Time as a continous functor
To recall from prior posts, a functor maps objects to objects and arrows to arrows between two categories. In other words, it is structure preserving. In the case of a monoidal category, suppose there is an arrow * from \(C\times C \rightarrow C\). Then a functor T makes the diagram below commute:
This is all fancy abstract math which has a simple physical interpretation when T corresponds to time evolution: the laws of physics do not change in time. Moreover it can be shown with a bit of effort and knowledge of C* algebras that Time as a functor = unitarity.
But what can we derive from the commutative diagram above? With the additional help of two more very simple and natural ingredients we will be able to reconstruct the complete formalism of quantum mechanics!!! Today I will introduce the first one: time is a continuous parameter. Just like in group theory adding continuity results in the theory of Lie groups we will consider continous functors and we will investigate what happens in the neighborhood of the identity element.
In the limit of time evolution going to zero T becomes the identity. For infinitesimal time evolution we can then write:
\(T = I + \epsilon D\)
We plug this back into the diagram commutativity condition \(T(A)*T(B) = T(A*B)\) and we obtain in first order the chain rule of differentiation:
\(D(A*B) = D(A)*B + A*D(B)\)
There is not a single kind of time evolution and \(D\) is not unique (think of various hamiltonians). There is a natural transformation between different time evolution functors and we can express D as an operation like this: \(D_A = A\alpha\) where \((\cdot \alpha \cdot)\) is a product.
\(\alpha : C\times C \rightarrow C\)
Then we obtain the Leibniz identity:
\(A\alpha (B * C) = (A\alpha B) * C + B * (A \alpha C)\)
This is extremely powerful, as it is unitarity in disguise. Next time we'll use the tensor product and the second ingredient to obtain many more mathematical consequences. Please stay tuned.