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In thermodynamical theory, if we have a set of states for example:

A) gas all in top left corner of box.

B) gas spread out near the left.

C) gas spread out to fill entire box.

We can give the states a time ordering given by the likely order in which these events occur. Using the thermodynamics rule "gas tends to spread out" to fill a vacuum.

For a timeless theory (e.g. quantum cosmology), there is a wavefunction which has an amplitude for every basis state.

If the basis states are as above, are we to give a time ordering to the basis states themselves or the particle state-vector, i.e. the wavefunction?

Further, if the time ordering of the basis states is purely psychological, how do we ensure probability theory works, which depends on time? Are there 'allowed' and 'disallowed' time-orderings of the basis states that ensure probability theory works? In quantum cosmology, I believe the value of some scalar field is used as a 'clock'.

For example if we had states: $\{|S_1\rangle, |S_2\rangle, |S_3\rangle, |S_4\rangle, |S_5\rangle\}$

And we said that basis states $|S_1\rangle$ and $|S_2\rangle$ are assigned a time of $0$. (By some definition of clock which is a property of the states).

and basis states $|S_3\rangle$ assigned a time of $1$.

and basis states $|S_4\rangle$ and $|S_5\rangle$ are assigned a time of $2$.

Then perhaps we would then be able to talk about for example $\Delta(S_1,S_5)$ being the amplitude for starting in state $S_1$ and ending in state $S_5$ after $2$ time steps.

But then we might have a state $\frac{1}{\sqrt{2}}|S_1\rangle+\frac{1}{\sqrt{2}}|S_3\rangle$ wouldn't make sense as it would involve basis states which are assigned different times. How would we make sense of such a state? Would we these have to be exluded as unphysical states?

And if $T(A)$ is the time we are assigning a particular basis state $|A\rangle$. How could we garuntee that the identity is satisfied:

$$T(A)>t>T(C) \implies \\ \sum\limits_{\{B|T(B)=t\}}\Delta(A,B)\Delta(B,C) = \Delta(A,C) $$

for our choice of time ordering? i.e. summing over all intermediate states that have been assigned a particular time.

Would the definition of $\Delta$ the have to depend on our function $T$ in order for this to be satisfied? What would the equation be for this?

My first guess if $\Psi(A)$ is the "wavefunction of the Universe". Then perhaps:

$$\Delta_t(A,B)=\Delta(A,B) \stackrel{?}{=} \frac{\overline{\Psi}(A){\Psi}(B)}{\sqrt{\sum\limits_{\{P|T(P)=T(A)\}}|\Psi(P)|^2 } \sqrt{ \sum\limits_{\{Q|T(Q)=T(B)\}}|\Psi(Q)|^2}} $$

where $t=T(B)-T(A)$. i.e. normalising the wavefunction for each time as specified by our clock. But this seems disagreeably separable. I think this is more like the amplitude for a state $A$ to collapse to a singularity and then expand out again to a state $B$!

Interestingly this seems to force diffeomorphism at least with respect to time, since the time definition can be arbitrary.

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  • $\begingroup$ First of all, I don't think it's correct to say that "probability theory depends on time." The fundamental objects in probability theory are random variables, which can vary over whatever parameter is applicable, for example, space (as in a random image), energy (as in a Monte Carlo solution of the time-independent Schrodinger equation), etc. $\endgroup$ – probably_someone May 16 at 21:14
  • $\begingroup$ @probably_someone Well maybe but you have to say what is the probability of A(x) given that I know A(x'). But this implies that the knowledge of A(x') precedes the knowledge of A(x). Therefor time is indirectly involved if you assume that information can only go from past to future. If you know both A(x) and A(x') then I don't see how probability enters into it. What you're calling random seems more like a measure of entropy or disorder. But it's just semantics. $\endgroup$ – zooby May 16 at 21:17
  • $\begingroup$ That's a particular part of probability theory, namely, statistical inference (and even then, you're talking more about how the theory is applied, rather than the structure of the theory itself). Not all parts of probability theory are like that. For example, consider a time-independent scalar field (like a temperature field) where the values at each point in space correspond to the behavior of a random variable. The information about the probability distribution is contained in space, not in time (in this case, in things like spatial correlations and fluctuations). $\endgroup$ – probably_someone May 16 at 21:26
  • $\begingroup$ Also, thermodynamics doesn't assign a time ordering to either macrostates or microstates. Thermodynamics as you have quoted it here works on systems that are in equilibrium (meaning that their macroscopic properties are time-independent), or close enough to it that we can treat the system as being in equilibrium. It can tell you the probability of being in a particular macrostate (this is where counting the number of microstates comes in), but this doesn't imply any time-ordering (note that quantities can still fluctuate with time, but the statistical behavior is still time-independent). $\endgroup$ – probably_someone May 16 at 21:32
  • $\begingroup$ Well we'll have to agree to disagree. You can tell me the probability of being in a particular state. But that implies you don't know what state I'm in. To check if you're probability is correct means time has moved forwards and now you have that knowledge. Probability is linked with knowledge and information and information only travels forwards in time. That's my view of things. Random variable theory is not strictly the same thing as probability theory. A distribution in space I'd call a correlation but it's just semenatics. $\endgroup$ – zooby May 16 at 22:00

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