Chapter 2 Classical field theory
2.1 Lagrangian formalism
The Lagrangian formalism for classical mechanics admits a straightforward generalization to classical field theory. To illustrate this formalism, we shall work on the \(d\)-dimensional Minkowski spacetime \(M=(\bbR ^d,\eta )\) and consider a field
\(\seteqnumber{0}{2.}{0}\)\begin{flalign} \phi : M= \bbR ^d \longrightarrow \mathcal {Q}=\bbR ^k~,~~x\longmapsto \phi (x) \end{flalign} that takes values in a \(k\)-dimensional real vector space \(\mathcal {Q}=\bbR ^k\). Note that such \(\phi \) is a multicomponent field, i.e.
\(\seteqnumber{0}{2.}{1}\)\begin{flalign} \phi (x) = \begin{pmatrix} \phi _1(x)\\ \phi _2(x)\\\vdots \\\phi _k(x) \end {pmatrix}\in \bbR ^k\quad , \end{flalign} and we shall use the index notation \(\phi _a(x)\), for \(a=1,\dots , k\), to denote the individual components.
An action functional for the field \(\phi \) has the general form
\begin{equation} S[\phi ] = \int _{\bbR ^d} \LL (\phi ,\partial \phi ) \,\dd x \quad , \end{equation}
where the integral is over the volume element
\(\seteqnumber{0}{2.}{3}\)\begin{flalign} \dd x \,:=\,\dd t\,\dd x^1\cdots \dd x^{d-1} \end{flalign} of spacetime \(\bbR ^d\). The integrand \(\LL (\phi ,\partial \phi )\) is called the Lagrangian density and it is assumed to be a real-valued local function, i.e. its value \(\LL (\phi ,\partial \phi )(x)\) at the point \(x\in \bbR ^d\) is a function of \(\phi (x)\) and of its partial derivatives \(\partial _\mu \phi (x) := \frac {\partial \phi (x)}{\partial x^\mu }\) at the same point \(x\). We use square brackets \(S[\phi ]\) to denote the argument of the action in order to emphasize that \(S\) is a functional, i.e. a function on the set of functions/fields \(\phi :\bbR ^d\to \bbR ^k\). The classical dynamics of the field \(\phi \) is defined by extremizing the action
\(\seteqnumber{0}{2.}{4}\)\begin{flalign} \delta S[\phi ] = 0\quad . \end{flalign} With a similar calculation as in the case of Lagrangian mechanics, one derives from this the Euler-Lagrange equations
for all \(a=1,\dots ,k\). For your convenience, let me briefly repeat this calculation
\(\seteqnumber{0}{2.}{6}\)\begin{flalign} \nn \delta S[\phi ] &= \int _{\bbR ^d} \bigg ( \frac {\partial \LL (\phi ,\partial \phi )}{\partial \phi _a} \,\delta \phi _a + \frac {\partial \LL (\phi ,\partial \phi )}{\partial (\partial _\mu \phi _a)} \,\delta (\partial _\mu \phi _a) \bigg )\,\dd x\\ \nn &=\int _{\bbR ^d} \bigg ( \frac {\partial \LL (\phi ,\partial \phi )}{\partial \phi _a} \,\delta \phi _a + \frac {\partial \LL (\phi ,\partial \phi )}{\partial (\partial _\mu \phi _a)} \,\partial _\mu (\delta \phi _a) \bigg )\,\dd x\\ \nn &=\int _{\bbR ^d} \bigg ( \frac {\partial \LL (\phi ,\partial \phi )}{\partial \phi _a} \,\delta \phi _a - \partial _\mu \bigg (\frac {\partial \LL (\phi ,\partial \phi )}{\partial (\partial _\mu \phi _a)}\bigg ) \delta \phi _a \bigg )\,\dd x\\ &=\int _{\bbR ^d} \bigg ( \frac {\partial \LL (\phi ,\partial \phi )}{\partial \phi _a} - \partial _\mu \bigg (\frac {\partial \LL (\phi ,\partial \phi )}{\partial (\partial _\mu \phi _a)}\bigg ) \bigg )~\delta \phi _a \,\dd x =0\quad , \end{flalign} where step three uses integration by parts together with the usual hypothesis that the variation \(\delta \phi _a\) vanishes at the boundary/infinity.
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Example 2.1 (Real Klein-Gordon field). Consider a real scalar field \(\Phi (x)\in \bbR \) together with the quadratic action functional
\(\seteqnumber{0}{2.}{7}\)\begin{flalign} \label {eqn:KGaction} S_{\mathrm {KG}}[\Phi ] := \int _{\bbR ^d} \bigg (-\frac {1}{2} \,\eta ^{\mu \nu }\,\partial _\mu \Phi \,\partial _\nu \Phi - \frac {m^2}{2}\,\Phi ^2\bigg )\,\dd x = \int _{\bbR ^d} -\frac {1}{2}\,\Big (\partial ^\mu \Phi \,\partial _\mu \Phi +m^2\, \Phi ^2\Big )\,\dd x\quad , \end{flalign} where \(\eta ^{\mu \nu }\) denotes the inverse Minkowski metric and \(m\geq 0\) is a parameter that will be interpreted later as the mass of a particle. The first summand is called the kinetic term and the second summand is called the mass term. The corresponding Euler-Lagrange equation (2.6) reads as
\(\seteqnumber{0}{2.}{8}\)\begin{flalign} \label {eqn:KGequation} - \partial ^2\Phi + m^2\,\Phi \,:= -\eta ^{\mu \nu } \partial _\mu \partial _\nu \Phi + m^2\,\Phi =0 \end{flalign} and it is called the Klein-Gordon equation. Note that (2.9) is a linear partial differential equation on the Minkowski spacetime, which is a consequence of the fact that the action functional (2.8) is quadratic in the field. One can introduce nonlinearities by adding higher-order terms to the action functional, e.g.
\(\seteqnumber{0}{2.}{9}\)\begin{flalign} \label {eqn:KGactioninteractions} S_{\mathrm {KG}+\mathrm {int}}[\Phi ] := \int _{\bbR ^d} \bigg (-\frac {1}{2}\, \partial ^\mu \Phi \,\partial _\mu \Phi - \frac {m^2}{2}\,\Phi ^2 - V(\Phi )\bigg )\,\dd x \end{flalign} for a potential function \(V(\Phi )\) that is a polynomial of degree \(\geq 3\) in \(\Phi \). The resulting Euler-Lagrange equation then reads as
\(\seteqnumber{0}{2.}{10}\)\begin{flalign} - \partial ^2\Phi + m^2\,\Phi + V^\prime (\Phi ) = 0\quad , \end{flalign} which is a nonlinear partial differential equation because the derivative \(V^\prime (\Phi )\) is by our hypothesis a polynomial of degree \(\geq 2\) in \(\Phi \).
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Example 2.2 (Electromagnetic potential). Consider a covector field \(A_\mu (x)\) and define the antisymmetric \((0,2)\)-tensor field
\(\seteqnumber{0}{2.}{11}\)\begin{flalign} \label {eqn:fieldstrength} F_{\mu \nu } := \partial _\mu A_\nu - \partial _\nu A_{\mu }\quad . \end{flalign} The physical interpretation of \(A_\mu \) is that of the electromagnetic potential and \(F_{\mu \nu }\) is the so-called field strength tensor. As a direct consequence of its definition, \(F_{\mu \nu }\) satisfies the identity
\(\seteqnumber{0}{2.}{12}\)\begin{flalign} \label {eqn:Maxwell1} \partial _\mu F_{\nu \rho } + \partial _\nu F_{\rho \mu } + \partial _\rho F_{\mu \nu } = 0\quad . \end{flalign} Consider further the quadratic action functional
\(\seteqnumber{0}{2.}{13}\)\begin{flalign} S_{\mathrm {MW}}[A] := \int _{\bbR ^d} -\frac {1}{4}\,\eta ^{\mu \rho }\,\eta ^{\nu \sigma }\,F_{\mu \nu }\,F_{\rho \sigma }\,\dd x = \int _{\bbR ^d} -\frac {1}{4}\,F^{\mu \nu }\,F_{\mu \nu }\,\dd x \end{flalign} whose Euler-Lagrange equations (2.6) read as
\(\seteqnumber{0}{2.}{14}\)\begin{flalign} \label {eqn:Maxwell2} \partial _\mu F^{\mu \nu } = 0\quad . \end{flalign} In spacetime dimension \(d=4\), the system of partial differential equations (2.13) and (2.15) is equivalent to Maxwell’s equations in vacuum, namely \(\nabla \cdot \mathbf {B} =0\), \(\nabla \times \mathbf {E} = -\frac {\partial \mathbf {B}}{\partial t}\), \(\nabla \cdot \mathbf {E} =0\) and \(\nabla \times \mathbf {B} = \frac {\partial \mathbf {E}}{\partial t}\). This example will be studied in more depth later in Chapter 6.
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Example 2.3 (Complex-valued fields). The Lagrangian formalism for field theory also works for complex-valued fields \(\phi : \bbR ^d\to \bbC ^k\). Indeed, decomposing each component \(\phi _a(x) =: \rho _a(x) + \ii \,\chi _a(x)\) into its real part \(\rho _a\) and imaginary part \(\chi _a\) allows us to regard \(\phi \) as a \(2k\)-dimensional real field
\(\seteqnumber{0}{2.}{15}\)\begin{flalign} \tilde {\phi } : \bbR ^d\longrightarrow \bbR ^{2k}~,~~x\longmapsto \tilde {\phi }(x) = \begin{pmatrix} \rho _1(x)\\\vdots \\\rho _k(x)\\ \chi _1(x)\\ \vdots \\\chi _k(x) \end {pmatrix}\quad . \end{flalign} The formulas from this section then apply to \(\tilde {\phi }\).
There is an equivalent, but simpler way to deal with complex-valued fields, which we are going to illustrate for a complex scalar field \(\Phi (x)\in \bbC \) whose complex conjugate we denote by \(\Phi ^\ast (x)\in \bbC \). Since the action functional is by definition real-valued, one has to combine \(\Phi (x)\) and \(\Phi ^\ast (x)\) in the Lagrangian density in order to obtain a real number. For example, the action functional
\(\seteqnumber{0}{2.}{16}\)\begin{flalign} \label {eqn:KGactioncomplexinteraction} S_{\mathrm {KG}_\bbC +\mathrm {int}}[\Phi ,\Phi ^\ast ] := \int _{\bbR ^d}\bigg (-\partial ^\mu \Phi ^\ast \,\partial _\mu \Phi - m^2\,\Phi ^\ast \,\Phi - V(\Phi ^\ast \,\Phi )\bigg )\,\dd x \end{flalign} achieves this by taking the absolute value of complex numbers. To obtain the Euler-Lagrange equations for this system, we treat \(\Phi \) and \(\Phi ^\ast \) as independent fields. The Euler-Lagrange equation for \(\Phi \) then reads as
\(\seteqnumber{1}{2.18}{0}\)\begin{flalign} \nn 0 &= \partial _\mu \bigg (\frac {\partial \LL }{\partial (\partial _\mu \Phi )}\bigg ) - \frac {\partial \LL }{\partial \Phi } = \partial _\mu \Big (-\partial ^\mu \Phi ^\ast \Big ) + m^2 \,\Phi ^\ast + V^\prime (\Phi ^\ast \,\Phi )\,\Phi ^\ast \\ &=-\partial ^2\Phi ^\ast + m^2 \,\Phi ^\ast + V^\prime (\Phi ^\ast \,\Phi )\,\Phi ^\ast \quad \end{flalign} and the one for \(\Phi ^\ast \) reads as
\(\seteqnumber{1}{2.18}{1}\)\begin{flalign} \nn 0 &= \partial _\mu \bigg (\frac {\partial \LL }{\partial (\partial _\mu \Phi ^\ast )}\bigg ) - \frac {\partial \LL }{\partial \Phi ^\ast } = \partial _\mu \Big (-\partial ^\mu \Phi \Big ) + m^2 \,\Phi + V^\prime (\Phi ^\ast \,\Phi )\,\Phi \\ &=-\partial ^2\Phi + m^2 \,\Phi + V^\prime (\Phi ^\ast \,\Phi )\,\Phi \quad . \end{flalign} Observe that the two equations (2.18) are the complex conjugates of each other. They also agree with the Euler-Lagrange equations that one would obtain by decomposing \(\Phi (x) = \rho (x) + \ii \,\chi (x)\) into its real and imaginary part. (This is a good exercise for you!) For a trivial potential \(V=0\), one obtains the so-called complex Klein-Gordon action
\(\seteqnumber{0}{2.}{18}\)\begin{flalign} \label {eqn:KGactioncomplex} S_{\mathrm {KG}_\bbC }[\Phi ,\Phi ^\ast ] := \int _{\bbR ^d} - \bigg (\partial ^\mu \Phi ^\ast \,\partial _\mu \Phi + m^2\,\Phi ^\ast \,\Phi \bigg )\,\dd x\quad , \end{flalign} whose Euler-Lagrange equations are the complex Klein-Gordon equations
\(\seteqnumber{0}{2.}{19}\)\begin{flalign} \label {eqn:KGequationcomplex} -\partial ^2\Phi + m^2 \,\Phi =0 \quad ,\qquad -\partial ^2\Phi ^\ast + m^2 \,\Phi ^\ast =0 \quad , \end{flalign} where again one follows from the other by complex conjugation.