5.3 Interaction and intersection
5.3.1 The mathematical description of interactions
5.3.1.2 Case study 2: Non-relativistic quantum scattering
Sub-atomic structure is most commonly investigated through scattering experiments and the description of these experiments is the primary use of QFT. This case study will present some of the basic concepts of non-relativistic scattering, which in the next section can easily be transferred into the full relativistic treatment of
scattering.9
For the elastic scattering of two spinless particles of mass m1 and m2 the complete
system can be described by the product of the wave functions of the two particles (~r1, ~r2, t) = 1(~r1, t) 2(~r2, t) . (5.3.8)
However, if the interaction only depends on the distance ~r = |~r1 ~r2| of the particles,
we can introduce the reduced mass µ =m1m2/(m1+m2), which reduces 5.3.8 to (~r, t).
Furthermore, the interaction can be approximated by a time-independent potential V (~r), and if the incident beam is switched on for a very long time compared to the time of the interaction, it reduces to the stationary wave function given by (~r, t) = (~r)e iEt/~, which solves the Schrödinger equation
✓ ~2
2µ + V ( ~r) ◆
(~r) = E (~r) . (5.3.9)
In experiments the incident wave is fairly well collimated, so that it does not interact with the detector before the scattering, but it has to be spatially large enough to keep the spreading of the wave small. However, if the interaction between the scattering particles is weak and only has an effect in a small region around the target, the incident wave can be approximated by a plane wave. This is the Born approximation:
in(~r) = ei~k0·~r.
The outgoing wave after the scattering, on the other and, has the form of a spherical wave
sc(~r) = f (✓, ')
ei~k·~r r
with an amplitude f(✓, '), dependent on the direction (✓, '), which is called the scattering amplitude. After the scattering event, the total wave function is therefore given by
(~r) = ei~k0·~r+ f (✓, ')e
i~k·~r
r . (5.3.10)
The probability current density of the wave function is given by the expression ~j(~r) = ~
2µi h
⇤⇣r ~ ⌘ ⇣r ~ ⇤⌘ i
to which for large r only the radial component contributes, which is for the incident and scattered wave respectively
9The following exposition is based on Bransden & Joachain (2000, ch. 12), Greiner & Reinhardt
Figure 5.1: Schematic illustration of a scattering experiment jin(r)=~ ~k0 µ jsc(r)=~~k µ |f(✓, ')|2 r2 . (5.3.11)
Now, an area of a detector has the size of r2d⌦ and therefore the quantity
dN = jrr2d⌦ = ~~k
µ |f(✓, ')|
2d⌦ (5.3.12)
can be interpreted as the number of particles N hitting an area of the detector per unit time. Furthermore, conjoining 5.3.11 with 5.3.12 and taking into account that in the case of elastic scattering ~k0= ~k, one can define the scattering cross section
d
d⌦ =|f(✓, ')|2
as the ratio of the number of scattered particles per unit area per unit time to the incident particle current density.
In order to find a solution to the scattering problem, the scattering amplitude can now be obtained from an integral formulation of the Schrödinger equation 5.3.9. First, since the wave function is stationary, the energy is given by E =~2~k2/
2µand
with the reduced potential U(~r) =2µ/~2V (~r) 5.3.9 can be reformulated as
⇣
+ ~k2⌘ (~r) = U (~r) (~r) . (5.3.13) A general solution to 5.3.13 consists of two parts, namely, a particular solution to the homogeneous part, which is just the incident wave, and a general solution, which can be found with the help Green’s function G0(~r ~r0). The general solution then
(~r) = in(~r) +
ˆ
G0(~r ~r0)U (~r0) (~r0)d3~r0 (5.3.14)
where the integral extends over the whole space. The role of the Green’s function G(~r ~r0), also called the propagator, can be understood from the following. In general, the Green’s function describes how one wave (~r0) relates to another wave
(~r), that is, for a free particle it will simply describe the evolution according to the free Schrödinger equation and different states will be related by the free Green’s function
(~r) = ˆ
G0(~r ~r0) (~r0)d3x .
Since it is assumed that for the scattering experiment the motion of the free particle is known completely, the question is whether the Green’s function for the interacting particles, G(~r ~r0), can be expressed by the free Green’s function. It
turns out that as a good approximation this is possible as
sc(~r) =
ˆ
G0(~r ~r0)U (~r0) (~r0)d3x
and thus the general solution to the scattering Schrödinger equation is given by 5.3.14.
The next step in finding the Green’s function explicitly is to note that the free Green’s function must be a solution to the homogeneous, that is, the free part of 5.3.13
⇣
+ ~k2⌘ ˆ G0(~r ~r0) (~r0)d3x = 0 .
One finds that this can be reformulated as (cf. Greiner & Reinhardt 2009, p. 25) ⇣
+ ~k2⌘G0(~r ~r0) = (~r ~r0)
This equation has two solutions
G+0(~r ~r0) = 1 4⇡ eik|~r ~r0| |~r ~r0| G0(~r ~r0) = 1 4⇡ e ik|~r ~r0| |~r ~r0|
corresponding to an outgoing wave from ~r and an incoming wave into ~r respectively. Since in the context of a scattering experiment only the outgoing wave is interesting, the general solution to 5.3.13 is now given by
(~r) = in(~r) 1 4⇡ ˆ eik|~r ~r0| |~r ~r0|U (~r0) (~r0)d 3~r0. (5.3.15)
An explicit solution to this can be found perturbatively with the Born series. Here the zero-order solution is given simply by the incident wave, which can then be inserted in the first-order solution and so on
0(~r) =ei~k0·~r 1(~r) = in(~r) 1 4⇡ ˆ eik|~r ~r0| |~r ~r0|U (~r0) 0(~r0)d 3~r0 n(~r) = in(~r) 1 4⇡ ˆ eik|~r ~r0| |~r ~r0|U (~r0) n 1(~r0)d3~r0
Furthermore, in the asymptotic limit of r ! 1 5.3.15 reduces to (~r) = ei~k~r 1 4⇡ eikr r ˆ e i~k~r0U (~r0) (~r0)d3~r0. (5.3.16) Upon comparison of 5.3.10 and 5.3.16 one can now see that the scattering amplitude in an integral formulation is given by
f (✓, ') = 1 4⇡
ˆ
e i~k~r0U (~r0) (~r0)d3~r0 = 1
4⇡h in| ˆU | i . (5.3.17) Here the matrix element h in| ˆU | i is usually called an element of the scattering
or S-matrix, which will receive more attention later. Generally, an element of the S-matrix Sf i gives the probability for the evolution of a state i into another state
f.
In the first Born approximation, the scattering cross section is then d d⌦ =|f(✓, ')| 2 = 1 16⇡2 ˆ e i~k~r0U (~r0) in(~r0)d3~r0 2 = 1 16⇡2 ˆ ei~q~r0U (~r0)d3~r0 2 with ~~q = ~⇣~k0 ~k ⌘
as the momentum transfer from the incident to the scattered state.
An interesting special case to end this section is that of the scattering of identical particles. I will present here only the simplest case of identical bosons. What makes scattering of identical particles special is that there is no way to distinguish between the two possible final states, a) where particle 1 is scattered in direction (✓, ') and particle 2 in the opposite direction (⇡ ✓, ' + ⇡), and case b) where particle 2 is scattered in direction (✓, ') and particle 1 in (⇡ ✓, ' + ⇡). Therefore, both possibilities have to be taken into account when calculating the scattering cross section.
In classical physics one would expect the cross section to be simply the sum of both cases
d
d⌦cl =|f(✓, ')| 2
However, in quantum physics the final state after scattering is a superposition of both cases and consequently the cross section will be
d
d⌦qm=|f(✓, ') + f(⇡ ✓, ' + ⇡)| 2
=|f(✓, ')|2+|f(⇡ ✓, ' + ⇡)|2+ 2Re [(✓, ')f⇤(⇡ ✓, ' + ⇡)]
One can see that the quantum physical scattering can clearly be distinguished from the classical case by an additional interference term.
5.3.1.3 Case study 3: Quantum electrodynamics
Classical field theory10 To begin with, it will be useful to start with the basics
of classical field theory. In particular, this should help to clarify the role of the Lagrangian in QFT. A field is a function that assigns a certain variable (~x, t) to every point in spacetime and, in the case of QFT, transforms under the Poincaré group. Consequently, the Lagrange function, L = T V,becomes a functional of the field and its first time derivative
L(t) = Lh (~r, t), ˙(~r, t)i.
Furthermore, since all Lagrange functions, appearing in fundamental physical theories today, are local, they can be expressed by a Lagrangian density11
L(t) = ˆ
L( , @µ )d3x (5.3.18)
which will henceforth be simply called the Lagrangian. This Lagrangian is local in so far that it only depends on one spatial vector x and not on any y 6= x. Nevertheless, the Lagrangian is in a certain sense not local in that it depends at every instant in time on the field values (x) over all of space.
The action integral is then given by S =
ˆ
L(t)dt = ˆ
L( , @µ )d4x
and from Hamilton’s principle of the least action follow the Euler-Lagrange equations as @L @ i @µ @L @µ i = 0 .
10The following exposition is based on Greiner & Reinhardt (1996, ch. 2) and Maggiore (2005, ch.
3).
11Here the usual relativistic notation will be used, in which @
µ=@/@xµwith spacetime coordinate
xand the signature metric ⌘µ⌫ = (+, , , ). ⇤ = @µ@µis the D’Alambert operator. The
The Euler-Lagrange equations are the equations of motion for the fields i. The most
common special cases of them are the Klein-Gordon equation for scalar particles and the Dirac equation for spin-1/2 particles. Hence, for a free real scalar field with mass
m the Lagrangian is L(x) = 1 2@µ @ µ 1 2m 2 2
from which Hamilton’s principle leads to the Klein-Gordon equation
⇤ + m2 = 0 . (5.3.19)
A simple solution to 5.3.19 is a plane wave e±ipx with p2 = m2.
The Dirac equation on the other hand is the equation of motion for spin-1/2
particles and therefore states are represented by spinors, that is, fields with four complex components satisfying certain transformation laws. However, for simplicity the spinor indices will be omitted. Using the Dirac or gamma matrices µ, the
notation of the Feynman slash /@ = µ@
µ and the Dirac adjoint ¯ = † 0 the Dirac
Lagrangian can be written as
LDirac= ¯ i /@ m (5.3.20)
from which in the usual way the Dirac equation follows as i /@ m = 0 .
Also useful for what is to come will be to take a look at the explicitly covariant formulation of electrodynamics. The electrodynamic field can be described by using a four-vector potential Aµ=⇣A0, ~A⌘, which is related to the electric and the magnetic
field strengths via
~
E = @ ~A
@t rA~ 0 ~
B =~r ⇥ ~A . Now, the field strength tensor can be defined as
Fµ⌫ = @µA⌫ @⌫Aµ.
The Lagrangian for the free electromagnetic field is then given by Le.m.= 1 4Fµ⌫F µ⌫ = 1 2 ⇣ ~ E2 B~2⌘ (5.3.21)
From which follows the equation of motion
A coupling of the electromagnetic field to an external source or charged matter field can be reached by introducing the conserved vector current jµ= q ¯ µ of a Dirac
field with charge q into the equation of motion 5.3.22 such that
@µFµ⌫ = jµ (5.3.23)
which is a consequence of the Lagrangian L = Le.m.+Lint. =
1 4Fµ⌫F
µ⌫ q ¯ µ A
µ. (5.3.24)
As a final comment, the transition from the Lagrange to the Hamilton formalism is made by introducing the canonically conjugate field
⇡(x) = @L @(@0 i)
.
A Legendre transformation of the Lagrangian then leads to the Hamiltonian density H(x) = ⇡i(x)@0 i(x) L
from which the total Hamiltonian follows as H =
ˆ Hd3x Furthermore, the equations of motion are now given as
˙ =@H @⇡ r @H @(r⇡) ˙⇡ =@H @ r @H @(r ).
Canonical quantisation formalism We will now take the leap from classical field theory to QFT. Field quantisation is an involved topic (cf. Greiner & Reinhardt 1996), that here has to be omitted completely for the sake of brevity. I will simply take it as a given result of the quantisation procedure that classical fields can be promoted to quantum fields in the following way.
A general classical solution to the Dirac equation can be written as (x) = ˆ d3p (2⇡)3p2E p X s=1,2
ap,sus(p)e ipx+ b⇤p,svs(p)eipx (5.3.25)
that is, as an expansion in a complete set of plane waves, with s being the degree of freedom of spin. This field is turned into a quantum field by imposing the equal time anticommutator relations
n ˆa(x, t), ˆ† b(y, t) o = ab (3)(x y) n ˆa(x, t), ˆb(y, t)o=nˆ† a(x, t), ˆ†b(y, t) o = 0 .
Here the anticommutator ensures that the described Fermions, that is, spin-1/2
particles conform to Fermi-Dirac statistics, as opposed to Bose-Einstein statistics for Bosons. Furthermore, the coefficients ap,s and bp,s are promoted to operators,
satisfying the equal time anticommutator relations n ˆ a(p, s), ˆa†(q, r)o=nˆb(p, s), ˆb†(q, r)o= (2⇡) sr (3)(p q) {ˆa(p, s), ˆa(q, r)} =nˆa†(p, s), ˆa†(q, r)o= 0 n ˆb(p, s), ˆb(q, r)o=nˆb†(p, s), ˆb†(q, r)o= 0 .
The operator ˆa†(p, s)is the creation operator for particles with U(1) charge Q = 1,
while ˆb†(p, s) is the creation operator for antiparticles with charge Q = 1. The
operators ˆa(p, s) and ˆb(p, s) are the annihilation operators respectively. The vacuum state can now be defined as
ˆ
a(p, s)|0i = ˆb(p, s) |0i = 0 .
One can then use the creation operators to build up any state in Fock space from the vacuum state
|p1,s, . . . pn,si = (2Ep1) 1/2 · · · (2Epn) 1/2 ˆ a†(p1, s)· · · ˆa†(pn, s)|0i (5.3.26)
with normalisation constants (2Epn) 1/2 .
The quantisation of the electromagnetic field on the other hand differs from that of the Dirac field, since the electromagnetic potential, being a gauge field, has more degrees of freedom. Furthermore, the fact that the photon has zero mass poses particular problems. However, the process can be simplified by working in the Coulomb gauge again, in which ~r · ~A = 0 and A0 = = 0. A general solution to
the classical wave equation for the electromagnetic potential 5.3.1 can be written as ~ A = ˆ d3p (2⇡)3p2! ~ p X =1,2 ⇥ ~✏(~p, )a~p, e ipx+ ~✏⇤(~p, )a⇤~p, eipx ⇤
with the polarisation vectors ~✏(~p, ). This field is now quantised, by again promoting the coefficients a~p, and a⇤~p, to operators, such that the operator field now reads
ˆ A(x) = ˆ d3p (2⇡)3p2! ~ p X =1,2 h ~✏(~p, )ˆa~p, e ipx+ ~✏⇤(~p, )ˆa†~p, eipx i .
The annihilation and creation operators ˆap,~ and ˆa†~p, now fulfil the equal time commutator relations h ˆ a~p, , ˆa†~q, 0 i =(2⇡)3 (3)(~p ~q) 0 ⇥ ˆ a~p, , ˆa~q, 0⇤= h ˆ a†~p, , ˆa†~q, 0 i = 0 . (5.3.27)
Interactions After the quantisation of the Dirac and the electromagnetic fields is done, we are now in a position to describe interactions between fermions involving the electromagnetic force. A good example for an electromagnetic interaction is the so- called Bhabha scattering, that is, the scattering of electrons with their antiparticles, the positron, through which muons and anti-muons are produced. Up to energies of 91.16 GeV, the mass of the Z Boson, this interaction can be treated entirely in quantum electrodynamics and any influence from the electroweak force can be neglected (cf. Greiner & Reinhardt 2009, p. 142). Similar to the Hamiltonian in the first case study, the Lagrangian now consists of parts for the free motion of the matter and force fields and one term that couples both
ˆ
LQED = ˆLDirac+ ˆLe.m.+ ˆLint. (5.3.28)
ˆ LDirac= ˆ i /¯ @ m ˆ (5.3.29) ˆ Le.m.= 1 4Fˆµ⌫Fˆ µ⌫ 1 2 ⇣ @µAˆµ ⌘2 (5.3.30) ˆ Lint.= q ˆ¯ µ ˆˆAµ. (5.3.31)
Most of the terms in this Lagrangian are already familiar from the presentation of classical field theory above. Only the second term on the right hand side of ˆLe.m.
is new and appears, because we are no longer working in the Coulomb gauge, but in the Feynman gauge. In what follows, it will be practical to use the Hamiltonian, derived from ˆLQED, which can also be divided into a free and an interaction part
ˆ
H = ˆH0+ ˆHint.. (5.3.32)
From our discussion of non-relativistic scattering above, we already know that the final goal is to calculate the cross section, which can be derived in a straightforward way from the S-matrix element
S =h out| ˆS| ini . (5.3.33)
Here again the operator ˆS is simply the time evolution operator, which can be written as ˆ S = lim t2!+1 lim t1! 1 e i ˆH(t2 t1)
Furthermore, ˆS is a unitary operator. This can be seen if we take S to be the probability amplitude, a complex number, of which the absolute square gives the probability P for the initial state to evolve into the final state
P =| h out| ˆS| ini |2 , P 2 R|0 P 1
If we now sum up over all possible outcomes n, the probability is always unity
P = 1 =X
n
| h n| ˆS| ini |2
which means that ˆS must be unitary operator.
The states, on the other hand, are usually constructed as momentum and spin eigenstates, long before and after the scattering event. It is most common in QFT to work in the Heisenberg picture, in which states are constant
| , tiH =| , 0iH =| iH and operator fields evolve in time
ˆH(t) = ei ˆHtˆH(0)e i ˆHt
However, it is always possible to transfer into the Schrödinger picture, in which states evolve in time and operator fields are constant
| , tiS =e i ˆHt| iH ˆS =e i ˆHtˆH(t)ei ˆHt.
One of the basic methods to evaluate the S-matrix is to apply the LSZ reduction formula, named after Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann, who came to realise that the S-matrix 5.3.33 can be reformulated in a way that reduces its calculation essentially to the evaluation of vacuum expectation values for time ordered products of field operators
h0|Tnˆ(x1) . . . ˆ(xn)
o
|0i (5.3.34)
where the T indicates time ordering, viz., t1 > t2 > . . . > tn. This is what is called
the n-point Green’s function, G (x1, . . . , xn), the role of which can be elucidated by
remembering how it was used in the previous case study of non-relativistic scattering. Given that with 5.3.26 any state can be build up from the vacuum, the possibility of the LSZ reduction should be intuitively clear. Now, keeping in mind that ˆS is the time evolution operator, evaluating the Green’s function amounts to calculating the time evolution under the condition that all momenta of the initial and final states are different, and means that all non-interacting particles are neglected.
The next problem one encounters is that interacting fields follow complicated