The book mentioned by Bora,

• F. Strocci: "An Introduction to the Mathematical Structur of Quantum Mechanics"

is indeed a good reference for an axiomatic approach to quantum mechanics where the observables of a given system are assumed to form a $C^*$-algebra. The physical motivation for this approach is that every detector is representable by an observable (a self-adjoint operator) that is bounded, because every detector has an upper bound of values it can measure. A detector is therefore a bounded function of an unbounded, essentially self-adjoint observable as the impuls operator in, say, the usual (space or impuls) representation of the Heisenberg comutation relations.

This is the physical motivation of the basic assumption of an axiomatic approach to quantum physics via $C^*$- algebras: The assumption is that a quantum system, whose observables are essentially self-adjoint - maybe unbounded - operators, can equivalently be described by an operator algebra containing all bounded functions of the original observables. (The function $e^{i s A}$ for a real number s and an essentially self-adjoint operator A is of course an example, as mentioned in the comment by Pieter).

For an explanation of how and why this works in quantum mechanics, see the book by Strocci. (The short answer is that the $C^*$ - algebra of a massive particle described by position and momentum as observables is the Weyl algebra which is generated by the bounded operators occuring in the Weyl communtation relations mentioned by Pieter.)

In axiomatic quantum field theory, the difference of bounded versus unbounded observables manifests itself in two different sets of axioms:

• the Wightman axioms use unbounded operators,

• the Haag-Kastler axioms use bounded operators.

In this setting it is not completely clear that both sets of axioms are equivalent, although the relations Wightman => Haag-Kastler and vice versa have been proven with certain additional "technical" assumptions, see for example:

• H.J. Borchers, Jakob Yngvason: “From quantum fields to local von Neumann algebras”, Rev.Math.Phys. Special issue, 1992, p.15-47.

("Quantum fields" refer to the Wightman axioms and "local von Neumann algebras" to the Haag-Kastler axioms.)

When you compare the situation in quantum field theory to the situation in quantum mechancis, you'll see that the situation is more complicated because in quantum field theory the von Neumann uniqueness theorem cannot be applied.

The book mentioned by Bora,

• F. Strocchi: "An Introduction to the Mathematical Structur of Quantum Mechanics"

is indeed a good reference for an axiomatic approach to quantum mechanics where the observables of a given system are assumed to form a $C^*$-algebra. The physical motivation for this approach is that every detector is representable by an observable (a self-adjoint operator) that is bounded, because every detector has an upper bound of values it can measure. A detector is therefore a bounded function of an unbounded, essentially self-adjoint observable as the impuls operator in, say, the usual (space or impuls) representation of the Heisenberg comutation relations.

This is the physical motivation of the basic assumption of an axiomatic approach to quantum physics via $C^*$- algebras: The assumption is that a quantum system, whose observables are essentially self-adjoint - maybe unbounded - operators, can equivalently be described by an operator algebra containing all bounded functions of the original observables. (The function $e^{i s A}$ for a real number s and an essentially self-adjoint operator A is of course an example, as mentioned in the comment by Pieter).

For an explanation of how and why this works in quantum mechanics, see the book by Strocci. (The short answer is that the $C^*$ - algebra of a massive particle described by position and momentum as observables is the Weyl algebra which is generated by the bounded operators occuring in the Weyl communtation relations mentioned by Pieter.)

In axiomatic quantum field theory, the difference of bounded versus unbounded observables manifests itself in two different sets of axioms:

• the Wightman axioms use unbounded operators,

• the Haag-Kastler axioms use bounded operators.

In this setting it is not completely clear that both sets of axioms are equivalent, although the relations Wightman => Haag-Kastler and vice versa have been proven with certain additional "technical" assumptions, see for example:

• H.J. Borchers, Jakob Yngvason: “From quantum fields to local von Neumann algebras”, Rev.Math.Phys. Special issue, 1992, p.15-47.

("Quantum fields" refer to the Wightman axioms and "local von Neumann algebras" to the Haag-Kastler axioms.)

When you compare the situation in quantum field theory to the situation in quantum mechancis, you'll see that the situation is more complicated because in quantum field theory the von Neumann uniqueness theorem cannot be applied.

The book mentioned by Bora,

• F. Strocci: "An Introduction to the Mathematical Structur of Quantum Mechanics"

is indeed a good reference for an axiomatic approach to quantum mechanics where the observables of a given system are assumed to form a $C^*$-algebra. The physical motivation for this approach is that every detector is representable by an observable (a self-adjoint operator) that is bounded, because every detector has an upper bound of values it can measure. A detector is therefore a bounded function of an unbounded, essentially self-adjoint observable as the impuls operator in, say, the usual (space or impuls) representation of the Heisenberg comutation relations.

This is the physical motivation of the basic assumption of an axiomatic approach to quantum physics via $C^*$- algebras: The assumption is that a quantum system, whose observables are essentially self-adjoint - maybe unbounded - operators, can equivalently be described by an operator algebra containing all bounded functions of the original observables. (The function $e^{i s A}$ for a real number s and an essentially self-adjoint operator A is of course an example, as mentioned in the comment by Pieter).

For an explanation of how and why this works in quantum mechanics, see the book by Strocci.

In axiomatic quantum field theory, the difference of bounded versus unbounded observables manifests itself in two different sets of axioms:

• the Wightman axioms use unbounded operators,

• the Haag-Kastler axioms use bounded operators.

In this setting it is not completely clear that both sets of axioms are equivalent, although the relations Wightman => Haag-Kastler and vice versa have been proven with certain additional "technical" assumptions, see for example:

• H.J. Borchers, Jakob Yngvason: “From quantum fields to local von Neumann algebras”, Rev.Math.Phys. Special issue, 1992, p.15-47.

("Quantum fields" refer to the Wightman axioms and "local von Neumann algebras" to the Haag-Kastler axioms.)

When you compare the situation in quantum field theory to the situation in quantum mechancis, you'll see that the situation is more complicated because in quantum field theory the von Neumann uniqueness theorem cannot be applied.

The book mentioned by Bora,

• F. Strocci: "An Introduction to the Mathematical Structur of Quantum Mechanics"

is indeed a good reference for an axiomatic approach to quantum mechanics where the observables of a given system are assumed to form a $C^*$-algebra. The physical motivation for this approach is that every detector is representable by an observable (a self-adjoint operator) that is bounded, because every detector has an upper bound of values it can measure. A detector is therefore a bounded function of an unbounded, essentially self-adjoint observable as the impuls operator in, say, the usual (space or impuls) representation of the Heisenberg comutation relations.

This is the physical motivation of the basic assumption of an axiomatic approach to quantum physics via $C^*$- algebras: The assumption is that a quantum system, whose observables are essentially self-adjoint - maybe unbounded - operators, can equivalently be described by an operator algebra containing all bounded functions of the original observables. (The function $e^{i s A}$ for a real number s and an essentially self-adjoint operator A is of course an example, as mentioned in the comment by Pieter).

For an explanation of how and why this works in quantum mechanics, see the book by Strocci. (The short answer is that the $C^*$ - algebra of a massive particle described by position and momentum as observables is the Weyl algebra which is generated by the bounded operators occuring in the Weyl communtation relations mentioned by Pieter.)

In axiomatic quantum field theory, the difference of bounded versus unbounded observables manifests itself in two different sets of axioms:

• the Wightman axioms use unbounded operators,

• the Haag-Kastler axioms use bounded operators.

In this setting it is not completely clear that both sets of axioms are equivalent, although the relations Wightman => Haag-Kastler and vice versa have been proven with certain additional "technical" assumptions, see for example:

• H.J. Borchers, Jakob Yngvason: “From quantum fields to local von Neumann algebras”, Rev.Math.Phys. Special issue, 1992, p.15-47.

("Quantum fields" refer to the Wightman axioms and "local von Neumann algebras" to the Haag-Kastler axioms.)

When you compare the situation in quantum field theory to the situation in quantum mechancis, you'll see that the situation is more complicated because in quantum field theory the von Neumann uniqueness theorem cannot be applied.