Is the Kronecker delta linear?

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Discussion Overview

The discussion revolves around the linearity of the Kronecker delta function and its role as a dual basis in a ket space. Participants explore the definitions and implications of linearity in different contexts, as well as the relationship between the Kronecker delta and Dirac delta functions.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant questions the linearity of the Kronecker delta and its ability to serve as a dual basis for a ket space.
  • Another participant suggests that the meaning of "linear" varies by context and indicates a potential misunderstanding between Dirac and Kronecker deltas.
  • A participant clarifies their understanding, noting that the Kronecker delta is defined only for certain basis vectors and is not a function, while the functionals with the Kronecker delta property are.
  • One participant poses a new question regarding the significance of basis functionals in relation to physical reality.
  • Another participant responds by emphasizing the importance of the Born rule in connecting basis functionals to physical outcomes, specifically in terms of probability coefficients.
  • A later reply indicates a misunderstanding about the terminology, clarifying that the focus is on basis functionals rather than basis vectors.

Areas of Agreement / Disagreement

Participants express differing views on the linearity of the Kronecker delta and its implications. There is no consensus on the definitions and significance of basis functionals versus basis vectors, indicating ongoing debate and exploration of these concepts.

Contextual Notes

Participants highlight the potential confusion arising from the different meanings of linearity and the distinction between functions and functionals. There are unresolved aspects regarding the definitions and implications of these mathematical constructs.

entropy1
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I don't see how the kronecker delta function is linear, and hence I don't see how it can serve as a dual basis for a ket space.

How is the kronecker delta linear? How can it serve as a dual basis?
 
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There are different meanings of "linear" depending on the context. It seems you are mixing them up, as well as Dirac and Kronecker. The functions spanning a function space don't have to be linear, but the space is a linear space, because it consists of linear superpositions of those functions.

In case of the Dirac delta you don't even have a function however. But the associated functional that extracts a single value from a test function is linear.

Look up these concepts, maybe that will clear up your confusion.
 
I think I understand now. Thanks. I mistook the Kronecker delta for being defined 0 for any vector which was not a particular basevector, but in fact it is only defined such for certain basevectors. And the Kronecker delta is not a function. The functionals with the Kronecker delta property are. :wink:
 
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New question:
What is the significance of a basis functional with respect to physical reality? :smile:
 
Last edited:
entropy1 said:
What is the significance of a basis functional with respect to physical reality? :smile:

I presume you mean basis vectors.

I think 'physical reality' is far too vague and undefined a concept to be of any value here.

Its value lies in the Born rule. If you expand the state |u> in terms of an observables O = Σyi |bi><bi| eigenvectors, |u> = Σ ci |bi>, the coefficients of that expansion, ci, give the probability of outcome i by |ci|^2.

Thanks
Bill
 
Last edited:
bhobba said:
I presume you mean basis vectors.
Nope, I ment basis functionals. But I guess I have to learn further to what eigenvectors are first. :wink:
 

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