Quantum field theory 1, lecture 12

7 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. The relativistic case will be discussed in full glory in the second part of the lecture course.

7.1 Scattering of non-relativistic bosons

Mode function expansion. Let us recall that one can expand fields in the operator picture as follows \begin{equation*} \varphi (t, \vec{x}) = \int _{\vec{p}} v_{\vec{p}}(t,\vec{x}) \, a_{\vec{p}}, \quad \quad \quad \varphi ^\dagger (t,\vec{x}) = \int _{\vec{p}} v^*_{\vec{p}}(t, \vec{x}) \, a^{\dagger }_{\vec{p}}, \end{equation*} with \(\int _{\vec{p}} = \int \tfrac{d^3 p}{(2\pi )^3}\), annihilation operators \(a_{\vec{p}}\), creation operators \(a^\dagger _{\vec{p}}\), and the mode functions \begin{equation*} v_{\vec{p}}(t,\vec{x}) = e^{-i\omega _{\vec{p}}t + i\vec{p}\vec{x}}. \end{equation*} The dispersion relation in the non-relativistic limit is \begin{equation*} \omega _{\vec{p}} = \frac{\vec{p}^2}{2m} + V_0. \end{equation*} Note that in contrast to the relativistic case, the expansion of \(\varphi (t, \vec{x})\) contains no creation operator and the one of \(\varphi ^* (t,\vec{x})\) no annihilation operator. This is a consequence of the absence of anti-particles.

Scalar product. For the following discussion, it is useful to introduce a scalar product between two functions of space and time \(f(t,\vec{x})\) and \(g(t,\vec{x})\), \begin{equation*} (f,g)_t = \int d^3 x \left \{f^*(t,\vec{x}) g(t,\vec{x}) \right \}. \end{equation*} 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 \begin{equation*} (v_{\vec{p}}, v_{\vec{p}\,'})_t = (2\pi )^3 \delta ^{(3)} (\vec{p}- \vec{p}\,'). \end{equation*} One can write \begin{equation*} \begin{split} a_{\vec{p}} = & (v_{\vec{p}}, \varphi )_t = \int d^3 x e^{i \omega _{\vec{p}}t-i\vec{p}\vec{x}} \varphi (t,\vec{x}), \\ a^{\dagger }_{\vec{p}} = & (v^*_{\vec{p}},{\varphi ^*})_t = \int d^3 x e^{i \omega _{\vec{p}}t-i\vec{p}\vec{x}} \varphi ^*(t,\vec{x}). \end{split} \end{equation*}

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

Incoming states. To construct the S-matrix, we first need incoming and out-going states. Incoming states can be constructed by the creation operator \begin{equation*} a^\dagger _{\vec p }(-\infty ) = \lim _{t \to -\infty } a^\dagger _{\vec p}(t). \end{equation*} For example, an incoming two-particle state would be \begin{equation*} |\vec p_1,\vec p_2 ; \text{in} \rangle = a^\dagger _{\vec p_1}(-\infty ) a^\dagger _{\vec p_2}(-\infty )|0\rangle . \end{equation*}

Bosonic exchange symmetry. We note as an aside point that these state automatically obey bosonic exchange symmetry \begin{equation*} |\vec p_1, \vec p_2; \text{in} \rangle = |\vec p_2, \vec p_1; \text{in} \rangle , \end{equation*} as a consequence of \begin{equation*} a^\dagger _{\vec p_1}(-\infty ) a^\dagger _{\vec p_2}(-\infty ) = a^\dagger _{\vec p_2}(-\infty ) a^\dagger _{\vec p_1}(-\infty ). \end{equation*}

Fock space. We note also general states of few particles can be constructed as \begin{equation*} |\psi ; \text{in} \rangle = C_0 |0\rangle + \int _{\vec{p}} C_1(\vec{p})\;|\vec{p}; \text{in} \rangle + \int _{\vec{p_1},\vec{p_2}} C_2(\vec{p_1},\vec{p_2}) |\vec{p_1},\vec{p_2}; \text{in} \rangle + \ldots \end{equation*} 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 \begin{equation*} \sum _\alpha |\alpha ; \text{in}\rangle \langle \alpha ; \text{in}| = \mathbb{1}, \end{equation*} 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 \begin{equation*} a^\dagger _{\vec{p}}(\infty ) = \lim _{t \to \infty } a^\dagger _{\vec{p}}(t). \end{equation*} For example \begin{equation*} |\vec{p}_1,\vec{p}_2; \text{out}\rangle = a^\dagger _{\vec{p}_1}(\infty )a^\dagger _{\vec{p}_2}(\infty ) |0\rangle . \end{equation*}

7.2 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 \(|\beta ; \text{out}\rangle \), \begin{equation*} S_{\beta \alpha } = \langle \beta ; \text{out} | \alpha ; \text{in} \rangle . \end{equation*} 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{equation*} \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} \end{equation*} We have used here the completeness of the out states \begin{equation*} \sum _j |\gamma ;\text{out}\rangle \langle \gamma ; \text{out}| = \mathbb{1}. \end{equation*}

Conservation laws. 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 \begin{equation*} H_2 + H_2 \to 4H. \end{equation*}

Connection between outgoing and incoming states. What is the connection between incoming and outgoing states? Let us write \begin{equation*} \begin{split} a_{\vec{p}}(\infty )- a_{\vec{p}}(-\infty ) &= \int _{-\infty }^\infty \partial _t a_{\vec{p}}(t)\\ &= i \int _{-\infty }^\infty dt \int d^3 x \; e^{i\omega _{\vec{p}}t - i\vec{p}\vec{x}} \left [-i\partial _t - \tfrac{\vec{\nabla }^2}{2m} +V_0\right ] \varphi (t,\vec{x}). \end{split} \end{equation*} 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 (\(px = -p^0 x^0 + \vec{p}\vec{x} = -p^0 t + \vec{p}\vec{x}\)), \begin{equation*} \varphi (t,\vec{x}) = \int \frac{dp^0}{2\omega }\frac{d^3\vec{p}}{(2\pi )^3} e^{ipx} \varphi (p), \end{equation*} this would read \begin{equation*} a_{\vec{p}}(\infty )-a_{\vec{p}}(-\infty ) = i\left [-p^0 + \frac{\vec{p}^2}{2m}+ V_0 \right ]\varphi (p). \end{equation*} In a similar way one finds \begin{equation*} \begin{split} a^\dagger _{\vec{p}}(\infty )-a^\dagger _{\vec{p}}(-\infty ) &= -i \int _{-\infty }^\infty dt \int d^3 x \; e^{-i\omega _{\vec{p}}t + i\vec{p}\vec{x}} \left [-i\partial _t - \tfrac{\vec{\nabla }^2}{2m} +V_0\right ] \varphi ^*(t,\vec{x})\\ &= -i\left [-p^0 + \frac{\vec{p}^2}{2m}+ V_0 \right ]\varphi ^*(p). \end{split} \end{equation*}

Relation between S-matrix elements and correlation functions. For concreteness, let us consider \(2 \to 2\) scattering with incoming state \begin{equation*} |\vec{p_1}, \vec{p_2}; \text{in}\rangle = a^\dagger _{\vec{p}_1}(-\infty ) a^\dagger _{\vec{p}_2}(-\infty )|0\rangle , \end{equation*} and out-going state \begin{equation*} |\vec{q_1}, \vec{q_2}; \text{out}\rangle = a^\dagger _{\vec{q}_1}(\infty ) a^\dagger _{\vec{q}_2}(\infty )|0\rangle . \end{equation*} The S-matrix element can be written as \begin{equation*} \begin{split} S_{\vec{q}_1\vec{q}_2,\vec{p}_1\vec{p}_2} &= \langle \vec{q}_1,\vec{q}_2; \text{out} | \vec{p}_1,\vec{p}_2;\text{in}\rangle \\ &= \langle 0 | T\{a_{\vec{q}_1}(\infty )\;a_{\vec{q}_2} (\infty )\; a^\dagger _{\vec{p}_1} (-\infty ) \;a^\dagger _{\vec{p}_2} (-\infty ) \}|0\rangle . \end{split} \end{equation*} We have inserted a time-ordering symbol but the operators are time-ordered already anyway.

Now, one can use \begin{equation*} a_{\vec{q}_1}(\infty ) = a_{\vec{q}_1}(-\infty ) + i\left [-q^0_1 + \frac{\vec{q}^2_1}{2m} + V_0\right ]\psi (q_1). \end{equation*} However, \(a_{\vec{q}_1}(-\infty )\) is moved to the right by time ordering and leads to a vanishing contribution because of \begin{equation*} a_{\vec{q}_1}(-\infty ) |0\rangle = 0. \end{equation*} So, effectively under time ordering, one can replace \begin{equation*} a_{\vec{q}_1}(\infty ) \to i\left [-q^0_1 + \frac{\vec{q}^2_1}{2m} + V_0\right ]\varphi (q_1). \end{equation*} By a similar argument, one can replace creation operators for \(t\to -\infty \) like \begin{equation*} a^\dagger _{\vec{p}_1}(-\infty ) \to i\left [-p^0_1 + \frac{\vec{p}^2_1}{2m} + V_0\right ]\varphi ^*(p_1). \end{equation*} The above argument is not fully correct. There is one contribution from the operators \(a_{\vec{q}}(-\infty )\) we have forgotten here. In fact, the replacements \(a_{\vec{q}_1} (\infty ) \to a_{\vec{q}_1} (-\infty )\) and \(a_{\vec{q}_2} (\infty ) \to a_{\vec{q}_2} (-\infty )\) give \begin{equation*} \langle 0|a_{\vec{q}_1} (-\infty )\;a_{\vec{q}_2} (-\infty )\; a^{\dagger }_{\vec{p}_1} (-\infty )\;a_{\vec{p}_2} (-\infty )|0\rangle . \end{equation*} We need to commute the annihilation operators to the right using the commutation relation \begin{equation*} \left [a_{\vec{q}} (-\infty ),a^{\dagger }_{\vec{p}} (-\infty ) \right ] = (2\pi )^3 \delta ^{(3)}(\vec{p}-\vec{q}). \end{equation*} This gives rise to a contribution to the S-matrix element \begin{equation*} (2\pi )^6 \left [\delta ^{(3)}(\vec{p}_1-\vec{q}_1)\;\delta ^{(3)}(\vec{p}_2-\vec{q}_2) + \delta ^{(3)}(\vec{p}_1-\vec{q}_2)\;\delta ^{(3)}(\vec{p}_2-\vec{q}_1)\right ]. \end{equation*} But this is just the “transition” amplitude for the case that no scattering has occurred! There is always this trivial part of the S-matrix and in fact one can write \begin{equation*} S_{\alpha \beta } = \delta _{\alpha \beta } + \text{contributions from interactions.} \end{equation*} Let us keep this in mind and concentrate on the contribution from interactions in the following.

Interacting part. We obtain thus for the S-matrix element \begin{equation*} \begin{split} & \langle \vec{q_1}, \vec{q_2}; \text{out}| \vec{p_1}, \vec{p_2}; \text{in}\rangle \\ & = i^4\left [-q^0_1 + \frac{\vec{q}^2_1}{2m} + V_0\right ] \left [-q^0_2 + \frac{\vec{q}^2_2}{2m} + V_0\right ] \left [-p^0_1 + \frac{\vec{p}^2_1}{2m} + V_0\right ] \left [-p^0_2 + \frac{\vec{p}^2_2}{2m} + V_0\right ] \\ & \quad \times \langle 0| T\{ \varphi (q_1)\varphi (q_2) \varphi ^*(p_1) \varphi ^*(p_2) \} |0\rangle . \end{split} \end{equation*} This shows how S-matrix elements are connected to time ordered 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 \begin{equation*} \left [-q^0 + \frac{\vec{q}^2}{2m}+V_0\right ] \to \left [-(q^0)^2 + \vec{q}^2 + m^2\right ], \end{equation*} 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 time-ordered correlation functions can be written as functional integrals, \begin{equation*} \langle 0| T\{\varphi (q_1) \varphi (q_2) \varphi ^*(p_1)\varphi ^*(p_2)\}|0\rangle = \frac{\int D\varphi \; \varphi (q_1)\varphi (q_2) \varphi ^*(p_1)\varphi ^*(p_2) \; e^{iS[\varphi ]}}{\int D\varphi \; e^{iS[\varphi ]}}. \end{equation*} We can now calculate S-matrix elements from functional integrals!