Category Hilb (Hilbert spaces)


Next: Hilb becomes a *-Category Up: Categories in Quantum Mechanics Previous: Category of Boolean subalgebras

Category Hilb (Hilbert spaces)

Given the collection of all possible Hilbert spaces it is possible, as shown by Baez in [1], to transform this collection into a Category in its own right by defining the following:
  • Objects of Hilb are defined as (arbitrary) Hilbert spaces
  • Morphisms in Hilb are identified as bounded linear operators between the various Hilbert spaces.
In order to rigorously prove that Hilb, as defined above is a category, we need to prove the following:
  1. composition condition
  2. identity law
  3. associative law

1) and 2) are straightforward to prove:
  1. given $ T:H\rightarrow H^1$ and $ G:H^1\rightarrow H^2$, we then get $ G\hspace{.01in}o\hspace{.01in} T:T:H\rightarrow H^2$.
  2. $ 1_H:H\rightarrow H$.
It is easy to deduce that, given the above, condition 3) follows.

It is possible to show that Hilb is a *-Category and a Monoidal Category (2.2 2.3).

Why are these extra requirements needed ?

The answer lies in the existence of the inner product and tensor product in the Hilbert space.
The problem that occur are the following:

  • linear operators do not preserve the inner product. This feature is irrelevant in the process of transforming the collections of Hilbert spaces in a category (form a mathematical point of view), but it is relevant for using the Hilbert space in the context of Quantum Mechanics.
  • In any "normal category" the tensor product would be equivalent to the Cartesian product (see definition of product), condition that does not agree in a Quantum Mechanical framework ,in which, the two can not be identified.
The extra properties of Hilb being a *-Category and a Monoidal category account for the inner product and tensor product respectively, as defined in Quantum Mechanics.
The exact details of how these two categories are implemented in Quantum Mechanics can be found in [1]. I will, hereby, reproduce a general summary of the ideas expressed in the above mentioned paper.

Next: Hilb becomes a *-Category Up: Categories in Quantum Mechanics Previous: Category of Boolean subalgebras
Cecilia Flori 2006-11-20