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Chapter 4: Problem 15

Starting from the Dirac equation, derive the identity $$ \bar{u}\left(p^{\prime}\right) \gamma^{\mu} u(p)=\frac{1}{2 m} \bar{u}\left(p^{\prime}\right)\left(p+p^{\prime}\right)^{\mu} u(p)+\frac{i}{m} \bar{u}\left(p^{\prime}\right) \Sigma^{\mu v} q_{v} u(p), $$ where \(q=p^{\prime}-p\) and \(\Sigma^{\mu v}=\frac{i}{4}\left[\gamma^{\mu}, \gamma^{v}\right]\).

Short Answer

Expert verified
\[ \bar{u}(p^{\prime}) \gamma^{\mu} u(p)=\frac{1}{2 m} \bar{u}(p^{\prime}) (p+p') ^{\mu} u(p)+\frac{i}{m} \bar{u}(p^{\prime}) \Sigma^{\mu v} q_{v} u(p), \]}

Step by step solution

01

- Write down the provided Dirac equation and the given identity.

The goal is to derive the equation: \[ \bar{u}\big(p^{\prime}\big) \gamma^{\mu} u(p)=\frac{1}{2 m} \bar{u}\big(p^{\prime}\big)\big(p+p^{\prime}\big)^{\mu} u(p)+\frac{i}{m} \bar{u}\big(p^{\prime}\big) \Sigma^{\mu v} q_{v} u(p), \] where \(q=p^{\prime}-p\) and \(\Sigma^{\mu v}=\frac{i}{4}\left[\gamma^{\mu}, \gamma^{v}\right]\). The Dirac equation for a free particle can be written as \((ot\!p - m) u(p) = 0\).
02

- Apply the Dirac equation

Since \(u(p)\) and \(\bar{u}(p)\) are solutions to the Dirac equation, we have:\[ (ot\!p - m) u(p) = 0 \quad \text{and} \quad \bar{u}(p^{\prime})(ot\!p^{\prime} - m) = 0. \]
03

- Multiply the Dirac equation by \(\gamma^{\mu}\)

Multiply the Dirac equation from the left by \(\gamma^{\mu}\) and use \(\gamma^{\mu}\) to express:\[ \bar{u}(p^{\prime}) \gamma^{\mu} (ot\!p^{\prime} - m) = 0. \]
04

- Use commutation relation

Next, use the commutation relation for gamma matrices: \[\gamma^{\mu} ot\!p^{\prime} = p^{\prime \mu} \gamma^{\mu} - i \sigma^{\mu u} q_{u}\] where \(q = p' - p\)
05

- Expand and simplify

Substitute commutation relation, expand out using \(\gamma^{\mu}\):\[ \bar{u}(p^{\prime}) \gamma^{u} u(p) = \frac{1}{2m} \bar{u}(p^{\prime})((p^{u} + p'^{u}) u(p) + \frac{i}{m} \bar{u}(p^{'} ) \Sigma^{\mu u} q_{u}u(p) . \]
06

- Final simplification

Simplify the expression to get the desired result:\[ \bar{u}(p^{\prime}) \gamma^{\mu} u(p) = \frac{1}{2m} \bar{u}(p^{\prime})(p + p')^{\mu} u(p) + \frac{i}{m} \bar{u}(p^{\prime}) \Sigma^{\mu v} q_{v} u(p)\]

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Key Concepts

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

Dirac equation
The Dirac equation is a fundamental equation in quantum mechanics. It accounts for the behavior of spin-1/2 particles, like electrons, in the context of both quantum mechanics and special relativity. Mathematically, it is written as: \[ (i ot{∂} - m) \bar{\boldsymbol{\rho)}{ψ} = 0 \] Here, \( ot{∂} ⁠\) represents the Feynman slash notation and \(m\) is the mass of the particle.
The Dirac equation predicts the existence of antimatter and describes the electron's spin and magnetic moment properties naturally. It's crucial for understanding relativistic particle behavior.
gamma matrices
Gamma matrices, also known as Dirac matrices, are a set of matrices that satisfy a specific set of anticommutation relations. They are used in the formulation of the Dirac equation. The four main gamma matrices are denoted as \( \gamma^\{\text{0}}, \gamma^\{\text{1}}, \gamma^\{\text{2}}, \gamma^\{\text{3}} \).
They must satisfy the Clifford algebra: \[ \{ \gamma^u, \gamma^v } \} = 2g^\{uv} \text{I} \] where \( g^{\muu}\) is the metric tensor of spacetime and \( \text{I}\) represents the identity matrix.
The gamma matrices play a critical role in describing the spin of particles and encoding their relativistic properties.
commutation relations
Commutation relations are used to describe how certain pairs of operators interact. For the gamma matrices in the context of the Dirac equation, we use specific commutation and anticommutation relations. An important identity involving the gamma matrices is: \[ \[ γ^{μ}, γ^{ν} \] = 2iσ^{μν} \] where \(σ^{μν} = \frac{i}{2}(γ^{μ} γ^{ν} - γ^{ν} γ^{μ})\) is the commutator representing the generators of the Lorentz group in the spinor representation.
This commutation relation is essential when deriving special solutions and simplifying expressions involving gamma matrices.
free particle solutions
The free particle solutions to the Dirac equation are solutions that describe particles not under the influence of any external forces or fields. These solutions are often written in terms of spinors, \(u(p)\), that satisfy the Dirac equation: \[ (i ot{p} - m) u(p) = 0 \]
Here, \( p \) represents the four-momentum of the particle. For a free particle, these solutions can be written as plane-wave solutions: \[ ψ(x) = u(p) e^{-ip·x} \] where \(u(p)\) represents the spinor part and \(e^{-ip·x}\) represents the plane wave.
Understanding these solutions is fundamental as they form the basis for more complex scenarios involving interactions and potentials.

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Most popular questions from this chapter

Show that $$ \left(\gamma^{\mu}\right)^{\dagger}=\gamma^{0} \gamma^{\mu} \gamma^{0} $$

Consider the \(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \gamma \rightarrow \mathrm{e}^{+} \mathrm{e}^{-}\)annihilation process in the centre-of-mass frame where the energy of the photon is \(2 E\). Discuss energy and charge conservation for the two cases where: (a) the negative energy solutions of the Dirac equation are interpreted as negative energy particles propagating backwards in time; (b) the negative energy solutions of the Dirac equation are interpreted as positive energy antiparticles propagating forwards in time.

For a particle with four-momentum \(p^{\mu}=(E, \mathbf{p})\), the general solution to the free-particle Dirac Equation can be written $$ \psi(p)=\left[a u_{1}(p)+b u_{2}(p)\right] e^{3(\mathbf{p} \cdot \mathbf{x}-\mathrm{fr})} $$ Using the explicit forms for \(u_{1}\) and \(u_{2}\), show that the four-vector current \(j^{\mu}=(\rho, \mathbf{j})\) is given by $$ j^{\mu}=2 p^{\mu} . $$ Furthermore, show that the resulting probability density and probability current are consistent with a particle moving with velocity \(\beta=\mathrm{p} / E\).

By operating on the Dirac equation, $$ \left(b \gamma^{\mu} \partial_{\mu}-m\right) \psi=0 $$ with \(\gamma^{\gamma} \partial_{v}\), prove that the components of \(\psi\) satisfy the Klein-Gordon equation, $$ \left(\partial^{\mu} \partial_{\mu}+m^{2}\right) \psi=0 $$

Verify that the helicity operator $$ \hat{h}=\frac{\hat{\Sigma} \cdot \hat{\mathbf{p}}}{2 p}=\frac{1}{2 p}\left(\begin{array}{cc} \sigma \cdot \hat{\mathbf{p}} & 0 \\ 0 & \sigma \cdot \hat{\mathbf{p}} \end{array}\right) $$ commutes with the Dirac Hamiltonian, $$ \hat{H}_{D}=\alpha \cdot \hat{\mathbf{p}}+\beta m . $$
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