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## 物理代写|量子场论代写Quantum field theory代考|Gauge transformations

The potentials are not unique, they are determined only up to gauge transformations
$$A_\mu \rightarrow A_\mu+\partial_\mu \chi$$
with arbitrary functions $\chi(x)$, leaving the field strengths (1.73) invariant because the partial derivatives commute. A restriction of this gauge degree of freedom is obtained by specifying a gauge condition. Of particular interest is the Lorentz gauge
$$\partial_\mu A^\mu=0,$$
which constitutes a relativistic-invariant condition, valid in all inertial frames.

The charge and current densities $\rho$ and $\vec{j}$ can be summarized in terms of a 4-vector, the 4-current,
$$\left(j^\mu\right)=\left(j^0, \vec{j}\right)=(\rho, \vec{j}) .$$
The current fulfills a continuity equation
$$\frac{\partial \rho}{\partial t}+\nabla \cdot \vec{j}=0$$
which corresponds to current conservation. In 4-dimensional notation the continuity equation is a vanishing 4-divergence,
$$\partial_\mu j^\mu=0,$$
and obviously invariant under Lorentz transformations. The total charge
$$Q=\int \mathrm{d}^3 x \rho=\int \mathrm{d}^3 x j^0$$
is a Lorentz invariant and, as a consequence of the continuity equation, also a conserved quantity.

## 物理代写|量子场论代写Quantum field theory代考|Maxwell’s equations

With the help of the field-strength tensor the set of Maxwell’s equations can be formulated in a covariant way as follows,
\begin{aligned} \partial_\mu F^{\mu \nu} &=j^\nu & \text { inhomogeneous equations } \ \epsilon^{\mu \nu \rho \sigma} \partial_\nu F_{\rho \sigma} &=0 & & \text { homogeneous equations. } \end{aligned}

The homogeneous equations are solved by expressing the field strengths in terms of the potentials. Insertion of Eq. (1.73) into the inhomogeneous equations yields coupled equations for the components of the 4-potential,
$$\partial_\mu\left(\partial^\mu A^\nu-\partial^\nu A^\mu\right)=\square A^\nu-\partial^\nu\left(\partial_\mu A^\mu\right)=j^\nu .$$
Choosing the Lorentz gauge for $A^\mu$ simplifies these equations considerably and one obtains decoupled equations,
$$A^\nu=j^\nu .$$
This set of inhomogeneous wave equations with the constraint (1.77) can be considered equivalent to the set of Maxwell’s equations.

# 量子场论代考

## 物理代写|量子场论代写Quantum field theory代考|Gauge transformations

$$A_\mu \rightarrow A_\mu+\partial_\mu \chi$$

$$\partial_\mu A^\mu=0,$$

$$\left(j^\mu\right)=\left(j^0, \vec{j}\right)=(\rho, \vec{j}) .$$

$$\frac{\partial \rho}{\partial t}+\nabla \cdot \vec{j}=0$$

$$\partial_\mu j^\mu=0,$$

$$Q=\int \mathrm{d}^3 x \rho=\int \mathrm{d}^3 x j^0$$

## 物理代写|量子场论代写Quantum field theory代考|Maxwell’s equations

$\partial_\mu F^{\mu \nu}=j^\nu \quad$ inhomogeneous equations $\epsilon^{\mu \nu \rho \sigma} \partial_\nu F_{\rho \sigma}=0 \quad$ homogeneous equations.

$$\partial_\mu\left(\partial^\mu A^\nu-\partial^\nu A^\mu\right)=\square A^\nu-\partial^\nu\left(\partial_\mu A^\mu\right)=j^\nu .$$

$$A^\nu=j^\nu .$$

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