This is a good summary. I've had problems linking to that page before. If the link doesn't work, do a Google search for "dirac-von neumann axioms" (without the quotes) and click the link to Strocchi's book at Google Books. It should take you directly to page 72.rpt said:
The axioms I linked to only treat time as fundamental. To replace time with spacetime, you should drop the axiom about the Schrödinger equation, and replace it with an axiom that says that there's a group homomorphism from the symmetry group of spacetime (which can be either Galilei spacetime or Minkowski spacetime, so the group is either the Galilei group or the Poincaré group) into the group of symmetries of the quantum system (bijections on the set of states). This recovers the Schrödinger equation, because the time translation group is a subgroup of the symmetry group of spacetime.rpt said:
Yes, the concept of state only makes sense if it's possible to prepare a system in different ways, and then perform a measurement on it. So a theory that involves states must include some concept of "time".rpt said:
Not really. The concept of "spacetime" becomes useful the moment we need to start talking about events and their coordinates. Spacetime is just the set of all events, and a coordinate system is a function that assigns a 4-tuple (t,x,y,z) to each event. Galilei spacetime is the spacetime of pre-relativistic physics. Minkowski spacetime is the spacetime of special relativity. Special relativity brings space and time together in the sense that it says that two events that you would describe as being separated only in space, I would describe as separated in space and time (assuming that you would also describe me as having a non-zero velocity component in the direction from the location of one of the events to the location of the other). Non-relativistic classical mechanics doesn't do anything like that, but the concept of spacetime is still useful there.rpt said:
I don't like to say things like that, because the theory doesn't. It doesn't talk about what "exists". It just tells you how to calculate probabilities of possibilities. However, the axioms of the theory are motivated by such ideas. For example, if there exists a mathematical "thing" that an observer S can use as a represent of the state of the system when he calculates the theory's predictions about results of experiments on it, then there must exist another such "thing" that an observer who's just like S but rotated clockwise by an angle θ can use to make those predictions. It's when we translate such ideas to mathematical language that we end up with the axiom that there's a group homomorphism from the symmetry group of spacetime into the group of symmetry transformations on the set of states.rpt said:
Yes, I would say that's accurate. At least the part before "in an environment". It's not clear what that part of the statement means, at least for someone who hasn't been reading the posts you're responding to (i.e. my posts).rpt said:
If you mean the state a very short time after the big bang, no one really knows. I doubt that a theory that describes things in terms of what can be measured, makes predictions in the form of probabilities of possibilities, and is sophisticated enough to incorporate the local symetries of spacetime, can be a different theory than QM. But since spacetime is likely to be something entirely different in a theory that can handle the really early times, it's unclear if quantum mechanics as we know it would survive.rpt said:
Space and time, with the properties your experiences tell you they have, are described by Galilei spacetime. That spacetime is simply the mathematical "thing" that makes our intuitive ideas about space and time precise. Space and time in the real world have different properties than that. We know this for sure because of experiments that show that special relativity (which uses Minkowski spacetime instead) make better predictions about results of experiments than theories based on Galilei spacetime. This is all we need to know that "space and time", as defined by our intuition, don't really exist in the real world.rpt said:
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The axioms of quantum physics are a set of fundamental principles that describe the behavior of particles at the microscopic level. These include the superposition principle, the uncertainty principle, and the wave-particle duality.
Unlike classical physics, which describes the behavior of macroscopic objects, quantum physics deals with the behavior of subatomic particles. The axioms of quantum physics allow for phenomena such as superposition and entanglement, which have no classical analogues.
The superposition principle states that a particle can exist in multiple states or locations simultaneously. This is a fundamental concept in quantum physics and has been demonstrated through experiments, such as the double-slit experiment.
The uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute precision. This is because the act of measuring one property affects the other. Quantum physics explains this by the wave-like nature of particles and the concept of wavefunction collapse.
While the axioms of quantum physics have been extensively tested and are widely accepted by the scientific community, there are still ongoing debates and research surrounding their interpretation and implications. Some physicists even propose alternative theories that challenge certain aspects of the axioms.