# (p.495) APPENDIX C SCATTERINGIN QUANTUM THEORY

# (p.495) APPENDIX C SCATTERINGIN QUANTUM THEORY

You don't understand anything until you learn it more than one way.M. Minsky

Here is a brief review of the description of scattering in quantum field theory. The purpose is to collect some useful formulæ and derive some results in detail to help avoid the confusion which can sometimes arise due to the alternative descriptions adopted in the literature to deal with this vast subject; the reader is directed to standard textbooks such as Newton (1982) or Weinberg (1995) for details.

# C.1 Scattering

Because of the impossibility of following the details of a microscopic interaction, all measurements in particle physics are performed by scattering experiments, so that the quantities of interest are the probability amplitudes for a given transition. In order to deal with interacting particles the Hamiltonian can be split in the form *H* = *H* _{0} + *H* _{int}, where *H* _{0} describes the free motion and *H* _{int} describes all the interactions among the particles, assumed to vanish in the remote past and future, when they are sufficiently far apart.

It is useful to work with states that have simple transformations under the Lorentz group: this is the case for the (rather uninteresting) single-particle states and for states consisting of several non-interacting particles, which are direct products of single-particle states. One therefore introduces the so-called ‘in’ and ‘out’ states, which are eigenstates of the *full* Hamiltonian

*α*stands for the full specification of number and type of particles, momenta, spins, etc.). For these states the particle content is specified at time

*t*= –∞ and

*t*= +∞respectively: e.g. the state |

*α*

_{(in)}〉 contains the set of particles specified by the quantum numbers

*α*at

*t*= –∞, and not necessarily at other times. Such (p.496) specification in the far past and future corresponds to

*free*states at such time |

*α*〉 = |

*i*,

*f*〉, since

*H*

_{int}→ 0 for

*t*→ ±∞:

^{164}. The connection between the free states and the ‘in’ and ‘out’ states which is appropriate for wave packets (superpositions of energy eigenstates) is given by the conditions

The amplitudes for finding in the remote future the system in the (free) final state |*f*〉, when it was prepared to be in the (free) initial state |*i*〉 in the remote past are the set of numbers of interest in scattering theory

*S*(scattering) matrix. In other words, the ‘in’ and ‘out’ states form two (complete) sets of basis states spanning the Hilbert space, and the coefficients

*S.*describe the change of basis among them, or how a given ‘in’ state is expanded in terms of ‘out’ states:

*S*=

_{fi}*δ*.

_{fi}The *S* matrix is unitary, since^{165}

(p.497)
The analogous relation *Σ _{γ} S_{γα}S*.

***=

_{γβ}*δ*(not equivalent to the previous one for infinite matrices) also holds. The unitarity of the

_{βα}*S*matrix is the expression of probability conservation:

*any*given state |

*i*

_{(in)}〉 whose content is specified in the remote past will evolve into

*some*of the states |

*f*

_{(out)}〉 whose content is specified in the remote future with unity probability:

*any*given state |

*f*

_{(out)}〉 should be obtainable from

*some*|

*i*

_{(in)}〉 state. This also requires

*operator S*is also introduced, which is defined to have matrix elements between

*free*states corresponding to the elements of the

*S*matrix

^{166}:

The unitarity of the *S* matrix implies that of the *S* operator:

*H*

_{0}is factored out, so that their time evolution only depends on

*H*

_{int}, while the operators

*O*evolve according to

*H*

_{0}:

(p.498)
(the superscript *(I)* denotes quantities in the interaction picture). The relations with Schrödinger and Heisenberg pictures are

*U*,

^{(I)}(t*t*

_{0}) describing time evolution from time

*t*

_{0}to

*t*in the interaction picture satisfies the equation

*U*

^{(I)}(t_{0},

*t*

_{0}) = 1, and its action on free states is

*U*transforms the free |

^{(I)}*i*,

*f*〉 (eigenstates of

*H*

_{0}) into |

*i*

_{(in)}〉, |

*f*

_{(out)}〉 (eigenstates of

*H*) so that

*S*operator is obtained

*U*and the time evolution of the states amounts to having already solved the scattering problem.

^{(I)}The explicit form of the time evolution operator in the interaction picture is

Another expression for the scattering operator is obtained (omitting the superscripts in the following) from the Lippmann–Schwinger integral equation for the ‘in’ states (Sakurai, 1985)

(p.499) thus defining the transition matrix, with elements

*T*, in analogy with (C.8), such that

^{167}

*M*is a Lorentz-invariant amplitude.

_{fi}The transition matrix elements give the differential transition rate (probability per unit time, normalized to the number of particles) as

*N*and

_{i}*N*are the number of particles in the initial and final states,

_{f}*V*the normalization volume and the last factor is the Lorentz-Invariant Phase Space volume element for the final state particles, denoted as

*d(*LIPS). For the decay of a particle of mass

*M*(

*N*= 1) the above expression gives the relativistic generalization of Fermi's golden rule

_{i}*m*

_{1},

*m*

_{2}(

*N*= 2) is

_{i}*S*matrix (C.9) gives the following relation for the transition matrix:

(p.500)
(valid for *Ei* = *Ef*), sometimes written formally as

*T*operator defined in eqn (C.16) trades the complication of the ‘in’ and ‘out’ states for that of having to deal with an operator which does not share the simple properties of the Hamiltonian, such as Hermiticity. Indeed the description of scattering can be done entirely in terms of free states, the ‘in’ and ‘out’ states being required only in the connection of the transition matrix elements with the Hamiltonian.

# C.2 Approximations

If the interaction is weak, i.e. the matrix elements of *H* _{int} are much smaller than those of *H* _{0}, then the ‘in’ states are not too different from free states, so that instead of the exact equation (C.15) a good approximation is

*Born approximation*: in this case, by comparing with eqn (C.16) the operator

*T*is seen to coincide with the interaction Hamiltonian

*H*int and is therefore Hermitian; this can also be seen from the unitarity relation (C.20) in the limit in which the right-hand side, of second order in

*T*, can be neglected. In this approximation therefore the following relation holds for the transition amplitudes

(p.501)
where the step function orders the interaction terms in time (note that *H* _{int} *(t)* and *H* _{int} *(t′)* need not commute if *t* ≢ *t′*). The above expression is often written as

^{168}T

*α*= 1, 2, …,

_{i}*n*; T reorders a product of time-labelled operators according to their times, so that the latest one occurs first:

*H*int is small enough so that the first few terms suffice. While this is often not the case, (e.g. when dealing with weak transitions among particles which also interact through strong or electromagnetic interactions) the formalism can be adapted to deal with the case in which the interaction contains two terms, one strong and one weak:

*α*

_{S(in,out)}〉 defined as what the ‘in’ and ‘out’ states would be if

*H*were absent, and obtaining

_{W}*HS*due to some selection rule (which makes the existence of

*H*relevant) the second term in eqn (C.29) vanishes and

_{W}*H*, the strong but ‘uninteresting’ part of the interaction. This leads to a picture in which, although (p.502) the complete interaction

_{S}*H*

_{int}is not weak, the first-order approximation in which the effect of

*H*on the states can be neglected in eqn (C.30) might still be valid:

_{W}*distorted Born approximation*, valid at first order in

*H*but to all orders in

_{W}*H*; it is the same as the previous case except that eigenstates of

_{S}*H*

_{0}+

*H*are used instead of free states (eigenstates of

_{S}*H*

_{0}). The difference is due to the presence of interactions induced by

*H*in the initial and final states, among the particles participating in the transition induced by

_{S}*H*; note however that for single particle states (e.g. the initial state for a decay process) there are no interactions and |

_{W}*i*

_{S(in)}〉 = |

*i*〉.

Introducing operators *T _{S}* ,

*T*for the strong and weak parts of the interaction:

_{W}_{W}gives ${S}_{S}^{\u2020}{S}_{S}=1$ and at first order

*T*in the Born approximation.

_{W}In the case of strongly interacting particles the first-order approximation in the weak interaction which induces a transition is not simply equivalent to taking the (Hermitian) weak Hamiltonian as the transition matrix.

## Notes:

(164) The above is also valid in the Heisenberg picture of motion, in which the states do not depend explicitly on time but contain the whole ‘history’ of time evolution. The ‘in’ states are those containing incoming plane waves and outgoing spherical waves, while the ‘out’ states contain outgoing plane waves and incoming spherical waves.

(165)
This proof requires the completeness of the ‘in’ and ‘out’ states: in the case at hand in which *all* interactions vanish at times *t* → ±∞these can be the scattering states, specified at those times as containing free particles, since no bound states can exist; in the general case in which bound states are possible they should also be included in the complete set.

(166)
Note that this is not the operator *Ŝ* which is sometimes defined to map an ‘in’ scattering state with a given plane-wave particle content at *t* = –∞into the ‘out’ state with the same particle content at *t* = +∞: |*α* _{(in)}〉 = ‘*Ŝ*|*α* _{(out)}〉; such an operator is guaranteed to exist (and be unitary) byWigner's theorem (Chapter 1) if the set of scattering states is complete, i.e. in absence of bound states.

(167)
Care should be taken as several alternative definitions of the transition matrix can be found in the literature, differing by 2*π* factors, the inclusion of delta functions or energy factors (see e.g. Donoghue *et al.* (1992) for a discussion); the one in eqn (C.15) is not Lorentz-invariant and keeps a close connection to the non-relativistic theory. Of course the final expressions for physical quantities such as cross sections or decay rates are the same in every notation.

(168)
Not to be confused with the time reversal operator *T*!