Quantization of Analysis
Kelvin Lui · Digital Commons - Colby (Colby College) · 2015
In quantum mechanics the replacement of complex vectors with operators is essential to “quantizing” space. Nonetheless, in many physics textbooks there is no justification for this action. Therefore in this thesis I will attempt to understand the mathematical formalism that allows for such a “replacement” to be rigorous. I will approach this topic by first defining a vector spaces and its dual space, a Hilbert space and a conjugate Hilbert space, and an operator space. Next, I will look at the algebraic tensor product of two vector spaces, two Hilbert spaces, and finally two operator spaces. Ultimately we will look at the completion of the tensor product with resect to the minimal norm and show that the minimal norm of a tensor has an analogous inequality to the Cauchy-Schwarz inequality.