Axioms of quantum mechanics

  • #1
dextercioby
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Main Question or Discussion Point

I propose this set of axioms and ask you to be bring arguments for it and arguments against it. What other axioms would you choose instead of the ones I wrote below ?

1. STATE DESCRIPTION:

All physical states of a quantum system are described mathematically by a set at most countable of positive numbers [itex] p_{k} [/itex], [itex] \sum_{k} p_{k} =1 [/itex] and unit norm vectors [itex] \psi_{k} [/itex] in a complex separable Hilbert space [itex] \mathcal{H} [/itex].

2. QUANTIZATION:

a) The physical observables of the quantum theory are described through linear self-adjoint operators on the Hilbert space of states.

b) For classical systems with Hamiltonians at most quadratic in momenta, the classical observables p, q are described by the closures (in the Hilbert space topology) of the following operators obeying the Born-Jordan commutation relations: [itex] [q,p] = i \hbar 1_{\mathcal{H}} [/itex] on the common dense everywhere domain of p and q.

3. CONNECTION BETWEEN MATHEMATICS AND MEASUREMENTS OF OBSERVABLES:

a) The possible values of all the observables being measured are the spectral values of the self-adjoint operators describing them.
b) Born rule: essentially this one http://en.wikipedia.org/wiki/Born_rule

4. DYNAMICS

a) The time-evolution of quantum states is governed by an observable called Hamiltonian denoted by H. The spectral values of the operator associated to it are the possible energy values of the system.
b) The time-evolution of a quantum state is given by the 1st order differential equation

[tex] \frac{d\Psi(t)}{dt} = \frac{1}{i\hbar} H \, \Psi (t) [/tex]

So comment upon them. Which is too narrow and admits generalizations ?

NOTES: There's another postulate for the description of multiparticle states. I didn't write it, because I think there's little possible debate around it. If you think that's wrong, please write the version you're familiar with.
 
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Answers and Replies

  • #2
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I propose this set of axioms and ask you to be bring arguments for it and arguments against it. What other axioms would you choose instead of the ones I wrote below ?
None.

1. STATE DESCRIPTION:

All physical states of a quantum system are described mathematically by a set at most countable of positive numbers [itex] p_{k} [/itex], [itex] \sum_{k} p_{k} =1 [/itex] and unit norm vectors [itex] \psi_{k} [/itex] in a complex separable Hilbert space [itex] \mathcal{H} [/itex].
Why a complex separable Hilbert space ? There is no a priori reason for it to be separable, neither are there compelling arguments why the complex numbers are sacrosant (why not the quaternions ?), nor is there any good reason why it should be a Hilbert space. Now, to actually understand why and by what you have to replace it, you have to try some things out (and there is no way an argumentation can be
given in even 3 full pages, I have already given a few though).
2. QUANTIZATION:

a) The physical observables of the quantum theory are described through linear self-adjoint operators on the Hilbert space of states.
Why hermitean operators, why not normal operators ? Just replace the measurement rule by saying that you measure the real part of the eigenvalue. The only important property is that the eigenstates are orthonormal.

b) For classical systems with Hamiltonians at most quadratic in momenta, the classical observables p, q are described by the closures (in the Hilbert space topology) of the following operators obeying the Born-Jordan commutation relations: [itex] [q,p] = i \hbar 1_{\mathcal{H}} [/itex] on the common dense everywhere domain of p and q.
Why start from a classical theory? There are plenty of quantization ambiguities and the Hamiltonian picture is known to be troublesome for QCD.

3. CONNECTION BETWEEN MATHEMATICS AND MEASUREMENTS OF OBSERVABLES:

a) The possible values of all the observables being measured are the spectral values of the self-adjoint operators describing them.
b) Born rule: essentially this one http://en.wikipedia.org/wiki/Born_rule
Why this simple probability interpretation? It is tightened to Hilbert space, but it is easy to come up with alternatives.

4. DYNAMICS

a) The time-evolution of quantum states is governed by an observable called Hamiltonian denoted by H. The spectral values of the operator associated to it are the possible energy values of the system.
No, it doesn't work like that. If you quantize GR, the Hamiltonian has to vanish, so energy is something very different that this.
b) The time-evolution of a quantum state is given by the 1st order differential equation

[tex] \frac{d\Psi(t)}{dt} = \frac{1}{i\hbar} H \, \Psi (t) [/tex]

So comment upon them. Which is too narrow and admits generalizations ?
No, chosing foliations break manifest general covariance, so one woud like a formulation which does not depend upon such global considerations. QM as it is formulated now, depends upon inertial systems, just like old Newtonian gravity does. One needs to generalize it in the same way Einstein dismissed Newtonian gravity.

NOTES: There's another postulate for the description of multiparticle states. I didn't write it, because I think there's little possible debate around it. If you think that's wrong, please write the version you're familiar with.
Even here, one must debate. The tensor product construction depends upon individual quantum systems. In a holistic universe such as GR dictates, the individual is a result of iteraction within the whole universe. To understand why the tensor product should fail as an effective construction in such generalized theory, see http://www.vub.ac.be/CLEA/aerts/publications/1978TensorProduct.pdf which clarifies where it comes from in the first place.
 
  • #3
dextercioby
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Why a complex separable Hilbert space ? There is no a priori reason for it to be separable, neither are there compelling arguments why the complex numbers are sacrosant (why not the quaternions ?), nor is there any good reason why it should be a Hilbert space. Now, to actually understand why and by what you have to replace it, you have to try some things out (and there is no way an argumentation can be
given in even 3 full pages, I have already given a few though).
1. I was sure you were going to address my post from the perspective of someone working in a far more advanced theory.
2. There's a standard argument for the separability of the state space: the spectral theorem for self-adjoint operators.

Why hermitean operators, why not normal operators ? Just replace the measurement rule by saying that you measure the real part of the eigenvalue. The only important property is that the eigenstates are orthonormal.
Ok, one can make that generalization, but what's the physical significance of the imaginary part of the eigenvalue ? If it has none, what's the benefit of this extension?

Why start from a classical theory?
I invite you to write down the Hamiltonian for a hydrogen atom without prior knowledge of classical mechanics.

There are plenty of quantization ambiguities and the Hamiltonian picture is known to be troublesome for QCD.
Quantization ambiguities are a true fact. Can you write down a quantization axiom which would evade them ? Oh, hold on, you think we shouldn't quantize anything at all, else why would you have written <Why start from a classical theory?>.

Could you, please, be consistent ?

Why this simple probability interpretation? It is tightened to Hilbert space, but it is easy to come up with alternatives.
Please, post a generalization of the probabilistic interpretation.

No, it doesn't work like that. If you quantize GR, the Hamiltonian has to vanish, so energy is something very different that this.
OK, my axioms are for unconstrained systems. How would you write down axioms for constrained systems ?

No, chosing foliations break manifest general covariance, so one woud like a formulation which does not depend upon such global considerations. QM as it is formulated now, depends upon inertial systems, just like old Newtonian gravity does. One needs to generalize it in the same way Einstein dismissed Newtonian gravity.
Agree here. I don't claim that I can attempt to quantize GR (not even in its free limit) with these axioms. I want to try them to simpler systems, like H-atom, free particle, particle in a square potential, you know, those you learn in school just to get a degree. :wink:

Even here, one must debate. The tensor product construction depends upon individual quantum systems. In a holistic universe such as GR dictates, the individual is a result of iteraction within the whole universe. To understand why the tensor product should fail as an effective construction in such generalized theory, see http://www.vub.ac.be/CLEA/aerts/publications/1978TensorProduct.pdf which clarifies where it comes from in the first place.
Again, the axioms are not meant to attempt a quantization of GR. I don't know of any.
 
  • #4
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1. I was sure you were going to address my post from the perspective of someone working in a far more advanced theory.
2. There's a standard argument for the separability of the state space: the spectral theorem for self-adjoint operators.
I don't recall that the spectral theorem depends upon cardinality. I guess it does not since the only thing you need are geometric considerations.

Ok, one can make that generalization, but what's the physical significance of the imaginary part of the eigenvalue ? If it has none, what's the benefit of this extension?
Well, it might get physical significance by means of the dynamics. The imaginary part is not just ''passive'', it can rotate and become real. For example, it would also modify the Heisenberg commutation relations if momentum and position were not Hermitean operators.
I invite you to write down the Hamiltonian for a hydrogen atom without prior knowledge of classical mechanics.
Easy, Weinberg does it in his book: introduction to QFT volume one. In his line of thought there are no classical fields anymore, everything is considered directly in terms creation/annihilation operators and priciples such as causality and cluster decompostion.

Quantization ambiguities are a true fact. Can you write down a quantization axiom which would evade them ? Oh, hold on, you think we shouldn't quantize anything at all, else why would you have written <Why start from a classical theory?>.

Could you, please, be consistent ?
I am consistent; I just argued that quantization ambiguities are another reason not to start from a classical theory.

Please, post a generalization of the probabilistic interpretation.
Is already partially in the book, will be thoroughly treated later on. Moreover, plenty of authors have already written about this in the standard literature.

OK, my axioms are for unconstrained systems. How would you write down axioms for constrained systems ?
There are no constrained systems to quantize, since I don't quantize a classical Lagrangian with a local symmetry. :tongue2:

Agree here. I don't claim that I can attempt to quantize GR (not even in its free limit) with these axioms. I want to try them to simpler systems, like H-atom, free particle, particle in a square potential, you know, those you learn in school just to get a degree. :wink:

Again, the axioms are not meant to attempt a quantization of GR. I don't know of any.
So then, you would consider these parts of QM as you understand it to be inadequate or not ?
 
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  • #5
A. Neumaier
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I propose this set of axioms and ask you to be bring arguments for it and arguments against it. What other axioms would you choose instead of the ones I wrote below ?
Your 2b), which gives a special status to p and q, already disqualifies your axiom system as being widely applicable. It covers neither the systems treated in quantum information theory (which work in a finite-dimensional Hilbert space) nor multi-particle systems with indistinguishable variables, let alone quantum field theories. In all these cases, p and q do not figure as observables.



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)

A3. A system is called _closed_ in a time interval [t1,t2]
if it satisfies the evolution equation
d/dt rho(t) = i/hbar [rho(t),H] for t in [t1,t2],
and _open_ otherwise. (hbar is Planck's constant, and is often set
to 1.) If nothing else is apparent from the context,
a system is assumed to be closed.

A4. Besides the energy H, certain other (densely defined, self-adjoint)
Hermitian operators (or vectors of such operators) are distinguished
as _observables_.
(E.g., the observables for a system of N distinguishable particles
conventionally include for each particle several 3-dimensional vectors:
the _position_ x^a, _momentum_ p^a, _orbital_angular_momentum_ L^a
and the _spin_vector_ (or Bloch vector) sigma^a of the particle with
label a. If u is a 3-vector of unit length then u dot p^a, u dot L^a
and u dot sigma^a define the momentum, orbital angular momentum,
and spin of particle a in direction u.)

A5. For any particular system, and every vector X of observables with
commuting components, one associates a time-dependent monotone
linear functional <.>_t defining the _expectation_
<f(X)>_t:=trace rho(t) f(X)
of bounded continuous functions f(X) at time t.
This is equivalent to a multivariate probability measure dmu_t(X)
(on a suitable sigma algebra over the spectrum spec(X) of X)
defined by
integral dmu_t(X) f(X) := trace rho(t) f(X) =<f(X)>_t.

A6. Quantum mechanical predictions amount to predicting properties
(typically expectations or conditional probabilities)
of the measures defined in Axiom A5 given reasonable assumptions
about the states (e.g., ground state, equilibrium state, etc.)



Discussion:

Axiom A6 specifies that the formal content of quantum mechanics is
covered exactly by what can be deduced from Axioms A1-A5 without
anything else added (except for restrictions defining the specific
nature of the states and observables), and hence says that
Axioms A1-A5 are complete.

The description of a particular closed system is therefore given by
the specification of a particular Hilbert space in A1, the
specification of the observable quantities in A4, and the
specification of conditions singling out a particular class of
states (in A6). Given this, everything else is determined by the theory,
and hence is (in principle) predicted by the theory.

The description of an open system involves, in addition, the
specification of the details of the dynamical law. (For the basics,
see the entry 'Open quantum systems' in the Theoretical Physics FAQ
at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html#open .)


In addition to these formal axioms one needs a rudimentary
interpretation relating the formal part to experiments.
The following _minimal_interpretation_ seems to be universally
accepted.

MI. Upon measuring at times t_l (l=1,...,n) a vector X of observables
with commuting components, for a large collection of independent
identical
(particular) systems closed for times t<t_l, all in the same state
rho_0 = lim_{t to t_l from below} rho(t)
(one calls such systems _identically_prepared_), the measurement
results are statistically consistent with independent realizations
of a random vector X with measure as defined in axiom A5.


Note that MI is no longer a formal statement since it neither defines
what 'measuring' is, nor what 'measurement results' are and what
'statistically consistent' or 'independent identical system' means.
Thus Axiom MI has no mathematical meaning. That's why it is already
part of the interpretation of formal quantum mechanics.

However, the terms 'measuring', 'measurement results', 'statistically
consistent', and 'independent' already have informal meaning in the
reality as perceived by a physicist. Everything stated in Axiom MI is
understandable by every trained physicist. Thus statement MI is not
for formal logical reasoning but for informal reasoning in the
traditional cultural setting that defines what a trained physicist
understands by reality.

Edit: This axiom system was taken from the entry 'Postulates for the
formal core of quantum mechanics' in the Theoretical Physics FAQ
at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html#postulates
 
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  • #6
Hurkyl
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There is no a priori reason for it to be separable
Er, so what? It's not like he's trying to axiomatize QM from a 1905 state of knowledge. :tongue:
 
  • #7
Fredrik
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Er, so what?
This is exactly what I was thinking. Separable spaces are easier to work with. That's why we try using a separable space first. If it doesn't work (if it makes us unable to impose other requirements that we want/need, or if it gives us a theory that makes predictions that don't agree with experiments), we start over.

When a thread titled "axioms of quantum mechanics" is posted in the "quantum physics" forum, I tend to assume that the OP wants to discuss quantum mechanics, not how QM should be changed to be consistent with Johan Noldus's ideas about "quantum" gravity.

I think I will post some comments about those axioms later, but I don't have time right now.
 
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  • #8
dextercioby
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When a thread titled "axioms of quantum mechanics" is posted in the "quantum physics" forum, I tend to assume that the OP wants to discuss quantum mechanics, not how QM should be changed to be consistent with Johan Noldus's ideas about "quantum" gravity.
What makes you think that I wanted to discuss something else than what is stated in the first post ? I already told Careful that it's not my intention to establish/propose/discuss an axiomatical structure useful for quantum gravity. That's his job.

Fredrik said:
I think I will post some comments about those axioms later, but I don't have time right now.
Please, do comment, if possible, both on my set and on Arnold's one.
 
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  • #9
dextercioby
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Your 2b), which gives a special status to p and q, already disqualifies your axiom system as being widely applicable. It covers neither the systems treated in quantum information theory (which work in a finite-dimensional Hilbert space) nor multi-particle systems with indistinguishable variables, let alone quantum field theories. In all these cases, p and q do not figure as observables.
2b) is not conflicting with 2a) in any way. I put it there to make a connection with the obsolete theory of classical mechanics in Hamilton formulation. An utility it has, you must agree, nor can it be derived. So why not include it in the axioms, just because it doesn't apply to 100% of quantum systems.

And I proposed an axiomatization of quantum mechanics, not quantum field theory.

1b) would be: For a quantum system made up of identical particles (question: how do we define a particle in QM ?), each particle's pure states being described by a complex separable Hilbert space, the space of physical pure states is either the symmetrized tensor product of uniparticle states (bosonic particles case) or the antisymmetrized tensor product of uniparticle states (fermionic particle case).
 
  • #10
Fredrik
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What makes you think that I wanted to discuss something else than what is stated in the first post ? I already told Careful that it's not my intention to establish an axiomatical structure useful for quantum gravity. That's his job.
Now you're really confusing me. Do you want to discuss QM or possible extensions of it? I was assuming the former, and that seems to be consistent with the second and third sentence in the quote above, but in the first sentence, you're suggesting that I'm wrong about what you want to discuss. :confused:

Please, do comment, if possible, both on my set and on Arnold's one.
I'll try to make time tomorrow. This is a topic that interests me too. I once started a similar thread, and I've been thinking that I should do it again, but I was planning to wait until I understand functional analysis better.
 
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  • #11
dextercioby
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Now you're really confusing me. That second sentence is essentially what I said, and you're still suggesting that I'm wrong about what you want to discuss. :confused:
Sorry, I didn't understand that you were only expressing your expectations, which in this case, coincide with mine. Now that you quoted me, I can't edit/delete that message. Nevermind.
 
  • #12
Fredrik
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OK, I see. :smile:

(I didn't mean to edit after you replied btw. You probably beat me to it just by a few seconds).
 
  • #13
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I once started a similar thread, and I've been thinking that I should do it again, but I was planning to wait until I understand functional analysis better.
One needs very little of functional analysis in order to understand the foundations of quantum mechanics (and indeed all of what is in a typical textbook on quantum mechanics).

You don't need any (beyond the definition of a Hilbert space and the trace of an operator) if you are prepared (like most physicists) to take for granted two nontrivial but intuitive functional analytic results:

1. The spectral theorem in the following form:

Theorem. (Gelfand & Maurin)
Given an arbitrary set of commuting self-adjoint operators defined on a (dense subspace of a) Hilbert space, there is always an isomorphic Hilbert space in which these operators are represented by multiplication with real-valued functions.

2. The Hille-Yosida theorem in the following form:

Theorem. (Hille & Yosida)
The exponential exp(itH) of a Hermitian linear operator H defined on a (dense subspace of a) Hilbert space exists if and only iff A is self-adjoint.

One may take the latter as a definition of self-adjoint; then there is nothing to prove.
 
  • #14
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Just to irritate everyone, remember that one can also use the following set of axioms which don't mention Hilbert space, the Schroedinger equation, or the Hamiltonian at all (just in case we get the impression that the Hilbert space version is fundamental). :surprised

This is the single-particle case for simplicity (the extension to many particles is obvious).

Axioms of non-relativistic quantum mechanics (single-particle case)

I. Particle

A particle is a point-like object localized in (three-dimensional) Galilean space with an inertial mass.

II. Wave field

A wave field is a physical process that propagates in (three-dimensional) Galilean space over time. A wave field is described by its wave function [tex]\Psi[/tex] which is a continuous, bounded function of the space and time coordinates.

III. Quantum system

A single particle quantum system consists of a quantum particle and its accompanying wave field, i.e. the set [tex]\{\Psi, x\}[/tex] where [tex]x[/tex] is the particle's position.

IV. Lagrangian density

A single-particle quantum system has a Lagrangian density [tex]L[/tex] which is expressed in terms of its wave function [tex]\Psi[/tex]:

[tex]
L = \frac{1}{2} i \hbar (\Psi^* \frac{d\Psi}{dt} - \frac{d\Psi^*}{dt} \Psi) - (\frac{\hbar^2}{2m}) (\nabla \Psi^*)\cdot(\nabla \Psi) - V\Psi^*\Psi
[/tex]

where [tex]V[/tex] is an external (classical potential), and [tex]m[/tex] is the particle's inertial mass.

V. Guidance condition
A quantum particle is guided by its wave field in accordance with the condition:

[tex]
\frac{dx}{dt} = \frac{\hbar}{2im} \nabla \log (\frac{\Psi}{\Psi^*})
[/tex]

And that's it. Some people supplement this with a statement about the 'quantum equilibrium condition' i.e. the probability density [tex]\rho(x)[/tex] of possible values of the initial particle position in an ensemble of similarly prepared quantum systems satisfies the Born rule condition [tex]\rho=|\Psi|^2[/tex]. However, it is possible to show how this arises naturally from the dynamics outlined above, so it isn't really an axiom.

From this perspective QM is a dynamical theory of particles trajectories rather than a statistical theory of observation, and probability is - as it almost certainly ought to be - just a tool for making rational inferences in situations of incomplete knowledge, rather than something truly fundamental.

Now I'm not saying which perspective is right, but it's important to remember that one can do this. If nothing else it's a good b***sh*t detector (i.e. is something you say based on the Hilbert space axioms still true with the above axioms? If not, it's just an opinion..)
 
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  • #15
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Just to irritate everyone, remember that one can also use the following set of axioms which don't mention Hilbert space or the Hamiltonian at all (just in case we get the impression that the Hilbert space version is fundamental). :surprised

This is the single-particle case for simplicity (the extension to many particles is obvious).
I'd like to see how you represent the Ising model with your axioms.
Or even a family of entangled qubits.
 
  • #16
Fredrik
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One needs very little of functional analysis in order to understand the foundations of quantum mechanics (and indeed all of what is in a typical textbook on quantum mechanics).

You don't need any (beyond the definition of a Hilbert space and the trace of an operator) if you are prepared (like most physicists) to take for granted two nontrivial but intuitive functional analytic results:

1. The spectral theorem in the following form:

Theorem. (Gelfand & Maurin)
Given an arbitrary set of commuting self-adjoint operators defined on a (dense subspace of a) Hilbert space, there is always an isomorphic Hilbert space in which these operators are represented by multiplication with real-valued functions.

2. The Hille-Yosida theorem in the following form:

Theorem. (Hille & Yosida)
The exponential exp(itH) of a Hermitian linear operator H defined on a (dense subspace of a) Hilbert space exists if and only iff A is self-adjoint.

One may take the latter as a definition of self-adjoint; then there is nothing to prove.
I'm bothered e.g. by the fact that not all observables have eigenvectors. How are you avoiding that issue? Where should I look for the proofs of those theorems? (Do I have to find a copy of Maurin's book?)

I'm currently working my way through the hierarchy of spectral theorems, and it will probably take a while before I study one for unbounded operators, but I intend to do that too, eventually.
 
  • #17
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Er, so what? It's not like he's trying to axiomatize QM from a 1905 state of knowledge. :tongue:
He is, I do it from a 2011 state of knowledge :rofl: Moreover, separable Hilbert spaces are not enough, but I guess you never heard about Guichardet. God bless ....
 
  • #18
dextercioby
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I'm bothered e.g. by the fact that not all observables have eigenvectors. How are you avoiding that issue? Where should I look for the proofs of those theorems? (Do I have to find a copy of Maurin's book?)

I'm currently working my way through the hierarchy of spectral theorems, and it will probably take a while before I study one for unbounded operators, but I intend to do that too, eventually.
You can find a critical review on the internet for Maurin's book of 1968. It contains several typos and omissions and some of the statements are not clearly spelled/proved.

I invite you to pick up the book by Gelfand. The general spectral theorem contains a small error which was addessed by the mathematician Gould later (thanks to strangerep for bringing this to my and everyone's attention).
 
  • #19
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I'd like to see how you represent the Ising model with your axioms.
Or even a family of entangled qubits.
Some guy did a thesis on it - see http://arxiv.org/abs/1012.4843" [Broken].
 
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  • #20
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I'm bothered e.g. by the fact that not all observables have eigenvectors. How are you avoiding that issue?
Traditionally, an observable is a self-adjoint operator. The Gelfand-Maurin theorem proves that there is a diagonal representation. _This_ is the relevant fact, not the existence of eigenvectors, which one hasn't in the continuous part of the spectrum.

If you take the components of position, you get the position representation.
If you take the components of momentum, you get the momentum representation.
In case of spin, one needs to add in both cases the operator J_3 to get a maximally commuting system and hence a (up to isomorphism) unique diagonal representation.

Where should I look for the proofs of those theorems? (Do I have to find a copy of Maurin's book?)
I don't know of a better source. I haven't looked at the proofs for many years, and write all this from memory....

The bounded case is reduced to Gelfand's work http://en.wikipedia.org/wiki/Gelfand_representation by noting that bounded commuting operators generate a commutative C^* algebra. One extends it to a maximal commutative C^* subalgebra B of the C^* algebra A of all bounded linear operators using Zorn's lemma (can perhaps be avoided if the Hilbert space is separable?), and then proceeds to show that A acts already on C_0(Phi_A). The unbounded case is easily reduced to the bounded case using Hille-Yosida.

My suggestion is that you first try to get the general feel of what is going on, and postpone the detailed proof for later (perhaps forever - it is interesting only if you want to do real research in that area).

Maurin probably formulates the theorem not as I do (I don't know whether my formulation is in the literature - though it actually might be in Maurin) but in terms of rigged Hilbert spaces (= Gelfand triples) version, where eigenvectors exist.

But for QM, this extension is not needed - except if one wants to have a rigorous version of the bra-ket calculus in case of a continuous spectrum. Indeed, one get the standard bra-ket heuristics for eigenkets from my formulation of the theorem in precisely the same way as it is introduced early on in the case of the position representation.
 
  • #21
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I have an axiom. It is not a strict mathematical one. Scientists tend to appeciate that the universe in its entirity is non-local. I believe it is local and non-local, depending on what situation we are considering.

Axiom

''Non-locality is a quantum phenomena. Non-locality should not have descriptions for macroscopic bodies. For large enough systems, locality is preserved.''
 
  • #22
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Some guy did a thesis on it - see http://arxiv.org/abs/1012.4843" [Broken].
I found there a pilot wave theory for qubits, but neither a discussion of the Ising model nor a mention of the Lagrangian that figures in your axiom IV; instead a heavy dependence on the Hamiltonian, which is absent from your axioms. And the Hilbert space, though not explicitly mentioned, is of course there - otherwise terms such as exp(itH) cannot even be defined.
 
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  • #23
A. Neumaier
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I have an axiom. It is not a strict mathematical one. Scientists tend to appeciate that the universe in its entirity is non-local. I believe it is local and non-local, depending on what situation we are considering.

Axiom

''Non-locality is a quantum phenomena. Non-locality should not have descriptions for macroscopic bodies. For large enough systems, locality is preserved.''
You post in the wrong forum. Jokes belong here: https://www.physicsforums.com/forumdisplay.php?f=198 [Broken]
 
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  • #24
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You post in the wrong forum. Jokes belong here: https://www.physicsforums.com/forumdisplay.php?f=198 [Broken]

That was harsh :)
 
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  • #25
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You post in the wrong forum. Jokes belong here: https://www.physicsforums.com/forumdisplay.php?f=198 [Broken]
Give me an example of a macroscopic body experiencing non-locality, and I'll take it back.
 
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