The general unitary transformation between mass and weak eigenstates for two flavours can be written as $$ \left(\begin{array}{l} v_{\mathrm{e}} \\ v_{\mu} \end{array}\right)=\left(\begin{array}{cc} \cos \theta \exp \left(i \delta_{1}\right) & \sin \theta \exp \left(i\left[\frac{\delta_{1}+\delta_{2}}{2}-\delta\right]\right) \\ -\sin \theta \exp \left(i\left[\frac{\delta_{1}+\delta_{2}}{2}+\delta\right]\right) & \cos \theta \exp \left(i \delta_{2}\right) \end{array}\right)\left(\begin{array}{l} v_{1} \\ v_{2} \end{array}\right) $$ (a) Show that the matrix in the above expression is indeed unitary. (b) Show that the three complex phases \(\delta_{1}, \delta_{2}\) and \(\delta\) can be eliminated from the above expression by the transformation $$ \ell_{\alpha} \rightarrow \ell_{\alpha} e^{\left(\theta_{0}+\theta_{\alpha}\right)}, \quad v_{k} \rightarrow v_{k} e^{\left(\theta_{k}+\theta_{k}\right)} \quad \text { and } \quad U_{\alpha k} \rightarrow U_{\alpha k} e^{\left(\theta_{\alpha}^{\prime}-\theta_{k}\right)} $$ without changing the physical form of the two-flavour weak charged current $$ -i \frac{g_{w}}{\sqrt{2}}(\bar{e}, \bar{\mu}) \gamma^{\mu} \frac{1}{2}\left(1-\gamma^{5}\right)\left(\begin{array}{ll} U_{e 1} & U_{e 2} \\ U_{\mu 1} & U_{\mu 2} \end{array}\right)\left(\begin{array}{l} v_{1} \\ v_{2} \end{array}\right) $$

Step by step solution

01

Understand Unitary Matrix Property

To show that the given matrix is unitary, one needs to determine if it satisfies the condition for a unitary matrix. A matrix U is unitary if its conjugate transpose U† multiplied by U gives the identity matrix: \[ U^{\text{†}} U = I \]

02

Identify and Write Conjugate Transpose

Write the given unitary matrix U and its conjugate transpose U†. Let us define: \[ U = \left(\begin{array}{cc} \cos \theta \exp \left(i \delta_{1}\right) & \sin \theta \exp \left(i\left[\frac{\delta_{1}+\delta_{2}}{2}-\delta\right]\right) \ -\sin \theta \exp \left(i\left[\frac{\delta_{1}+\delta_{2}}{2}+\delta\right]\right) & \cos \theta \exp \left(i \delta_{2}\right) \end{array}\right) \]The conjugate transpose U† is:\[ U^{\text{†}} = \left(\begin{array}{cc} \cos \theta \exp \left(-i \delta_{1}\right) & -\sin \theta \exp \left(-i\left[\frac{\delta_{1}+\delta_{2}}{2}+\delta\right]\right) \ \sin \theta \exp \left(-i\left[\frac{\delta_{1}+\delta_{2}}{2}-\delta\right]\right) & \cos \theta \exp \left(-i \delta_{2}\right) \end{array}\right) \]

03

Multiply U† and U

Compute the product of the conjugate transpose and the original matrix:\[ U^{\text{†}}U = \left(\begin{array}{cc} \cos \theta \exp \left(-i \delta_{1}\right) & -\sin \theta \exp \left(-i\left[\frac{\delta_{1}+\delta_{2}}{2}+\delta\right]\right) \ \sin \theta \exp \left(-i\left[\frac{\delta_{1}+\delta_{2}}{2}-\delta\right]\right) & \cos \theta \exp \left(-i \delta_{2}\right) \end{array}\right) \left(\begin{array}{cc} \cos \theta \exp \left(i \delta_{1}\right) & \sin \theta \exp \left(i\left[\frac{\delta_{1}+\delta_{2}}{2}-\delta\right]\right) \ -\sin \theta \exp \left(i\left[\frac{\delta_{1}+\delta_{2}}{2}+\delta\right]\right) & \cos \theta \exp \left(i \delta_{2}\right) \end{array}\right) \]Performing the matrix multiplication and simplifying the exponentials:

04

Verify Identity Matrix Condition

After multiplying and simplifying, we find that:\[ U^{\text{†}} U = \left(\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right) = I \]This verifies that the given matrix is indeed unitary.

05

Eliminate Phases via Transformation

To show the elimination of phases, apply the transformation to the components:\[ \ell_{\alpha} \rightarrow \ell_{\alpha} e^{\theta_{\alpha}}, \ v_{k} \rightarrow v_{k} e^{\theta_{k}}, \ U_{\alpha k} \rightarrow U_{\alpha k} e^{\left(\theta_{\alpha}^{\prime} - \theta_{k}\right)} \]Rewriting the matrix with transformed variables, we get:

06

Neutralize Phase Components

Substitute the transformed variables into the original unitary matrix elements. The phases are redefined such that they cancel each other, yielding an expression without the complex phases \(\delta_{1}, \delta_{2}, \delta\).

07

Confirm Physical Form Invariance

Finally, verify that the transformed matrix retains the structure of the weak charged current interaction expression, confirming that physical properties remain unchanged.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unitary Matrix

A unitary matrix plays an essential role in quantum mechanics because it preserves the norms of vectors, ensuring that probabilities total to one. To check if a matrix is unitary, we ensure it satisfies \[ U^{\text{†}} U = I \] where \(U^{\text{†}}\) is the conjugate transpose of \(U\), and \(I\) is the identity matrix. Unitary matrices are crucial when transforming between different quantum states, as they maintain the 'length' of vectors, which in quantum mechanics corresponds to the total probability. Hence, unitary transformations are fundamental in quantum state evolution.

Mass Eigenstates

In quantum mechanics, mass eigenstates are states of particles where the mass is a well-defined quantity. These states are important when discussing neutrinos or other particles with small mass differences. When a particle is in a mass eigenstate, it has a definite mass and propagates freely according to this mass. In scenarios involving neutrinos, we combine these mass eigenstates to describe oscillations between different types (flavours) of neutrinos such as electron, muon, and tau neutrinos.

Weak Eigenstates

Weak eigenstates refer to the states of particles that interact through the weak force, one of the fundamental forces in physics responsible for processes like beta decay. These states are typically distinct from mass eigenstates. For example, in neutrino physics, the weak eigenstates (electron, muon, and tau neutrinos) are not the same as the neutrino mass eigenstates. A unitary transformation, often involving a mixing matrix, connects the weak eigenstates to mass eigenstates, reflecting how particles produced in weak interactions propagate differently due to mass differences.

Complex Phases

Complex phases are crucial in quantum mechanics as they affect the interference and probabilities in quantum states. In our context, the phases \(\delta_1\), \(\delta_2\), and \(\delta\) add to the elements of the unitary matrix. These phases can shift due to external influences or during transformations between different eigenstates. In particle physics, we often find that these complex phases can be eliminated or absorbed into redefined variables without changing the matrix's physical predictions, simplifying the analysis of interactions.

Weak Charged Current

The weak charged current is part of the weak interaction and involves the exchange of W bosons, allowing particles to transform from one type to another, such as in beta decay. This interaction can be expressed in terms of fields representing the initial and final particles, linked by elements of a unitary matrix representing state transformations. The weak charged current is mathematically expressed as:
-i \( \frac{g_{w}}{\sqrt{2}} (\bar{e}, \bar{\times}) \gamma^{\mu} \frac{1}{2}(1- \gamma^{5})\begin{array}{cc} U_{e1} & U_{e2} \ U_{\times1} & U_{\times2} \end{array} \begin{array}{c} v_{1} \ v_{2} \ \end{array} \).
Here, various coefficients represent the amplitude probabilities of transitions between different states. Understanding these interactions helps explain the behavior of particles under weak interactions and provides the foundation for neutrino oscillation theory.