Ken G said:
That's actually not true expressly because of the inadequacy of the concept of "superposition" in regard to a macro system.
If you stick by the axioms of quantum theory (which you are free to do so or not, but I'm looking at the *toy world* in which these axioms are considered true), then EVERY state of the system is described by an element of a projective hilbert space. There's no distinction between "macro" and "micro" states. EVERY state.
Now, if you assume that this is not applicable to certain kinds of systems, then you're playing with *another* toyworld. It will then follow different rules, but for sure, you cannot say that it is purely described by the axioms of quantum mechanics. And then you have the difficulty of explaining what is "micro" and what is "macro" and what applies where.
So, for sake of argument, I stick to this toy world in which the axioms of quantum mechanics are strictly valid. By definition then, the physical state is given by a state vector. And from here on, we go further.
Many people think a "superposition" is a fundamental state, but it's not-- the fundamental state is called a "pure state", and what "superposition" really means is a relationship between two non-commuting measurements-- the measurement that first prepared the initial pure state, and the later measurement you are using the concept of superposition to predict. So if there is no "initial measurement" that prepared the system in a known state, then there is no such thing as the "superposition". The idea breaks down right away, even before consideration of any unitarity of the operators.
This is in a Copenhagen like view, where you have a classical world with "quantum gates" or whatever, where systems are classically prepared, then "plunge into the quantum world", and re-emerge classically when they are observed.
But clearly in that view, not everything is at all times described by the axioms of quantum mechanics.
Put differently, once has to assume the macro system is describable as a pure state before one can even apply your argument-- but that assumption is borne out by no experiment. I see it very similar to the pre-quantum view that particles had an exact position and velocity, we just didn't have the precision to specify them. But that had never been shown to be true by experiment, and in fact, turned out to not be true-- we were just taking our own theories too seriously.
To me, the exercise is to take the theory TOTALLY seriously, in its toy world.
And yes, in the toy world of classical physics, particles DO have perfectly well defined positions and momenta.
I wouldn't shout that, it simply isn't true. Energy is only very nearly exactly conserved by anything dynamical, because of the finite lifetime of the system.
Uh, but a system with a finite lifetime doesn't violate the conservation of energy! It simply wasn't in a pure energy eigenstate - otherwise it could not evolve, and hence not have a finite lifetime.
The classic mistake of "classical" physics is to take its principles as if they were absolute statements of reality, yet when we go to the quantum realm, we find they are not. Why would we think we can do that in reverse-- to claim that a macro system can be in a pure state even though we have no idea how to accomplish that feat, or even to demonstrate that we accomplished it?
Because in the toy world defined by the axioms of quantum mechanics, that's what postulated!
In terms of the "correspondence principle", this means if we are to take that as a scientific principle, it must be demonstrable, which means the principle should actually be stated "aggregating quantum principles as we aggregate the quantum systems into a classical one cannot make a false prediction about the classical system"-- but that doesn't establish that a classical system can be in a pure state, because no experiment will either refute or establish that pure state. My answer to the "cat paradox" is very simple: cats cannot be in pure states, and coupling them to pure states ends the purity of the quantum state-- not the converse. Again, that's the whole point of coupling quantum systems to measurement devices that we can count on to behave classically.
That's the Copenhagen view. But it leaves you with the unsatisfied impression that there is no available description for the link between quantum theory (which is valid microscopically, and clearly not macroscopically here) and classical theory which does the opposite. It is simply by the big distance between "micro" and "macro" that we don't seem to be bothered by what actually makes nature "jump theories" in between.
In such a viewpoint, there's no need to talk about things like decoherence. At a certain point, you simply DECIDE to say that now, we switch to classical, no more superpositions. You can do that whenever you feel like not following through the quantum interactions anymore. A photon interacting with an electron can be "classical" or "quantum" according to how much pain you want to give yourself.
You can call a photo-electric effect a "measurement", and if you stop there, that can be good enough. You can also call it a quantum-mechanical interaction, and careful experimenting might give you some interference effects. So if you decide to study that, it is still "in the quantum world", but if you don't bother, well then it was in fact already classical.
) and also the "relational interpretation" by Rovelli.
