What is this unitary invariant?

In summary, the article discusses the use of eigenvalues of the Dirac operator to determine the shape of a drum. However, the missing information is the unitary invariant, which the author calls spec_,N (M). This is a set of real numbers that is invariant and can be represented by a nxn unitary matrix. The unit sphere is then mapped to the projective plane Pn. The article also mentions the use of a CKM matrix, which has 4 free real parameters (3 Euler's angles and a CP-violating phase). These parameters are also seen in the Cabibbo-Kobayashi-Maskawa matrix, where different parametrizations can be used.
  • #1
Heidi
418
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
  • #2
It seems that all is here.
 
  • Like
Likes vanhees71
  • #3
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.
 
  • #4

FAQ: What is this unitary invariant?

What is a unitary invariant?

A unitary invariant is a property of a matrix or operator that remains unchanged under unitary transformations. In other words, if a matrix \( A \) has a certain property that is preserved when \( A \) is conjugated by a unitary matrix \( U \) (i.e., \( UAU^\dagger \)), then that property is a unitary invariant.

Why are unitary invariants important in quantum mechanics?

Unitary invariants are crucial in quantum mechanics because the evolution of quantum states is governed by unitary operators. Properties that remain invariant under unitary transformations are essential for understanding the fundamental characteristics of quantum systems, such as energy levels and transition probabilities.

Can you give an example of a unitary invariant?

An example of a unitary invariant is the trace of a matrix. For any matrix \( A \), the trace \( \text{Tr}(A) \) remains the same under a unitary transformation \( U \), i.e., \( \text{Tr}(UAU^\dagger) = \text{Tr}(A) \). Other examples include eigenvalues and the determinant of a matrix.

How do unitary invariants relate to eigenvalues?

Eigenvalues of a matrix are unitary invariants. This means that if a matrix \( A \) is transformed by a unitary matrix \( U \) to \( UAU^\dagger \), the eigenvalues of \( A \) remain unchanged. This property is particularly useful in various applications, including stability analysis and quantum mechanics.

What is the role of unitary invariants in linear algebra?

In linear algebra, unitary invariants are used to classify and understand the properties of matrices. They help in identifying matrices that are similar under unitary transformations and play a key role in simplifying problems involving matrix computations, such as diagonalization and singular value decomposition.

Similar threads

A Twistors and celestial holography
Replies
14
Views
4K
  • Poll
Methods of Mathematical Physics by Hilbert and Courant
Replies
1
Views
5K
Mathematica This Week's Finds in Mathematical Physics (Week 247)
Replies
1
Views
3K
Mathematica This Week's Finds in Mathematical Physics (Week 266)
Replies
1
Views
3K
Mathematica This Week's Finds in Mathematical Physics (Week 230)
Replies
1
Views
2K
Mathematica This Week's Finds in Mathematical Physics (Week 230)
Replies
1
Views
3K

Hot Threads

Recent Insights

Back
Top