# Little fundamental question

1. Aug 10, 2009

### 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?

2. Aug 10, 2009

### 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.

3. Aug 10, 2009

### Bob_for_short

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 HR and Hr. It is a probability property to be represented as a product of independent probabilities, to be exact.

4. Aug 10, 2009

### 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.

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