How Do You Write the Neutrino Mixing Matrix Us from Equations 3 and 4?

In summary, a neutrino mixing matrix is a mathematical tool that describes the mixing of different types of neutrinos. It is important in understanding the properties and behavior of these elusive particles and plays a crucial role in the study of neutrino oscillations. The matrix is typically written in a 3x3 form with complex elements representing the probability amplitudes for flavor changes. These elements have physical meaning as they represent the likelihood of neutrinos changing between different flavors. The values of the elements are determined through experiments measuring neutrino oscillations.
  • #1
SuperStringboy
74
0
Please look at equation 3 and 4 of this paper

http://arxiv.org/abs/0707.2481v1

I am facing problem to write the matrix Us

Can anybody help me to write the complete matrix?
 
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  • #2
I think I understand how to write it, but I feel like I'm getting some different minus signs. Basically you want the SO(5) rotation group, and matrices for one direction about another (plane, or whatever its called).

If you look at http://reference.wolfram.com/mathematica/ref/RotationMatrix.html
Under Applications, they show how to generate the matrix form for a rotation in SO[N]. Then you take these and multiply them how they have it in the paper, order obviously matters.

I do:
Code: 
SO[n_] := Map[RotationMatrix[\[Theta], #] &,    Subsets[Table[UnitVector[n, i], {i, n}], {2}]];
SO5MAP = Map[ MatrixForm, SO[5]];
\[Theta][a_, b_] := Subscript[\[CapitalTheta], a, b]
R12 = SO5MAP[[1]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 2]};
R13 = SO5MAP[[2]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 3]};
R23 = SO5MAP[[5]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      2, 3]};
R14 = SO5MAP[[3]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 4]};
R15 = SO5MAP[[4]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      1, 5]};
R24 = SO5MAP[[6]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      2, 4]};
R25 = SO5MAP[[7]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      2, 5]};
R34 = SO5MAP[[8]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      3, 4]};
R35 = SO5MAP[[9]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      3, 5]};
R45 = SO5MAP[[10]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][      4, 5]};
ROT = R45.(R35.(R34.(R25.(R24.(R15.(R14.(R23.(R13.R12))))))));
ROT /. {Cos[Subscript[\[CapitalTheta], a_, b_]] -> Subscript[c, a, b],     Sin[Subscript[\[CapitalTheta], a_, b_]] -> Subscript[s, a, b]} //   Simplify // MatrixForm

The output looks like

[tex]
\left(
\begin{array}{ccccc}
c_{1,2} c_{1,3} c_{1,4} c_{1,5} & -c_{1,3} c_{1,4} c_{1,5} s_{1,2} & -c_{1,4} c_{1,5} s_{1,3} & -c_{1,5} s_{1,4} & -s_{1,5} \\
c_{2,3} c_{2,4} c_{2,5} s_{1,2}-c_{1,2} \left(c_{2,4} c_{2,5} s_{1,3} s_{2,3}+c_{1,3} \left(c_{2,5} s_{1,4} s_{2,4}+c_{1,4} s_{1,5} s_{2,5}\right)\right) & c_{1,2} c_{2,3} c_{2,4} c_{2,5}+s_{1,2} \left(c_{2,4} c_{2,5} s_{1,3} s_{2,3}+c_{1,3} \left(c_{2,5} s_{1,4} s_{2,4}+c_{1,4} s_{1,5} s_{2,5}\right)\right) & -c_{1,3} c_{2,4} c_{2,5} s_{2,3}+s_{1,3} \left(c_{2,5} s_{1,4} s_{2,4}+c_{1,4} s_{1,5} s_{2,5}\right) & -c_{1,4} c_{2,5} s_{2,4}+s_{1,4} s_{1,5} s_{2,5} & -c_{1,5} s_{2,5} \\
c_{3,5} \left(s_{1,2} \left(c_{3,4} s_{2,3}-c_{2,3} s_{2,4} s_{3,4}\right)+c_{1,2} \left(c_{2,3} c_{3,4} s_{1,3}+\left(-c_{1,3} c_{2,4} s_{1,4}+s_{1,3} s_{2,3} s_{2,4}\right) s_{3,4}\right)\right)-\left(c_{1,2} c_{1,3} c_{1,4} c_{2,5} s_{1,5}+\left(c_{2,3} c_{2,4} s_{1,2}-c_{1,2} \left(c_{2,4} s_{1,3} s_{2,3}+c_{1,3} s_{1,4} s_{2,4}\right)\right) s_{2,5}\right) s_{3,5} & c_{3,5} \left(c_{1,2} c_{3,4} s_{2,3}+s_{1,2} \left(c_{1,3} c_{2,4} s_{1,4}-s_{1,3} s_{2,3} s_{2,4}\right) s_{3,4}-c_{2,3} \left(c_{3,4} s_{1,2} s_{1,3}+c_{1,2} s_{2,4} s_{3,4}\right)\right)-\left(c_{2,4} \left(c_{1,2} c_{2,3}+s_{1,2} s_{1,3} s_{2,3}\right) s_{2,5}+c_{1,3} s_{1,2} \left(-c_{1,4} c_{2,5} s_{1,5}+s_{1,4} s_{2,4} s_{2,5}\right)\right) s_{3,5} & s_{1,3} \left(c_{2,4} c_{3,5} s_{1,4} s_{3,4}+\left(c_{1,4} c_{2,5} s_{1,5}-s_{1,4} s_{2,4} s_{2,5}\right) s_{3,5}\right)+c_{1,3} \left(c_{2,3} c_{3,4} c_{3,5}+s_{2,3} \left(c_{3,5} s_{2,4} s_{3,4}+c_{2,4} s_{2,5} s_{3,5}\right)\right) & c_{2,5} s_{1,4} s_{1,5} s_{3,5}+c_{1,4} \left(-c_{2,4} c_{3,5} s_{3,4}+s_{2,4} s_{2,5} s_{3,5}\right) & -c_{1,5} c_{2,5} s_{3,5} \\
c_{4,5} \left(s_{1,2} \left(c_{2,3} c_{3,4} s_{2,4}+s_{2,3} s_{3,4}\right)+c_{1,2} \left(c_{1,3} c_{2,4} c_{3,4} s_{1,4}+s_{1,3} \left(-c_{3,4} s_{2,3} s_{2,4}+c_{2,3} s_{3,4}\right)\right)\right)-\left(s_{1,2} \left(c_{3,4} s_{2,3} s_{3,5}+c_{2,3} \left(c_{2,4} c_{3,5} s_{2,5}-s_{2,4} s_{3,4} s_{3,5}\right)\right)+c_{1,2} \left(s_{1,3} \left(-c_{2,4} c_{3,5} s_{2,3} s_{2,5}+\left(c_{2,3} c_{3,4}+s_{2,3} s_{2,4} s_{3,4}\right) s_{3,5}\right)+c_{1,3} \left(c_{1,4} c_{2,5} c_{3,5} s_{1,5}-s_{1,4} \left(c_{3,5} s_{2,4} s_{2,5}+c_{2,4} s_{3,4} s_{3,5}\right)\right)\right)\right) s_{4,5} & c_{4,5} \left(c_{3,4} \left(-c_{1,3} c_{2,4} s_{1,2} s_{1,4}+\left(c_{1,2} c_{2,3}+s_{1,2} s_{1,3} s_{2,3}\right) s_{2,4}\right)+\left(-c_{2,3} s_{1,2} s_{1,3}+c_{1,2} s_{2,3}\right) s_{3,4}\right)-\left(c_{3,5} \left(c_{2,4} \left(c_{1,2} c_{2,3}+s_{1,2} s_{1,3} s_{2,3}\right) s_{2,5}+c_{1,3} s_{1,2} \left(-c_{1,4} c_{2,5} s_{1,5}+s_{1,4} s_{2,4} s_{2,5}\right)\right)+\left(c_{1,2} c_{3,4} s_{2,3}+s_{1,2} \left(c_{1,3} c_{2,4} s_{1,4}-s_{1,3} s_{2,3} s_{2,4}\right) s_{3,4}-c_{2,3} \left(c_{3,4} s_{1,2} s_{1,3}+c_{1,2} s_{2,4} s_{3,4}\right)\right) s_{3,5}\right) s_{4,5} & c_{4,5} \left(-c_{3,4} \left(c_{2,4} s_{1,3} s_{1,4}+c_{1,3} s_{2,3} s_{2,4}\right)+c_{1,3} c_{2,3} s_{3,4}\right)-\left(c_{3,5} \left(-c_{1,4} c_{2,5} s_{1,3} s_{1,5}+\left(-c_{1,3} c_{2,4} s_{2,3}+s_{1,3} s_{1,4} s_{2,4}\right) s_{2,5}\right)+\left(c_{2,4} s_{1,3} s_{1,4} s_{3,4}+c_{1,3} \left(c_{2,3} c_{3,4}+s_{2,3} s_{2,4} s_{3,4}\right)\right) s_{3,5}\right) s_{4,5} & c_{2,5} c_{3,5} s_{1,4} s_{1,5} s_{4,5}+c_{1,4} \left(c_{3,5} s_{2,4} s_{2,5} s_{4,5}+c_{2,4} \left(c_{3,4} c_{4,5}+s_{3,4} s_{3,5} s_{4,5}\right)\right) & -c_{1,5} c_{2,5} c_{3,5} s_{4,5} \\
c_{4,5} \left(s_{1,2} \left(c_{3,4} s_{2,3} s_{3,5}+c_{2,3} \left(c_{2,4} c_{3,5} s_{2,5}-s_{2,4} s_{3,4} s_{3,5}\right)\right)+c_{1,2} \left(s_{1,3} \left(-c_{2,4} c_{3,5} s_{2,3} s_{2,5}+\left(c_{2,3} c_{3,4}+s_{2,3} s_{2,4} s_{3,4}\right) s_{3,5}\right)+c_{1,3} \left(c_{1,4} c_{2,5} c_{3,5} s_{1,5}-s_{1,4} \left(c_{3,5} s_{2,4} s_{2,5}+c_{2,4} s_{3,4} s_{3,5}\right)\right)\right)\right)+\left(s_{1,2} \left(c_{2,3} c_{3,4} s_{2,4}+s_{2,3} s_{3,4}\right)+c_{1,2} \left(c_{1,3} c_{2,4} c_{3,4} s_{1,4}+s_{1,3} \left(-c_{3,4} s_{2,3} s_{2,4}+c_{2,3} s_{3,4}\right)\right)\right) s_{4,5} & c_{4,5} \left(c_{3,5} \left(c_{2,4} \left(c_{1,2} c_{2,3}+s_{1,2} s_{1,3} s_{2,3}\right) s_{2,5}+c_{1,3} s_{1,2} \left(-c_{1,4} c_{2,5} s_{1,5}+s_{1,4} s_{2,4} s_{2,5}\right)\right)+\left(c_{1,2} c_{3,4} s_{2,3}+s_{1,2} \left(c_{1,3} c_{2,4} s_{1,4}-s_{1,3} s_{2,3} s_{2,4}\right) s_{3,4}-c_{2,3} \left(c_{3,4} s_{1,2} s_{1,3}+c_{1,2} s_{2,4} s_{3,4}\right)\right) s_{3,5}\right)+\left(c_{3,4} \left(-c_{1,3} c_{2,4} s_{1,2} s_{1,4}+\left(c_{1,2} c_{2,3}+s_{1,2} s_{1,3} s_{2,3}\right) s_{2,4}\right)+\left(-c_{2,3} s_{1,2} s_{1,3}+c_{1,2} s_{2,3}\right) s_{3,4}\right) s_{4,5} & c_{4,5} \left(c_{3,5} \left(-c_{1,4} c_{2,5} s_{1,3} s_{1,5}+\left(-c_{1,3} c_{2,4} s_{2,3}+s_{1,3} s_{1,4} s_{2,4}\right) s_{2,5}\right)+\left(c_{2,4} s_{1,3} s_{1,4} s_{3,4}+c_{1,3} \left(c_{2,3} c_{3,4}+s_{2,3} s_{2,4} s_{3,4}\right)\right) s_{3,5}\right)+\left(-c_{3,4} \left(c_{2,4} s_{1,3} s_{1,4}+c_{1,3} s_{2,3} s_{2,4}\right)+c_{1,3} c_{2,3} s_{3,4}\right) s_{4,5} & -c_{4,5} \left(c_{2,5} c_{3,5} s_{1,4} s_{1,5}+c_{1,4} \left(c_{3,5} s_{2,4} s_{2,5}+c_{2,4} s_{3,4} s_{3,5}\right)\right)+c_{1,4} c_{2,4} c_{3,4} s_{4,5} & c_{1,5} c_{2,5} c_{3,5} c_{4,5}
\end{array}
\right)
[/tex]

again, I am not sure where there is a sign difference.
 
  • #3
Thanks a loooooooooooot ! :)
 
  • #4
Ah actually what you need to do is take the transpose of each of those matrices, then you get what's right:

R12 = Transpose[SO5MAP[[1]]] /. {MatrixForm[x_] :> x} /. { \[Theta] -> \[Theta][1, 2]};

The transpose is what you want, and then apply those in order. The signs will be correct then.
 
  • #5
Thanks again. I realized that too. So did not check recently.
 

FAQ: How Do You Write the Neutrino Mixing Matrix Us from Equations 3 and 4?

What is a neutrino mixing matrix?

A neutrino mixing matrix is a mathematical tool used to describe the mixing of different types of neutrinos. It is a complex matrix that relates the flavor states of neutrinos to their mass states.

Why is the neutrino mixing matrix important?

The neutrino mixing matrix helps us understand the properties and behavior of neutrinos, which are elusive particles that have important implications for our understanding of the universe. It also plays a crucial role in the study of neutrino oscillations, which can provide insight into the nature of neutrino mass.

How is the neutrino mixing matrix written?

The neutrino mixing matrix is typically written in the form of a 3x3 matrix, with each element representing the probability amplitude for a neutrino of a specific flavor to be measured as a neutrino of a different flavor. The elements of the matrix are complex numbers and are often denoted by symbols such as θ and δ.

What is the physical meaning of the elements in the neutrino mixing matrix?

The elements in the neutrino mixing matrix represent the probability amplitudes for neutrinos to change between different flavor states. This means that as a neutrino travels through space, it can change from one type (or flavor) to another, and the elements of the matrix tell us the likelihood of this happening.

How is the neutrino mixing matrix determined?

The values of the elements in the neutrino mixing matrix are determined through experiments that measure neutrino oscillations. By studying the patterns of neutrino flavor changes, scientists can calculate the values of the matrix elements and gain a better understanding of the properties of neutrinos.

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