Quantum field theory 1, lecture 03

Classical field theory

Relativistic scalar field theory

Classical action

Let us consider the classical action for a real scalar field \[\begin{split} S[\phi] = & \int dt \int d^3 x \, \mathscr{L}(\phi, \partial_\mu \phi) = \int dt \int d^3 x \left\{ \frac{1}{2} \dot \phi^2 - \frac{1}{2} (\boldsymbol \nabla \phi)^2 - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{4!} \phi^4 \right\} \\ = & \int dt \int d^3 x \, \left\{ - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{4!} \phi^4 \right\} \end{split}\label{eq:actionRealScalar}\] We have suppressed here the argument, but the field \(\phi\) is to be understood as a (real) function of time and space \[\phi = \phi(x) = \phi(t, \mathbf{x}).\] The integrand \(\mathscr{L}\) is known as the Lagrange density. We represent time derivatives by dots, i. e. \[\dot \phi = \frac{\partial}{\partial t}\phi(t, \mathbf{x}),\] and the spatial gradient by \(\boldsymbol{\nabla}\phi(t, \mathbf{x})\). In the following we use the abbreviation \[\int_x = \int d^4 x = \int dt \int d^3x.\] It is sometimes convenient to restrict the integrals to some intervals in time, for example \(t_\text{i} \leq t \leq t_\text{f}\), and the spatial integral could be restricted to some volume \(V\). In that case one must specify what boundary conditions the field \(\phi(t, \mathbf{x})\) is supposed to satisfy on the boundary \(\partial V\). Some common choices are Dirichlet boundary conditions, where the field is fixed to some value on the boundary, or Neumann boundary conditions, where the normal derivative of the field is fixed, or periodic boundary conditions.

The square brackets in \(S[\Phi]\) indicate that the action depends on the fields in a functional way, which means it depends not on single numbers but on the entire set of functions of space \(\phi(t,\mathbf{x})\), with \((t,\mathbf{x})\in \mathbb{R}^d\) where we usually take \(d=1+3\). We also use the Minkowski metric with mainly plus signature, \(g_{\mu\nu} = \text{diag}(-1,+1,+1,+1)\).

Fields as vectors

One can consider \(\phi(t,\mathbf{x})\) as a vector in a (real) vector space of infinite dimension where components are labeled by time \(t\) and the spatial position \(\mathbf{x}\). In particular, linear superpositions of field configurations \(\phi(t, \mathbf{x}) = \lambda_1 \phi_1(t,\mathbf{x}) + \lambda_2 \phi_2(t, \mathbf{x})\) are again field configurations. It is sometimes useful to think about a field theory as limit of a discrete lattice model where the positions \(\mathbf{x}\) and times \(t\) are restricted to discrete points on some lattice. When space and time are furthermore constrained to a finite region of spacetime, the number of spacetime positions \((t, \mathbf{x})\) becomes finite. A field configuration is then specified by a set of real numbers, one per spacetime lattice site. This would be a finite set of numbers in the lattice scheme but it becomes infinitely large in the continuum and infinite volume limits.

Functional spaces

How regular should one assume field configurations \(\phi(t, \mathbf{x})\) to be? It is tempting to assume that they should be continuous and at least differentiable once so that the action in eq. \(\eqref{eq:actionRealScalar}\) is well defined. Assumptions of this kind are sometimes justified in classical physics, but in the statistical and quantum field theoretic formalism we discuss below one also needs to work with very irregular field configurations that are not even continuous.

Variational principle and equations of motion

Variation of the action

One can obtain the classical equation of motion by the principle of stationary action \[\delta S = 0.\] For the action in \(\eqref{eq:actionRealScalar}\) this yields \[\begin{split} \delta S = & \int_x \left\{ \dot \phi (x) \, \delta\dot \phi(x) - \boldsymbol \nabla \phi(x) \, \delta\boldsymbol\nabla\phi(x) - m^2 \phi(x) \, \delta \phi(x) - \frac{\lambda}{3!} \phi(x)^3 \, \delta\phi(x) \right\} \\ = & \int_x \left\{ - g^{\mu\nu}\partial_\mu \phi(x) \, \delta\partial_\nu \phi(x) - m^2 \phi(x) \, \delta \phi(x) - \frac{\lambda}{3!} \phi(x)^3 \, \delta\phi(x) \right\}. \end{split}\] Note that on the right hand side the variation of the field \(\delta \phi(x)\) occurs, but also the variation of the derivatives \(\delta \partial_\mu \phi(x)\).

Partial integration

To deal with the time derivative of the variation \(\delta \dot \phi(x)\) we integrate by parts, \[\begin{split} \int_{t_\text{i}}^{t_\text{f}} dt \int_V d^3 x \left\{ \dot \phi(x) \, \delta \dot \phi(x) \right\} = & \int_V d^3 x \left\{ \dot \phi(t_\text{f}, \mathbf{x}) \, \delta \phi(t_\text{f}, \mathbf{x}) - \dot \phi(t_\text{i}, \mathbf{x}) \, \delta \phi(t_\text{i}, \mathbf{x}) \right\} \\ & - \int_{t_\text{i}}^{t_\text{f}} dt \int_V d^3 x \left\{ \ddot \phi(x) \, \delta \phi(x) \right\} \end{split}\] The first term on the right hand side is a boundary term at the final time \(t_\text{f}\) and initial time \(t_\text{i}\). Similarly one can deal with the spatial gradient term, \[\begin{split} \int_{t_\text{i}}^{t_\text{f}} dt \int_V d^3 x \left\{ - \boldsymbol \nabla \phi(x) \, \delta \boldsymbol \nabla \phi(x) \right\} = & - \int_{t_\text{i}}^{t_\text{f}} dt \int_{\partial V} d^2 x \, \mathbf{n} \left\{ \boldsymbol \nabla \phi(x) \, \delta \phi(x) \right\} \\ & + \int_{t_\text{i}}^{t_\text{f}} dt \int_V d^3 x \left\{ \boldsymbol \nabla \cdot \boldsymbol \nabla \phi(x) \, \delta \phi(x) \right\}. \end{split}\] The first term on the right hand side is a surface integral with outward-pointing normal vector \(\mathbf{n}\) on the boundary \(\partial V\).

Fixed boundary conditions

We consider now fully fixed or constrained field configurations on the boundaries of the spacetime volume at \(t_\text{i}\), \(t_\text{f}\) and the spatial boundary \(\partial V\). This means that the variation \(\delta \phi(x)\) is supposed to vanish there, \(\delta\phi(x)=0\). For example we could demand at initial and final time \[\phi(t_\text{i}, \mathbf{x}) = \phi_\text{i}(\mathbf{x}), \quad\quad\quad \phi(t_\text{f}, \mathbf{x}) = \phi_\text{f}(\mathbf{x}), \label{eq:initialFinalBoundaryConditions}\] and take the spatial volume \(V\) to be all of \(\mathbb{R}^3\) so that there is actually no spatial boundary.

Combining terms yields then \[\delta S = \int_x \left\{ - \ddot \phi(x) + \boldsymbol \nabla^2 \phi(x) - m^2 \phi(x) - \frac{\lambda}{3!} \phi(x)^3 \right\} \delta \phi(x).\] For this to vanish for arbitrary variation \(\delta \phi(x)\) inside the spacetime volume we need \[- \ddot \phi(x) + \boldsymbol \nabla^2 \phi(x) - m^2 \phi(x) - \frac{\lambda}{3!} \phi(x)^3 = 0, \label{eq:01eom}\] which is the classical equation of motion. Together with the boundary conditions \(\eqref{eq:initialFinalBoundaryConditions}\) the classical field \(\phi(t, \mathbf{x})\) is actually fully fixed through this differential equation. Indeed, this equation of motion is a quasi-linear hyperbolic partial differential equation of second order. A solution is determined by initial data in the form of a configuration of the field \(\phi(t_\text{i}, \mathbf{x}) = \phi_\text{i}(\mathbf{x})\) and its first time derivative, on some Cauchy hypersurface like \(t=t_\text{i}\), or alternatively by field configurations at initial and final time as in eq. \(\eqref{eq:initialFinalBoundaryConditions}\).

Maxwell theory

Gauge field and field strength tensor

Another example for a classical field theory is Maxwell theory. The field is here the four-vector potential, or gauge field \(A_\mu(x) = (-\Phi(x), \mathbf{A}(x))\). We work in unites where \(c=1\) and with Minkowski space metric \(\eta_{\mu\nu} = \text{diag}(-1,+1,+1,+1)\). The electric and magnetic fields are given by \[\mathbf{E}(x) = - \frac{\partial}{\partial t} \mathbf{A}(x) - \boldsymbol \nabla \Phi(x), \quad\quad\quad \mathbf{B}(x) = \boldsymbol \nabla \times \mathbf{A}(x),\] and can be combined into the field strength tensor \[F_{\mu\nu}(x) = \partial_\mu A_\nu(x) - \partial_\nu A_\mu(x) = \begin{pmatrix} 0 & -E_1 & -E_2 & -E_3 \\ E_1 & 0 & -B_3 & B_2 \\ E_2 & B_3 & 0 & -B_1 \\ E_3 & -B_2 & B_1 & 0 \end{pmatrix}. \label{eq:EMFieldStrengthTensor}\] Note that this immediately implies the homogeneous Maxwell equations, \[\varepsilon^{\mu\nu\rho\sigma} \partial_\nu F_{\rho\sigma}(x) = \varepsilon^{\mu\nu\rho\sigma} \partial_\nu [\partial_\rho A_\sigma(x) - \partial_\sigma A_\rho(x)] = 0,\] following from the fact that \(\partial_\nu \partial_\rho A_\sigma(x) = \partial_\rho \partial_\mu A_\sigma(x)\). Here we are using the completely antisymmetric Levi-Civita symbol \(\varepsilon^{\mu\nu\rho\sigma}\) in \(d=1+3\) dimensions.

One can also write the homogeneous Maxwell equations in the familiar form \(\boldsymbol{\nabla}\cdot \mathbf{B}=0\) and \(\partial_t \mathbf{B} + \boldsymbol{\nabla}\times \mathbf{E} = 0\).

Gauge invariance

We note immediately that the field strength tensor is antisymmetric, \(F_{\mu\nu} = - F_{\nu\mu}\). It is invariant under gauge transformations \[A_\mu(x) \to A_\mu(x) + \partial_\mu \alpha(x), \label{eq:gaugeTransformation}\] where \(\alpha(x)\) is an arbitrary scalar function with \(\partial_\mu\partial_\nu \alpha(x) = \partial_\nu \partial_\mu \alpha(x)\). This also implies that \(A_\mu(x)\) is not fully fixed by the measurable electric and magnetic fields, but only up to gauge transformations.

Action

The action for classical Maxwell theory in in the presence of some electromagnetic current \(J^\mu(x) = (\rho(x), \mathbf{J}(x))\) given by \[\begin{split} S[A_\mu] = & \int d^d x \left\{ -\frac{1}{4} F^{\mu\nu}(x) F_{\mu\nu}(x) - J^\mu(x) A_\mu(x) \right\} \\ = & \int d^d x \left\{ -\frac{1}{2} \partial^\mu A^\nu(x) \partial_\mu A_\nu(x) + \frac{1}{2} \partial^\mu A^\nu(x) \partial_\nu A_\mu(x) - J^\mu(x) A_\mu(x) \right\}. \end{split}\label{eq:actionMaxwell}\] Note that this is invariant under gauge transformations \(\eqref{eq:gaugeTransformation}\), up to a boundary term, when the current is conserved, \(\partial_\mu J^\mu(x) = 0\). For the first term this is clear because the field strength tensor is invariant, and for the second term it follows through partial integration.

One can expand the action \(\eqref{eq:actionMaxwell}\) in the form \[\begin{split} S = & \int d^d x \left\{ \frac{1}{2} \partial_k A_0 \partial_k A_0 - \partial_0 A_k \partial_k A_0 + \frac{1}{2} \partial_0 A_k \partial_0 A_k - \frac{1}{2} \partial_j A_k \partial_j A_k + \frac{1}{2} \partial_j A_k \partial_k A_j - J^0 A_0 - J^k A_k \right\}, \end{split}\label{eq:actionMaxwell2}\] where Latin indices \(j,k=1,2,3\) run over spatial components.

Variation

To vary the action we need the variation of the field strength tensor, expressed in terms of the variation of the gauge field, \[\delta F_{\mu\nu}(x) = \partial_\mu \delta A_\nu(x) - \partial_\nu \delta A_\mu(x).\] Performing then partial integrations and using the anti-symmetry property of \(F_{\mu\nu}\) yields \[\begin{split} \delta S = & \int d^d x \left\{ - \frac{1}{2} F^{\mu\nu}(x) \delta F_{\mu\nu}(x) - J^\mu(x) \delta A_\mu(x) \right\} \\ = & \int d^d x \left\{ \left[ \partial_\mu F^{\mu\nu}(x) - J^\nu(x) \right] \delta A_\nu(x) \right\}. \end{split}\] Accordingly, the principle of stationary action implies the inhomogeneous Maxwell equations \[\partial_\mu F^{\mu\nu}(x) = J^\nu(x).\] One can also write these in the familiar form \(\boldsymbol{\nabla}\cdot \mathbf{E}=\rho\) and \(-\partial_t \mathbf{E} + \boldsymbol{\nabla}\times \mathbf{B} = \mathbf{J}\).

Gross-Pitaevskii theory

Action

We also discuss an example for a non-relativistic classical field theory. It describes a Bose-Einstein condensate of bosons with repulsive contact interaction. The action is \[S[\phi] = \int dt \int d^3 x \left\{ \frac{i \hbar}{2} [\phi^* \partial_t \phi - \phi \partial_t \phi^*] - \frac{\hbar^2}{2m} \boldsymbol \nabla \phi^* \boldsymbol \nabla \phi - V \phi^* \phi - \frac{\lambda}{2} (\phi^* \phi)^2 \right\}. \label{eq:actionGrossPitaevskii}\] The field \(\phi(t, \mathbf{x})\) is here complex. We have included an external potential \(V(t, \mathbf{x})\) which could be an optical trap, for example, and \(\lambda \geq 0\) is the interaction parameter. Note that despite the presence of the imaginary unit \(i\) the action is real. This becomes more explicit when it is rewritten in terms of real fields \(\varphi_1\) and \(\varphi_2\) defined through \(\phi = [\varphi_1 + i \varphi_2]/\sqrt{2}\).

Variation and equation of motion

For a complex field one can either consider the real and imaginary part as independent fields, and vary with respect to them, or one can vary with respect to \(\phi\) and \(\phi^*\). In the latter case one obtains two complex conjugate equations of motion, so it is enough to do one of these variations. Specifically, variation with respect to \(\phi^*\) yields, up to boundary terms, \[\delta S = \int dt \int d^3 x \left\{ \left[ i \hbar \partial_t \phi + \frac{\hbar^2}{2m} \boldsymbol \nabla^2 \phi - V \phi - \lambda \phi^* \phi \phi \right] \delta \phi^* \right\}.\] The corresponding equation of motion is a kind of non-linear Schrödinger equation, the Gross-Pitaevskii equation \[i \hbar \partial_t \phi(t, \mathbf{x}) = - \frac{\hbar^2}{2m} \boldsymbol \nabla^2 \phi(t, \mathbf{x}) + V(t, \mathbf{x}) \phi(t, \mathbf{x}) + \lambda \phi^*(t, \mathbf{x}) \phi(t, \mathbf{x}) \phi(t, \mathbf{x}). \label{eq:GrossPitaevskiiEquation}\] Because this is a differential equation of first order in time, initial conditions can be posed in the form \(\phi(t_\text{i}, \mathbf{x}) = \phi_\text{i}(\mathbf{x})\). As a complex equation this fixes both the real and imaginary part \(\varphi_1(t_\text{i}, \mathbf{x})\) and \(\varphi_2(t_\text{i}, \mathbf{x})\), so the number of initial conditions is the same as for the relativistic real scalar field.

Hamiltonian formalism

Conjugate momentum field

As familiar from classical mechanics, one may also introduce a Hamiltonian description which is connected to the Lagrangian description through a Legendre transform. As a first step in that direction one defines a conjugate momentum field. One starts here from the Lagrange function at some given time \(t\), \[L[\Phi(t), \dot \Phi(t)] = \int d^3 x \, \mathscr{L}(\Phi(t, \mathbf{x}), \partial_\mu \Phi(t, \mathbf{x})).\] For a field theory, the Lagrange function is a functional of the field \(\Phi(t, \mathbf{x})\) at fixed time \(t\), with spatial position \(\mathbf{x}\) treated similar to an index in mechanics. In other words, we can see the field theory as a mechanical system in the continuum limit with one degree of freedom per spatial position \(\mathbf{x}\). The difference is that \(\mathbf{x}\) is being integrated over instead of the sum over different mechanical degrees of freedom familiar from mechanics.

The field \(\Phi(t, \mathbf{x})\) has a canonical conjugate momentum field \(\Pi(t, \mathbf{x})\), which is defined through \[\Pi(t, \mathbf{x}) = \frac{\partial \mathscr{L}(\Phi(t, \mathbf{x}), \partial_\mu \Phi(t, \mathbf{x}))}{\partial \dot \Phi(t, \mathbf{x})}.\]

Examples

Let us consider the actions we have introduced already.

• For the real relativistic scalar field with action as in \(\eqref{eq:actionRealScalar}\) one finds the momentum field conjugate to the real scalar field \(\phi(t, \mathbf{x})\) to be \[\pi(t, \mathbf{x}) = \dot \phi(t, \mathbf{x}).\]

• For the Maxwell theory with action as in \(\eqref{eq:actionMaxwell}\) one finds the momentum field conjugate to \(A_k(t, \mathbf{x})\) to be minus the electric field, \[\pi_k(t, \mathbf{x}) = \frac{\partial \mathscr{L}}{\partial \dot A_k(t, \mathbf{x})} = \partial_0 A_k(t, \mathbf{x}) - \partial_k A_0(t, \mathbf{x}) = F_{0k}(t, \mathbf{x}) = - E_k(t, \mathbf{x}).\] In contrast, the field \(A_0(t, \mathbf{x})\) has a vanishing conjugate momentum field, \[\pi_0(t, \mathbf{x}) = \frac{\partial \mathscr{L}}{\partial \dot A_0(t, \mathbf{x})} = 0.\]

• Finally, for the Gross-Pitaevskii theory with action as in \(\eqref{eq:actionGrossPitaevskii}\) one finds the momentum field conjugate to the complex scalar field \(\phi(t, \mathbf{x})\) to be \[\pi_\phi(t, \mathbf{x}) = \frac{\partial \mathscr{L}}{\partial \dot \phi(t, \mathbf{x})} = \frac{i\hbar}{2} \phi^*(t, \mathbf{x}),\] while the conjugate momentum field to the complex conjugate field \(\phi^*(t, \mathbf{x})\) is \[\pi_{\phi^*}(t, \mathbf{x}) = \frac{\partial \mathscr{L}}{\partial \dot \phi^*(t, \mathbf{x})} = \frac{-i\hbar}{2} \phi(t, \mathbf{x}).\]

Hamiltonian as Legendre transform

The Hamiltonian (at some time \(t\)) is now given by the functional Legendre transform \[\begin{split} H[\Phi(t), \Pi(t)] = & \int d^3 x \, \mathscr{H}(\Phi(t, \mathbf{x}), \boldsymbol{\nabla}\Phi(t, \mathbf{x}), \Pi(t, \mathbf{x})) \\ = & \int d^3 x \sup_{\dot \Phi(t, \mathbf{x})} \left\{ \Pi(t, \mathbf{x}) \dot\Phi(t, \mathbf{x}) - \mathscr{L}(\Phi(t, \mathbf{x}), \partial_\mu \Phi(t, \mathbf{x}))\right\}. \end{split}\] It is to be understood as a functional of \(\Phi(t, \mathbf{x})\) and \(\Pi(t, \mathbf{x})\) at fixed time \(t\) but with \(\mathbf{x}\) playing the role of an index. Through the Legendre transform the time derivative of the field \(\dot \Phi(t, \mathbf{x})\) is replaced by the conjugate momentum field \(\Pi(t, \mathbf{x})\).

Examples

For the three theories introduced above we find

• For the real relativistic scalar field with action as in \(\eqref{eq:actionRealScalar}\) the Hamiltonian is \[H = \int d^3 x \left\{ \frac{1}{2} \pi(t, \mathbf{x})^2 + \frac{1}{2} (\boldsymbol \nabla \phi(t, \mathbf{x}))^2 + V(\phi(t, \mathbf{x})) \right\}, \label{eq:01HamiltonianFieldTheory}\] where we have introduced the potential \[V(\phi) = \frac{1}{2} m^2 \phi^2 + \frac{\lambda}{4!} \phi^4.\]

• For the Gross-Pitaevskii theory with action as in \(\eqref{eq:actionGrossPitaevskii}\) one finds the Hamiltonian to be \[H = \int d^3 x \left\{ \frac{\hbar^2}{2m} \boldsymbol \nabla \phi^*(t, \mathbf{x}) \cdot \boldsymbol{\nabla} \phi(t, \mathbf{x}) + V(t, \mathbf{x}) \phi^*(t, \mathbf{x}) \phi(t, \mathbf{x}) + \frac{\lambda}{2} \phi^*(t, \mathbf{x})^2 \phi(t, \mathbf{x})^2 \right\}. \label{eq:01HamiltonianGrossPitaevskii}\]

Note that in both theories the Hamiltonian density \(\mathscr{H}\) is bounded from below for \(\lambda > 0\). For the Maxwell theory the Hamiltonian description is more involved, essentially because \(A_0\) has vanishing conjugate momentum field. We will not discuss this here.

Functional differentiation

In the following we need the notion of a functional derivative. For a functional \(I[\phi]\) with variation \[\delta I[\phi] = \int d^d x \left\{ f(x) \delta\phi(x) \right\},\] one defines \[\frac{\delta I[\phi]}{\delta \phi(x)} = f(x).\] More generally, when the variation is of the form \[\delta I[\phi] = \int d^d x \left\{ f(x) \delta\phi(x) + g^j(x) \frac{\partial}{\partial x^j}\delta\phi(x) + h^{jk}(x) \frac{\partial^2}{\partial x^j \partial x^k}\delta\phi(x) + \ldots \right\}\] one needs to first perform partial integrations. Usually the series on the right hand side terminates after one, two or three terms.When \(\delta\phi(x)\) is assumed to vanish on the boundaries one can drop the boundary terms arising from the partial integration. This yields \[\delta I[\phi] = \int d^d x \left\{ \left[ f(x) - \frac{\partial}{\partial x^j}g^j(x) + \frac{\partial^2}{\partial x^j \partial x^k} h^{jk}(x) - \ldots \right] \delta \phi(x) \right\},\] and the functional derivative is thus \[\frac{\delta I[\phi]}{\delta \phi(x)} = f(x) - \frac{\partial}{\partial x^j}g^j(x) + \frac{\partial^2}{\partial x^j \partial x^k} h^{jk}(x) - \ldots.\]

As an example consider the functional \(I[\phi] = \phi(y)\). One can write this as \[I[\phi] = \int d^dx \delta^{(d)}(x-y) \phi(x),\] and thus the functional derivative is here \[\frac{\delta I[\phi]}{\delta \phi(x)} = \frac{\delta \phi(y)}{\delta \phi(x)} = \delta^{(d)}(x-y).\] The definition of the functional derivative depends slightly on the context, in particular it is sometimes used for integrals over space and sometimes for integrals over space and time. With a bit of care it gets clear from the context what is the right definition in a given context.

Poisson brackets

In classical mechanics one introduces Poisson brackets to describe time evolution or symmetry transformations in the Hamiltonian formalism. This can also be done in a classical field theory. The Poisson bracket between two functionals \(A[\Phi(t), \Pi(t)]\) and \(B[\Phi(t), \Pi(t)]\) of the fields and conjugate momenta at some given fixed time \(t\) is defined as \[\left\{ A, B \right\} = \int d^3 x \left\{ \frac{\delta A}{\delta \Phi(t,\mathbf{x})} \frac{\delta B}{\delta \Pi(t,\mathbf{x})} - \frac{\delta A}{\delta \Pi(t,\mathbf{x})} \frac{\delta B}{\delta \Phi(t,\mathbf{x})} \right\}.\] The functional derivatives are here defined for three-dimensional integrals over space.

In particular, by taking the Poisson bracket with the Hamiltonian, one can obtain the time derivative of a functional \(A[\Phi(t), \Pi(t)]\) without explicit time dependence, along the solution to the equation of motion, \[\frac{d}{dt} A[\Phi(t), \Pi(t)] = \{ A, H \}. %+ \frac{\partial}{\partial t} A[\phi(t), \pi(t)]. \label{eq:PoissonBracketTimeDerivative}\] For a functional \(A[\Phi(t), \Pi(t)]\) with explicit time dependence one has to add the partial time derivative of \(A\) to the right hand side.

Relativistic scalar theory as example

For the real relativistic scalar field theory the Hamiltonian is given in eq. \(\eqref{eq:01HamiltonianFieldTheory}\). To calculate the Poisson brackets we need the functional derivatives \[\frac{\delta H[\pi(t), \phi(t)]}{\delta \pi(t, \mathbf{x})} = \pi(t, \mathbf{x}),\] and \[\frac{\delta H[\pi(t), \phi(t)]}{\delta \phi(t, \mathbf{x})} = - \boldsymbol{\nabla}^2 \phi(t, \mathbf{x}) + m^2 \phi(t, \mathbf{x}) + \frac{\lambda}{3!} \phi(t, \mathbf{x})^3.\]

As a first example let us take \(A[\pi(t), \phi(t)]=\phi(t,\mathbf{y})\). Here we find \(\delta A/\delta \phi(t,\mathbf{x}) = \delta^{(3)}(\mathbf{x}-\mathbf{y})\) and \(\delta A/ \delta \pi(t, \mathbf{x}) = 0\) such that eq. \(\eqref{eq:PoissonBracketTimeDerivative}\) gives \[\frac{d}{dt} A[\phi(t), \pi(t)] = \dot \phi(t, \mathbf{y}) = \pi(t,\mathbf{x}). \label{eq:PoissonBracketDefinitionClassicalFieldTheory}\] This is consistent with the definition of the conjugate momentum field for this theory.

Similarly, for the choice \(A[\phi(t), \pi(t)] = \pi(t,\mathbf{y})\) we have \(\delta A/\delta \phi(t,\mathbf{x}) = 0\) and \(\delta A/ \delta \pi(t, \mathbf{x}) = \delta^{(3)}(\mathbf{x}-\mathbf{y})\) such that eq. \(\eqref{eq:PoissonBracketTimeDerivative}\) gives \[\frac{d}{dt} A[\phi(t), \pi(t)] = \dot \pi(t, \mathbf{y}) = \boldsymbol{\nabla}^2 \phi(t, \mathbf{y}) - m^2 \phi(t, \mathbf{y}) - \frac{\lambda}{3!} \phi(t, \mathbf{y})^3.\] This is the equation of motion previously obtained from variation of the action.

Fundamental Poisson brackets

Based on the definitions one may easily check the fundamental Poisson brackets \[\{ \phi(t, \mathbf{x}) , \phi(t, \mathbf{y}) \} = \{ \pi(t, \mathbf{x}) , \pi(t, \mathbf{y}) \} = 0, \quad\quad\quad \{ \phi(t, \mathbf{x}) , \pi(t, \mathbf{y}) \} = \delta^{(3)}(\mathbf{x} - \mathbf{y}).\] These can be taken as a starting point for “canonical quantization”, which is a heuristic transition from a classical field theory to a quantum field theory.