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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 ?

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.

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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