Three axioms of quantum mechanics

Introduction

Often, quantum mechanics is presented as if it were a strange theory, exceeding even the human understanding. However if it is true that this theory is very badly included/understood by the majority of the laymen, it is rather well included/understood by the initiates. Why such a difference in comprehension? Perhaps simply because of many different things one be known as on this one, perhaps owing to the fact that its formalism is of an abstraction ever equalized before by a theory wanting to describe the " world observable". Or simply because this theory goes against the bases to the scientific thought, these even which enabled us to build our representation of the world. Indeed quantum mechanics poses large problems with the concept of the determinism such as we knew it before the 20th century. With its advent, it is necessary for us to rebuild the concepts of measurement, reproducibility of an experiment, and even of determinism…

The model of the three axioms is a rigorous approach which leads to the idea that the space of the states is a vector space (often a space of Hilbert) thing which is postulated by other approaches. The knowledge of the theorem of Stones and the theorem of Noether (probably the two most important theorems of quantum mechanics, the being useful first has contruire the idea of temporal evolution, the second the concept of quantity of mounvement) carries out without too much difficulty to the rebuilding the usual postulates of quantum mechanics (see Postulats of quantum mechanics).

3 axioms

In order to have the vocabulary of the three axioms, it is necessary to introduce some basic concepts. Indeed, in opposed to the six postulates of quantum mechanics, these three axioms even rest on concepts closely related to the experiment with measurement. (it can be interesting to read Problème of quantum measurement on this subject)

Questions, Properties and Space of the states

The concept of question rises from the idea from measurement. A question $\backslash \; alpha\; ~$ is a measurement on a physical system whose answer is true or false.

To illustrate this idea, we can make the experiment of following thought:

the physical system consists of a car which runs on the highway. The question is: Does it roll to 130 km/h? Our measuring device is a radar. The answer is true, so on the radar, one reads 130 km/h, it is false if not.

From any question $\backslash \; alpha\; ~$, it is possible to define a question reverses $\backslash \; tilde\; \{\backslash \; alpha\}$. In our experiment of thought, the question reverses would be: does the car run at a speed different from 130 km/h?

The $Q~$ unit is the whole of all the questions which one can put on the studied system.

One can build a relation of préordre (Relation of order) on the questions:

our experiment of thought, we Take again define two questions:

$\backslash alpha~$ : does the car run at a speed ranging between 120km/h and 140km/h?

$\backslash beta~$ : does the car run to 130km/h?

One notices that $\backslash \; alpha~$ is always true if $\backslash \; beta~$ is true, one will say that $\backslash \; alpha~$ is weaker than $\backslash \; beta~$

Notation:

Using this relation of préordre, it is possible to create a Relation of equivalence:

one will say $\backslash \; alpha~$ is equivalent to $\backslash \; beta~$, if and only if and

notation: $\backslash alpha~\; \backslash sim\; \backslash beta~$

The Classe of equivalence $a~$ of a Question $\backslash \; alpha~$ is a Property.

A property is known as current if the questions associated with this one are true, on the contrary if they are false, one says that the property is potential.

We define $L~$ as the whole of all the properties of the system.

A remarkable thing is, that without any other assumption, we can already have certain information on the structure of $L~$. Indeed, the relation of préordre on $Q~$ imposes the fact that $L~$ is partially ordered. And thus $L~$ is always a complete lattice, i.e.:

$\backslash forall\; E~=\backslash big\backslash \{\; a~\_i\; |\; a~\_i\; \backslash \; in\; L~,\; I\; \backslash \; in\; J~\; \backslash \; big\; \backslash \}$ there exists $a~,\; b~\; \backslash \; in\; L~$ such as:

if $x~\; \backslash \; in\; L~$ then:

$x~\; >\; a~\_i\; \backslash \; forall\; I\; \backslash \; in\; L~\; \backslash \; Longleftrightarrow\; x~>\; b~$

$a~,\; b~$ is respectively the lower and higher limit subset $E~$.

States and Property-states

A state $E~$ is a subset of $L~$ such as the property $p~$ is current when all the properties contained in $E~$ are current. One can thus define a state as follows:

Such a property $p~$ entirely defines $E~$ and is called property-state.

Atoms

A property $p\; \backslash \; in\; L$ is called atom if:

$p\; \backslash \; neq\; 0,\; \backslash \; quad\; p$ is different from the property minimal defined by the question $\backslash \; tilde\; \{I\}$ (the reverse of the commonplace question)

and that:

if $p$ is an atom of $L$ then $p$ is a property-state.

Representation of Cartan

That is to say $S$ the whole of all the states possible of the system. We can define an application $\backslash \; mu$ of $L$ in $P\; (S)$ the whole of the part of S.

$\backslash \; driven:\; \backslash \; to\; \backslash \; driven\; has\; =\; \backslash \; left\; \backslash \; \{E\; has\; |\; has\; \backslash \; in\; E,\; E\; \backslash \; in\; S\; \backslash \; right\; \backslash \}$

This application is called the morphism of Cartan, and $P\; (S)$ is called the representation of Cartan.

Moreover, this application is injective and preserves the order and the lower limit.

Concept of orthogonality

One says of two states $E\_1,\; E\_2\; \backslash \; subset\; L$ which they are orthogonal (notation: $E\_1\; \backslash \; perp\; E\_2$) if there exists a question $\backslash \; alpha$ such as:

$\backslash \; alpha$ is true for $E\_1$ and $\backslash \; tilde\; \{\backslash \; alpha\}$ is true for $E\_2$

It is said that two property $a,\; B\; \backslash \; in\; L$ which it are orthogonal (notation: $a\; \backslash \; perp\; b$) if all the states $\backslash \; driven\; a$ orthogonal with the states $\backslash \; are\; driven\; b$:

$a\; \backslash \; perp\; b$ if and only if $\backslash \; driven\; has\; \backslash \; perp\; \backslash \; driven\; b$

Traditional system

Axiom 0:

We will say that a question $\backslash \; alpha~$ is traditional if, for each state $E~$, or $\backslash \; alpha~$ is true, or $\backslash \; tilde\; \{\backslash \; alpha\}$ is true. And we will say of a system which it respects the traditional prejudice so for each property $a~$ it exists at least a traditional question $\backslash \; alpha~$

This axiom entirely determines the structure of $L~$. Moreover if the system satisfies axiom 0 then the morphism of Cartan is surjective and $L~$ and $P~\; (S)$ are isomorphous

Generalization: quantum systems

Axiom I

Any property-states is an atom of $L$

This axiom means simply that if two states $E\_1,\; E\_2\; \backslash \; subset\; L$ are different then the relation $E\_1\; \backslash \; subset\; E\_2$ is excluded.

This axiom is in fact a physical law which goes back to Aristote: if the system changes state, that it thus passes from the state $E\_1$ to $E\_2$, it grows rich by new properties which are brought up to date, and it loses of them necessarily others. Thus $E\_1$ cannot be entirely contained in $E\_2$

Axiom II

For each state $E$ given there exists at least a question which is true if and only if the state of the system is orthogonal with $E$

By taking account of axiom I axiom II means that:

For each property-state $p$ atom of $L$, there exists a property $p\text{'}$ which is current if and only if the state is orthogonal with $p$

One can define an application of $L$ in $L$ which applies $p$ to $p\text{'}$.

formulation of this application in the representation of Cartan:

$\backslash \; driven\; (p\text{'})\; =\; \backslash \; driven\; (p)\; ^\; \backslash \; perp$

notation: $A^\; \{\backslash \; perp\}\; =\; \backslash \; big\; \backslash \; \{E\; |\; E\; \backslash \; in\; S,\; E\; \backslash \; perp\; \backslash \; eta\; \backslash \; quad\; \backslash \; forall\; \backslash \; eta\; \backslash \; in\; has\; \backslash \; big\; \backslash \}$

Axiom III

the application of $L$ in $L$: $p\; \backslash \; to\; p\text{'}$ is surjective.

Structure of the space of the states

Just like axiom 0, axioms I, II, III entirely determine the structure of the space of the states.

Theorem:

If the system satisfies axioms 1,2 and 3 then:

* the morphism of Cartan determines an isomorphism between $L$ and $(S,\; \backslash \; perp)$

* $L$ is a complete lattice, filled atoms

* the application defined in axiom III is a orthocomplementation

the space of the states is a lattice complete, filled atoms and provided with a orthocomplementation

Spaces of Hilbert

We wish to show that the space generated by the rays of a space of Hilbert is appropriate perfectly to describe a space of the states.

Theorem:

Is H a space of Hilbert, the whole of all the subspaces $G$ contained in $H$ is a lattice complete, orthocomplémenté and filled atom.

Demonstration:

*treillis complete:

the closed subspaces $G$ can be ordered by inclusion. They form a complete lattice because the intersection of closed units is still a closed unit.

*Orthocomplementé:

the application which with $G$ makes correspond $G^\; \backslash \; perp$ defines a orthocomplementation. Indeed:

* Filled atoms:

the atoms of this lattice are the subspaces of dimension 1 (i.e. the rays) and all the subspaces are generated by the rays which it contain.

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