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A fully relativistic treatment of the quantum mechanics of particles requires the introduction of quantum field theory, that is to say, the quantum mechan­ ics of systems with an infinite number of degrees of freedom. This is because the relativistic equivalence of mass and energy plus the quantum possibility of fluctuations imply the existence of (real or virtual) creation and annihilation of particles in unlimited numbers. In spite of this, there exist processes, and energy ranges, where a treat­ ment in terms of ordinary quantum mechanical tools is appropriate, and the approximation of neglecting the full field-theoretic description is justified. Thus, one may use concepts such as potentials, and wave equations, clas­ sical fields and classical currents, etc. The present text is devoted precisely to the systematic discussion of these topics, to which we have added a gen­ eral description of one- and two-particle relativistic states, in particular for scattering processes. A field-theoretic approach may not be entirely avoided, and in fact an introduction to quantum field theory is presented in this text. However, field theory is not the object per se of this book; apart from a few examples, field theory is mainly employed to establish the connection with equivalent potentials, to study the classical limit of the emission of radiation or to discuss the propagation of a fermion in classical electromagnetic fields.



1. Relativistic Transformations. The Lorentz Group

A rotation1 may be specified by a vector, θ , in such a way that (Fig. 1.1.1) the rotation axis lies along θ , the rotation angle being θ = |θ|, and the direction of the rotation determined by the corkscrew rule. If we denote the rotation by R(θ ), it acts upon a vector r according to
$$r \to r = R\left( \theta \right)r = \left( {\cos \theta } \right)r + \left( {1 - \cos \theta } \right)\frac{{\theta r}}{{\theta ^2 }}\theta + \frac{{\sin \theta }}{\theta }\theta \times r;$$
for θ infinitesimal,
$$R\left( \theta \right)r = r + \theta \times r + O\left( {\theta ^2 } \right).$$
Francisco J. Ynduráin

2. The Klein-Gordon Equation. Relativistic Equation for Spinless Particles

When Schrödinger developed his nonrelativistic wave equation, he also proposed a relativistic generalization. This equation, studied in greater detail by Klein and Gordon (whose name it now bears) can be immediately obtained from the relation (1.7.4) for a free particle,
$$p.p = m^2 c^2,$$
with the substitutions suggested by the correspondence principle,
$$p0 = \frac{1}{c}E \to i\hbar\partial _t ,\,\,\,P \to -i\hbar\nabla.$$
Francisco J. Ynduráin

3. Spin 1/2 Particles

Given a positive-definite operator, such as (m2c4 + P2c2), there is a mathematical theorem that guarantees that there is one, and only one, square root that is also positive definite, denoted by +(m2c4 + P2c2)1/2. Other square roots become possible if we give up positive definiteness. This may appear to spoil the theory by allowing negative energies; but, if the operator is Hermitean, states corresponding to negative energies will be orthogonal to positive-energy states and a sensible physical theory is obtained if we restrict ourselves to the latter. We can, moreover, ensure manifest covariance by looking for an equation not only linear in ∂t but also linear in the space derivatives; that equation, we expect, will describe relativistic spin 1/2 particles, such as the electron1. We then use a multicomponent wave function2, \(\mathop \Psi \limits_ \sim \), and look for an equation linear in the P μ , the Dirac equation,
$$ih{\partial _t}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\Psi } \left( {r,t} \right) = {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{H} _0}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\Psi } = - ihc\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{a} \nabla \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\Psi } \left( {r,t} \right) + m{c^2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\beta } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\Psi } \left( {r,t} \right),$$
where the free Dirac Hamiltonian \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{H} _0 \) satisfies
$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{H} _0^2 = {m^2}{c^4} + {c^2}{P^2};$$
$${{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{H} }_0} = ihc\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{a} \nabla + m{c^2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\beta }$$
Francisco J. Ynduráin

4. Dirac Particle in a Potential

The study of a Dirac particle in a spherical well is similar to that of the nonrelativistic case, up to a few complications. First, we must work with two coupled equations. Second, we have to distinguish between scalar potentials and “vector” potentials, for which the potential is the fourth component of a Minkowski four-vector. We will study the first case only; the second may be found in the treatise of Greiner, Müller and Rafelski (1985). We then consider a scalar well (more precisely, a barrier) with potential
$$V_S (r) = \left\{ \begin{gathered} \upsilon _0 ,r > R, \hfill \\ 0,r < R. \hfill \\ \end{gathered} \right.$$
Francisco J. Ynduráin

5. Massive Particles with Spin 1. Massless Spin 1 Particle: Photon Wave Functions. Particles with Higher Spins (3/2, 2, …)

A nonrelativistic particle with spin 1 can be described by a three-component wave function, V(r, t) or, in momentum space, V(p, t). Under Lorentz transformations a three-vector will develop a fourth component; therefore, to describe a relativistic particle with spin 1 (and mass m ≠ 0) we will need a four-vector, V μ (x). This wave function has one component too many, so we will have to subject it to a supplementary condition. As we shall see in a moment, the one leading to correct interpretation is that of (four-) transversality, ∂ · V(x) = 0. V(x) will also have to verify the Klein—Gordon equation, so that we have, in natural units ћ = c = 1,
$$\left( {\partial \cdot \partial + m^2 } \right)V_\mu(x) = 0,\,\,\partial \cdot V_\mu = 0.$$
Francisco J. Ynduráin

6. General Description of Relativistic States

In spite of the successes of the Dirac equation and of its usefulness in the construction of relativistic quantum fields (to be discussed later, in Sect. 8), there is little doubt that the wave function formalism for relativistic particles is not quite satisfactory. First of all, the meaning of the variables r and t in a wave function Ψ(r,t) is unclear; as we will show, r does not represent the position for a Dirac particle, and in fact a position operator does not even exist, strictu senso, for a photon. As for t, the interpretation of it as the time becomes less clear when we have several particles: which time? The proper time of each of the particles? Time as measured in the centre of mass reference system?
Francisco J. Ynduráin

7. General Description of Relativistic Collisions: S Matrix, Cross-sections and Decay Rates. Partial Wave Analyses

Let us consider two free particles (which for simplicity we take to be distinguishable), A, B, with masses m A , m B . A state of these two particles can be specified by giving the momenta P A , P B and spin quantum numbers (for example, the helicities) to be denoted by α, β: we thus write it as
$$\left| {p_A ,\alpha ;p_B ,\beta } \right\rangle ,\,\,\,p_{A0} \equiv \sqrt {m^2 _A + P^2 _A } ,\,\,\,p_{B0} \equiv \sqrt {m^2 _B + P^2 _B } $$
with normalization
$$\begin{gathered} \left\langle {p{\prime _A},\alpha \prime ;p{\prime _B},\beta \prime |{p_A},\alpha ;{p_B},\beta } \right\rangle = {\delta _{\alpha \alpha \prime }}2{p_{A0}}\delta \left( {{{\text{p}}_A} - {\text{p}}{\prime _A}} \right) \hfill \\ \times {\delta _{\beta \beta }}\prime 2{p_{B0}}\delta \left( {{{\text{p}}_B} - {\text{p}}{\prime _B}} \right) \hfill \\ \end{gathered}$$
Francisco J. Ynduráin

8. Quantization of the Electromagnetic Field. Interaction of Radiation with Matter

A very useful technique when considering the quantized version of field theory (in particular, the quantum theory of electromagnetic fields) is that of normal, or Wick, products of operators, which will now be described.
Francisco J. Ynduráin

9. Quantum Fields: Spin 0, 1/2, 1. Covariant Quantization of the Electromagnetic Field

As we have stated several times, it is a fact that a theory based upon the wave function formalism, and with interactions given by potentials, does not provide a consistent description of physical reality. There are a number of reasons for this. Some are empirical: in any process at high energy, particles are created; therefore a wave function formalism, where the number of particles stays constant in time, will not be appropriate. Moreover, even if particles cannot be created because the energy is not sufficient, they may appear as quantum fluctuations provided the time they are present, Δt, and the energy fluctuation, ΔE, satisfy the Bohr-Heisenberg relation
$$\Delta t\,\Delta E \sim \hbar.$$
Francisco J. Ynduráin

10. Interactions in Quantum Field Theory. Nonrelativistic Limit. Reduction to Equivalent Potential

It is impossible to solve interactions such as those introduced in Sect. 9.8 exactly. To study them we therefore have to resort to approximation methods, notably to expansions in powers of the parameter characterizing the strength of the interaction, e for the electromagnetic interactions, \({\text{gY}},{{\text{g}}_{\text{s}}}\) in other cases. This method, which will be described in some detail, gives excellent results for scattering problems in electromagnetic interactions of elementary particles. Here the expansion parameter is effectively the fine structure constant, α ≃ 1/137, which is small: hence, in all but exceptional situations, one or at most two terms in the perturbation expansion give very accurate results.
Francisco J. Ynduráin

11. Relativistic Collisions in Field Theory. Feynman Rules. Decays

The initial and final states for this process are,
$$ [\left| i \right\rangle {\text{ = }}{{\hat b}^ + }\left( {{p_1},{\lambda _1}} \right){{\hat d}^ + }\left( {{p_{2,}},{\lambda _2}} \right)\left| 0 \right\rangle ,\langle f| = \langle 0|\hat \alpha \left( {{k_1},{\eta _1}} \right)\hat \alpha \left( {{k_2},{\eta _2}} \right) $$
where the η are the helicities of the photons and we define photon states without a factor \( 1/\sqrt {n!} \), as we did for electron states. The interaction Hamiltonian is still given by (10.4.1). In the Born approximation,
$$ \begin{gathered} \langle f|\hat S{\left| i \right\rangle _{Born}} = \frac{{{i^2}{e^2}}}{{2!}}\int {{d^4}{x_1}{d^4}{x_2}\sum\limits_{\mu \upsilon } {{g_\mu }_\mu {g_\upsilon }_\upsilon \langle 0|\hat \alpha \left( {{k_1},{\eta _1}} \right)\hat \alpha } } \left( {{k_2},{\eta _2}} \right) \hfill \\ \times T(:\hat \bar \psi \left( {{x_1}} \right){\gamma _\upsilon }\hat \psi :{{\hat A}_\mu }\left( {{x_1}} \right){{\hat A}_\nu }\left( {{x_2}} \right):\hat \bar \psi \left( {{x_2}} \right){\gamma _\nu }\hat \psi \left( {{x_2}} \right)\left. : \right) \hfill \\ x{{\hat b}^ + }\left( {{p_1},{\lambda _1}} \right){{\hat d}^ + }\left( {{p_2},{\lambda _2}} \right)\left| {\left. 0 \right\rangle .} \right. \hfill \\ \end{gathered} $$
Francisco J. Ynduráin

12. Relativistic Interactions with Classical Sources

We will now consider the scattering of a particle by an external, fixed classical field. For definiteness we consider an electromagnetic field; the results can be immediately generalized to any other potential (e.g., a Yukawa potential). Then, we treat the corresponding four-potential, A μ cl (x), as a given, known c number function. It is convenient (although not necessary) to imagine a very heavy particle, which we denote by a cross, as in Fig. 12.1.1, to be the Source of the potential.
Francisco J. Ynduráin


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