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Erik Verlinde AN INTRODUCTION TO QUANTUM FIELD THEORY Erik Verlinde Theory Division, CERN, CH-1211 Geneva 23, Switzerland Abstract The basic concepts and ideas underlying quantum eld theory are presented, together with a brief discussion of Lagrangian eld theory, symmetries and gauge invariance. The use of the Feyn- man rules to calculate scattering amplitudes and cross sections is illustrated in the context of quantum electro dynamics. 1. INTRODUCTION Quantum eld theory provides a successful theoretical framework for describing elementary particles and their interactions. It combines the theory of sp ecial relativity and quantum mechanics and leads to a set of rules that allows one to compute physical quantities that can b e compared with high energy exp eriments. The physical quantityof most interest in high energy exp eriments is the cross-section . It determines directly the number of events that an exp erimentalist will b e able to see, and also is in a direct way related to the quantum mechanical probabilities that can b e computed with the help of eld theory. In these notes I will present the basic concepts and ideas underlying quantum eld theory. Starting from the quantum mechanics of relativistic particles I will intro duce quantum eld op erators and explain how these are used to describ e the interactions among elementary particles. These notes also contain a brief discussion of Lagrangian eld theory, symmetries and gauge invariance. Finally, after presenting a heuristic derivation of the Feynman rules, I'll illustrate in the context of quantum electro dynamics how to calculate the amplitude that determines the cross section for a simple scattering pro cess. But b efore we turn to the quantum mechanics, let us rst discuss the meaning of the cross section in more familiar classical terms. 1.1 From Events to Cross Sections For concreteness, let us consider an exp eriment with two colliding particle b eams in which one b eam consists of, say, electrons and the other b eam of p ositrons. An exp erimen- talist can now collect and count the events in which an electron and a p ositron collide and pro duce several other particles. The number of events N that o ccur in a certain amount of time dep ends on various exp erimental details, in particular the densities and of 1 2 the particles inside the b eams and of course the particle velo cities ~v and ~v . 1 2 To nd out how these parameters a ect the number of events, let us rst consider the probability that an event o ccurs inside a a small volume V and within a time interval t. In this case we can assume that the densities and velo cities are constant. Now imagine that one of the particles, say the electron, has a nite cross-section as seen from the p ersp ective of the other particle, the p ositron. After a time t each electron has moved with resp ect to the p ositrons over a distance j~v ~v jt, where ~v ~v is the 1 2 1 2 relativevelo city, and in this way it has swept out a part of space with a volume equal to j~v ~v jt. The probability that in this time the electron has collided with a p ositron 1 2 is equal to the numb er of p ositrons inside this volume, whichis j~v ~v jt. Finally, 2 1 2 to obtain the total number N of collisions wehavetomultiply this with the number of electrons inside the volume V ; this number is V . In this waywe nd that the 1 number of events that o ccur in V and t is N = j~v ~v jV t (1) 1 2 1 2 In a realistic situation the densities (~x; t) of the particles and their velo cities are not i constant in space and time. Instead of the velo city ~v it is also more appropriate to use ~ the current density j that describ es the ux of the particles p er unit of area. It is de ned ~ by j = ~v where ~v is the lo cal velo city at a certain p oint in space and time. By rep eating the ab ove argumentation we can express the number of events in an in nitesimal volume 3 ~ ~ dV and time interval dt as dN = j j j jd ~xdt. Then byintegrating of space and 2 1 1 2 time we obtain the following expression for the total number of events N Z N = dt L(t); (2) 1) where Z 3 ~ ~ L(t)= d xj (~x; t)j (~x; t) (~x; t)j (~x; t)j: (3) 1 2 2 1 All exp erimental details that in uence the number of events are combined in the time integral of L(t), which is called the integrated luminosity. The integrated luminositydoes not dep end on the kind of pro cess that one is interested in, and so it can b e measured by lo oking at a reference pro cess for which the cross section is accurately known. With this knowledge one can then determine the cross sections for all other pro cesses. These cross- sections are indep endent of the exp erimental details, and thus can b e used to compare the data that come from di erent exp eriments. They are also the relevant quantities that one wishes to compute using quantum eld theory. 1.2 Quantum Mechanical Probabilities Just to give an idea of how the cross section is related to a quantum mechanical amplitude, let us assume that b efore the collision the densities and velo cities of the par- ticles inside the b eams are accurately known. In terms of quantum mechanics this means 1) A more careful analysis shows that the integrand has an additional relativistic con- 1 ~ ~ ~ ~ tribution jj j j. This term can b e dropp ed in a situation where j and j are 1 2 1 2 c (approximately) parallel. that one has sp eci ed the initial quantum state j i of the colliding particles. Similarly, i by measuring the energies and directions of the particles that are b eing pro duced by the collisions one selects a certain nal wave-function j i that describ es the state of the f out-going particles. According to the rules of quantum mechanics the probablity W for this event to happ en is given by the absolute valued squared of the overlap b etween the nal state and the initial state 2 W = jh j ij i!f i f Quantum eld theory provides the to ols to compute the amplitudes h j i, and hence i f determine the quantum mechanical probabilities W . In fact, the quantity that is most directly computed is the probability W that a collision takes place p er unit of volume and time for a situation with constant particle densities and .For this situation W 1 2 may b e identi ed with the ratio N=V t, which according to equation (1) gives the cross section times the pro duct of the densities and relativevelo city. All this will b e explained in more detail in the following sections. 2. RELATIVITY AND QUANTUM MECHANICS The elementary particles that are used in high energy exp eriments havevelo cities close to the sp eed of light, and should therefore b e describ ed by the theory of sp ecial relativity.For relativistic particles it is convenient to sp ecify their motion, instead by their velo city ~v , in terms of their momentum m~v q (4) ~p = ~v 2 1 ( ) c An imp ortant reason for this is that during collision the total momentum of the particles will b e conserved. The total energy of the particles will also b e conserved. In sp ecial relativity the energy of a particle with momentum ~p is given by q 2 2 2 4 E = ~p c + m c (5) 2 For a particle at rest this b ecomes Einstein's famous relation E = mc . 2.1 Relativistic WaveFunctions In quantum mechanics the energy E b ecomes identi ed with the Hamiltonian op er- ator H . The time evolution of the wave function for a single particle is, as usual, governed by the Schrodinger equation in which the Hamiltonian H acts as a di erential op erator. The form of this di erential op erator can b e read of from (5) by replacing the momentum ~ ~ ~p by the op erator p = ih r. This gives ih@ (~x;t)= H (~x; t) (6) t with q 2 2 2 4 ~ H = h r + m c : (7) An imp ortant di erence with the non-relativistic case is, however, that the Hamiltonian involves a square ro ot. Still the Schrodinger equation can easily b e solved by expanding the wave-function in terms of plane waves ~ ~ ik~xi! (k )t (~x;t)=e : ~ k ~ These plane waves are the eigenfunctions of the momentum op erators p with eigenvalue ~ ~ p~ =hk and are also energy eigenstates with eigenvalue E =h! (k ). The most general 2) solution to the Schrodinger equation can now b e written as Z 3 ~ d k 1 ~ ~ ik ~xi! (k )t ~ (~x;t)= e (k ); (8) 3 ~ (2 ) 2! (k ) where q 2 2 ~ ~ ! (k )= k +m : (9) To simplify the formulas we put here and in the followingh = c = 1. When the momentum ~ ~ p~ is well sp eci ed the wave-function (k ) will b e strongly p eaked near k = ~p.
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