Little fundamental question

  • Context: Graduate 
  • Thread starter Thread starter facenian
  • Start date Start date
  • Tags Tags
    Fundamental
Click For Summary

Discussion Overview

The discussion revolves around the nature of the state space for a composite quantum system formed by two subsystems. Participants explore whether the direct product of the state spaces of the subsystems should be considered a postulate or a logical consequence of more fundamental principles. The conversation touches on theoretical implications in quantum mechanics and classical mechanics.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant questions whether the direct product space H_a ⊗ H_b for a composite system C is a postulate or a consequence of something more fundamental.
  • Another participant argues that using the direct product reflects the full capacity of the structure, allowing for interactions within a unified framework.
  • A different viewpoint suggests that the concept of direct product space is not fundamental or postulate-like, using classical mechanics as an analogy to illustrate decoupling in systems.
  • One participant presents a theorem from the "quantum logic" approach that supports the use of direct product spaces, citing three axioms that are considered more fundamental.

Areas of Agreement / Disagreement

Participants express differing views on whether the direct product of state spaces is a postulate or a consequence of fundamental principles. There is no consensus on this issue, and multiple competing perspectives are presented.

Contextual Notes

The discussion includes references to classical mechanics and quantum mechanics, highlighting the complexity of the relationship between subsystems and their composite states. The implications of the axioms presented are not fully resolved, and the discussion remains open to interpretation.

facenian
Messages
433
Reaction score
25
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 [tex]H_a \otimes H_b[/tex]. Shouldn't this be considered a postulate? or it is a logical consecuence of something more fundamental?
 
Physics news on Phys.org
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.
 
facenian said:
When a system C is composed of two subsystems A and B, each described by state spaces Ha and Hb, then we are told that the state space corresponding to C is the direct product space [tex]H_a \otimes H_b[/tex]. 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 HR and Hr. It is a probability property to be represented as a product of independent probabilities, to be exact.
 
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.
 

Similar threads

  • · Replies 8 ·
Replies
8
Views
1K
  • · Replies 10 ·
Replies
10
Views
3K
  • · Replies 77 ·
3
Replies
77
Views
9K
  • · Replies 6 ·
Replies
6
Views
3K
  • · Replies 16 ·
Replies
16
Views
4K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 87 ·
3
Replies
87
Views
9K
  • · Replies 81 ·
3
Replies
81
Views
10K