What is this unitary invariant?

  • Context: Graduate 
  • Thread starter Thread starter Heidi
  • Start date Start date
  • Tags Tags
    Invariant Spectra
Click For Summary
SUMMARY

The discussion centers on the concept of a unitary invariant, specifically denoted as spec_,N (M), in relation to the eigenvalues of the Dirac operator. The participants explore the nature of this invariant, questioning whether it represents a set of real numbers and its invariance properties. The conversation also references the construction of a 3x3 CKM matrix, highlighting its four free real parameters: three Euler angles and a CP-violating phase, which are essential in various parametrizations of the standard model.

PREREQUISITES
  • Understanding of eigenvalues and the Dirac operator in quantum mechanics.
  • Familiarity with unitary matrices and their properties.
  • Knowledge of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and its significance in particle physics.
  • Basic comprehension of complex numbers and their geometric interpretations.
NEXT STEPS
  • Research the mathematical properties of unitary invariants in quantum mechanics.
  • Study the implications of eigenvalues in the context of the Dirac operator.
  • Explore different parametrizations of the CKM matrix and their physical interpretations.
  • Learn about the role of Euler angles in particle physics and their applications in various models.
USEFUL FOR

Physicists, mathematicians, and students interested in quantum mechanics, particularly those focusing on particle physics and the mathematical frameworks underlying the standard model.

Heidi
Messages
420
Reaction score
40
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.
 
Last edited by a moderator:
Physics news on Phys.org
It seems that all is here.
 
Reply
  • Like
Likes   Reactions: vanhees71
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.
 

Similar threads

Graduate Twistors and celestial holography
  • · Replies 21 ·
Replies
21
Views
6K
  • Poll Poll
Methods of Mathematical Physics by Hilbert and Courant
  • · Replies 1 ·
Replies
1
Views
6K
  • Poll Poll
Calculus Ordinary Differential Equations by Vladimir I. Arnold
  • · Replies 1 ·
Replies
1
Views
8K
Mathematica This Week's Finds in Mathematical Physics (Week 247)
  • · Replies 1 ·
Replies
1
Views
4K
Mathematica This Week's Finds in Mathematical Physics (Week 266)
  • · Replies 1 ·
Replies
1
Views
4K