Chapter 4 Interacting quantum Klein-Gordon field

Interacting QFTs are much richer than the free QFTs from Chapter 3, and consequently also harder to describe. This chapter discusses the perturbative quantization of the interacting real Klein-Gordon field from Example 2.1 and uses these techniques to compute scattering amplitudes for quantum particle interactions.

4.1 Perturbation techniques

What characterizes an interacting QFT is that its underlying classical action functional contains terms that are of polynomial degree \(\geq 3\), which implies that the Euler-Lagrange equations are nonlinear partial differential equations. For instance, for a real scalar field \(\Phi \), one can consider as in Example 2.1 a potential term, say \(V(\Phi ) = \sum _{n\geq 3} \frac {\lambda _n}{n!}\,\Phi ^n \) with some parameters \(\lambda _n\in \bbR \) called the coupling constants, and write down the action

\begin{flalign} S[\Phi ] = S_0[\Phi ] + S_{\mathrm {int}}[\Phi ] = \int _{\bbR ^d} -\frac {1}{2}\,\Big ( \partial ^\mu \Phi \,\partial _\mu \Phi + m^2\,\Phi ^2 \Big ) \,\dd x - \sum _{n\geq 3} \frac {\lambda _n}{n!}\,\int _{\bbR ^d} \Phi ^n \, \dd x\quad . \end{flalign} The corresponding Hamiltonian reads as

\begin{flalign} \nn H[\Pi ,\Phi ] &= H_{0}[\Pi ,\Phi ] + H_{\mathrm {int}}[\Pi ,\Phi ]\\ &= \int _{\bbR ^{d-1}} \frac {1}{2} \Big ( \Pi ^2 + (\nabla \Phi )^2 + m^2\,\Phi ^2 \Big ) \,\dd \mathbf {x} + \sum _{n\geq 3} \frac {\lambda _n}{n!} \int _{\bbR ^{d-1}}\Phi ^n\,\dd \mathbf {x}\quad . \end{flalign} In both expressions we use a subscript \({}_0\) to denote the quadratic parts, i.e. the action functional and Hamiltonian for the free Klein-Gordon field.

The same canonical quantization procedure as in Section 3.1 can be applied to this system, up to the point where we want to solve Heisenberg’s equations. The issue there is that the higher-order powers \(\Phi ^n\) in the interaction Hamiltonian \(H_{\mathrm {int}}\) make the commutators \([H_{\mathrm {int}},\Pi ]\simeq \Phi ^{n-1}\) nonlinear expressions in the \(\Phi \)’s, hence Heisenberg’s equations are nonlinear differential equations whenever an interaction term is present. Making mathematical sense of such nonlinear differential equations for operators, which turn out to suffer from divergences that can not be cured by using only a simple normal ordering prescription as in Definition 3.1, and determining solutions is an extremely difficult problem that has so far only been successfully addressed in very special cases.

When a nonlinear equation is too complicated, it is always a natural first step to work perturbatively by expanding order-by-order in the nonlinearities, with the hope to capture at least some features of the nonlinear equation. This is indeed the approach taken in most of the QFT literature and consequently also in our module. Ignoring for the moment all potential divergences that can arise in this expansion (these are treated later in Chapter 8 with renormalization techniques), what one does in the context of QFT is to write down a formal solution to Heisenberg’s equation for the field operator

\begin{flalign} \label {eqn:interactionHeisenberg} \Phi (x)=e^{\ii \,t\,[H_0 + H_{\mathrm {int}},-]}\, \Phi (\mathbf {x}) = e^{\ii \,t\,(H_0 + H_{\mathrm {int}})}\,\Phi (\mathbf {x})\, e^{-\ii \,t\,(H_0 + H_{\mathrm {int}})} \end{flalign} and expand this expression order-by-order in the coupling constants \(\lambda _n\), hence in the interaction Hamiltonian \(H_{\mathrm {int}}\), that control the nonlinearities.

We can manipulate the expression (4.3) a bit further in order to bring it to a simpler and more useful form. The main idea is to factorize off the time evolution induced by the free Hamiltonian \(H_0\), which we have explicitly solved in Section 3.1. This motivates us to rewrite (4.3) as follows

\begin{flalign} \nn \Phi (x) &= e^{\ii \,t\,(H_0 + H_{\mathrm {int}})}\, e^{-\ii \,t\,H_0} \,e^{\ii \,t\,H_0}\,\Phi (\mathbf {x})\, e^{-\ii \,t\,H_0}\,e^{\ii \,t\,H_0}\, e^{-\ii \,t\,(H_0 + H_{\mathrm {int}})}\\ &=e^{\ii \,t\,(H_0 + H_{\mathrm {int}})}\, e^{-\ii \,t\,H_0}\, \Phi _0(x)\, \,e^{\ii \,t\,H_0}\, e^{-\ii \,t\,(H_0 + H_{\mathrm {int}})}\quad ,\label {eqn:interactingKGHeisenbergpre} \end{flalign} where \(\Phi _0(x)\) is the free Heisenberg picture field operator from (3.20), which I repeat here for your convenience

\begin{flalign} \label {eqn:freeKGHeisenbergsub0} \Phi _0(x) = \int _{\bbR ^{d-1}} \frac {1}{\sqrt {2\omega _{\mathbf {k}}}}~ \bigg (a(\mathbf {k}) \,e^{\ii \,k\,x}+ a^\dagger (\mathbf {k})\, e^{-\ii \,k\,x}\bigg )\,\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\quad . \end{flalign} Note that normal ordering of \(H_0\) is not required here since \([H_0,-] = [\noor {H_0},-]\) give the same right-hand sides of the free Heisenberg equation. You might now be tempted to identify the operator \(e^{\ii \,t\,H_0}\,e^{-\ii \,t\,(H_0 + H_{\mathrm {int}})}\) with the simpler one \(e^{-\ii \,t\, H_{\mathrm {int}}}\), but unfortunately this does not work because \([H_0,H_{\mathrm {int}}]\neq 0\) are noncommuting operators, hence the product of exponential functions is given by the Baker-Campbell-Hausdorff formula and not simply by summing the exponents.

There is however a clever trick to express the operator \(e^{\ii \,t\,H_0}\,e^{-\ii \,t\,(H_0 + H_{\mathrm {int}})}\) in a more useful way, which is called Dyson’s formula and derived as follows: Taking the time derivative of this operator, one finds that

\begin{flalign} \nn \frac {\dd }{\dd t}\bigg (e^{\ii \,t\,H_0}\,e^{-\ii \,t\,(H_0 + H_{\mathrm {int}})}\bigg ) &=- \ii \, e^{\ii \,t\,H_0}\,H_{\mathrm {int}}\,e^{-\ii \,t\,(H_0 + H_{\mathrm {int}})}\\ \nn &=- \ii \, e^{\ii \,t\,H_0}\,H_{\mathrm {int}}\, e^{-\ii \,t\,H_0}\,e^{\ii \,t\,H_0}\,e^{-\ii \,t\,(H_0 + H_{\mathrm {int}})}\\ &=- \ii \, H_{\mathrm {int}}(t)\,e^{\ii \,t\,H_0}\,e^{-\ii \,t\,(H_0 + H_{\mathrm {int}})}\quad , \end{flalign} where we would like to emphasize that the time evolution

\begin{flalign} \label {eqn:interactionHamiltoniant} H_{\mathrm {int}}(t) =e^{\ii \,t\,H_0}\,H_{\mathrm {int}}\, e^{-\ii \,t\,H_0} = \sum _{n\geq 3}\frac {\lambda _n}{n!}\int _{\bbR ^{d-1}} \big (\Phi _0(x)\big )^n\,\dd \mathbf {x} \end{flalign} with respect to the free Hamiltonian \(H_0\) can be expressed in terms of the free Heisenberg picture field operator \(\Phi _0(x)\). From the definition of the time-ordered product \(\TO \) in Section 3.4, it is easy to check that the unique solution of the differential equation

\begin{flalign} \label {eqn:Dysonformula1} \frac {\dd }{\dd t} U(t,t_0) = -\ii \, H_{\mathrm {int}}(t) \, U(t,t_0)\quad ,\qquad U(t_0,t_0) = 1 \quad , \end{flalign} for some arbitrary initial time \(t_0\in \bbR \), is given in the case where \(t\geq t_0\) is later than the initial time by the time-ordered exponential

which is defined by the following series

\begin{flalign} \TO \exp \bigg ( -\ii \int _{t_0}^t H_{\mathrm {int}}(t^\prime )\,\dd t^\prime \bigg ) = \sum _{n=0}^\infty \frac {(-\ii )^n}{n!}\, \int _{t_0}^t\cdots \int _{t_0}^t \TO \Big (H_{\mathrm {int}}(t_1)\cdots H_{\mathrm {int}}(t_n)\Big )~\dd t_1\cdots \dd t_n \quad . \end{flalign} In the case where \(t\leq t_0\) is earlier than \(t_0\), the solution \(U(t,t_0)\) of (4.7a) can be expressed in terms of the adjoint

where the right-hand side is defined by the time-ordered exponential (4.7b) since \(t_0\geq t\). One can prove that this family of operators satisfies the following composition identity

\begin{flalign} \label {eqn:Dysonformulacompo} U(t_2,t_1)\,U(t_1,t_0) = U(t_2,t_0) \end{flalign} for all \(t_2,t_1,t_0\in \bbR \), hence they behave like unitary time evolution operators. With these preparations, we can now express the interacting Heisenberg picture field operator \(\Phi (x)\) from (4.4a) in terms of the free one \(\Phi _0(x)\) as follows

where the argument \(0\) is due to our choice of initial time \(t_0=0\) at which we have set up the Hamiltonian formalism. This formula shows that the new features arising from perturbative interactions can be encoded in the family of operators \(U(t,t_0)\) from (4.7), which is constructed from data given by the underlying free QFT, namely the free quantum fields \(\Phi _0(x)\), which enter the free time evolution of the interaction Hamiltonian (4.6), and their time-ordered products \(\TO \).

It is important to emphasize that interactions do not only effect the field operators but also the vacuum state of the QFT. Indeed, the vacuum state \(\ket {0}\in \HH \) for the free QFT from Section 3.3 is not an eigenstate of the full Hamiltonian \(H = H_0 + H_{\mathrm {int}}\), hence it cannot be the vacuum of the interacting QFT. The argument is quite simple: The interaction Hamiltonian \(H_{\mathrm {int}}\) contains terms of the form \(\Phi ^n\), for \(n\geq 3\), which when expanded in the creation and annihilation operators leads to nonvanishing terms like \(a^{\dagger }(\mathbf {k}_1)\cdots a^{\dagger }(\mathbf {k}_n)\) that consist only of creation operators. This means that the eigenstate equation \(H\ket {0} = H_0\ket {0} + H_{\mathrm {int}}\ket {0} = E\,\ket {0}\) cannot hold true for any eigenvalue \(E\), because the left-hand side consists of a superposition of multiparticle states while the right-hand side has particle number zero.

There exists a very useful and nice result, called the Gell-Mann and Low Theorem, that allows us to compute the vacuum state of the interacting theory \(\ket {\Omega }\in \HH \) in terms of the vacuum state of the free theory \(\ket {0}\in \HH \) . The key idea is to assume that in the far future \(t\to +\infty \) and in the far past \(t\to -\infty \) the interactions can be neglected, which is technically implemented by a so-called adiabatic switching of the interaction Hamiltonian. This allows us to identify asymptotically in the far future/past the interacting theory with the free one. Using the operators (4.7), we then define the states

where again the argument \(0\) is due to our choice of initial time \(t_0=0\) at which we have set up the Hamiltonian formalism. To interpret these states, it is useful to allow for a general initial time \(t_0\in \bbR \), for which they read as \(\ket {\Omega ^\pm } = U(t_0,\pm \infty )\ket {0}\). If we choose \(t_0\) to be in the far future/past, i.e. \(t_0\to \pm \infty \), this simplifies to \(\ket {\Omega ^\pm } = U(\pm \infty ,\pm \infty )\ket {0} = \ket {0}\). Hence, we obtain the following physical interpretation: \(\ket {\Omega ^\pm } \) is the state that coincides with the free vacuum state in the far future/past. Since the far future/past are often called the out/in regions, one says that \(\ket {\Omega ^+}\) is the out vacuum and \(\ket {\Omega ^-}\) is the in vacuum. With some analytical efforts and care (see e.g. here or Example 8.1 in the textbook by Greiner/Reinhardt), one can show that both states \(\ket {\Omega ^\pm }\) are eigenstates of the full Hamiltonian \(H=H_0 + H_{\mathrm {int}}\) with the same eigenvalue as the interacting vacuum state \(\ket {\Omega }\). Assuming that the vacuum state is nondegenerate, it follows that all three states are the same, up to complex phases

The phase difference \(e^{\ii (\alpha ^+ -\alpha ^-)}\) can be computed from (4.10) and (4.7)

\begin{flalign} e^{\ii (\alpha ^+ -\alpha ^-)} =e^{\ii (\alpha ^+-\alpha ^-)} \,\braket {\Omega }{\Omega } = \braket {\Omega ^+}{\Omega ^-} = \expect {0}{U(+\infty ,-\infty )}{0} \quad , \end{flalign} where we also used the standard normalization condition \(\braket {\Omega }{\Omega }=1\) of the interacting vacuum.

Combining (4.9), (4.10) and (4.11), one obtains nice formulas to compute vacuum expectation values of operators in the interacting QFT in terms of vacuum expectation values in the free QFT. Let us illustrate this for the time-ordered \(n\)-point functions. Consider \(\expect {\Omega }{\TO \big (\Phi (x_1)\,\Phi (x_2)\cdots \Phi (x_n)\big )}{\Omega }\) and assume without loss of generality that \(x_1,\dots ,x_n\in \bbR ^d\) are already time-ordered. (This avoids notational clutter coming from the time-ordering permutation (3.55).) We can then write

\begin{flalign} \nn \expect {\Omega }{\TO \big (\Phi (x_1)\,&\Phi (x_2)\cdots \Phi (x_n)\big )}{\Omega } = \expect {\Omega }{\Phi (x_1)\,\Phi (x_2)\cdots \Phi (x_n)}{\Omega }\\ \nn & =\expect {\Omega }{U(0,t_1)\,\Phi _0(x_1)\,U(t_1,t_2)\,\Phi _0(x_2)\, \cdots \Phi _0(x_n)\,U(t_n,0)}{\Omega }\\ \nn &= e^{-\ii (\alpha ^+ -\alpha ^-)}\,\expect {\Omega ^+}{U(0,t_1)\,\Phi _0(x_1)\,U(t_1,t_2)\,\Phi _0(x_2)\, \cdots \Phi _0(x_n)\,U(t_n,0)}{\Omega ^-}\\ &=\frac {\expect {0}{U(+\infty ,t_1)\,\Phi _0(x_1)\,U(t_1,t_2)\,\Phi _0(x_2)\, \cdots \Phi _0(x_n)\,U(t_n,-\infty )}{0}}{\expect {0}{U(+\infty ,-\infty )}{0}}\quad , \end{flalign} where we frequently used the composition property of the \(U\)-operators (4.8). Observe that the expression in the last line comes naturally time-ordered, hence we can apply the time-ordering \(\TO \) without changing it. This allows us to rewrite this expression in the following more compact form

\begin{flalign} \nn U(+\infty ,t_1)\,\Phi _0(x_1)\,&U(t_1,t_2)\,\Phi _0(x_2)\, \cdots \Phi _0(x_n)\,U(t_n,-\infty ) \\ \nn & = \TO \Big (U(+\infty ,t_1)\,\Phi _0(x_1)\,U(t_1,t_2)\,\Phi _0(x_2)\, \cdots \Phi _0(x_n)\,U(t_n,-\infty )\Big )\\ \nn & = \TO \Big (\Phi _0(x_1)\,\Phi _0(x_2)\, \cdots \Phi _0(x_n)\,U(+\infty ,t_1)\,U(t_1,t_2)\,\cdots \,U(t_n,-\infty )\Big )\\ \nn & = \TO \Big (\Phi _0(x_1)\,\Phi _0(x_2)\, \cdots \Phi _0(x_n)\,U(+\infty ,-\infty )\Big )\\ & = \TO \bigg (\Phi _0(x_1)\,\Phi _0(x_2)\, \cdots \Phi _0(x_n)\, e^{-\ii \int _{-\infty }^\infty H_{\mathrm {int}}(t) \,\dd t}\bigg )\quad . \end{flalign} Using that

\begin{flalign} -\int _{-\infty }^\infty H_{\mathrm {int}}(t)\,\dd t = - \sum _{n\geq 3}\frac {\lambda _n}{n!} \int _{\bbR ^{d}} \big (\Phi _0(x)\big )^n\,\dd x = S_{\mathrm {int}}[\Phi _0] \end{flalign} is simply the interaction term of the action functional, evaluated on the free quantum field \(\Phi _0\), we can write the result in the following neater way

This is called the Gell-Mann and Low reduction formula and it is one of the most important formulas in QFT. Indeed, this formula achieves something truly amazing: It reduces the problem of computing time-ordered expectation values of interacting field operators \(\Phi (x)\) in the interacting vacuum state \(\ket {\Omega }\) to computing time-ordered expectation values of free field operators \(\Phi _0(x)\) in the free vacuum state \(\ket {0}\).