Graduate What is this unitary invariant?

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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
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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.
 
Last edited by a moderator:
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It seems that all is here.
 
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.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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