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
411
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.
 
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  • #2
It seems that all is here.
 
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  • #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

1. What is a unitary invariant?

A unitary invariant is a mathematical quantity that remains unchanged under unitary transformations. These transformations are operations that preserve the length and angles of vectors in a vector space. In other words, a unitary invariant is a property that is independent of the basis used to describe a system.

2. How is a unitary invariant different from other invariants?

Unlike other invariants, which remain unchanged under all types of transformations, unitary invariants only remain unchanged under unitary transformations. This makes them particularly useful in quantum mechanics, where unitary transformations play a crucial role in describing the evolution of quantum systems.

3. What are some examples of unitary invariants?

Some common examples of unitary invariants include the trace of a matrix, the determinant of a matrix, and the eigenvalues of a matrix. In quantum mechanics, the expectation value of an observable is also a unitary invariant.

4. How are unitary invariants used in physics?

In physics, unitary invariants are used to describe the properties of systems that are invariant under unitary transformations. This is particularly important in quantum mechanics, where unitary transformations are used to describe the evolution of quantum states and the measurement of observables.

5. Can unitary invariants be used to simplify calculations?

Yes, unitary invariants can be used to simplify calculations in quantum mechanics. By using unitary invariants, physicists can choose a basis that simplifies the problem at hand, making it easier to solve and understand. This is particularly useful in complex quantum systems, where traditional methods may be too computationally intensive.

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