In today's world, Dirac–von Neumann axioms is a topic that has gained great relevance and interest. Its impact has been felt in different aspects of society, from politics to popular culture. In this article, we will explore in detail the different nuances and perspectives surrounding Dirac–von Neumann axioms, analyzing its influence on the modern world and its role in shaping contemporary mindsets and dynamics. Through an exhaustive and multidimensional analysis, we aim to shed light on this topic and understand its importance in the current context. Additionally, we will examine possible future implications and possible avenues to address the challenges that Dirac–von Neumann axioms presents in our ever-changing world.
Formulation of quantum mechanics on a Hilbert Space
The Dirac–von Neumann axioms can be formulated in terms of a C*-algebra as follows.
The bounded observables of the quantum mechanical system are defined to be the self-adjoint elements of the C*-algebra.
The states of the quantum mechanical system are defined to be the states of the C*-algebra (in other words the normalized positive linear functionals ).
The value of a state on an element is the expectation value of the observable if the quantum system is in the state .
Example
If the C*-algebra is the algebra of all bounded operators on a Hilbert space , then the bounded observables are just the bounded self-adjoint operators on . If is a unit vector of then is a state on the C*-algebra, meaning the unit vectors (up to scalar multiplication) give the states of the system. This is similar to Dirac's formulation of quantum mechanics, though Dirac also allowed unbounded operators, and did not distinguish clearly between self-adjoint and Hermitian operators.
Strocchi, F. (2008), "An introduction to the mathematical structure of quantum mechanics. A short course for mathematicians", An Introduction to the Mathematical Structure of Quantum Mechanics. Series: Advanced Series in Mathematical Physics, Advanced Series in Mathematical Physics, 28 (2 ed.), World Scientific Publishing Co., Bibcode:2008ASMP...28.....S, doi:10.1142/7038, ISBN9789812835222, MR2484367