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# 物理代写|电动力学作业代写Electrodynamics代考|Principle of Least Action in Field Theory

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## 物理代写|电动力学作业代写Electrodynamics代考|Hypersurfaces in Minkowski Space-Time

In this section we introduce a few notions regarding hypersurfaces in four dimensions, which will prove useful later on.

Parameterizations of hypersurfaces. By definition a hypersurface $\Gamma$ in a fourdimensional Minkowski space-time is a subset – more precisely a submanifold of $\mathbb{R}^{4}$, of dimension three. In parametric form a hypersurface is described by four functions of three parameters
$$y^{\mu}(\boldsymbol{\lambda})$$
where $\boldsymbol{\lambda}$ denotes the triple $\left{\lambda^{a}\right}$, with $a=1,2,3$. Alternatively one may represent a hypersurface $\Gamma$ in implicit form in terms of a single scalar function $f(x)$, through the relation
$$x^{\mu} \in \Gamma \quad \Leftrightarrow \quad f(x)=0 .$$
We can step from one representation to the other by inverting, for example, the spatial coordinates $\mathbf{y}(\boldsymbol{\lambda})$ of (3.11) to determine the parameters $\boldsymbol{\lambda}$ as functions of the spatial coordinates $\mathbf{x}$, i.e. by inverting the functions $\mathbf{x}=\mathbf{y}(\boldsymbol{\lambda}) \rightarrow \boldsymbol{\lambda}(\mathbf{x})$, and by setting then
$$f(x)=f\left(x^{0}, \mathbf{x}\right)=x^{0}-y^{0}(\boldsymbol{\lambda}(\mathbf{x}))$$
This function satisfies, in fact, the identity
$$f(y(\boldsymbol{\lambda}))=0$$

## 物理代写|电动力学作业代写Electrodynamics代考|Relativistic Invariance

So far we have made no assumptions about the invariance properties of the considered field theory. In this section we analyze some important aspects of the principle of least action, in the particular case of a relativistic field theory.

Principle of least action and manifest covariance. As explained in Chap. 1 , in a relativistic field theory we expect the equations of motion to be manifestly covariant. In the variational framework manifest covariance is, actually, realized in a natural way, if the fields are organized into tensor multiplets, and the Lagrangian $\mathcal{L}$ is a fourscalar. In fact, in this case the Euler-Lagrange equations (3.8) are automatically manifestly covariant. In a relativistic theory we require, therefore, the Lagrangian to be invariant under the Poincaré transformations $x^{\prime}=\Lambda x+a$, more precisely we demand the equality $$\mathcal{L}\left(\varphi^{\prime}\left(x^{\prime}\right), \partial^{\prime} \varphi^{\prime}\left(x^{\prime}\right)\right)=\mathcal{L}(\varphi(x), \partial \varphi(x))$$
to be fulfilled for all $\Lambda_{\nu}^{\mu} \in O(1,3)$, and for all $a^{\mu}$. If this equality holds, we can ask whether the action (3.6) is a scalar, as postulated in the introduction of this chapter. Actually, from the expression (3.6) there emerges an obvious obstacle to the invariance of $I$ : whereas the measure of the integral is invariant,
$$d^{4} x^{\prime}=|\operatorname{det} \Lambda| d^{4} x=d^{4} x$$
the integration domain is not, since the time variable is integrated over a finite interval. Nevertheless, it is not difficult to overcome this obstruction. In fact, it is sufficient to substitute in (3.6) the space-like hyperplanes $t=t_{1}$ and $t=t_{2}$, which bound the four-dimensional integration domain, with two generic infinitely extended and non-intersecting space-like hypersurfaces $\Gamma_{1}$ and $\Gamma_{2}$. In fact, a hyperplane at constant time is a particular space-like hypersurface, which after a Poincaré transformation is no longer a hyperplane at constant time, while remaining a space-like hypersurface. In order to overcome the above obstacle we, therefore, replace the expression (3.6) with the generalized action
$$I[\varphi]=\int_{\Gamma_{1}}^{\Gamma_{2}} \mathcal{L}(\varphi(x), \partial \varphi(x)) d^{4} x$$
which, thanks to the relations (3.25) and (3.26), is indeed a relativistic invariant,
$$I^{\prime}=\int_{\Gamma_{1}^{\prime}}^{\Gamma_{2}^{\prime}} \mathcal{L}\left(\varphi^{\prime}\left(x^{\prime}\right), \partial^{\prime} \varphi^{\prime}\left(x^{\prime}\right)\right) d^{4} x^{\prime}=\int_{\Gamma_{1}}^{\Gamma_{2}} \mathcal{L}(\varphi(x), \partial \varphi(x)) d^{4} x=I$$

## 物理代写|电动力学作业代写ELECTRODYNAMICS代考|Lagrangian for Maxwell’s Equations

In this section we illustrate the variational method, retrieving the equations of the electromagnetic field through the principle of least action. In principle, our aim is thus to interpret equations (2.21) and (2.22) as the Euler-Lagrange equations relative to an appropriate Lagrangian. The first question to be addressed is the choice of the Lagrangian fields $\varphi_{r}$. Since equations (2.21) and (2.22) correspond altogether to eight equations, a priori we should hence introduce as many Lagrangian fields, namely eight. The natural choice $\varphi_{r}=F^{\mu \nu}$, which, by the way, would have the advantage of introducing only observable fields, is therefore precluded, for the Maxwell tensor corresponds not to eight, but only to six independent fields, i.e. E and $\mathbf{B}$, see in particular Problem 3.9. This strategy must hence be abandoned, and we must search for an alternative one. Such an alternative strategy consists in proceeding as anticipated in Sect. 2.2.4: we first solve the Bianchi identity in terms of a vector potential $A_{\mu}$, setting
$$F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$$
and then choose as Lagrangian fields the four components of this potential
$$\varphi_{r}=A_{\mu}$$
with the identification $r=\mu=0,1,2,3$. According to this approach the principle of least action should give rise to the Maxwell equation
$$\partial_{\mu} F^{\mu \nu}-j^{\nu}=0$$

## 物理代写|电动力学作业代写ELECTRODYNAMICS代考|HYPERSURFACES IN MINKOWSKI SPACE-TIME

$$y^{\mu}(\boldsymbol{\lambda})$$

Xμ∈Γ⇔F(X)=0.

$$f(x)=f\left(x^{0}, \mathbf{x}\right)=x^{0}-y^{0}(\boldsymbol{\lambda}(\mathbf{x}))$$

$$f(y(\boldsymbol{\lambda}))=0$$

## 物理代写|电动力学作业代写ELECTRODYNAMICS代考|RELATIVISTIC INVARIANCE

d4X′=|这⁡Λ|d4X=d4X

$$I[\varphi]=\int_{\Gamma_{1}}^{\Gamma_{2}} \mathcal{L}(\varphi(x), \partial \varphi(x)) d^{4} x$$
which, thanks to the relations (3.25) and (3.26), is indeed a relativistic invariant,
$$I^{\prime}=\int_{\Gamma_{1}^{\prime}}^{\Gamma_{2}^{\prime}} \mathcal{L}\left(\varphi^{\prime}\left(x^{\prime}\right), \partial^{\prime} \varphi^{\prime}\left(x^{\prime}\right)\right) d^{4} x^{\prime}=\int_{\Gamma_{1}}^{\Gamma_{2}} \mathcal{L}(\varphi(x), \partial \varphi(x)) d^{4} x=I$$

## 物理代写|电动力学作业代写ELECTRODYNAMICS代考|LAGRANGIAN FOR MAXWELL’S EQUATIONS

$$F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$$

$$\varphi_{r}=A_{\mu}$$

$$\partial_{\mu} F^{\mu \nu}-j^{\nu}=0$$

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