What is this unitary invariant?

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    Invariant Spectra
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Discussion Overview

The discussion revolves around the concept of a unitary invariant related to the eigenvalues of the Dirac operator, as presented in a specific article. Participants explore the nature of this invariant, its mathematical properties, and its implications in the context of matrix representations, particularly in relation to CKM matrices.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether the unitary invariant referred to as spec_,N (M) is a set of real numbers or something else, expressing confusion about the term "unitary."
  • Another participant inquires about the reasons behind the invariance of this quantity.
  • There is a mention of constructing a nxn unitary matrix similar to a CKM matrix, with a discussion on the equivalence of lines when multiplied by a complex number with modulus equal to 1.
  • A participant references a specific CKM matrix structure and asks about the parameters involved, noting the presence of 4 free real parameters in a 3x3 CKM matrix.
  • Another participant points to a Wikipedia article discussing the CKM matrix, highlighting that different parametrizations exist for the Euler's angles in the standard model.

Areas of Agreement / Disagreement

The discussion contains multiple competing views and questions regarding the nature of the unitary invariant and the parameters of the CKM matrix, indicating that no consensus has been reached.

Contextual Notes

Participants express uncertainty about the definitions and implications of the unitary invariant and the parameters of the CKM matrix, which may depend on specific mathematical frameworks or interpretations.

Heidi
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Hi Pfs
I am reading this article:
https://arxiv.org/abs/0810.2091
It is know that hearing the possible frequencies emitted by a drum are not enough to know its shape.
Here the frequencies are the eigenvalues of the Dirac operator.
the missing information is the unitary invariant of the title.
The autor call it spec_,N (M)
I have several questions:
Is it a set of real numbers or something else ? (the word unitary..)
why is it invariant?
I see that he builds a nxn unitary matrix (like a ckm matrix)
Lines are equivalent when multiplied by a complex with module = 1. This means that each line is equivalent to its opposite. So the unit sphere is mapped to the projective plane Pn. I do not undertand what is done with the comums....
thanks.
 
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It seems that all is here.
 
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We have on page 24 a CKM matrix:
1 1 1
1 z z*
1 z* z
where z = exp(2i pi /3)
In a 3*3 ckm matrix we have 4 free real parameters: 3 Euler's angles and a cp violating phase.
What are they here?
thanks.
 

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