P&S Exercise 3.4 Majorana Fermions Derivative of ##\chi##

  • #1
diracsgrandgrandson
6
1
Homework Statement
Peskin and Schroeder Exercise 3.4 Majorana Fermions part b wants me to show that the variation of the action ##S## with respect to ##\chi^\dagger## gives the Majorana equation.
Relevant Equations
The action is given by: $$S = \int d^4 x \left[ \chi^\dagger i \sigma \cdot \partial \chi + \frac{im}{2} \left( \chi^T \sigma^2 \chi - \chi^\dagger \sigma^2 \chi^* \right) \right]$$

The Majorana equation is $$i \bar{\sigma} \cdot \partial \chi - im \sigma^2 \chi^* = 0$$
I am stuck at the final part where one is supposed to show that the derivative of the second term of the action gives the mass term in the Majorana equation. For $$\chi^T\sigma^2\chi = -(\chi^\dagger\sigma^2\chi^*)^*$$ we get $$\frac{\delta}{\delta\chi^\dagger}(\chi^\dagger\sigma^2\chi^*)^*$$ which is supposed to give $$\sigma^2\chi^*.$$ I don't see how. Suppose $$f(\chi) = \chi^*,$$ and now $$\frac{d}{d\chi}f(\chi) = \frac{d\chi^*}{d\chi}$$ which would be zero due to the field and its complex conjugate being zero.
 
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  • #2
diracsgrandgrandson said:
I am stuck at the final part where one is supposed to show that the derivative of the second term of the action gives the mass term in the Majorana equation.
You can get the Majorana equation by varying ##S## with respect to ##\chi_1^*## and ##\chi_2^*##.

Note that ##\chi^T \sigma^2 \chi## does not contain either ##\chi_1^*## or ##\chi_2^*##.
So, this expression in the Lagrangian will not contribute when doing the variation with respect to ##\chi_1^*## and ##\chi_2^*##.

Write out the expression ##\chi^\dagger \sigma^2 \chi^*## explicitly in terms of ##\chi_1^*## and ##\chi_2^*##. Then you can look at its variation with respect to ##\chi_1^*## and ##\chi_2^*##.
 
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  • #3
TSny said:
Write out the expression χ†σ2χ∗ explicitly in terms of χ1∗ and χ2∗. Then you can look at its variation with respect to χ1∗ and χ2∗.
Doesn't that give zero?

$$
\begin{align*}
\chi^\dagger \sigma^2 \chi^* &=
\begin{pmatrix}
\chi_1^* & \chi_2^*
\end{pmatrix}
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}
\begin{pmatrix}
\chi_1^* \\
\chi_2^*
\end{pmatrix} \\
&= \begin{pmatrix}
\chi_1^* & \chi_2^*
\end{pmatrix}
\begin{pmatrix}
- i \chi_2^* \\
i \chi_1^*
\end{pmatrix} \\
&= i (-\chi^*_1\chi^*_2+\chi^*_2\chi_1^*)
\end{align*}
$$

which gives zero for each derivative.
 
  • #4
According to the problem statement in the textbook, ##\chi_1## and ##\chi_2## are to be treated as anticommuting quantities (Grassmann numbers) with the following properties

##\chi_1 \chi_2## = -##\chi_2 \chi_1 \,\,## and ##\,\, (\chi_1 \chi_2)^* \equiv \chi_2^* \chi_1^* = - \chi_1^* \chi_2^*##
 
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  • #5
Ahh I see, that clarifies it, thank you. I find the problem is formulated in a very confusing way, at least for a beginner like me :)
 
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  • #6
Yes, it's a difficult subject. I'm also a beginner. I've been a beginner for years. :oldsmile:
 
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FAQ: P&S Exercise 3.4 Majorana Fermions Derivative of ##\chi##

What is the significance of Majorana fermions in condensed matter physics?

Majorana fermions are significant in condensed matter physics because they are quasiparticles that are their own antiparticles. They have potential applications in fault-tolerant quantum computing due to their non-abelian statistics, which can help in creating more stable qubits that are less susceptible to decoherence.

How do you derive the equation for the Majorana fermion field ##\chi##?

The derivation of the Majorana fermion field ##\chi## typically involves starting from the Dirac equation and imposing the Majorana condition, which requires that the field be equal to its own charge conjugate. This leads to a real-valued field operator that can be expressed in terms of creation and annihilation operators. The specific steps depend on the representation used and the context of the problem.

What are the mathematical properties of the Majorana fermion field ##\chi##?

The Majorana fermion field ##\chi## has the property that it is its own antiparticle, meaning ##\chi = \chi^{\dagger}##. This implies that the field is real, and it satisfies the Majorana condition. Additionally, Majorana fermions obey Fermi-Dirac statistics and their creation and annihilation operators follow specific anticommutation relations.

Why is the derivative of the Majorana fermion field ##\chi## important?

The derivative of the Majorana fermion field ##\chi## is important because it appears in the kinetic term of the Lagrangian for Majorana fermions. This term is crucial for understanding the dynamics of the field and how it propagates through space-time. The derivative also plays a role in the equations of motion derived from the Lagrangian.

Can you provide an example of calculating the derivative of the Majorana fermion field ##\chi##?

To calculate the derivative of the Majorana fermion field ##\chi##, one typically starts with the field expression in terms of creation and annihilation operators. For example, if ##\chi(x)## is expressed as ##\chi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}} \left( a(p) e^{-ipx} + a^{\dagger}(p) e^{ipx} \right)##, the space-time derivative ##\partial_\mu \chi(x)## can be computed by differentiating under the integral sign, leading to terms involving the momentum ##p_\mu##. The exact form will depend on the specific representation and normalization conventions used.

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