QFT treats particles as excited states also called quanta of their underlying fields , which are—in a sense—more fundamental than the basic particles. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding fields. Each interaction can be visually represented by Feynman diagrams , which are formal computational tools, in the process of relativistic perturbation theory.
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 field s 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. 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,  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 initial, so-called "bare", parameters mass, electric charge, etc. To cancel the apparently infinite parameters, one has to introduce additional, infinite, "counterterms" into the Lagrangian.
This systematic computational procedure is known as renormalization and can be applied to arbitrary order in perturbation theory. 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. 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. Dyson proved in that this is only possible for a small class of theories called "renormalizable theories", of which QED is an example. However, most theories, including the Fermi theory of the weak interaction , are "non-renormalizable".
Any perturbative calculation in these theories beyond the first order would result in infinities that could not be removed by redefining a finite number of physical quantities.
The second major problem stemmed from the limited validity of the Feynman diagram method, which is based on a series expansion in perturbation theory. In order for the series to converge and low-order calculations to be a good approximation, the coupling constant , in which the series is expanded, must be a sufficiently small number. In contrast, the coupling constant in the strong interaction is roughly of the order of one, making complicated, higher order, Feynman diagrams just as important as simple ones.
There was thus no way of deriving reliable quantitative predictions for the strong interaction using perturbative QFT methods. With these difficulties looming, many theorists began to turn away from QFT.
Some focused on symmetry principles and conservation laws , while others picked up the old S-matrix theory of Wheeler and Heisenberg. QFT was used heuristically as guiding principles, but not as a basis for quantitative calculations. In , Yang Chen-Ning and Robert Mills generalised the local symmetry of QED, leading to non-Abelian gauge theories also known as Yang-Mills theories , which are based on more complicated local symmetry groups. Unlike photons, these gauge bosons themselves carry charge.
Sheldon Glashow developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in This theory, nevertheless, was non-renormalizable. By combining the earlier theory of Glashow, Salam, and Ward with the idea of spontaneous symmetry breaking, Steven Weinberg wrote down in a theory describing electroweak interactions between all leptons and the effects of the Higgs boson.
This book describes, in clear terms, the Why, What and the How of Quantum Field Theory. The raison d'etre of QFT is explained by starting from the dynamics of. Editorial Reviews. Review. “The readership of this book consists of graduate students and.
His theory was at first mostly ignored,   : 6 until it was brought back to light in by Gerard 't Hooft 's proof that non-Abelian gauge theories are renormalizable. The electroweak theory of Weinberg and Salam was extended from leptons to quarks in by Glashow, John Iliopoulos , and Luciano Maiani , marking its completion.
Harald Fritzsch , Murray Gell-Mann , and Heinrich Leutwyler discovered in that certain phenomena involving the strong interaction could also be explained by non-Abelian gauge theory.
Quantum chromodynamics QCD was born. In , David Gross , Frank Wilczek , and Hugh David Politzer showed that non-Abelian gauge theories are " asymptotically free ", meaning that under renormalization, the coupling constant of the strong interaction decreases as the interaction energy increases. Similar discoveries had been made numerous times previously, but they had been largely ignored. These theoretical breakthroughs brought about a renaissance in QFT.
The full theory, which includes the electroweak theory and chromodynamics, is referred to today as the Standard Model of elementary particles. The s saw the development of non-perturbative methods in non-Abelian gauge theories. These objects are inaccessible through perturbation theory. Supersymmetry also appeared in the same period. The first supersymmetric QFT in four dimensions was built by Yuri Golfand and Evgeny Likhtman in , but their result failed to garner widespread interest due to the Iron Curtain.
Supersymmetry only took off in the theoretical community after the work of Julius Wess and Bruno Zumino in Among the four fundamental interactions, gravity remains the only one that lacks a consistent QFT description. Various attempts at a theory of quantum gravity led to the development of string theory ,  : 6 itself a type of two-dimensional QFT with conformal symmetry.
Although quantum field theory arose from the study of interactions between elementary particles, it has been successfully applied to other physical systems, particularly to many-body systems in condensed matter physics. Historically, the Higgs mechanism of spontaneous symmetry breaking was a result of Yoichiro Nambu 's application of superconductor theory to elementary particles, while the concept of renormalization came out of the study of second-order phase transitions in matter.
Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first quasiparticle — phonons. Lev Landau claimed that low-energy excitations in many condensed matter systems could be described in terms of interactions between a set of quasiparticles.
The Feynman diagram method of QFT was naturally well suited to the analysis of various phenomena in condensed matter systems. Gauge theory is used to describe the quantization of magnetic flux in superconductors, the resistivity in the quantum Hall effect , as well as the relation between frequency and voltage in the AC Josephson effect. A classical field is a function of spatial and time coordinates. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time.
Hence, it has infinite degrees of freedom. Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the photoelectric effect are best explained by discrete particles photons , rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields.
Canonical quantisation and path integrals are two common formulations of QFT. The simplest classical field is a real scalar field — a real number at every point in space that changes in time. Suppose the Lagrangian of the field is. Applying the Euler—Lagrange equation on the Lagrangian:  : This is known as the Klein—Gordon equation. The Klein—Gordon equation is a wave equation , so its solutions can be expressed as a sum of normal modes obtained via Fourier transform as follows:. The quantisation procedure for the above classical field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator.
Note that x is the displacement of a particle in simple harmonic motion from the equilibrium position, which should not be confused with the spatial label x of a field. The commutation relation between the two is. Their commutation relations are:  : Although the field appearing in the Lagrangian is spatially continuous, the quantum states of the field are discrete. While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles.
The latter space is known as a Fock space , which can account for the fact that particle numbers are not fixed in relativistic quantum systems. The preceding procedure is a direct application of non-relativistic quantum mechanics and can be used to quantise complex scalar fields, Dirac fields ,  : 52 vector fields e. In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes.
To perform calculations on any realistic interacting theory, perturbation theory would be necessary. The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a quartic interaction term could be introduced to the Lagrangian of the real scalar field:  : The path integral formulation of QFT is concerned with the direct computation of the scattering amplitude of a certain interaction process, rather than the establishment of operators and state spaces.
The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let H be the Hamiltonian i. The initial and final conditions of the path integral are respectively. In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.
Now we assume that the theory contains interactions whose Lagrangian terms are a small perturbation from the free theory. This expression, known as the two-point correlation function or the two-point Green's function , represents the probability amplitude for the field to propagate from y to x. In canonical quantisation, the two-point correlation function can be written as:  : This equation is useful in that it expresses the field operator and ground state in the interacting theory, which are difficult to define, in terms of their counterparts in the free theory, which are well defined.
In the path integral formulation, the two-point correlation function can be written as:  : According to Wick's theorem , any n -point correlation function in the free theory can be written as a sum of products of two-point correlation functions. For example,. Since correlation functions in the interacting theory can be expressed in terms of those in the free theory, only the latter need to be evaluated in order to calculate all physical quantities in the perturbative interacting theory. This is known as the Feynman propagator for the real scalar field. Correlation functions in the interacting theory can be written as a perturbation series.
Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a Feynman diagram. Points labelled with x and y are called external points, while those in the interior are called internal points or vertices there is one in this diagram. The product of factors corresponding to every element in the diagram, divided by the "symmetry factor" 2 for this diagram , gives the expression for the term in the perturbation series. In order to compute the n -point correlation function to the k -th order, list all valid Feynman diagrams with n external points and k or fewer vertices, and then use Feynman rules to obtain the expression for each term.
To be precise,. Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles". In realistic applications, the scattering amplitude of a certain interaction or the decay rate of a particle can be computed from the S-matrix , which itself can be found using the Feynman diagram method. Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing n loops are referred to as n -loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction.
Feynman rules can be used to directly evaluate tree-level diagrams. The renormalisation procedure is a systematic process for removing such infinities. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant.
A different approach, called renormalised perturbation theory, is to use physically meaningful quantities from the very beginning. The Lagrangian density becomes:. As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. In this way, meaningful finite quantities are obtained. It is only possible to eliminate all infinities to obtain a finite result in renormalisable theories, whereas in non-renormalisable theories infinities cannot be removed by the redefinition of a small number of parameters.
ISBN The disillusionment with QFT as a basis for the theory of elementary particles was also premature.
What was missing was many ingredients, including the identification of the underlying gauge symmetry of the weak interactions, the concept of spontaneous symmetry breaking that could explain how this symmetry was hidden, the identification of the fundamental constituents of the nucleons as colored quarks, the discovery of asymptotic freedom which explained how the elementary colored constituents of hadrons could be seen at short distances yet evade detection through confinement, and the identification of the underlying gauge symmetry of the strong interactions.
Once these were discovered, it was but a short step to the construction of the standard model, a gauge theory modeled on QED , which opened the door to the understanding of mesons and nucleons. Tian Yu Cao 25 March Conceptual Foundations of Quantum Field Theory. Cambridge University Press. In parallel with the changes it brought in our attitude toward symmetries, the birth of the Standard Model marked changes also in our attitude toward quantum field theory.
After 't Hooft's breakthrough in , it became clear that the old problem of infinities in the weak interactions had been solved by the use of spontaneous symmetry breaking to give masses to the W and Z particles. Then the asymptotic freedom of quantum chromodynamics gave us a framework in which we could actually calculate something about the strong interactions - not everything, but at least something.
But in scoring these victories, quantum field theory was preparing the way for a further change in our attitude, in which quantum field theory would lose its central position. Lillian Hoddeson 13 November External links [ edit ]. Wikipedia has an article about: Quantum field theory. Categories : Science stubs Physics.