On Physics from the beginning

We start with states and observables.

We assume, without definition, the following concepts: there exists a system (in a more general sense than dynamical systems) which is made of states, and we can make measurements of different properties (observables) of the system, obtaining real values which depend on which state the system is. The system changes of state (with respect to a parameter $t \in \mathbb N$ or $t\in \mathbb R$ which we call time) according to a dynamical law.

All this can be modeled, at least to my knowledge, in two ways:

Mathematical model 1 (MM1)

We have a set $\Omega=\{a,b,c,d,\ldots\}$. Its elements are called the pure states of the system. We call this set the phase space.

The observables are nothing but functions $f: \Omega \to \mathbb{R}$. The measurement process consists, simply, of evaluating the function. It is assumed that the measurement is "clean" in the sense that does not modify the state in which the system is.

If we have "observables enough" we can have a bijection of $\Omega$ with a "numerical set".

Probabilities. You can consider that the system is not in a pure state but in a mixed one, a kind of generalized state. As if you have a big box with lots of copies of the system, every of them in a different state. You repeat a measurement several times and you obtain several values, with a distribution of frequencies or probability. To my knowledge, there are two reasons to do this:

1. You are working with too many degrees of freedom (this is the trick of Classical Statistical Mechanics for avoiding too many dimensions); or

2. You assume that the measurement is not one hundred percent precise, and you repeat again and again obtaining a probability distribution. I think it is the same "considering that the state is distributed" than "considering that the measurement procedure is repeated once and again to yield a distribution".

Algebraic formulation: the observables (functions) live inside a complex commutative algebra $\mathcal F$. They can be seen like the main characters of the story, since it is what "we can access". If you can't measure the difference between two systems, you have no right to treat them as different. In mathematics there are lots of examples where a space is recovered from its algebra of functions, or at least from its sheaves. This is the spirit of Algebraic Geometry. Important example: a space is locally compact Hausdorff iff its algebra of continuous functions is a commutative c-star algebra, and reciprocally any commutative c-star algebra is the algebra of continuous functions on some space (the Gelfand spectrum). See @strocchi2008introduction page 15, Gelfand-Naimark theorem. In Classical Mechanics the observables constitute a commutative algebra $\mathcal F$ and the Gelfand spectrum $\Omega$ (multiplicative linear functionals on $\mathcal F$) are the pure states (@strocchi2008introduction page 13). The mixed states are the normalized positive linear functionals. The Riesz-Markov representation theorem let us assure that a mixed state $\omega$ in this sense determines a unique probability distribution $\mu_{\omega}$ on the Gelfand spectrum (see @strocchi2008introduction page 13) such that for an observable $f$

$$ \omega(f)=\int_{\Omega} f d\mu_{\omega},\quad \mu_{\omega}(\Omega)=\omega(1)=1 $$

Dynamics. Once we have the bijection with the numerical set, we usually formulate the dynamical law like a continuous or discrete dynamical system. I have to develop this yet.

Symmetries: within the natural transformation of the phase space (they could be bijections, if it simply a set, or symplectomorphism if we are in a symplectic manifold) we choose the ones which do preserve the dynamical law (for example Hamiltonian symmetrys) and obtain the symmetry group of a physical system.

Mathematical model 2 (MM2)

The system consist of a Hilbert space $(\mathcal{H},\langle -,- \rangle)$ whose elements of length 1 (or its rays, I am not sure) are the states of the system.

The observables are Hermitian operators $\hat Q: \mathcal H \to \mathcal H$. The measurement process, provided we are in state $|\psi \rangle \in \mathcal H$, consists of

obtaining the basis of eigenvectors $\{|q_i\rangle\}_i$ (with eigenvalues $q_i$) of the operator $\hat Q$.

randomly selecting one of them, according to the probability distribution $P(|q_i\rangle)=|\langle q_i|\psi\rangle|^2$ , that is, $P(|q_i\rangle)=|c_i|^2$ where $|\psi\rangle=c_1|q_1\rangle+c_2|q_2\rangle+\cdots$

applying the operator to the selected $|q_i\rangle$ and project the result over $|q_i\rangle$. In general we will call to the operation $\langle \psi |\hat Q |\psi \rangle$ the expected value of $\hat Q$ with respect to $|\psi\rangle$. In the case of eigenvectors this operation returns the eigenvalue.

The measurement has changed the state in which the system is, unless it were in one of the eigenvectors.

There is something called complete set of commuting observables. If you need more that one observable everything gets complicated, because you have eigenvalues with several eigenvectors (eigenspace of dimension greater than 1) and you maybe have to introduce tensor product to separate then and so on... Think of $x$ coordinate and $y$ coordinate in a discrete setup, for example. Key idea: commuting Hermitian operators have a common basis of eigenvectors. I have to think it more. For the moment, suppose we only need 1 observable.

Probability is inside the model from the beginning .... it can be added the other sense of probability......... I think

The observables can be embedded in a c-star algebra $\mathcal A$ which in general is not commutative. I guess that the Gelfand spectrum in this case is the original Hilbert space (pure states). The set of observables corresponds to the subset of $*$-invariant elements of $\mathcal{A}$.

Dynamical law: since states have length 1, the evolution needs to be yielded by unitary operators (a 1-parameter subgroup, indeed). The Stone's theorem assures that there is a self adjoint operator generating this 1-parameter subgroup, which is called the Hamiltonian.

Symmetries: symmetry group of a physical system.

Summary

So to recap, we can say that:

1. A system (classical system or quantum system) is given by a c-star algebra $\mathcal{A}$. A subset of $\mathcal{A}$ (the $*$-invariants elements) are the observables.

2. Systems can be set on pure states or mixed states. The latter ones are interpreted as a kind of "combination" of the former ones, or better said, probability distributions of them. They are all positive normalized linear functionals on $\mathcal{A}$. The pure states are required to be also multiplicative (also called characters).

3. In the case of a classical system the algebra is commutative. The multiplicative functionals constitute the Gelfand spectrum of $\mathcal{A}$, which is a compact topological space representing the pure states. The rest of the normalized functionals correspond to probability distributions defined on the Gelfand spectrum of $\mathcal{A}$. For pure states, the probability distribution is typically described by a delta function concentrated at a single point in the phase space. This corresponds to the fact that the system is in a definite state, with the position and momentum of each particle in the system being well-defined.

4. In the case of quantum systems, the algebra is not commutative. The multiplicative functionals (pure states) constitute again the Gelfand spectrum of $\mathcal{A}$, which in many important cases, is a complex projective space with an underlying Hilbert space. They can be also interpreted as one-dimensional projections in the algebra, while the general normalized functionals (mixed states) are represented by density matrices, which are positive semi-definite operators with trace equal to 1. I have to think this yet.

Quantum physics as a generalization of classical physics

We can embed MM1 into MM2 in the following way:

$\Omega$ goes to $\mathbb P \mathbb C^n$, si cardinal de $\Omega$ es $n$

$f:\Omega \to \mathbb C$ goes to the linear map

$$ M_f:\mathbb C^n \longmapsto \mathbb C^n $$

defined like the diagonal matrix whose entries are the values of $f$.

and so on... (I have to write this better)

But, why do we care about this embedding of MM1 into MM2? Why don't we settle for MM1? See this note: Why did physics go quantum.

To review and maybe incorporate: as it is said in this MO answer the point of QM is to see the algebra of functions as "immersed" in a bigger not necessarily commutative algebra.

Probability theory

Indeed, Quantum Mechanics can be approached from the point of view of probability theory. QM would be a new approach to probabilities, Quantum probability, extending classical probability. See probability theory#Quantum probability.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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