a first book of quantum field theory pdf

A first book of quantum field theory pdf

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Concepts and Mathematical Foundations

Quantum Field Theory David Tong

Quantum field theory

For students interested in high energy theory, exposure to QFT at any early stage is slowly becoming the standard for top American graduate schools.

Concepts and Mathematical Foundations

In theoretical physics , quantum field theory QFT is a theoretical framework that combines classical field theory , special relativity and quantum mechanics , [1] : xi. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles.

QFT treats particles as excited states also called quanta of their underlying quantum fields , which are more fundamental than the particles. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory in quantum mechanics. As a successful theoretical framework today, quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century.

Its development began in the s with the description of interactions between light and electrons , culminating in the first quantum field theory— quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the s with the invention of the renormalization procedure.

A second major barrier came with QFT's apparent inability to describe the weak and strong interactions , to the point where some theorists called for the abandonment of the field theoretic approach. The development of gauge theory and the completion of the Standard Model in the s led to a renaissance of quantum field theory.

Quantum field theory is the result of the combination of classical field theory , quantum mechanics , and special relativity. The force of gravity as described by Newton is an " action at a distance "—its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with Richard Bentley , however, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact.

However, this was considered merely a mathematical trick. Fields began to take on an existence of their own with the development of electromagnetism in the 19th century.

Michael Faraday coined the English term "field" in He introduced fields as properties of space even when it is devoid of matter having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines of force". This description of fields remains to this day.

The theory of classical electromagnetism was completed in with Maxwell's equations , which described the relationship between the electric field , the magnetic field , electric current , and electric charge.

Maxwell's equations implied the existence of electromagnetic waves , a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the speed of light.

Action-at-a-distance was thus conclusively refuted. Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in atomic spectra , nor for the distribution of blackbody radiation in different wavelengths.

He treated atoms, which absorb and emit electromagnetic radiation, as tiny oscillators with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as quantum harmonic oscillators. This process of restricting energies to discrete values is called quantization. This implied that the electromagnetic radiation, while being waves in the classical electromagnetic field, also exists in the form of particles.

In , Niels Bohr introduced the Bohr model of atomic structure, wherein electrons within atoms can only take on a series of discrete, rather than continuous, energies. This is another example of quantization. The Bohr model successfully explained the discrete nature of atomic spectral lines. In , Louis de Broglie proposed the hypothesis of wave—particle duality , that microscopic particles exhibit both wave-like and particle-like properties under different circumstances. In the same year as his paper on the photoelectric effect, Einstein published his theory of special relativity , built on Maxwell's electromagnetism.

New rules, called Lorentz transformation , were given for the way time and space coordinates of an event change under changes in the observer's velocity, and the distinction between time and space was blurred. Two difficulties remained. Quantum field theory naturally began with the study of electromagnetic interactions, as the electromagnetic field was the only known classical field as of the s.

Through the works of Born, Heisenberg, and Pascual Jordan in —, a quantum theory of the free electromagnetic field one with no interactions with matter was developed via canonical quantization by treating the electromagnetic field as a set of quantum harmonic oscillators.

In his seminal paper The quantum theory of the emission and absorption of radiation , Dirac coined the term quantum electrodynamics QED , a theory that adds upon the terms describing the free electromagnetic field an additional interaction term between electric current density and the electromagnetic vector potential. Using first-order perturbation theory , he successfully explained the phenomenon of spontaneous emission. According to the uncertainty principle in quantum mechanics, quantum harmonic oscillators cannot remain stationary, but they have a non-zero minimum energy and must always be oscillating, even in the lowest energy state the ground state.

Therefore, even in a perfect vacuum , there remains an oscillating electromagnetic field having zero-point energy. It is this quantum fluctuation of electromagnetic fields in the vacuum that "stimulates" the spontaneous emission of radiation by electrons in atoms. Dirac's theory was hugely successful in explaining both the emission and absorption of radiation by atoms; by applying second-order perturbation theory, it was able to account for the scattering of photons, resonance fluorescence , as well as non-relativistic Compton scattering.

Nonetheless, the application of higher-order perturbation theory was plagued with problematic infinities in calculations. In , Dirac wrote down a wave equation that described relativistic electrons—the Dirac equation. Although the results were fruitful, the theory also apparently implied the existence of negative energy states, which would cause atoms to be unstable, since they could always decay to lower energy states by the emission of radiation.

The prevailing view at the time was that the world was composed of two very different ingredients: material particles such as electrons and quantum fields such as photons.

Material particles were considered to be eternal, with their physical state described by the probabilities of finding each particle in any given region of space or range of velocities. On the other hand, photons were considered merely the excited states of the underlying quantized electromagnetic field, and could be freely created or destroyed.

It was between and that Jordan, Eugene Wigner , Heisenberg, Pauli, and Enrico Fermi discovered that material particles could also be seen as excited states of quantum fields. Just as photons are excited states of the quantized electromagnetic field, so each type of particle had its corresponding quantum field: an electron field, a proton field, etc. Given enough energy, it would now be possible to create material particles. Building on this idea, Fermi proposed in an explanation for beta decay known as Fermi's interaction.

Atomic nuclei do not contain electrons per se , but in the process of decay, an electron is created out of the surrounding electron field, analogous to the photon created from the surrounding electromagnetic field in the radiative decay of an excited atom. It was realized in by Dirac and others that negative energy states implied by the Dirac equation could be removed by assuming the existence of particles with the same mass as electrons but opposite electric charge. This not only ensured the stability of atoms, but it was also the first proposal of the existence of antimatter.

Indeed, the evidence for positrons was discovered in by Carl David Anderson in cosmic rays. With enough energy, such as by absorbing a photon, an electron-positron pair could be created, a process called pair production ; the reverse process, annihilation, could also occur with the emission of a photon.

This showed that particle numbers need not be fixed during an interaction. Historically, however, positrons were at first thought of as "holes" in an infinite electron sea, rather than a new kind of particle, and this theory was referred to as the Dirac hole theory.

Robert Oppenheimer showed in that higher-order perturbative calculations in QED always resulted in infinite quantities, such as the electron self-energy and the vacuum zero-point energy of the electron and photon fields, [6] suggesting that the computational methods at the time could not properly deal with interactions involving photons with extremely high momenta.

A series of papers was published between and by Ernst Stueckelberg that established a relativistically invariant formulation of QFT. In , Stueckelberg also independently developed a complete renormalization procedure. Unfortunately, such achievements were not understood and recognized by the theoretical community.

Faced with these infinities, John Archibald Wheeler and Heisenberg proposed, in and respectively, to supplant the problematic QFT with the so-called S-matrix theory.

Since the specific details of microscopic interactions are inaccessible to observations, the theory should only attempt to describe the relationships between a small number of observables e.

In , Richard Feynman and Wheeler daringly suggested abandoning QFT altogether and proposed action-at-a-distance as the mechanism of particle interactions. By ignoring the contribution of photons whose energy exceeds the electron mass, Hans Bethe successfully estimated the numerical value of the Lamb shift. However, this method was clumsy and unreliable and could not be generalized to other calculations.

The breakthrough eventually came around when a more robust method for eliminating infinities was developed by Julian Schwinger , Feynman, Freeman Dyson , and Shinichiro Tomonaga. The main idea is to replace the calculated values of mass and charge, infinite though they may be, by their finite measured values. This systematic computational procedure is known as renormalization and can be applied to arbitrary order in perturbation theory. However, the mass and charge observed in experiments are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite.

On the other hand, the mass and charge appearing in the theory are… the values modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental values for them phenomenologically This procedure is called the renormalization of mass and charge… After long, laborious calculations, less skillful than Schwinger's, we obtained a result By applying the renormalization procedure, calculations were finally made to explain the electron's anomalous magnetic moment the deviation of the electron g -factor from 2 and vacuum polarisation.

These results agreed with experimental measurements to a remarkable degree, thus marking the end of a "war against infinities".

At the same time, Feynman introduced the path integral formulation of quantum mechanics and Feynman diagrams. Each diagram can be interpreted as paths of particles in an interaction, with each vertex and line having a corresponding mathematical expression, and the product of these expressions gives the scattering amplitude of the interaction represented by the diagram. It was with the invention of the renormalization procedure and Feynman diagrams that QFT finally arose as a complete theoretical framework.

While renormalization was accepted by most physicists as both legitimate and necessary, Schwinger was not happy. At a talk given at the International Symposium on the History of Particle Physics at Fermilab in , he said, The pressure to account for those [experimental] results had produced a certain theoretical structure that was perfectly adequate for the original task, but demanded simplification and generalization; a new vision was required… Like the silicon chip of more recent years, the Feynman diagram was bringing computation to the masses… But eventually one has to put it all together again, and then the piecemeal approach loses some of its attraction… Quantum field theory must deal with Bose-Einstein fields and Fermi-Dirac fields on a fully equivalent footing… There was my challenge.

In fact, he devoted his Nobel speech in to describing this work, just as Einstein had talked about Relativity in his Nobel speech and not the photoelectric effect theory that he got the award for. The relativistic quantum theory of fields was born some thirty-five years ago through the paternal efforts of Dirac, Heisenberg, Pauli and others. It was a somewhat retarded youngster, however, and first reached adolescence seventeen years later, an event which we are gathered here to celebrate.

But it is the subsequent development and more mature phase of the subject that I wish to discuss briefly today. This use of Hilbert space leads to the concept of field quanta:.

Each quantum is a holistic unit of field that cannot be subdivided. An electron is a quantized ripple of the electron quantum field, which acts as a particle because it travels holistically with its conserved quantities always sustained as a unit. Despite the success of Schwinger's theory in answering the paradoxes and mysteries of Quantum Mechanics, [16] it is now largely overlooked or forgotten. One of the reasons is that the idea of instantaneous collapse is troubling to many physicists, including Einstein, who called it spooky action at a distance.

However, it is an experimental fact, [17] nor does it violate the principle of Relativity, because no information is transmitted in the process. Removing a field before it has had a chance to do anything, or changing the spin or other property of a field before it has changed anything is not the same as changing something that has already happened.

Another reason is that this later work of Schwinger was not well understood in the physics community. And so a tragedy ensued. Given the tremendous success of QED, many theorists believed, in the few years after , that QFT could soon provide an understanding of all microscopic phenomena, not only the interactions between photons, electrons, and positrons. Contrary to this optimism, QFT entered yet another period of depression that lasted for almost two decades. The first obstacle was the limited applicability of the renormalization procedure.

In perturbative calculations in QED, all infinite quantities could be eliminated by redefining a small finite number of physical quantities namely the mass and charge of the electron.

Quantum Field Theory David Tong

After some decades of work a satisfactory theory of quantum gravity is still not available; moreover, there are indications that the original field theoretical approach may be better suited than originally expected. There, to first approximation, one is left with the problem of quantum field theory on Lorentzian manifolds. Surprisingly, this seemingly modest approach leads to far reaching conceptual and mathematical problems and to spectacular predictions, the most famous one being the Hawking radiation of black holes. Skip to main content Skip to table of contents. Advertisement Hide. This service is more advanced with JavaScript available. Front Matter Pages i-ix.

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Quantum field theory

In theoretical physics , quantum field theory QFT is a theoretical framework that combines classical field theory , special relativity and quantum mechanics , [1] : xi. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states also called quanta of their underlying quantum fields , which are more fundamental than the particles. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory in quantum mechanics.

Erich Poppitz. Peeter's education includes a B. Eng ECE electromagnetics , a lot of self-study, and non-degree study of most of the interesting 4th year UofT undergrad physics courses. Peeter's day job is software development. He has over 20 years of experience with low level systems programing, operating system and hardware abstraction and exploitation, concurrency, and large scale refactoring.

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