Again, we use objects in this category to describe physical systems and morphisms to describe
physical processes.
One reason quantum theory seems ‘weird’ to some people is that in this theory, we ’glom together’
two physical systems using the tensor product of Hilbert spaces, which is not the ‘product’ in the
sense just described!
I.e., given Hilbert spaces Xand Y, we have this new Hilbert space X⊗Y, but there are generally
not any interesting morphisms
p1:X⊗Y→X
p2:X⊗Y→Y
For example, we use the vector ψ⊗φ∈X⊗Yto describe to describe a state of the system X⊗Y
where the first subsystem is in the state ψand the second subsystem is in the state φ. But, there
are no linear operators as above that pick out these states:
p1(ψ⊗φ) = ψ
p2(ψ⊗φ) = φ
for all ψ∈X,φ∈X. Even more importantly, we can’t find p1, p2making X⊗Yinto the product
of Xand Y: that is, operators such that for all f:Z→X,g:Z→Y,∃hf, gi:Z→X⊗Ysuch that
Z
f
{{w
w
w
w
w
w
w
w
wg
##
G
G
G
G
G
G
G
G
G
hf,gi
XX⊗Y
p1
oop2
//Y
commutes.
This has important consequences. For example, in a category with products, we can always
“duplicate” a system: i.e. we have a morphism
∆X:X→X×X.
We get this as follows:
X
1x
{{w
w
w
w
w
w
w
w
w1x
##
G
G
G
G
G
G
G
G
G
∆x
XX×X
p1
oop2
//X
In the case of Set, we have
∆X:X→X×X
x7→ (x, x).
But in Hilb we do not have any interesting linear operators
∆X:X→X⊗X.
For example,
ψ7→ ψ⊗ψ
is not linear. Wooters and Zurek proved a theorem making this issue precise: “you can not clone a
quantum”.
In fact, the right way of glomming together classical systems is also not the Cartesian product,
but some kind of ‘tensor product’ of Poisson manifolds!
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