Foundations of Quantum Field Theory

The following sections are included:

• Dedication
• Preface
• Contents

Quantum Field Theory is a natural outgrowth of non-relativistic Quantum Mechanics, combining it with the Principles of Special Relativity and particle production at sufficiently high energies. We therefore devote this introductory chapter to recalling some of the basic principles of Quantum Mechanics which are either shared or not shared with Quantum Field Theory.

In this chapter we first discuss the realization of the homogeneous Lorentz transformations in four-dimensional space-time, as well as the corresponding Lie algebra. From here we obtain all finite dimensional representations, and in particular the explicit form of the matrices representing the boosts for the case of spin = 1/2, which will play a fundamental role in Chapter 3. The Lorentz transformation properties of massive and zero-mass 1-particle in Hilbert space (and their explicit realization in Chapter 9) lie at the heart of the Fock space representation (second quantization) in Chapter 9. It is assumed that the reader is already familiar with the essentials of the Special Theory of Relativity and of Group Theory.

In this chapter we engage in the search for a relativistic wave equation for a spin zero particle moving in an external potential, reducing to the ordinary Schrödinger equation in the non-relativistic limit. We begin by looking at the case of a free particle. We adhere to the familiar quantum mechanical principles, as long as we can. We shall encounter a number of difficulties which will lead us to eventually abandon the usual probability interpretation.

We begin this chapter by obtaining the relativistic “Schrödinger” equation for a free spin-1/2 field by following first the historical approach, and then presenting a derivation based on Lorentz covariance and space-time parity alone. This leads us to a four-component wave equation which is first-order in space and time coordinates. We present the solution of this equation for three choices of basis: The Dirac, Weyl (or chiral), and Majorana representations. The latter representation is shown to be particularly useful for the case of Majorana fermions, i.e. fermions which are their own anti-particles. We show that the Dirac equation allows for the notion of a probability density after suitable interpretation of the negative energy states.

As is well known, the electromagnetic field can be interpreted on the quantum level as a flux of quanta, called photons. In fact, this interpretation first arose in connection with Planck’s formula describing the spectrum of black-body radiation. As Maxwell’s equations show, these quanta propagate with the velocity of light in all inertial frames, so there exists no rest frame we can associate with them. Photons can thus be viewed as “massless” particles. According to our discussion in Section 6 of Chapter 2, the respective 1-particle states should thus transform according to a one-dimensional representation of the little group…

In the case of the scalar field we have seen that there was no possibility for a probability interpretation in the sense of non-relativistic Quantum Mechanics. In the case of Dirac spin one-half particles, such a probabilistic interpretation is possible if their electromagnetic interaction is restricted to external electromagnetic fields.

In this chapter we take the first steps towards a quantum field theoretic formulation of the interaction of scalar bosons, fermions and photons. To this end we first consider the case of non-interacting particles. The essential step consists in replacing the fields of the Klein–Gordon, Dirac and Maxwell equations by operator-valued fields living in a Fock space. This process is referred to as second Quantization. The equations of motion for these operators can be derived from a lagrangian density via a variational principle. This will constitute the backbone of our general quantization procedure. It allows to rewrite the equations of motion for the operator-valued fields in the form of Hamilton equations, where the Hamiltonian is obtained from the Lagrange density via the usual Legendre transform. The 1-particle wave function is recovered in this second-quantized formulation by considering the appropriate matrix elements of the operators in Fock space.

Given some classical field equations, we show in this chapter how to systematically arrive at the corresponding quantum theory satisfying the correspondence principle. The approach is essentially the same as in non-relativistic quantum mechanics and presupposes that the equations of motion can be derived from a Lagrange density via a variational principle. The quantization is then realized in phase space by going over to the Hamiltonian formulation via the usual Legendre transformation. Although there have been attempts of quantization without invoking the existence of a lagrangian, the canonical approach has the great advantage of providing a systematic construction of conserved “currents”, and hence of the generators of symmetry transformations. It also provides a powerful method for discovering and implementing the constraints of the theory in question. Since most interesting theories in particle physics correspond to constrained Hamiltonian systems, we shall have occasion to exemplify the power of the canonical approach for constrained Hamiltonian systems as developed by Dirac.

We have seen in Sections 3 and 4 of Chapter 8 how to construct, following Dirac, the generator of local symmetries in the case of hamiltonian systems with so-called first class constraints (gauge theories). They are characterized by the fact that the infinitesimal parameters of the symmetry transformations depend on the space-time coordinates and may thus take different values at different space-time points…

The experimental physicist concerns himself with the scattering of particles on a target by recording the rate at which different particles are registered in his counter and by measuring the mass, momenta, spin and polarization of these particles. From his measurements he can indirectly infer the existence of unstable particles (resonances) which have been produced during the interaction. All this information is contained in the so-called differential scattering cross-section. In terms of this quantity, he can then compare his results with the theoretical predictions…

The exact computation of vacuum expectation values of field operators is only possible for some particular model quantum field theories such as the Thirring model as well as QED and chiral gauge theories in 1+1 dimensions. In all these examples the (1+1)-dimensionality of space-time plays a crucial role. The S-matrix, on the other hand, can be computed exactly for a much larger class of two-dimensional quantum field theories, including the Gross–Neveu, non-linear sigma and CPⁿ models. The reason is that in two space-time dimensions the allowed scattering processes are strongly restricted by the existence of an infinite number of conservation laws, which make the theory integrable, as far as on-shell processes are concerned. In 3+1 dimensions we can compute the Green functions and S-matrix only perturbatively as some power series expansion in the coupling constant or as a “loop” expansion in powers of Planck’s constant ℏ…

In the previous chapter we have repeatedly used the representation of 1-loop diagrams in terms of integrals over the Feynman parameters ranging over the finite domain (0,1). The resulting integrals had the virtue of exhibiting explicitly the Lorentz-covariant structure with respect to the external momenta, the result having been of the form

The derivation of the Feynman rules in Chapter 11 has been based on operator methods in the interaction picture. The rules allow for the perturbative computation of the Green functions. The corresponding perturbative series is known to be at best an asymptotic series in the coupling constant with zero radius of convergence around the origin of the complex coupling-constant plane. It thus reflects the true content of the theory only as long as it provides a good approximation in the sense of an asymptotic series for sufficiently small coupling constant. It furthermore presupposes the existence of a unique vacuum given by the Fock vacuum. This appears to be the case for Quantum Electrodynamics (QED), where perturbation theory provides an excellent description of the experimental results. As we know however, this reflects only partially, in general, a much more complex situation. Thus it is known that the QCD vacuum is highly non-trivial, and is in fact a condensate of quark–antiquark pairs, very much reminiscent of the superconducting ground state in superconductors known to be populated by spin-zero electron–electron (Cooper) pairs. Many of the modern developments in Quantum Field Theory relating to such non-perturbative phenomena would have been missed had one adhered to perturbation theory in the formulation of Chapter 11. In fact, for a very small class of models in 1+1 dimensions (Thirring model, QED₂) exact solutions have been constructed using operator methods; they exhibit non-trivial properties lying outside the scope of perturbation theory. From here one learned that Quantum Field Theories in 3+1 dimensions can be expected to exhibit highly non-trivial non-perturbative features. It was only the functional approach to QFT which opened the possibility for studying such non-perturbative phenomena in “realistic” theories. This approach, dispensing of the notion of an underlying “Fock space”, will be the subject of this chapter. We shall show how one recovers the perturbative series embodied by Eq. (11.21), and prepare the ground for new approximation schemes paying tribute to “non-perturbative” phenomena, as exemplified in Chapter 20.

The fully dressed propagators and vertex-functions form the building blocks of a general diagram in Quantum Field Theory. Their respective perturbation series as obtained from (11.21) can be summed graphically into the form of highly nonlinear integral equations, which only admit a perturbative solution, as we show in this chapter.

In the preceding chapter we have obtained the non-linear integral equations satisfied by the (unrenormalized) 2- and 3-point functions of QED. We proceed now to calculate these quantities in second order of perturbation theory. This will involve as a first step the regularization of divergent integrals and the use of some technical tools. For didactic purposes we shall illustrate these tools as we go along. It should also be kept in mind that the results for the 2- and 3-point functions are gauge dependent, since under a gauge transformation the photon and fermion fields transform as in (13.88). We shall use the so-called Feynmann gauge (see (13.98)), for which the photon-propagator takes the particularly simple form

The individual terms contributing to the perturbative expansion of a general Green function are represented by Feynman diagrams. As we have seen in Chapter 15, the integrals representing diagrams involving at least one loop are generally divergent, and are thus a priori meaningless. We may nevertheless isolate the finite part of a diagram by subtracting a suitable polynomial involving infinite constants. Since this procedure does not define a priori a unique finite part, there remains an ambiguity in the choice of this polynomial. If we do this for each loop-diagram separately, we appear to generate, in general, an infinite number of undetermined constants as we consider diagrams to all orders of perturbation theory. If this were so, the theory would be meaningless…

If there were no dimensional parameters in the theory, then dimensional arguments alone (plus translational invariance) would uniquely fix, up to a normalization constant, the form for the 2-point function of a scalar field to be

In Section 1 of Chapter 16 we imposed suitable conditions in order to define the wave function renormalization constant, as well as the renormalized mass and coupling as a function of the cutoff and the bare parameters. We have done this without introducing further dimensioned parameters into the theory. This is an option we are free to take, but need not be the most adequate one from the point of view of perturbation theory. Thus from the point of view of the convergence of perturbation theory it could turn out to be advantageous to define the coupling constant in ϕ⁴-theory, instead of (17.64), by

The so-called spontaneous breaking of a continuous symmetry plays a central role in systems with an infinite number of degrees of freedom such as condensed matter, where it is responsible for phase transitions, as well as in QFT, where it is known as the Higgs mechanism in the Weak Interactions. A fundamental theorem due to Goldstone plays here a central role.

In the past chapters we have concentrated on the vacuum expectation value of the product of field operators, and in particular on connected Green functions. The basic constituents of a general Feynman diagram are however the proper Green functions. Since they are interlaced by simple propagators within a general Feynman diagram, their associated momentum-space integrals can be carried out independently. The generator of the proper Green functions will provide the definition of the effective potential to arbitrary order in ℏ, as we shall see.

The following section is included:

• Index