Prof. Dr. Stefan Floerchinger

Researcher and professor for quantum field theory.

Scattering

In this section we will discuss a rather useful concept in quantum field theory – the S-matrix. It describes situations where the incoming state is a perturbation of a symmetric (homogeneous and isotropic) vacuum state in terms of particle excitations and the outgoing state similarly. We are interested in calculating the transition amplitude, and subsequently transition probability, between such few-particle states. An important example is the scattering of two particles with a certain center-of-mass energy. This is an experimental situation in many high energy laboratories, for example at CERN. The final states consists again of a few particles (although “few” might be rather many if the collision energy is high). Another interesting example is a single incoming particle, or resonance, that can be unstable and decay into other particles. For example \(\pi^+ \to \mu^+ + \nu_\mu\). As we will discuss later on in more detail, particles as excitations of quantum fields are actually closely connected with symmetries of space-time, in particular translations in space and time as well as Lorentz transformations including rotations. (In the non-relativistic limit, Lorentz transformations are replaced by Galilei transformations). The standard application of the S-matrix concept assumes therefore that the vacuum state has these symmetries. The S-matrix is closely connected to the functional integral. Technically, this connection is somewhat simpler to establish for non-relativistic quantum field theories. This will be discussed in the following.

Mode expansion and Fock space

Mode function expansion

Let us write the non-relativistic bosonic fields as \[\varphi(t, \mathbf{x}) = \int_{\mathbf{p}} v_{\mathbf{p}}(t,\mathbf{x}) \, a_{\mathbf{p}}(t), \quad\quad\quad \varphi^*(t,\mathbf{x}) = \int_{\mathbf{p}} v^*_{\mathbf{p}}(t, \mathbf{x}) \, a^{\dagger}_{\mathbf{p}}(t),\] with \(\int_{\mathbf{p}} = \int \tfrac{d^3 p}{(2\pi)^3}\) and the mode functions \[v_{\mathbf{p}}(t,\mathbf{x}) = e^{-i\omega_{\mathbf{p}}t + i\mathbf{p}\mathbf{x}}.\] While we plan to work in the in-out functional integral formalism, let us note that in an operator picture \(a_{\mathbf{p}}(t)\) and \(a^{\dagger}_{\mathbf{p}}(t)\) would be annihilation and creation operators for particles with momentum \(\mathbf p\) and frequency \[\omega_{\mathbf{p}} = \frac{\mathbf{p}^2}{2m} + V_0.\] Note that in contrast to the relativistic case, the expansion of the non-relativistic field \(\varphi(t, \mathbf{x})\) contains no creation operator and the one of \(\varphi^* (t,\mathbf{x})\) no annihilation operator. This is a consequence of the absence of anti-particles in the non-relativistic theory.

Scalar product

For the following discussion, it is useful to introduce a scalar product between two functions of space and time \(f(t,\mathbf{x})\) and \(g(t,\mathbf{x})\), \[(f,g)_t = \int d^3 x \left\{f^*(t,\mathbf{x}) g(t,\mathbf{x}) \right\}.\] The integer goes over the spatial coordinates at fixed time \(t\). Note that if \(f\) and \(g\) were solutions of the non-relativistic, single-particle Schrödinger equation, the above scalar product were actually independent of time \(t\) as a consequence of unitarity in non-relativistic quantum mechanics.

Normalization of mode functions

The mode functions are normalized with respect to this scalar product as \[(v_{\mathbf{p}}, v_\mathbf{q})_t = (2\pi)^3 \delta^{(3)} (\mathbf{p}- \mathbf{q}).\] One can write \[\begin{split} a_{\mathbf{p}}(t) = & (v_{\mathbf{p}}, \varphi)_t = \int d^3 x e^{i \omega_{\mathbf{p}}t-i\mathbf{p}\mathbf{x}} \varphi(t,\mathbf{x}), \\ a^{\dagger}_{\mathbf{p}}(t) = & (v^*_{\mathbf{p}}, {\varphi^*})_t = \int d^3 x e^{-i \omega_{\mathbf{p}}t+i\mathbf{p}\mathbf{x}} \varphi^*(t,\mathbf{x}). \end{split}\]

Time dependence of creation annihilation and creation operators

The right hand side depends on time \(t\) and it is instructive to take the time derivative, \[\begin{split} \partial_t a_{\mathbf{p}}(t) &= \int d^3 x \;e^{i\omega_{\mathbf{p}}t - i\mathbf{p}\mathbf{x}}[\partial_t + i\omega_{\mathbf{p}}] \varphi(t,\mathbf{x})\\ &= \int d^3 x \;e^{i\omega_{\mathbf{p}}t - i\mathbf{p}\mathbf{x}}\left[\partial_t + i\left(\frac{\mathbf{p}^2}{2m} + V_0\right)\right] \varphi(t,\mathbf{x})\\ &= \int d^3 x \;e^{i\omega_{\mathbf{p}}t - i\mathbf{p}\mathbf{x}}\left[\partial_t + i\left(-\frac{\boldsymbol{\nabla}^2}{2m} + V_0\right)\right] \varphi(t,\mathbf{x}). \end{split}\] We used here first the dispersion relation and expressed then \({\mathbf{p}^2}\) as a derivative acting on the mode function (it acts here to the left). In a final step one can use partial integration to make the derivative operator act to the right, \[\partial_t a_{\mathbf{p}}(t) = i \int d^3 x \;e^{i\omega_{\mathbf{p}}t - i\mathbf{p}\mathbf{x}}\left[-i\partial_t -\frac{\boldsymbol{\nabla}^2}{2m} + V_0\right] \varphi(t,\mathbf{x}).\] This expression confirms that \(a_{\mathbf{p}}(t)\) were time-independent if \(\varphi(t,\mathbf{x})\) were a solution of the one-particle Schrödinger equation. More general, it is a time-dependent, however. In a similar way one finds (exercise) \[\partial_t a^\dagger_{\mathbf{p}}(t) = -i \int d^3 x \;e^{-i\omega_{\mathbf{p}}t + i\mathbf{p}\mathbf{x}}\left[i\partial_t -\frac{\boldsymbol{\nabla}^2}{2m} + V_0\right] \varphi^*(t,\mathbf{x}).\]

Incoming states

To construct the S-matrix, we first need incoming and out-going states. Incoming states can be constructed by the creation operator \[a^\dagger_{\mathbf{p} }(-\infty) = \lim_{t \to -\infty} a^\dagger_{\mathbf{p}}(t).\] For example, an incoming two-particle state would be \[|\mathbf{p}_1,\mathbf{p}_2 ; \text{in} \rangle = a^\dagger_{\mathbf{p}_1}(-\infty) a^\dagger_{\mathbf{p}_2}(-\infty)|0\rangle.\]

Bosonic exchange symmetry

We note as an aside point that these state automatically obey bosonic exchange symmetry \[|\mathbf{p}_1, \mathbf{p}_2; \text{in} \rangle = |\mathbf{p}_2, \mathbf{p}_1; \text{in} \rangle,\] as a consequence of \[a^\dagger_{\mathbf{p}_1}(-\infty) a^\dagger_{\mathbf{p}_2}(-\infty) = a^\dagger_{\mathbf{p}_2}(-\infty) a^\dagger_{\mathbf{p}_1}(-\infty).\]

Fock space

We note also general states of few particles can be constructed as \[|\psi; \text{in} \rangle = C_0 |0\rangle + \int_{\mathbf{p}} C_1(\mathbf{p})\;|\mathbf{p}; \text{in} \rangle + \int_{\mathbf{p}_1,\mathbf{p}_2} C_2(\mathbf{p}_1,\mathbf{p}_2) |\mathbf{p}_1,\mathbf{p}_2; \text{in} \rangle + \ldots\] This is a superposition of vacuum (0 particles), 1-particle states, 2-particle states and so on. The space of such states is known as Fock space. In the following we will sometimes use an abstract index \(\alpha\) to label all the states in Fock space, i. e. \(|\alpha; \text{in} \rangle\) is a general incoming state. These states are complete in the sense such that \[\sum_\alpha |\alpha; \text{in}\rangle \langle \alpha; \text{in}| = \mathbb{1},\] and normalized such that \(\langle \alpha; \text{in}| \beta; \text{in}\rangle = \delta_{\alpha\beta}.\)

Outgoing states

In a similar way to incoming states one can construct outgoing states with the operators \[a^\dagger_{\mathbf{p}}(\infty) = \lim_{t \to \infty} a^\dagger_{\mathbf{p}}(t).\] For example \[|\mathbf{p}_1,\mathbf{p}_2; \text{out}\rangle = a^\dagger_{\mathbf{p}_1}(\infty)a^\dagger_{\mathbf{p}_2}(\infty) |0\rangle.\] We consider usually transition amplitudes where outgoing states appear as a “bra”, i. e. in the form \[\langle \mathbf{p}_1,\mathbf{p}_2; \text{out}| = \langle 0 | a_{\mathbf{p}_1}(\infty)a_{\mathbf{p}_2}(\infty).\] One can read this in the sense that existing particles get annihlilated at asymptotically large times before the state becomes the vacuum again.

The S-matrix

S-matrix

The S-matrix denotes now simply the transition amplitude between incoming and out-going general states \(|\alpha; \text{in} \rangle\) and \(|\beta; \text{out}\rangle\), \[S_{\beta\alpha} = \langle \beta; \text{out} | \alpha; \text{in} \rangle.\] Because \(\alpha\) labels all states in Fock space, the S-matrix is a rather general and powerful object. It contains the vacuum-to-vacuum transition amplitude as well as transition amplitudes between all particle-like excited states.

Unitarity of the S-matrix

Let us first prove that the scattering matrix is unitary, \[\begin{split} (S^\dagger S)_{\alpha\beta} &= \sum_\gamma (S^\dagger)_{\alpha\gamma} S_{\gamma\beta}\\ &= \sum_j {\langle \gamma; \text{out}| \alpha; \text{in} \rangle}^* \, \langle \gamma; \text{out}| \beta; \text{in}\rangle\\ &= \sum_j \langle \alpha; \text{in}|\gamma;\text{out}\rangle \langle \gamma; \text{out}| \beta;\text{in}\rangle \\ &= \langle \alpha; \text{in}| \beta; \text{in} \rangle \\ &= \delta_{\alpha\beta}. \end{split}\] We have used here the completeness of the out states \[\sum_j |\gamma;\text{out}\rangle \langle \gamma; \text{out}| = \mathbb{1}.\]

Decomposition of S-matrix

It is useful to decompose the \(S\)-matrix as \[S_{\alpha\beta} = \delta_{\alpha\beta} + \text{contributions from interactions.}\] The first part \(\delta_{\beta\alpha}\) is just the transition amplitude for the case that no scatering has occurred, i. e. the outgoing state is the same as the incoming state. For example, the \(S\)-matrix element for \(2\to 2\) scattering \(\langle\mathbf{q}_1,\mathbf{q}_2; \text{out} | \mathbf{p}_1,\mathbf{p}_2;\text{in}\rangle\) has a contribution \[(2\pi)^6 \left[\delta^{(3)}(\mathbf{p}_1-\mathbf{q}_1)\;\delta^{(3)}(\mathbf{p}_2-\mathbf{q}_2) + \delta^{(3)}(\mathbf{p}_1-\mathbf{q}_2)\;\delta^{(3)}(\mathbf{p}_2-\mathbf{q}_1)\right].\] This is amplitude that momenta did not change, symmetrized in a way that respects bosonic exchange symmetry. The contribution from interactions (actual scattering) is more interesting and we concentrate on it in the following.

Conservation laws, elastic and inelastic collisions

The S-matrix respects a number of conservation laws such as for energy and momentum. There can also be conservation laws for particle numbers, in particular also in the non-relativistic domain. One distinguishes between elastic collisions where particle numbers do not change, e.g. \(2 \to 2\), and inelastic collisions, such as \(2 \to 4\). In a non-relativistic theory, such inelastic processes can occur for bound states, for example two \(H_2\) - molecules can scatter into their constituents \[H_2 + H_2 \to 4H.\]

The S-matrix and correlation functions

Connection between outgoing and incoming states

What is the connection between incoming and outgoing states? Let us write \[\begin{split} a_{\mathbf{p}}(\infty)- a_{\mathbf{p}}(-\infty) &= \int_{-\infty}^\infty \partial_t a_{\mathbf{p}}(t)\\ &= i \int_{-\infty}^\infty dt \int d^3 x \; e^{i\omega_{\mathbf{p}}t - i\mathbf{p}\mathbf{x}} \left[-i\partial_t - \tfrac{\boldsymbol{\nabla}^2}{2m} +V_0\right] \varphi(t,\mathbf{x}). \end{split}\] Annihilation operators at asymptotically large incoming and outgoing times differ by an integral over space-time of the Schrödinger operator acting on the field. In momentum space with (\(px = -p^0 x^0 + \mathbf{p}\mathbf{x} = -p^0 t + \mathbf{p}\mathbf{x}\)), \[\varphi(t,\mathbf{x}) = \int \frac{dp^0}{2\omega}\frac{d^3\mathbf{p}}{(2\pi)^3} e^{ipx} \varphi(p),\] this would read \[a_{\mathbf{p}}(\infty)-a_{\mathbf{p}}(-\infty) = i\left[-p^0 + \frac{\mathbf{p}^2}{2m}+ V_0 \right]\varphi(p).\] In a similar way one finds \[\begin{split} a^\dagger_{\mathbf{p}}(\infty)-a^\dagger_{\mathbf{p}}(-\infty) &= -i \int_{-\infty}^\infty dt \int d^3 x \; e^{-i\omega_{\mathbf{p}}t + i\mathbf{p}\mathbf{x}} \left[-i\partial_t - \tfrac{\boldsymbol{\nabla}^2}{2m} +V_0\right] \varphi^*(t,\mathbf{x})\\ &= -i\left[-p^0 + \frac{\mathbf{p}^2}{2m}+ V_0 \right]\varphi^*(p). \end{split}\]

Relation between S-matrix elements and correlation functions

To create particles in the initial state we can use \(a^\dagger_\mathbf{p}(-\infty)\). In contrast, \(a_\mathbf{p}(-\infty)\) gives a vanishing contribution when it acts on the incoming vacuum \(|0 \rangle\). For the final state we can similarly use \(a_\mathbf{p}(\infty)\) to annihilate particles, while \(a^\dagger_\mathbf{p}(\infty)\) gives a vanishing contribution when it acts on \(\langle 0 |\) from the right.

So, effectively, one can replace \[a_{\mathbf{p}}(\infty) \to i\left[-p^0 + \frac{\mathbf{p}^2}{2m} + V_0\right]\varphi(p)\] and similarly \[a^\dagger_{\mathbf{p}}(-\infty) \to i\left[-p^0 + \frac{\mathbf{p}^2}{2m} + V_0\right]\varphi^*(p).\] This allows to reduce S-matrix elements to correlation functions in the in-out functional integral formalism.

Lehmann-Symanzik-Zimmermann (LSZ) reduction formula

As a concrete example, we obtain for the S-matrix element of \(2\to 2\) scattering \[\begin{split} & \langle \mathbf{q}_1, \mathbf{q}_2; \text{out}| \mathbf{p}_1, \mathbf{p}_2; \text{in}\rangle \\ & = i^4\left[-q^0_1 + \frac{\mathbf{q}^2_1}{2m} + V_0\right] \left[-q^0_2 + \frac{\mathbf{q}^2_2}{2m} + V_0\right] \left[-p^0_1 + \frac{\mathbf{p}^2_1}{2m} + V_0\right] \left[-p^0_2 + \frac{\mathbf{p}^2_2}{2m} + V_0\right] \\ & \quad \times \langle 0| \varphi(q_1)\varphi(q_2) \varphi^*(p_1) \varphi^*(p_2) |0\rangle. \end{split}\] This shows how S-matrix elements are connected to correlation functions. This relation is known as the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula, here applied to non-relativistic quantum field theory.

Relativistic scalar theories

Let us mention here that for a relativistic theory the LSZ formula is quite similar but one needs to replace \[\left[-q^0 + \frac{\mathbf{q}^2}{2m}+V_0\right] \to \left[-(q^0)^2 + \mathbf{q}^2 + m^2\right],\] and for particles \(\varphi(q) \to \phi(q)\), \(\varphi^*(q) \to \phi^*(q),\) while for anti-particles \(\varphi(q) \to \phi^*(-q)\), \(\varphi^*(q) \to \phi(-q)\).

Correlation functions from functional integrals

The (formally time-ordered) correlation functions can be written as functional integrals in the in-out formalism, \[\langle0| \varphi(q_1) \varphi(q_2) \varphi^*(p_1)\varphi^*(p_2) |0\rangle = \frac{1}{Z} \int D\varphi \; \varphi(q_1)\varphi(q_2) \varphi^*(p_1)\varphi^*(p_2) \; e^{iS[\varphi]} .\] We can now calculate S-matrix elements from functional integrals!

Partition function

Let us now consider a non-relativistic theory with the action \[S[\varphi] = \int dt d^3 x \left\{\varphi^*\left(i\partial_t + \frac{\nabla^2}{2m} -V_0 + i \epsilon \right)\varphi -\frac{\lambda}{2}(\varphi^*\varphi)^2\right\}.\] Compared to equation \(\eqref{eq:nonrelativisticactionScalar}\) we have rescaled the interaction parameter, \(\tfrac{\lambda}{4m^2} \to \lambda\) and included the \(i\epsilon\) term needed for the in-out formalism. We introduce now the partition function in the presence of source terms \(J\) as \[Z[J] = \int D\varphi \; \exp\left[iS[\varphi]+ i \int_x \left\{J^*(x)\varphi(x) + J(x)\varphi^*(x)\right\}\right],\] with \(x = (t,\mathbf{x})\) and \(\int_x = \int dt \int d^3 x.\)

Perturbation theory for partition function

Let us write the partition function formally as \[Z[J] = \int D \varphi\; \exp\left[-i\frac{\lambda}{2}\int_x\left(-i\frac{\delta}{\delta J(x)}\right)^2 \left(-i\frac{\delta}{\delta J^*(x)}\right)^2\right]\; \exp\left[iS_2[\varphi]+ i\int\left\{J^*\varphi + \varphi^*J\right\}\right],\] where the quadratic action is \[S_2[\varphi] = \int_x \varphi^* \left(i \partial_t + \frac{\boldsymbol{\nabla}^2}{2m}-V_0 + i \epsilon\right)\varphi.\] Note that when acting on the source term in the exponent, every functional derivative \(-i\frac{\delta}{\delta J(x)}\) results in a field \(\varphi^*(x)\) and so on. In this way, the quartic interaction term has been separated and written in terms of derivatives with respect to the source field. We can now pull it out of the functional integral and write \[Z[J] = \exp\left[-i\frac{\lambda}{2}\int_x\left(-i\frac{\delta}{\delta J(x)}\right)^2 \left(-i\frac{\delta}{\delta J^*(x)}\right)^2\right] Z_2[J],\] with the partition function for the quadratic theory \[Z_2[J] = \int D\varphi \; e^{i S_2[\varphi] + i\int\left\{J^*\varphi+\varphi^*J\right\}}.\] The latter is rather easy to evaluate this in momentum space, as we have seen previously. Gaussian integration yields \[Z_2[J] = \exp\left[i\int_p J^*(p)\left(-p^0 + \tfrac{\mathbf{p}^2}{2m}+V_0+i\epsilon\right)^{-1} J(p)\right].\]

Relating functional derivatives in position and momentum space

In the following it will be useful to write also the interaction term in momentum space. One may use \[\begin{split} & \frac{\delta}{\delta J(x)}= \int d^4 p \frac{\delta J(p)}{\delta J(x)}\frac{\delta}{\delta J(p)} = \int \frac{d^4 p}{(2\pi)^4} e^{-ipx} (2\pi)^4 \frac{\delta}{\delta J(p)} \\ & = \int \frac{d^4 p}{(2\pi)^4} e^{-ipx} \delta_{J(p)} = \int_p e^{-ipx}\delta_{J(p)}. \end{split}\] Here we defined the abbreviation \[\delta_{J(p)} = (2\pi)^4 \frac{\delta}{\delta J(p)}.\] In a similar way \[\frac{\delta}{\delta J^*(x)} = \int_p e^{ipx} \delta_{J^*(p)}.\] We used also \[\int_x e^{ipx} = (2\pi)^4 \delta^{(4)} (p).\]

Perturbation series

One finds for the partition function \[\begin{split} Z[J] &= \exp \left[-i \frac{\lambda}{2} \int_x \left(\tfrac{\delta}{\delta J(x)}\right)^2 \left(\tfrac{\delta}{\delta J^* (x)}\right)^2\right] Z_2[J]\\ &= \exp \left[-i \frac{\lambda}{2}\int_{k_1...k_4}\left\{(2\pi)^4 \delta^4(k_1+k_2-k_3-k_4) \delta_{J(k_1)}\delta_{J(k_2)}\delta_{J^*(k_3)}\delta_{J^*(k_4)}\right\}\right] \\ & \quad \times \exp\left[i\int_p J^*(p)\left(-p^0 + \tfrac{\mathbf{p}^2}{2m}+V_0 - i \epsilon \right)^{-1} J(p)\right]. \end{split} \label{eq:partitionFunctionPerturbativeSeries}\] One can now expand the exponential to obtain a formal perturbation series in \(\lambda\), similar to what we have seen previously for statistical field theories.

S-matrix element

Let us now come back to the S-matrix element for \(2\to 2\) scattering \[\begin{split} & \langle \mathbf{q}_1,\mathbf{q}_2; \text{out}| \mathbf{p}_1,\mathbf{p}_2;\text{in}\rangle \\ & =i^4 \left[-q^0_1+\tfrac{\mathbf{q}^2_1}{2m}+V_0\right]\left[-q^0_2+\tfrac{\mathbf{q}^2_2}{2m}+V_0\right]\left[-p^0_1+\tfrac{\mathbf{p}^2_1}{2m}+V_0\right]\left[-p^0_2+\tfrac{\mathbf{p}^2_2}{2m}+V_0\right] \\ & \quad\quad \times \left(\frac{1}{Z[J]} \delta_{J^*(q_1)}\delta_{J^*(q_2)}\delta_{J(p_1)}\delta_{J(p_2)} Z[J]\right)_{J=0}. \end{split}\] If we now insert the perturbation expansion for Z[J], we can concentrate on the contribution at order \(\lambda^1=\lambda\), because at order \(\lambda^0 = 1\) we have only the trivial S-matrix element for no scattering that we already discussed.

Order \(\lambda\)

At order \(\lambda\) we have different derivatives acting on \(Z_2 [J]\),

  • \(\delta_{J(p_1)}\) for incoming particles with momentum \(p_1\)

  • \(\delta_{J^*(q_1)}\) for outgoing particle with momentum \(q_1\)

  • \(\delta_{J(k)}\) and \(\delta_{J^*(k)}\) for the interaction term.

Propagator

At the end, all these derivatives are evaluated at \(J=J^*=0\). Therefore, there must always be derivatives \(\delta_J\) and \(\delta_J^*\) acting together on one integral appearing in \(Z_2[J]\). Note that \[%\begin{split} \delta_{J(p_1)}\delta_{J^*(q_1)}\left[i\int_p J^*(p)\left(-p^0 + \tfrac{\mathbf{p}^2}{2m}+V_0 - i \epsilon \right)^{-1} J(p)\right] =i G(p) (2\pi)^4 \delta^{(4)}(p_1-q_1). %\end{split}\] with the non-relativistic propagator in momentum space \[G(p) = \frac{1}{-p_1^0 + \tfrac{\mathbf{p}^2_1}{2m}+V_0-i \epsilon}.\]

Momentum conservation

If two derivatives representing external particles would hit the same integral in \(Z_2[J]\), one would have no scattering because \(\mathbf{p}_1 =\mathbf{q}_1\) and as a result of momentum conservation then also \(\mathbf{p}_2 =\mathbf{q}_2\). This is no real scattering. Only if a derivative representing an incoming or outgoing particle is combined with a derivative from the interaction term, this is avoided.

Resulting contribution to S-matrix

By doing the algebra one finds at order \(\lambda\) the term for scattering \[\langle\mathbf{q}_1,\mathbf{q}_2; \text{out}| \mathbf{p}_1,\mathbf{p}_2;\text{in}\rangle = -i \frac{\lambda}{2}4 \, (2\pi)^4 \delta^{(4)} (q_1 +q_2 -p_1 -p_2).\] The factor \(4=2\times2\) comes from different ways to combine functional derivatives with sources.

Momentum conservation

The overall Dirac function makes sure that the incoming four-momentum equals the out-going four-momentum, \[p^{\text{in}}= p_1 + p_2 = q_1+q_2= p^{\text{out}}.\]

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