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Introductory Lectures on Quantum Field Theory
[article]

Luís Alvarez-Gaumé, Miguel Angel Vázquez-Mozo

2010

In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been selected because they appear frequently in current applications to particle physics and string theory. * Luis.Alvarez-Gaume@cern.ch † Miguel. Vazquez-Mozo@cern.ch, vazquez@usal.es Indeed, this is what happens if E − m > V 0 . In this case both p 1 and p 2 are real and we have a partly reflected and a partly transmitted wave. In the same way, if V 0 −
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... < E − m < V 0 , then p 2 is imaginary and there is total reflection. However, in the case when V 0 > 2m and the energy is in the range 0 < E − m < V 0 − 2m, a completely different situation arises. In this case one finds that both p 1 and p 2 are real and therefore 5 INTRODUCTORY LECTURES ON QUANTUM FIELD THEORY 5 the incoming wave function is partially reflected and partially transmitted across the barrier. This is a shocking result, since it implies that there is a non-vanishing probability of finding the particle at any point across the barrier with negative kinetic energy (E − m − V 0 < 0)! This weird result is known as Klein's paradox. As with the negative energy states, the Klein paradox results from our insistence on giving a single-particle interpretation to the relativistic wave function. In fact, a multiparticle analysis of the paradox [13] shows that what happens when 0 < E − m < V 0 − 2m is that the reflection of the incoming particle by the barrier is accompanied by the creation of particle-antiparticle pairs out of the energy of the barrier (note that for this to happen it is required that V 0 > 2m, the threshold for the creation of a particle-antiparticle pair). This particle creation can be understood by noticing that the sudden potential step in Fig. 3 localizes the incoming particle with mass m at distances smaller than its Compton wavelength λ = 1/m. This can be seen by replacing the square potential by another one where the potential varies smoothly from 0 to V 0 > 2m on distance scales larger than 1/m. This case was worked out by Sauter shortly after Klein pointed out the paradox [15] . He considered a situation where the regions with V = 0 and V = V 0 are connected by a region of length d with a linear potential V (x) = V 0 x/d. When d > 1/m he found that the transmission coefficient is exponentially small. 1 The creation of particles is impossible to avoid whenever one tries to locate a particle of mass m within its Compton wavelength. Indeed, from the Heisenberg uncertainty relation, we find that if ∆x ∼ 1/m, the fluctuations in the momentum will be of order ∆p ∼ m, and fluctuations in the energy of order ∆E ∼ m (2.14) can be expected. Therefore, in a relativistic theory, the fluctuations of the energy are enough to allow the creation of particles out of the vacuum. In the case of a spin-1 2 particle, the Dirac sea picture shows clearly how, when the energy fluctuations are of order m, electrons from the Dirac sea can be excited to positive energy states, thus creating electron-positron pairs.

doi:10.5170/cern-2006-015.1
fatcat:h2rem7bwargr5krzt6umicntk4