Quantum field theory 1, lecture 14
7.4 From the S-matrix to a cross-section
Transition propability. Let us start from an S-matrix element in the form \begin{equation*} \langle \beta ;\text{out}|\alpha ;\text{in}\rangle = (2\pi )^4 \delta ^{(4)} (p^{\text{out}}-p^{\text{in}}) \, i \,{\cal T} \end{equation*} with transition amplitude \(\cal T\) which may depend on the momenta itself. (For \(2 \to 2\) scattering of non-relativistic bosons, and at lowest order in \(\lambda \), we found simply \({\cal T} = - 2\lambda \).) Let us now discuss how one can relate S-matrix elements to actual scattering cross-sections that can be measured in an experiment. We start by writing the transition probability from a state \(\alpha \) to a state \(\beta \) as \begin{equation*} P= \frac{|\langle \beta ;\text{out}|\alpha ;\text{in}\rangle |^2}{\langle \beta ;\text{out}|\beta ;\text{out}\rangle \langle \alpha ;\text{in}|\alpha ;\text{in}\rangle }. \end{equation*}
Transition rate. The numerator contains a factor \begin{equation*} \left [(2\pi )^4 \delta ^{(4)} (p^{\text{out}}-p^{\text{in}})\right ]^2 = (2\pi )^4 \delta ^{(4)} (p^{\text{out}}-p^{\text{in}}) (2\pi )^4 \delta ^{(4)}(0). \end{equation*} This looks ill defined but becomes meaningful in a finite volume \(V\) and for finite time interval \(\Delta T\). In fact \begin{equation*} (2\pi )^4 \delta ^4 (0) = \int d^4 x \, e^{i0 x} = V\Delta T. \end{equation*} For the transition rate \(\dot{P} = \frac{P}{\Delta T}\) we can therefore write \begin{equation*} \dot{P} = \frac{V(2\pi )^4 \delta ^{(4)} (p^{\text{out}}-p^{\text{in}})|{\cal T}|^2}{\langle \beta ; \text{out}|\beta ; \text{out}\rangle \langle \alpha ; \text{in}| \alpha ; \text{in}\rangle }. \end{equation*}
Normalization of incoming and outgoing states. Moreover, for incoming and outgoing two-particle states, their normalization is obtained from \begin{equation*} \begin{split} \langle \vec{p}_1,\vec{p}_2;\text{in}| \vec{q}_1,\vec{q}_2; \text{in}\rangle &= \lim _{\vec{q}_j \to \vec{p}_j}\langle \vec{p}_1,\vec{p}_2;\text{in}| \vec{p}_1,\vec{p}_2; \text{in}\rangle \\ &= \lim _{\vec{q}_j \to \vec{p}_j}\left [(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 )\right ]\\ &= \left [(2\pi )^3 \delta ^{(3)}(0)\right ]^2 \\ &= V^2. \end{split} \end{equation*}
Counting of momentum states. In a finite volume \(V= L^3\), and with periodic boundary conditions, the final momenta are of the form \begin{equation*} \vec{p} = \frac{2\pi }{L}(m,n,l), \end{equation*} with some integer numbers \(m,n,l\). We can count final states according to \begin{equation*} \sum _{m,n,l} = \sum _{m,n,l} \Delta m \Delta n \Delta l = L^3 \sum _{m,n,l}\frac{\Delta p_1 \Delta p_2 \Delta p_3}{(2\pi )^3}. \end{equation*} In the continuum limit this becomes \begin{equation*} V \int \frac{d^3p}{(2\pi )^3}. \end{equation*} The differential transition rate has one factor \(V d^3p/(2\pi )^3\) for each final state particle.
Differential transition rate. For \( 2\to 2 \) scattering, \begin{equation*} d\dot{P} = (2\pi )^4 \delta ^{(4)}(p^{\text{out}}-p^{\text{in}}) |{\cal T}|^2 \frac{1}{V} \frac{d^3 q_1}{(2\pi )^3}\frac{d^3 q_2}{(2\pi )^3}. \end{equation*} This can be integrated to give the transition rate into a certain region of momentum states.
Flux of incoming particles. We can go from the transition probability to a cross-section by dividing through the flux of incoming particles \begin{equation*}{\cal F} = \frac{1}{V} v = \frac{2 |\vec{p}_1|}{m V}. \end{equation*} Here we have a density of one particle per volume \(V\) and the relative velocity of the two particles is \(v= 2|\vec{p}_1| / m\), in the center-of-mass frame where \(|\vec{p}_1| = |\vec{p}_2|,\) for identical particles with equal mass \(m\).
Differential cross-section. This cancels the last factor \(V\) and we find for the differential cross-section \begin{equation*} d\sigma = \frac{|{\cal T}|^2 m}{2|\vec{p}_1|}(2\pi )^4 \delta ^{(4)}(p^\text{out}-p^\text{in}) \frac{d^3 q_1}{(2\pi )^3}\frac{d^3 q_2}{(2\pi )^3}. \end{equation*}
Phase space integrals. In the center-of-mass frame one has also \(\vec{p}^{\text{in}} = \vec{p}_1 + \vec{p}_2 = 0 \) and accordingly \begin{equation*} \delta ^{(4)}(p^\text{out}-p^\text{in}) = \delta (E^\text{out}-E^\text{in}) \, \delta ^{(3)}(\vec{q}_1 + \vec{q}_2). \end{equation*} The three-dimensional part can be used to perform the integral over \(\vec{q}_2\). In doing these integrals over final state momenta, a bit of care is needed because the two final state particles are indistinguishable. An outgoing state \(|\vec{q}_1, \vec{q}_2; \text{out} \rangle \) equals the state \(|\vec{q}_2, \vec{q}_1;\text{out}\rangle \). Therefore, in order to count only really different final states, one must divide by a factor \(2\) if one simply integrates \(d^3 q_1\) and \(d^3 q_2\) independently. Keeping this in mind, we find for the differential cross-section after doing the integral over \(\vec{q}_2\), \begin{equation*} d\sigma = \frac{|{\cal T}|^2 m}{2 |\vec{p}_1|(2\pi )^2} \delta (E^{\text{out}}-E^{\text{in}}) d^3q_1. \end{equation*}
Magnitude and solid angle. We can now use \begin{equation*} d^3 \vec{q}_1 = |\vec{q}_1|^2 d|\vec{q}_1|\; d\Omega _{q_1} \end{equation*} where \(d\Omega _{q_1}\) is the differential solid angle element. Moreover \begin{equation*} E^{\text{out}} = \frac{\vec{q}_1^2}{2m} + \frac{\vec{q}_2^2}{2m} + 2V_0 = \frac{\vec{q}_1^2}{m} + 2V_0 , \end{equation*} and \begin{equation*} \frac{dE^\text{out}}{d|\vec{q}_1|} = 2\frac{|\vec{q}_1|}{m}. \end{equation*} With this, and using the familiar relation \(\delta (f(x))=\delta (x-x_0) / |f^\prime (x_0)|\), one can perform the integral over the magnitude \(|\vec{q}_1|\) using the Dirac function \(\delta (E^{\text{out}}-E^{\text{in}})\). This yields \(|\vec q_1|=|\vec p_1|\) and \begin{equation*} d\sigma = \frac{|{\cal T}|^2 m^2}{ 16\pi ^2}\;d\Omega _{q_1}. \end{equation*}
Total cross-section. For the simple case where \(\cal T\) is independent of the solid angle \(\omega _{q_1}\), we can calculate the total cross-section. Here we must now take into account that only half of the solid angle \(4\pi \) corresponds to physically independent configurations. The total cross-sections is therefore \begin{equation*} \sigma = \frac{|{\cal T}|^2 m^2}{8 \pi }. \end{equation*} In a final step we use \({\cal T} = -2\lambda \) to lowest order in \(\lambda \) (equivalent to the Born approximation in quantum mechanics) and find here the cross-section \begin{equation*} \sigma = \frac{\lambda ^2 m^2}{2\pi }. \end{equation*}
Dimensions. Let us check the dimensions. The action \begin{equation*} S = \int dt\;d^3x \left \{\varphi ^*\left (i\partial _t + \tfrac{\vec{\nabla ^2}}{2m}-V_0\right ) \varphi - \tfrac{\lambda }{2}(\phi ^*\phi )^2\right \} \end{equation*} must be dimensionless. The field \(\varphi \) must have dimension \begin{equation*} [\varphi ] = \text{length}^{-\tfrac{3}{2}}. \end{equation*} The interaction strength \(\lambda \) must accordingly have dimension \begin{equation*} [\lambda ] = \frac{\text{length}^3}{\text{time}}. \end{equation*} Because \begin{equation*} \left [\tfrac{\vec{\nabla }^2}{2m}\right ] = \frac{1}{\text{time}}, \end{equation*} one has \([m] = \frac{\text{time}}{\text{length}^2}\) and therefore \([\lambda m] = \text{length}.\) It follows that indeed \begin{equation*} [\sigma ] = \text{length}^2 \end{equation*} as appropriate for a cross-section.
8 Fermions
So far we have discussed bosonic fields and bosonic particles as their excitations. Let us now turn to fermions. Fermions as quantum particles differ in two central aspects from bosons. First, they satisfy fermionic statistics. Wave functions for several particles are anti-symmetric under the exchange of particles and occupation numbers of modes can only be 0 or 1. Second, fermionic particles have half integer spin, i. e. (1/2), (3/2), and so on, in contrast to bosonic particles which have integer spin (0), (1), (2) and so on. Both these aspects lead to interesting new developments. Half-integer spin in the context of relativistic theories leads to a new and deeper understanding of space-time symmetries and fermionic statistics leads to a new kind of functional integral based on anti-commuting numbers. The latter appears already for functional integral representations of non-relativistic quantum fields. We will start with this second-aspect and then turn to aspects of space-time symmetry for relativistic theories later on.
8.1 Non-relativistic fermions
Pauli spinor fields. In non-relativistic quantum mechanics, particles with spin \(1/2\) are described by a variant of Schrödinger’s equation with two-component fields. The fields are so-called Pauli spinors with components describing spin-up and spin-down parts with respect to some axis. One can write this as \begin{equation*} \Psi (t,\vec{x}) = \begin{pmatrix} \psi _\uparrow (t,\vec{x}) \\ \psi _\downarrow (t,\vec{x}) \end{pmatrix} \end{equation*} We also use the notation \(\psi _a (t,\vec{x})\) where \(a= 1,2\) and \begin{equation*} \psi _1(t,\vec{x})= \psi _\uparrow (t,\vec{x}),\quad \quad \quad \psi _2(t,\vec{x})= \psi _\downarrow (t,\vec{x}). \end{equation*}
Pauli equation. The Pauli equation is a generalisation of Schrödinger’s equation (neglecting spin-orbit coupling), \begin{equation*} \left [\left (-i\partial _t - \tfrac{\vec \nabla ^2}{2m}+V_0\right )\mathbb{1}+\mu _B \, \vec{\sigma } \cdot \vec{B}\right ]\Psi (t, \vec{x}) = 0 , \end{equation*} or equivalently \begin{equation*} \left [\left (-i\partial _t - \tfrac{\vec \nabla ^2}{2m}+V_0\right )\delta _{ab}+\mu _B \, \vec \sigma _{ab} \cdot \vec{B}\right ]\psi _b(t, \vec{x}) = 0. \end{equation*} Here we use the Pauli matrices \begin{equation*} \sigma _1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma _2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma _3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \end{equation*} and \(\vec{B} = (B_1, B_2, B_3)\) is the magnetic field, while \(\mu _B\) is the magneton that quantifies the magnetic moment.
Attempt for an action. Based on this, one would expect that the quadratic part of an action for a non-relativistic field describing spin-\(1/2\) particles is of the form \begin{equation*} S_2 \overset{?}{=} \int dt d^3 x \left \{-\Psi ^\dagger \left [\left (-i\partial _t - \tfrac{\vec{\nabla }^2}{2m}+V_0\right )\mathbb{1} + \mu _B \, \vec \sigma \cdot \vec{B}\right ]\Psi \right \} \end{equation*} However, we also need to take care of fermionic (anti-symmetric) exchange symmetry, such that for fermionic states \begin{equation*} |\vec{p}_1, \vec{p}_2; \text{in} \rangle = -|\vec{p}_2, \vec{p}_1; \text{in} \rangle . \end{equation*} To this aspect we turn next.
8.2 Grassmann fields
Grassmann variables. So-called Grassmann variables are generators \(\theta _i\) of an algebra, and they are anti-commuting such that \begin{equation*} \theta _i \theta _j + \theta _j \theta _i = 0. \end{equation*} An immediate consequence is that \({\theta _j}^2 = 0\).
Basis. If there is a finite set of generators \({\theta _1, \theta _2, \ldots , \theta _n},\) one can write general elements of the Grassmann algebra as a linear superposition (with coefficients that are ordinary complex (or real) numbers) of the following basis elements \begin{align*} & 1, \\ & \theta _1, \theta _2, \ldots , \theta _n, \\ & \theta _1\theta _2, \theta _1\theta _3,\ldots , \theta _{2} \theta _3, \theta _{2} \theta _4,\ldots , \theta _{n-1} \theta _n, \\ & \ldots \\ & \theta _1\theta _2\theta _3 \cdots \theta _n. \end{align*}
There are \(2^n\) such basis elements, because each Grassmann variable \(\theta _j\) can be either present or absent.
Grade of monomial. To a monomial \(\theta _{j_1}\cdots \theta _{j_q}\) one can associate a grade \(q\) which counts the number of generators in the monomial. For \(A_p\) and \(A_q\) being two such monomials one has \begin{equation*} A_p A_q = (-1)^{p \cdot q} A_q A_p. \end{equation*} In particular, the monomials of even grade \begin{align*} & 1, \\ & \theta _1\theta _2, \theta _1\theta _3,\ldots , \theta _{2} \theta _3,\ldots , \theta _{n-1} \theta _n,\\ & \ldots \end{align*}
commute with other monomials, be the latter of even or odd grade.
Grassmann parity. One can define a Grassmann parity transformation \(P\) that acts on all generators according to \begin{equation*} P(\theta _j) = -\theta _j, \quad \quad \quad P^2 = \mathbb{1}. \end{equation*} Even monomials are even, odd monomials are odd under this transformation. The parity even part of the algebra, spanned by the monomials of even grade, constitutes a sub-algebra. Because its elements commute with other elements of the algebra they behave “bosonic”, while elements of the Grassmann algebra that are odd with respect to \(P\) behave “fermionic”.