The discussion revolves around proposed axioms for quantum mechanics, examining their foundations, implications, and potential alternatives. Participants explore the mathematical structure of quantum states, the nature of observables, the connection between mathematics and measurements, and the dynamics of quantum systems. The conversation includes both theoretical and conceptual aspects.
Participants express multiple competing views on the proposed axioms, with significant disagreement on foundational aspects such as the choice of mathematical framework, the interpretation of observables, and the implications of quantization methods. The discussion remains unresolved, with no consensus reached.
Limitations include the lack of consensus on the necessity of separability in Hilbert spaces, the appropriateness of classical mechanics as a starting point for quantum theories, and the implications of different interpretations of quantum dynamics. The discussion highlights the complexity and ambiguity inherent in foundational questions of quantum mechanics.
A. Neumaier said:...
QuantumClue said:You seem to be retorting everyone else but me. Surely it is a requisit to reply to people if they talk to you, especially when you make wild assertions about the other posters work.
QuantumClue said:I asked you to show an example of a macroscopic body experiencing non-locality and I would take my comment back. If you can't do this, just say.
A. Neumaier said:I only admitted that it is possible to treat entangled qubits with pilot waves.
But the paper you cited didn't verify your axioms. It featured no Lagrangian but a Hamiltonian, and it made use of Hilbert space.
Pretending that pilot waves avoid Hamiltonians and Hilbert space is just that - pretense.
It requires _all_ the standard stuff and in addition things that are superfluous for working with QM.
Poor prospects for a good axiom system.
A. Neumaier said:You are new to the forum and don't know the rules yet. Nobody is obliged to answer.
I told you that you post in the wrong forum, and I don't take back my comment.
You neither understand the purpose of this thread nor the meaning of the term ''axiom''.
zenith8 said:My sole point is that it is possible to have fundamentally different sets of axioms than the ones proposed by the OP, and still have a theory which agrees with observations. I thought this might be an interesting point to make in the context of the thread.
QuantumClue said:Explain please why this post was intended as a joke. And explain again please why you don't take back your post. If this is about some kind of laughing matter, I wouldn't mind a laugh myself. Please, explain.
QuantumClue said:An axiom is a postulate. My sentance was a postulate And since this thread is about axioms, I decided to share mine.
which was... ''Non-locality is a quantum phenomena. Non-locality should not have descriptions for macroscopic bodies. For large enough systems, locality is preserved. ''
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bigubau said:1. How do define non-locality is a concise and understandable manner ?
2. What does <For large enough systems> mean ?
a) An axiom must not have vague or unprecise statements.
b) How do you relate your statement (assumingly cured from vagueness) to the the other axioms which form the mathematical and experimental nucleus of the theory ?
c) Is it logically independent from the axioms in my post or the ones proposed by prof. Neumaier ?
A. Neumaier said:Axioms form the foundations of a theory or discipline. They summarize in a compact way the assumptions that need to be made in order to be able to derive everything else from it.
Given that, it should be easy for you to realize that what you proposed could at best be regarded as a joke, if not as a sign of basic incompetence.
With your current state of knowledge (as displayed by the few postings you made so far) you are better advised in this forum to learn from it and to ask questions rather than to propose answers (which are not likely to be well-received).
Remember that the web forgets nothing. People will forever be able to read about your follies, even if you don't recognize them now as such...
To learn more about the meaning of axioms in science, read http://en.wikipedia.org/wiki/Axiom
bigubau said:So your statement has more of a philosophical value. That settles it, I guess.
bigubau said:OFF-TOPIC NOTE:
I don't want this thread to turn into a/another battlefield with personal remarks. Not to mention rude/offending. This is a moderated forum, after all, so it could only cause harm to the participants. So please, attack the words and not the person.
QuantumClue said:No I explained reasons why my axiom holds. I said it becomes philosophical when you want to discuss something like the definition of something, when it is purely a mathematical conjecture.
QuantumClue said:If you want a definition of non-locality, you look for the philosophical interpretations which have been drawn by different scientists. You will also find each scientists either share the same interpretation, or will prefer another postulation.
QuantumClue said:My axiom has underlying assertions that it is a quantum phenomenon, which is associated to the similarity of the wave function and quantum tunnelling as being also quantum phenomena. After a certain threshold, the wave function cannot be viewed, and quantum tunnelling after the same threshold cease to be operative for large enough systems. On the same arguement, you do not witness non-locality at macroscopic levels. It is purely a quantum phenomena.
QuantumClue said:Then I asked the professor to explain why the statement was wrong, or intended to be a joke. Remember?
bigubau said:But an axiom of quantum mechanics, seen as a theoretical science, cannot have a philosophical content, but an operational and a mathematical one. Namely it introduces/defines concepts, links these through logical connectors and uses its defining property to made deductions, or theorems.
What mathematical conjecture are you talking about ?
I don't want to venture into philosophy. I'd rather stick to physics and mathematics. Interpretations of a theory are already in the realms of philosophy. I don't venture there, I'm just asking you to state an axiom which meets the standard requirements of mathematics.
This part is completely as in 110% wrong.
He was harsh and offensive on you, but at least I give him credit on one part: please, be humble and come here to learn, so seek answers rather than offer solutions when you don't posess the necessary knowledge of the topics being discussed.
Words are merely the mask of the person. In some cases, it is more important to read what is not responded to than to notice a few idle sentences meant to divert the attention from the unspoken word. To avoid this and out of sincere respect for the full range of thougts of the person, I respond to everything within a single message.bigubau said:after all, so it could only cause harm to the participants. So please, attack the words and not the person.
QuantumClue said:Philosophy is used when making interpretations of science. You seem to be denying we don't draw speculations on the meaning of mathematics.
QuantumClue said:Bells Inequalities. This where the idea of non-locality is drawn from.
QuantumClue said:So would I. I am a undergraduate of physics, so I am very interesting in drawing the mathmatical side of things.
QuantumClue said:It ironic, saying something is 110% wrong, when it is even wrong to speculate 110% even exists.
QuantumClue said:Would you please elaborate on how my contentions above are incorrect?
bigubau said:I retract my statement, yours it 100% correct.
QuantumClue said:Well if the professor will not explain his arguement, perhaps you will? Put your money where your mouth is, explain how my paragraph was incorrect. Saying it ''just is'' is about as helpful as an ashtray on a motorcycle.
bigubau said:It's ok, it's nothing to debate/refute about your paragraph, except probably that wavefunctions can never be view, felt, nor measured. They are only mathematical objects, just like the sign + is.
QuantumClue said:And by the way, we can view the wave function, or effects thereof: http://www.dailymail.co.uk/sciencet...-mechanics-shown-work-visible-world-time.html
bigubau said:Experimentalists observe phenomena, they measure physical quantities. We can never view, nor measure wave functions/density operators as they are simply mathematical tools to describe reality. I think of a quantum states as being a part of reality.
We will always measure (or determine from numerical analysis of experimental tests of quantum mechanics) probabilities or spectral values of self-adjoint operators, because, as it follows from the axiomatization I proposed in post 1 of the thread, they are the only items assuring the connection between mathematics (functional analysis) and experiment.
A. Neumaier said:Here is an axiom system fully covering current mainstream quantum mechanics and quantum field theory (but not various speculations beyond the standard model). It covers both the nonrelativistic case and the relativistic case.
There are six basic axioms:
A1. A generic system (e.g., a 'hydrogen molecule')
is defined by specifying a Hilbert space K whose elements
are called state vectors and a (densely defined, self-adjoint)
Hermitian linear operator H called the _Hamiltonian_ or the _energy_.
A2. A particular system (e.g., 'the ion in the ion trap on this
particular desk') is characterized by its _state_ rho(t)
at every time t in R (the set of real numbers). Here rho(t) is a
Hermitian, positive semidefinite (trace class) linear operator on K
satisfying at all times the conditions
trace rho(t) = 1. (normalization)
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You might find this interesting, but there is a somewhat justifiable reason not to use them. For a non-separable Hilbert space the fact that a Lie group has a representation on the Hilbert space doesn't imply that the Lie Algebra has a representation. So even though rotations might be represented by a group of unitary operators, angular momentum wouldn't be a well defined operator.Fredrik said:This is exactly what I was thinking. Separable spaces are easier to work with. That's why we try using a separable space first.
DarMM said:You might find this interesting, but there is a somewhat justifiable reason not to use them. For a non-separable Hilbert space the fact that a Lie group has a representation on the Hilbert space doesn't imply that the Lie Algebra has a representation. So even though rotations might be represented by a group of unitary operators, angular momentum wouldn't be a well defined operator.
bigubau said:1. Is it apparent to me, or you introduce two different description of states, one through <state vectors> and the other through <states \rho (t)> ? Are they both necessary, thus independent of each other ?
2. How are these two these axioms used to describe the physical states of a helium atom ?
3. How does A2 apply to the simplest possible system, the nonrelativistic free massive particle ?
In the theory of representations of Lie groups on Hilbert spaces, the separability property allows you prove the existence of a representation of the Lie algebra. However without separability you cannot complete the proof, so there is no guarantee that the Lie algebra has a representation. There are several example theories where this is the case.bigubau said:Can you post or send a reference to a mathematical proof for that ? Thanks!
DarMM said:In the theory of representations of Lie groups on Hilbert spaces, the separability property allows you prove the existence of a representation of the Lie algebra. However without separability you cannot complete the proof, so there is no guarantee that the Lie algebra has a representation.
A. Neumaier said:[...]just replace Axiom A1 by the following improved version.
A1. A generic system (e.g., a 'hydrogen molecule') is defined by
specifying a Hilbert space K and a (densely defined, self-adjoint)
Hermitian linear operator H called the _Hamiltonian_ or the _energy_.
A. Neumaier said:2. Here you need also Axiom A4. with three particles (alpha, e, e'). But e and e' are indistinguishable, so only symmetric functions of the labels e and e' are observable. H is given by the standard atomic Hamiltonian one can find in any textbook.
A. Neumaier said:3. H= p^2/2m.
DarMM said:In the theory of representations of Lie groups on Hilbert spaces, the separability property allows you prove the existence of a representation of the Lie algebra. However without separability you cannot complete the proof, so there is no guarantee that the Lie algebra has a representation. There are several example theories where this is the case. [...] There isn't really a proof, since it is a description of what occurs in a case where another proof (representations on separable Hilbert spaces) fails.