What is the definition of physically meaningful? I asked a question recently where I wanted to know whether it was physically meaningful to talk about the arrow of time in other universes.
Although many people apparently have an intuitive notion of what is meant by "physically meaningful", the only real context I have found is in terms of "physically meaningful observables" where in a quantum mechanical context we talk about the possible eigenvalues associated with eigenstates of an operator:
$H |\psi\rangle = \lambda|\psi\rangle$
The observables being possible outcomes of experiments or other operations on the state vector.  So in the context of the arrow of time, where there are claims of the possibility of a vector time, one would like to think of some possible operation that would produce set of possible values for that time.  So the first question one might ask is, "what operation can I perform to observe the vector values of the arrow of time?"  
It would seem that there is some difference between defining such an operation and the possibility of it actually being executable.  So one might think of a thing being physically meaningful if one can define a set of executable operations (executabality being determined by some set of constraints).  
However, I suspect there is a better definition, or a better way to conceptualize what is meant by "physically meaningful", and I would like to read what those are. 
 A: A conceptual description of any aspect of the world is "physically meaningful" if one can conceive of an empirical test to distinguish between the consequences of that description and its negation.
However, whilst certain tests might be theoretically satisfactory, they might not be very practical. One can test for the existence of magnetic monopoles, but the a priori probability of detecting one is negligible, thus it is difficult in practice to tell whether magnetic monopoles exist or do not, even though their existence is "physically meaningful" in the sense given above.
A: Physically meaningful means something that is, in principle, testable. That effectively means some measurement is possible that could answer a given question. 
In the case of the arrow of time, we might say that we could, in principle, measure how heat flows as a way of determining changes in entropy in order to make a comparison between another universe's arrow of time to ours. Of course, this presumes tons of other stuff that would be relevant to that specific question (equal numbers of time dimensions, entropy change as a measure of the arrow of time, laws governing entropy being similar between universes, the ability to measure this in more than one universe from a meaningful perspective, etc.), which is what makes that question interesting, but it underscores the basic idea of "physically meaningful" as testable or measurable.
Much simpler examples that don't involve postulating other universes make the point more clearly. For example, we know it is not physically meaningful to talk about how close a radioactive atom is to decaying. By the above definition, this means that it is not possible, even in principle, to make measurements that allow you to answer the question "How soon before this specific atom decays?" It is only physically meaningful to talk about whether the atom has decayed or not, since this is something that can be measured, i.e. we can answer the question, "Has this atom decayed yet?"
A: This talk begins to cover some of this question in detail at around 6:30.  I haven't completed viewing it but it is moving further into the rabbit hole. Found also here.
Just the first of many notes.  I find the analogy of N=4 SYM to the harmonic oscillator very fascinating.  Also the implication that physical reality is about the structure of scattering amplitudes is fantastic.  

This entry was given down check, however, I was intending to work iteratively.I will continue to update periodically as I move forward. 2/1/2011

Update one
To answer this question we must define three key components first:
1) Physical System
2) Data
3) Interconnection
A physical system is a type of system that deals with analyzing the physical universe.  Physical is having to do with physics which has to do with analyzing nature which has to do with phenomena which has to do with "any event that is observable, however commonplace it might be, even if it requires the use of instrumentation to observe, record, or compile data concerning it."
Data is the "lowest level of abstraction from which information and then knowledge are derived"; suggesting that data is the most fundamental element associated with an observation.  Starting with a set of data, nothing about a physical system can be said using that data unless the data and physical system have some interconnection.  
With those three components: physical system, data and interconnection; we might be able to explore a definition of physically meaningful.

To be continued, despite the trolls...

