Hilb becomes a Monoidal Category


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Hilb becomes a Monoidal Category

The reason that in the context of Quantum Mechanics it is not possible to identify the Tensor product with the Cartesian Product, as defined in general Category Theory is the following: imagine we have two Hilbert spaces H and $ H^{'}$ and we define $ H\otimes H^{'}$, then, according to the definition of Cartesian Product we would have two projection arrows $ pr_1:H\rightarrow H\otimes H^{'}$ and $ pr_2: H^{'}\rightarrow H\otimes H^{'}$. But, in the context of Quantum Mechanics, these projections do not exist!
This contradiction requires a new definition of the Tensor Product in Quantum Mechanics in Categorical terms. J. Baez showed this can be achieved if we transform the category Hilb into a Monoidal Category (2.2). This transformation implies that the tensor product can now be identified with the functor

$\displaystyle \otimes: Hilb\times Hilb\rightarrow Hilb$    

such that, given $ H, H^{'}\in Hilb$ then,

$\displaystyle \otimes (H, H^{'}) = H\otimes H^{'} \in Hilb$    

Moreover given $ T:H\rightarrow H^{'}$ and $ T_1:H_1\rightarrow H_1^{'}$, it is now possible to define

$\displaystyle T\otimes T_1: H\otimes H_1\rightarrow H^{'}\otimes H_1^{'}$    

Given this definition of Tensor product it is easy to see that the above mentioned contradiction does not occur.


Next: Category nCob in General Up: Categories in Quantum Mechanics Previous: Hilb becomes a *-Category

Cecilia Flori 2006-11-21