Little fundamental question

[facenian]

Hi
when a system C is composed of two subsystems A and B each described by states spaces Ha and Hb then we are told that the state space corresponding to C is the direc produc space $$H_a \otimes H_b$$. Shouldn't this be considered a postulate? or it is a logical consecuence of something more fundamental?

 

[GPPaille]

A postulate has more profound implication on the physics itself. Making one state corresponding to two states with a direct product is using the full capacity of a structure, like making a big 2n-dimensional vector in lagrangian mechanics. And it offers the possibility to create interactions between every particle in a system using only one common structure.

 

[Bob_for_short]

When a system C is composed of two subsystems A and B, each described by state spaces H_a and H_b, then we are told that the state space corresponding to C is the direct product space $$H_a \otimes H_b$$. Shouldn't this be considered a postulate? or it is a logical consequence of something more fundamental?

Nothing is fundamental here or postulate-like. Consider, for example, a system of two classical particles. Their coordinates r₁ and r₂ are coupled with the Newton equations. Introduce the center of inertia R and relative coordinates r= r₁ -r₂. Their equations are decoupled as if they were two non-interacting systems. In QM the total wave function factorizes and the total Hamiltonian is a (direct) sum of Hamiltonians H_R and H_r. It is a probability property to be represented as a product of independent probabilities, to be exact.

 

[meopemuk]

The statement about direct product space can be proved within the "quantum logic" approach. This theorem is derived from 3 axioms, which are assumed to be "more fundamental". Loosely speaking, these three axioms are:

1. Propositions (yes-no experiments) about constituent systems a and b make sense in the compound system a+b. The logical structure is preserved.
2. Propositions about constituent systems a and b are independent (the corresponding projection operators commute).
3. If we have full information about constituents a and b, then we have full information (atomic proposition) about the compound system a+b.

You can find more details in

Matolcsi, T. "Tensor product of Hilbert lattices and free orthodistributive
product of orthomodular lattices", Acta Sci. Math. Szeged 37 (1975), 263-272.

Aerts, D.; Daubechies, I. "Physical justification for using the tensor product
to describe two quantum systems as one joint system."
Helv. Phys. Acta 51 (1978), 661-675.