2.3 Hamiltonian formalism
The passage from the Lagrangian formalism to the Hamiltonian formalism in classical field theory is analogous to the case of classical mechanics. For your convenience, let me briefly recall the relevant steps.
Consider as in Section 2.1 a multicomponent field \(\phi _a(x)\) on the \(d\)-dimensional Minkowski spacetime \((\bbR ^d,\eta )\) with Lagrangian density \(\LL (\phi ,\partial \phi )\). The canonical momentum associated with \(\phi _a\) is defined by
\(\seteqnumber{0}{2.}{49}\)\begin{flalign} \label {eqn:fieldcanonicalmomenta} \pi ^a \,:=\, \frac {\partial \LL (\phi ,\dot {\phi },\nabla \phi )}{\partial \dot {\phi }_a} \quad , \end{flalign} where in the argument of \(\LL \) we have splitted the partial derivatives \(\partial \phi \) into the time derivative \(\dot {\phi }\) and the space derivatives \(\nabla \phi \). Assuming that this equation can be solved for \(\dot {\phi }_a = \dot {\phi }_a(\pi ,\phi ,\nabla \phi )\) as a function of the \(\pi \)’s, the \(\phi \)’s and the \(\nabla \phi \)’s, we define the Hamiltonian density by
\begin{equation} \HH \big (\pi ,\phi ,\nabla \phi \big ) \,:= \,\pi ^a\,\dot {\phi }_a(\pi ,\phi ,\nabla \phi ) - \LL \big (\phi ,\dot {\phi }(\pi ,\phi ,\nabla \phi ), \nabla \phi \big ) \quad . \end{equation}
Note that the Hamiltonian density is a local function of \(\pi \), \(\phi \) and \(\nabla \phi \), i.e. its value at the point \(x\) is determined by \(\pi (x)\), \(\phi (x)\) and \(\nabla \phi (x)\) at the same point \(x\). The Hamiltonian is then defined by integrating
the Hamiltonian density over space \(\mathbf {x}\in \bbR ^{d-1}\), where we recall that the square bracket notation \(H[\pi ,\phi ]\) is used to emphasize that \(H\) is a functional. To obtain Hamilton’s equations in field theory, we have to take functional derivatives of the Hamiltonian. These are defined as follows: In analogy to varying an action functional, we consider
\(\seteqnumber{0}{2.}{52}\)\begin{flalign} \nn \delta H[\pi ,\phi ] &= \int _{\bbR ^{d-1}}\bigg (\frac {\partial \HH \big (\pi ,\phi ,\nabla \phi \big )}{\partial \pi ^a} \,\delta \pi ^a + \frac {\partial \HH \big (\pi ,\phi ,\nabla \phi \big )}{\partial \phi _a} \,\delta \phi _a + \frac {\partial \HH \big (\pi ,\phi ,\nabla \phi \big )}{\partial (\partial _i\phi _a)}\,\partial _i(\delta \phi _a) \bigg )\,\dd \mathbf {x}\\ &= \int _{\bbR ^{d-1}}\bigg (\frac {\partial \HH \big (\pi ,\phi ,\nabla \phi \big )}{\partial \pi ^a} \,\delta \pi ^a + \bigg (\frac {\partial \HH \big (\pi ,\phi ,\nabla \phi \big )}{\partial \phi _a} - \partial _i \bigg (\frac {\partial \HH \big (\pi ,\phi ,\nabla \phi \big )}{\partial (\partial _i\phi _a)}\bigg )\bigg )\,\delta \phi _a \bigg )\,\dd \mathbf {x}\quad . \end{flalign} The functional derivative along \(\pi ^a\) is then the coefficient in front of \(\delta \pi ^a\) and the functional derivative along \(\phi _a\) is the coefficient in front of \(\delta \phi _a\). The following notations are often used
\(\seteqnumber{0}{2.}{53}\)\begin{flalign} \frac {\delta H[\pi ,\phi ]}{\delta \pi ^a} = \frac {\partial \HH \big (\pi ,\phi ,\nabla \phi \big )}{\partial \pi ^a}\quad ,\qquad \frac {\delta H[\pi ,\phi ]}{\delta \phi _a} = \frac {\partial \HH \big (\pi ,\phi ,\nabla \phi \big )}{\partial \phi _a} - \partial _i \bigg (\frac {\partial \HH \big (\pi ,\phi ,\nabla \phi \big )}{\partial (\partial _i\phi _a)}\bigg )\quad . \end{flalign} Hamilton’s equations in field theory then read as
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Remark 2.12. The Hamiltonian formalism for field theory has both advantages and disadvantages in comparison to the Lagrangian formalism. Its main advantage is that it serves as a bridge to QFT, since canonical quantization is typically described starting from the Hamiltonian formalism. Its main disadvantage is that Poincaré symmetry is not manifest in this approach. In fact, in order to derive Hamilton’s equations (2.55), we had to make an arbitrary split between time and space \(x = (t,\mathbf {x})\) and treat these concepts very differently. For instance, the canonical momenta (2.50) are determined from the time derivatives \(\dot {\phi }_a\), the Hamiltonian (2.52) involves an integration over only the space coordinates \(\mathbf {x}\), and Hamilton’s equations (2.55) involve taking only time derivatives. Since Poincaré transformations do in general mix between time and space coordinates, these quantities are not manifestly Poincaré invariant. It therefore requires additional investigations whether or not a field theory that is formulated in the Hamiltonian formalism admits Poincaré symmetry. We will later carry out these investigations in the context of QFT.
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Example 2.13 (Hamiltonian for the real Klein-Gordon field). Consider the real Klein-Gordon field from Example 2.1. Splitting the derivatives in the action functional (2.8) into time and space components, we can write
\(\seteqnumber{0}{2.}{55}\)\begin{flalign} S_{\mathrm {KG}}[\Phi ] = \int _{\bbR ^d}\frac {1}{2}\Big (\dot {\Phi }^2 - (\nabla \Phi )^2 - m^2\,\Phi ^2\Big )\,\dd x\quad . \end{flalign} For the canonical momentum (2.50) we find
\(\seteqnumber{0}{2.}{56}\)\begin{flalign} \Pi = \dot {\Phi } \end{flalign} and the Hamiltonian (2.52) is given by
\(\seteqnumber{0}{2.}{57}\)\begin{flalign} \label {eqn:KGHamiltonian} H[\Pi ,\Phi ] = \int _{\bbR ^{d-1}} \frac {1}{2} \,\Big (\Pi ^2 + (\nabla \Phi )^2 + m^2\,\Phi ^2\Big )\,\dd \mathbf {x}\quad . \end{flalign} Hamilton’s equations (2.55) for the real Klein-Gordon field then read as
\(\seteqnumber{0}{2.}{58}\)\begin{flalign} \dot {\Phi } = \Pi \quad ,\qquad \dot {\Pi } = -m^2\,\Phi + \nabla ^2\Phi \quad . \end{flalign} This is the first-order form of the Klein-Gordon equation \(-\partial ^2\Phi + m^2\,\Phi =0\), as can be seen by inserting the first into the second equation and using that \(\partial ^2 = \eta ^{\mu \nu }\partial _\mu \partial _\nu = -\frac {\partial ^2}{\partial t^2} + \nabla ^2\).
It would be a good exercise for you to derive the Hamiltonian for the complex Klein-Gordon field from Example 2.3.
Further reading
For more details about classical field theories, see e.g. Nastase (Chapter 1), Greiner/Reinhardt (Chapter 2), Maggiore (Chapter 3) and Schwartz (Chapter 3) from our reading list in Section 1.3.