This is a set of Essays on the Foundations of Quantum Field theory that continues the topics of the textbook into the domain of relativistic quantum field theory.
One of the lessons learned in the textbook was that classical relativistic field theory suffers from many limitations, although the classical relativisitic theory took us far beyond the scope of Newtonian physics. Many of the limitations of classical field theory are addressed when the theory is unified with quantum mechanics. In particular, the unification of relativity, field theory and quantum mechanics leads to theories where each particle carrying a conserved charge is accompanied by an anti-particle having the same mass but opposite charge, where particle creation and destruction occurs, and where the spin statistics theorem, which shows that half integral particles satisfy Fermi statistics and integral spin particles satisfy Bose statistics, is predicted.
Additional Essays consider other topics in the fundamentals of relativistic quantum field theory like,
1. The Schwinger Effect (pair production in the vacuum due to external electric fields)
2. The Unruh Effect (thermal radiation from an accelerating vacuum)
3. The Casimir Effect (attraction/repulsion between objects in a quantum vacuum)
4. Solitary Waves and Solitons
5. Instantons and Theta Vacua
6. Monopoles in Quantum Field Theory
7. Asymptotic Freedom
8. Parton Distribution Functions
9. Confinement in Model Field Theories and Quantum Chromodynamics
10. Short distance properties of field theories, including renormalization theory,
11. Ideas on the unification of relativistic quantum mechanics and gravitation.
We begin with particles and anti-particle….
Abstract
This Essay, “Particles and Anti-Particles” explains how locality, causality and special relativity imply, in the context of quantum field theory, that each charged particle must be accompanied by an anti-particle of opposite charge but equal mass. The Essay consists of two parts: “Background” which presents preliminary ideas on non-relativistic harmonic motion by introducing creation and annihilation operators, illustrating them with coherent states, and then presents “second quantization” in the context of non-relativistic many body quantum mechanics. These ideas set the stage for the next portion of the Essay which presents the primary result, that relativistic quantum field theory predicts the existence of anti-particles. We will see that current conservation, locality and causality are the crucial ingredients here.
The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. An undergraduate physics course on quantum mechanics. The fundamentals of quantum field theory will be developed within these Essays.
Abstract
We consider the vacuum state of a scalar quantum field theory in Minkowski space time from the perspective of an accelerating observer and find that she observes a thermal bath of scalar particles at a temperature proportional to her proper acceleration. This is the Unruh effect. The mechanics of Rindler space time are developed in order to obtain this effect. Similarities to Hawking radiation are drawn.
In preparation for this discussion we begin with a simpler problem: The quantum mechanics of a driven harmonic oscillator. We use coherent states to describe its scattering states and find the probability distribution of the excitations in the final state.
In order to discuss problems in quantum field theory, we include several appendices on background subjects. One appendix introduces canonical quantization and Lagrangian methods for field theory. Noether’s Theorem is introduced and the relation between symmetries and conservation laws is obtained. Another appendix introduces Bogoliubov transformations which are used in the body of this Essay to relate the vacuum states of Minkowski and Rindler space times and equip us to solve for the Unruh temperature. Bogoliubov transformations will be useful in later Essays on the Casimir effect, the Schwinger effect and spontaneous symmetry breaking. A final appendix introduces the Euler Gamma function which is used in the calculation of the Bogoliubov transformation used to derive the Unruh temperature.
The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. An undergraduate physics course on quantum mechanics. The fundamentals of quantum field theory will be developed within these Essays.
Abstract
Uncharged grounded conducting parallel plates experience a mutual attractive force which is a quantum, relativistic effect: the force is proportional to Planck’s constant and the speed of light. It was calculated in the early days of relativistic quantum field theory (1948). In fact, it was originally analyzed as a limiting case of the retarded van der Waal’s force between dielectric plates. Casimir made the fascinating observation that if the plates had sufficiently high dielectric constants, then for some physical effects the plates’ effect on the electromagnetic field could be replaced by boundary conditions and the force between the plates could be calculated just by considering the quantum zero point fluctuations of the electromagnetic field in this environment. In this case the attractive force is called the Casimir effect. These considerations indicated that the force can be calculated either from 1. the direct electromagnetic interactions between the plates (van der Waals), or 2. the spatial dependence of the energy stored in the vacuum fluctuations of the electromagnetic field. The heuristic derivation of the Casimir effect is presented here. The derivation is analyzed, critically assessed and it’s physical and unphysical elements are discussed.
The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. An undergraduate physics course on quantum mechanics. The fundamentals of quantum field theory will be developed within these Essays.
Abstract
Quantum Electrodynamics predicts that a classical electric field of sufficient strength will produce pairs of electrons and positrons when applied to the vacuum. This is a tunneling effect, in the sense of ordinary quantum mechanics, and it illustrates the field theoretic nature of the vacuum state: its virtual fluctuations, electron-positron pairs, can be materialized by external classical sources. The critical electric field needed for the Schwinger effect is estimated and is found to be reachable with modern laser technology. Two Appendices in this Essay lay the groundwork for Schwinger’s prediction: The WKBJ approximate description of quantum tunneling and the Bogoliubov transformation that makes possible the construction of the vacuum state in the presence of the external electric field.
The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. An undergraduate physics course on quantum mechanics. The fundamentals of quantum field theory will be developed within these Essays.
- Statistical Field Theory, Landau-Ginzburg Theory, Spontaneous Symmetry Breaking, Symmetry and Conservations Laws, Goldstone Bosons and the Higgs Mechanism
Abstract
This lecture consists of two parts: An introduction to Statistical Field theory, Landau-Ginzburg Theory and the Goldstone Mechanism in condensed matter physics, and second, elementary particle field theory, symmetries and conservation laws, the Goldstone Theorem and the Higgs mechanism of the Standard Model. We begin with a description of Statistical Field Theory for condensed matter systems and work through various illustrations including elastic systems and magnetic materials. Phase transitions, phase diagrams and critical behavior are introduced and these concepts are illustrated with magnetic systems and the ferro-paramagnetic phase transition. The critical indices of second order phase transitions are defined and the correlation length is seen as the source of the singularities characterizing the bulk thermodynamics of continuous phase transitions. The Landau-Ginzburg Theory of Statistical Field Theory is introduced and illustrated for magnetic systems. Mean Field Theory is illustrated for magnetic systems and values of the critical indices are obtained. Spontaneous symmetry breaking of continuous symmetries is illustrated within the context of mean field theory. The Goldstone Theorem and Goldstone modes are illustrated for superfluidity. This exercise is done in d-dimensions and the Mermin-Wagner Theorem, the absence of spontaneous symmetry breaking and Goldstone modes in two dimensions, is illustrated. We then turn to relativistic field theories of elementary particles and find analogous phenomena. Continuous symmetries and conservation laws are studied in general by developing Noether’s Theorem. Symmetries and their realizations are contrasted in Quantum Mechanics and Quantum Field Theory. In field theory we find two natural modes of symmetry realization in a theory’s low energy spectrum: the Wigner-Weyl mode and the spontaneous symmetry breaking mode. The Goldstone Theorem is illustrated in theories with spontaneous symmetry breaking. The “Mexican Hat” potential is illustrated in several contexts. These considerations are generalized to Abelian Gauge Theories with charged scalar fields and the Higg’s mechanism is developed. The Higg’s mechanism leads to massive vector fields and massive scalar fields. The spontaneous symmetry breaking mechanism underlying the Higg’s mechanism is seen to alter the theory’s long-distance features, the theory’s spectroscopy, while leaving its high energy, short distance behavior unaffected. This leads to the great success of the Standard Model where the Higg’s mechanism gives masses to the vector mesons carrying the weak force as well as providing masses to other fundamental fields while leaving the photon massless. At the same time, the Higg’s mechanism leaves the short distance, gauge invariant behavior of the Standard Model unaffected, thus guaranteeing its renormalizability. This underlies the calculational success of the Standard Model.
The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. Undergraduate physics courses on quantum mechanics and statistical mechanics.
The Dirac Equation, Anti-Particles and the Quantum Vacuum
Abstract
This lecture introduces the Dirac equation, the relativistic equation for spin-1/2 point-like particles and fields. The Hamiltonian and the covariant forms of the equation are obtained and discussed. Dirac’s “Hole Theory” re-interpretation of the theory’s vacuum is introduced and discussed. The physical origins of pair production and charge renormalization are discussed from this perspective. The Charge Conjugation transformation is introduced to implement Dirac’s interpretation of positrons as the absence of negative energy electrons in the quantum vacuum. The theory is seen to be intrinsically multi-particle as a consequence of the unification of special relativity and quantum mechanics. The low energy properties of the Dirac equation are derived and the free field gyromagnetic ratio of the spin-1/2 electron is found and compared to the Pauli equation of non-relativistic quantum mechanics. The covariance of the equation is established and the implementation of Lorentz transformations in spinor space is developed.
The Chiral Anomaly and The Dirac Sea
Abstract
We consider axial symmetries and the conservation of the axial current in 1+1 and 3+1 dimensional massless quantum electrodynamics. Although the axial vector current is predicted to be conserved in the classical versions of these theories as a consequence of continuous axial rotations of the fermion field, these symmetries and conservation laws are broken in the quantum theories. We find that the response of the Dirac sea to an external electric field leads to this “axial anomaly”. The exact expressions for the divergence of the axial currents in 1+1 and 3+1 dimensions are derived. In 3+1 dimensions the degeneracy of the theory’s Landau levels plays an important role in the final result. Other approaches to understanding and calculating this and other axial anomalies are presented and discussed briefly.
Prerequisite: This lecture continues topics started in Supplementary Lecture 19.
Bosonization in 1+1 Dimensions and Solving the Schwinger Model
Abstract
We derive the duality map between fermions and bosons in 1+1 dimensions and apply this method to solve the Schwinger model, quantum electrodynamics in 1+1 dimensions. For massless interacting fermions, the solution is a free but massive neutral boson. The physics of the model is discussed from the perspective of quantum choromodynamics and other non-abelian gauge theories. The Schwinger model illustrates the physics of flux tubes, confinement, the Higgs mechanism, the chiral anomaly, chiral symmetry breaking, mass generation and theta-vacua in a simple setting.
Prerequisite: This lecture continues topics started in Supplementary Lectures 19 and 20.
Quantum Electrodynamics in a Strong External Magnetic Field
Abstract
We consider charged particles of mass and charge propagating in a uniform magnetic field . We begin with classical non-relativistic motion, then turn to non-relativistic quantum mechanics, then relativistic quantum mechanics and, finally, our goal: Quantum Electrodynamics (QED) in a strong, uniform external magnetic field, . The stationary states of these problems are Landau levels. Their energy spectra and degeneracies are computed. We argue that some aspects of the full 3+1 dimensional QED problem reduce to 1+1 dimensional QED (Schwinger model) which is soluble. The static potential between heavy charged impurities is strongly screened and the screening length can be computed from mass generation in 1+1 dimensional QED, in the idealized case of massless electrons. These considerations show that, for the limiting case of massless electrons, 3+1 dimensional QED in a strong external field exists in a Higgs phase where the electrostatic potential is screened to a Yukawa form.
How Hot is an Accelerating Spacecraft (and a Black Hole) ?
Abstract
In relativistic quantum field theory, the vacuum state contains quantum fluctuations which have important implications for atomic physics (the Lamb shift in hydrogen spectroscopy), elementary particle physics (electron-positron creation in quantum electrodynamics) and gravity (the Unruh effect and Hawking radiation). The Unruh effect was studied in Supplementary Lecture 15 and we found that the accelerated vacuum (Rindler vacuum) is a thermal ensemble of inertial states with a linear relation between the acceleration and temperature. In this lecture we consider a small accelerating spacecraft ( a “detector”) and find the relation between its state and that of an inertial detector in a thermal bath. The Unruh effect is rediscovered in a different and simpler fashion. The relation of these calculations to the Hawking effect is discussed and various related puzzles such as “firewalls” in black hole environments, the violation of conservation laws by black holes and black hole evaporation are introduced and commented upon.
Nuclear Physics Theory and Spontaneously Broken Chiral Symmetry
Abstract
Low energy experimental nuclear physics involves mesons, baryons and their interactions which determine the properties of nuclei. Low energy theoretical nuclear physics involves the vacuum state, the realization of vector and axial symmetries, mesons, baryons and their interactions which determine the properties of nuclei. The symmetries of low energy, few-body nuclear physics include isospin and axial isospin. The mode of symmetry realization in the low energy spectrum is argued to be algebraic for the isospin symmetry but the axial symmetry is argued to be spontaneously broken by the vacuum state. A triplet of pions emerges as Nambu-Goldstone bosons and their interactions with nucleons are described by Partial Conservation of Axial Current (PCAC) and the Goldberger-Treiman relation.. These physical concepts are captured in the linear and non-linear sigma models which are introduced and developed in this lecture. The mass of the nucleon is seen to be fundamentally nonperturbative and is a consequence of spontaneous axial symmetry breaking. The construction of the linear and non-linear sigma models is guided by properties of the underlying theory of quarks and gluons, Quantum Chromodynamics (QCD). Further developments in the modern era of nuclear physics include heavy ion collisions and hot nuclear matter