a constant rate" rates are comparisons (same root as ratio). All clocks tick at a constant rate with respect to themselves. We must start with the base clock to compare other changing quantities to in order to define a rate.
In any event this quantum time is itself
parametric in nature just as mentioned in the relativistic case above. Hence time is not a physical observable in standard QM.
Note however that in the treatment of position as an observable automatically breaks the time and space unification of special relativity and its generalization. In using time parametrically we must in the relativistic setting also treat space similarly.
If we take the reverse course and seek to treat time as an observable of a system then our quantum systems can no longer be thought of as corresponding to classical objects (in the large scale of action limit) but rather to classical events. Consider e.g. when doing collision experiments then how one deduces where and when two particles collided based on the tracks of their products. Also consider how one might need to modify the interpretation of "bras" and "kets" not as state vectors but as "event vectors" some of which are not localized either in space or in time.
The Lorentz invariance of which you speak isn't there in finite dimensional QM e.g. particle spin. The relativistic treatment is rather more involved and typically jumps straight into a quantified theory (quantification = variable number of instances of the base system i.e. many particle theory = field theory). I think to treat it properly we must look beyond unitary QM and develop a satisfactory pseudo-unitary formulation. That's an open project as far as I've been able to determine.
And the third where the direction of time is given by the direction of entropy increase of the system. And a also relating to the time symmetry of the underlying laws.
Be careful here. The entropy arrow is only meaningful when we speak of systems in isolation. The entropy of my leftovers certainly drops when I stick them in the freezer.
This then opens a possible can of worms as to whether we can speak of "systems in isolation" meaningfully without invoking time. Certainly we can't control or know about a "system in isolation" unless and until we make prior and posterior observations (breaking isolation) of that system. This then requires we speak of "isolation for a period of time" in some sense. I don't believe we can glean as much understanding of time from considering entropy as we can glean understanding of entropy by considering time...if that makes any sense. In short I think it better to consider time apart from the 2nd law first, get a handle on it and then come back to the thermodynamic issues.
(However at the heart of any measurement process especially at the quantum level are thermodynamic considerations... Just look at what the physicist's do in the lab, how much cooling equipment is required to do precision quantum measurements!)
In the end, to my mind, it all comes down to causal structure which we infer and assume as we interpret what it means to observe and effect measurements i.e. to affect and be affected by nature (which itself is the epistemological foundation of any knowledge we assert in physics). I don't think we can reduce it further but must treat it in the same way mathematicians treat undefined terms... as primaries which we understand in application from common experience and use then to define other terms.
Well that's my 2cents worth. (Make the check payable to James Baugh ;)
