Should tensor sum be used in matrix mechanics?

Click For Summary

Discussion Overview

The discussion revolves around the application of tensor sums in matrix mechanics, particularly in the context of the Bell operator and its implications for nonlocal realism. Participants explore the relationship between eigenvalues, measurement results, and the use of different summation methods in quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents the Bell operator and discusses how using the usual matrix sum can lead to eigenvalues smaller than 1, which challenges nonlocal realism.
  • Another participant suggests that using the Kronecker sum resolves the issue of eigenvalues being smaller than 1, raising questions about their experimental meaning.
  • A participant reflects on the philosophical implications of "realism" in the context of measurement results, suggesting it relates to a digitalization or binarization process.
  • One participant summarizes that matrix mechanics could operate within a forbidden band, implying a sub-local realistic interpretation.
  • A participant questions the terminology used regarding "matrix sum" and "eigenvalues," pointing out that the original formula involves numbers rather than matrices.
  • Another participant clarifies that the ##AB## values can be interpreted as covariance operators for two spin 1/2 particles and expresses difficulty in calculating eigenvalues without computational assistance.
  • A question is raised about the relationship between locality violations and the concept of an "under locality value."

Areas of Agreement / Disagreement

Participants express differing views on the implications of using tensor sums versus traditional matrix sums, and there is no consensus on the interpretation of eigenvalues or the philosophical implications of realism in this context.

Contextual Notes

Participants note the complexity of the mathematical framework and the potential for misinterpretation of terms, particularly regarding the nature of the Bell operator and the definitions of locality and realism.

Who May Find This Useful

This discussion may be of interest to those studying quantum mechanics, particularly in the areas of matrix mechanics, nonlocality, and the philosophical implications of measurement in quantum theory.

jk22
Messages
732
Reaction score
25
Suppose the Bell operator ##B=|AB(1,2)+AB(1,3)+AB(2,3)|##

With ##AB\in{1,-1}##

Nonlocal realism implies ##B\in{1,3}##

However using usual matrix sum one eigenvalues for the result of measurement can be smaller than 1, implying nonlocal realism cannot explain the quantum result.

However if the Kronecker sum is used this fact disappears.

So what would these eigenvalues mean experimentally ?
 
Physics news on Phys.org
Can't answer the question itself, but as a tip: in LaTeX code the set notation with curly brackets ##\left\{1,-1\right\}## is written as "\left\{1,-1\right\}".
 
Thanks for the tip. I thought a bit about my OP

I used "realism", which is a big word from philosophy, but technically I think it's a threshold switch that digitalizes or binarizes the results that makes that difference. So nothing deep.
 
As summary matrix mechanics were possible in this forbidden band ##]1,-1[## hence sub-local realistic ?
 
jk22 said:
using usual matrix sum one eigenvalues for the result of measurement can be smaller than 1

What "matrix sum" and "eigenvalues" are you talking about? The formula you give in the OP only involves numbers, not matrices.
 
  • Like
Likes   Reactions: jk22
Good point, it is if the ##AB##s are raised to the rank of covariance operators of two spin 1/2 particles.

I.e. ##A\rightarrow \vec{n}\cdot\vec{\sigma}## ?

But I could calculate the eigenvalue only with a CAS (computers are useful but it looks like acedia)

Why is that as there is a violation of locality there is also an under locality value ?
 
Last edited:

Similar threads

  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 10 ·
Replies
10
Views
3K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 6 ·
Replies
6
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 9 ·
Replies
9
Views
2K
  • · Replies 30 ·
2
Replies
30
Views
3K
  • · Replies 1 ·
Replies
1
Views
2K