# Quantum field theory

**Quantum field theory** (QFT) is a mathematical theory in physics which modernizes quantum mechanics from its early-20th-century formulation. In the areas of its domain (electromagnetic, weak nuclear, and strong nuclear forces), it is arguably the most precisely verified theory in all of physics. It is so close to being the "theory of everything" that it is part of the advanced physics curriculum everywhere. But since it does not integrate gravity into its formulation, it isn't actually the ultimate "theory of everything".

QFT describes the interactions between subatomic particles, such as electrons, protons, quarks and photons.

The differences between basic QM and Field Theory are these: in QM, the interactions between more than two particles are increasingly difficult to model, and the creation and destruction of particles cannot be modeled at all. In contrast, Field Theory can describe states containing arbitrary numbers of particles of different energies, masses, charges and types. Field Theory also provides an elegant framework for describing the interactions between particles, and the creation of new particles and destruction of old ones-- for example, the emission and absorption of photons by electrons, and vice versa.^{[1]}

Like all physical theories since the mid-1930's, QFT is "Lorentz invariant". That is, it is consistent with Special Relativity. However, as discussed below, it does not explain gravity in a manner consistent with the curvature of spacetime in General Relativity.

QFT bases the mathematical work on an assumption that elementary particles can be treated as "point-like objects of zero intrinsic size."^{[2]}

For a given interaction of particles, field theoretical calculations generally cannot be solved in a closed, analytical form-- that is, the predicted probabilities cannot be described by one simple equation; however, physicists have developed numerous approximation methods which produce estimates of ever-increasing precision (compared to experiment), with the precision depending upon how much mathematical work is put into the analysis.^{[3]}

Field theoretical calculations, while extremely labor-intensive, have yielded predictions of unparalleled precision compared to experimental measurements. The most famous of these is the anomalous magnetic moment of the electron, a very difficult calculation that has (so far) yielded a prediction^{[4]} accurate to one part in a billion compared to experiment^{[5]}

Field theoretical calculations can be extremely labor-intensive, if math is employed by hand; or computer time-intensive, if software approximations are used. Computational attempts to model the field theoretical equations of the strong nuclear force within protons, neutrons^{[6]} and mesons^{[7]} are among the most intensive computational research projects ever attempted, but have produced good agreement with experiment.

Field theoretical calculations become increasingly difficult as the number of (incoming or outgoing) particles increases; for very large numbers of particles (e.g. macroscopic liquids and solids), the methods of Solid State Physics (also based on QM) are employed instead.

The most naive application of Quantum Field Theory to any theory of gravity is known to be unworkable. It has a well-known (in theoretical physics circles) "cosmological constant problem."^{[8]}
This is in contrast to the other three forces of nature (electromagnetic, weak nuclear, strong nuclear), which have extensive and impressive experimental confirmation^{[9]}. No alternative formulations of quantum gravity have experimental confirmation. While Special Relativity is an intrinsic part of quantum field theory, the prevailing theory of gravity (the curvature of spacetime under General Relativity) has not been reconciled. Integrating this fourth force of nature into quantum field theory is an extremely active area of theoretical research. Current research in the field involves such things as supersymmetry and string theory

## Contents

## The Forces of Nature

In the Standard Model of physics, there are four forces of nature: electromagnetic, weak nuclear, strong nuclear, and gravity.^{[10]}

The electromagnetic force models interactions between electrically charged particles, and historically resulted from a unification of the electrical and magnetic fields, which were once thought to be separate fields.

The weak nuclear force^{[11]} is most well-known for mediating radioactive atomic decays, in which (for example) a proton in a nucleus will turn into a neutron (which remains in the nucleus), and a positron and neutrino, which are emitted. Non-nuclear particles such as electrons also participate in the weak force. Neutrinos only participate in the weak force, and have extremely low mass, making their observation very difficult. The weak nuclear force only exerts force when particles are extremely close together.

The strong nuclear force holds together the protons and neutrons in an atomic nucleus, and the quarks within protons, neutrons, and mesons.^{[12]} Because protons all have the same charge, they repel each other strongly, and the strong nuclear force is necessary to overcome this and hold them together in a nucleus; likewise for the quarks inside protons, neutrons and mesons. Unlike electromagnetism, which can extend over long distances, the strong nuclear force only exerts force when particles are extremely close together; but at close range, it is enormously stronger than electromagnetism.

Field theory can model the first three forces with a high degree of precision and success, but fails for gravity^{[13]}, for reasons described below. Gravity is by far the weakest of all the forces of nature, when measured in absolute units. This may seem to us to be paradoxical. Electromagnetism (for example) appears to us to be weaker than gravity, because most matter we encounter is nearly equal in positive and negative charges, so that the opposing charges nearly cancel at long distances. However, with gravity, there is no observed "negative mass" to cancel out the effect of positive mass, so the gravitational forces of particles, while individually very tiny, are cumulatively enormous over long distances. (Antimatter has positive mass, but opposite properties to matter.)

## Field Bosons Mediate Action At a Distance

How is it that particles affect each other-- for example, the repulsion of like charges, or attraction of unlike charges? Field theory can incorporate "action at a distance" models by the direct incorporation of an external potential field, e.g. V(x) where x is the distance from a fixed, unmoving charge, and V is the potential energy of the interaction. However, such models are of limited use because x becomes dependent on the motion of the other particle, if the other particle moves; and in addition this method ignores fluctuations of virtual particles expected from Heisenberg's Uncertainty Principle. Thus, most commonly field theory employs other methods.

The forces of nature are mediated by particles called field bosons. Particles exert forces on each other, which appears to be action at a distance, by passing back and forth between them virtual field bosons, which exchange their momenta. As an analogy, consider two people on ice skates on an ice rink, playing football. The first throws a football, and the reaction force of the throw pushes him backward. The second catches the football, and the catch pushes her backward. The two ice skaters are now "repelled" and moving away from each other, with the football as mediator of the force. Thus, virtual bosons mediate attractive and repulsive forces.

Bosons in general are particles of integral spin (0, 1, 2, etc.) "Spin" here refers to quantum mechanical angular momentum of a particle spinning about its axis. In quantum mechanics, all elementary particles can only have angular momenta which are multiples or half-multiples of Planck's constant. In particular, the first three forces (but not gravity) are considered to be mediated by bosons of spin 1.

The electromagnetic force is mediated by the photon, a particle of light, which is massless and uncharged. The theory which describes the interactions of charged particles and photons is called Quantum Electrodynamics (QED)^{[14]}, attributed primarily to Richard Feynman and Julian Schwinger.

The weak nuclear force is mediated by particles called W and Z bosons. These are very massive particles, as massive as a heavy atomic nucleus. The W particles are charged (W+ and W-) and Z is uncharged.

The electromagnetic and weak forces have been mathematically unified into a single formalism, called Electroweak Theory.^{[15]} The unification means that photons, W and Z bosons are all considered to be different aspects of a more fundamental doublet of field bosons. The electroweak theory can describe electromagnetic and weak phenomena with fewer tuneable free parameters-- as the single most important goal of physics is to describe all forces with as few free parameters as possible. Electroweak Theory has been extremely successful, and predicted the existence and approximate mass of the Z boson before its observation in experiments.

The strong nuclear force is mediated by particles called gluons. Particles such as quarks are said to have "color charge", which gives them an ability to exchange gluons, in the same way that particles with electrical charge can exchange photons. The theory that describes quarks and gluons is called Quantum Chromodynamics, or QCD.^{[16]} A major complication in QCD is that the gluons themselves have "color charge", unlike, say, photons which have no electrical charge. This makes QCD calculations extremely difficult. On the other hand, it has the advantage of eliminating the "screening problem" identified by Landau. Also, the coupling between quarks and gluons (color charge) is, measured in absolute units, much larger than the electrical charges of charged particles like electrons.

These differences make QCD much harder than QED. However, all the forces of nature involve their own theoretical challenges.

## Perturbative and Non-Perturbative Methods

A difference between QED/Electroweak and QCD is that the coupling of electron and photons (electrical charge) is much less than 1 when measured in absolute units, while the color charges of quarks in QCD is greater than 1. This means that predictions in QED can be approximated by a set of mathematical methods called "perturbative"^{[17]}, while QCD calculations often requires methods called "non-perturbative."

As a simple case, consider two electrons rushing toward each other. They should be repelled by like charges. In a QED perturbative calculation, this repulsion, while very complex, is modelled as a series of ever more complicated Feynman graphs each representing different exchanges of intermediate virtual particles. With each step in the calculation, more and more intermediate (internal) particles are exchanged, and the graphs grow more and more complicated. The zeroth. order approximation is that the outgoing particles are the same as the incoming (no change). The first order approximation is that one virtual photon is passed from one electron to the other. Graphs for the second order approximation are: the electrons exchange two virtual photons; OR, one incoming electron emits a photon, which splits into an electron/position pair, the pair recombines to form a photon, which is finally absorbed by the other incoming electron. Each new graph is considered an additional "perturbation" of the zeroth. approximation (no change). By the addition of many such graphs, predictions of extraordinary precision can be computed^{[18]}; but this is very labor-intensive.

In QCD, perturbative methods are of limited use because the color charge is so strong, that each new graph contributes more than the previous one, so the series may not terminate. Consequently, the interactions of quarks are often modeled by other means, e.g. the use of very powerful computer simulations called "Lattice Gauge QCD" simulations. In these simulations, space and time are approximated as a lattice of points separated by fixed distances. The quarks and gluons do not have fixed positions on the lattice, as they are quantum mechanical wave functions; rather, the field of each quark and gluon must be continually computed all over the lattice, the value at each lattice point continually changing due to the strong interactions of the wave functions. These methods require enormous amounts of time on powerful supercomputers, but have produced several important recent successes.^{[19]}^{[20]}

## Experimental Successes

Calculations involving field theory are universal throughout high-energy particle physics. Here we will briefly summarize a few more notable successes.

As stated above, one the greatest achievements of QED is the very labor intensive, but extremely precise, calculation of the angular magnetic moment of the electron. In classical quantum mechanics, the magnetic moment of the electron, called *g*, should be exactly 2.0. The very small deviation from 2 is called the anomalous magnetic moment, and can be experimentally measured to extremely high precision. The very labor-intensive theoretical predictions^{[21]} from QED match the experimental measurement to one part in a billion^{[22]}, a precision unparalleled in all of science. Also, the magnetic moment of the muon can be predicted to one part in a billion.

Also, as stated above, the Electroweak Theory, a unification of QED and weak force, predicted the existence and approximate mass of the Z boson before its observation.^{[23]}

QCD has successfully described jet events at particle colliders. In these events, a particle and antiparticle are smashed together in a collider; the resulting energy then turns into a quark and anti-quark pair. The quark and anti-quark each then split into vast, complex showers of other high-energy particles, called "jets", which are seen as showers of particles in opposing directions in the particle detector. If the quark and anti-quark produce showers immediately, it is called a two-jet event. In about 10% of all cases, a quark or anti-quark will emit a gluon, which then splits into still more particles, thus displaying a three-jet event. QCD successfully models the probability and momentum distributions of the jets.^{[24]}

Non-perturbative methods have recently been very successfully used in QCD. Lattice gauge QCD calculations, requiring vast computer power, have computed the masses of the proton, neutron^{[25]}^{[26]}, by computing the energy of the interaction E of the quarks and gluons on a lattice, and then employing Einstein's equation m = E/c2. These calculations compute the masses of several important particles entirely *a priori,* with no tuneable free parameters input to the model, except the strong coupling constant.

## Field Theory and Relativity

Although Quantum Field Theory is fully compatible with, and in fact requires Special Relativity, the most naive application of field theory to General Relativity is known to be unworkable. This is generally considered to be the most difficult outstanding problem in Physics.

The incompatibility is due both to the complications of curved space-time required by GR, as well as from presumed properties of gravitons. Gravitons are hypothetical bosons which, in the most naive application of Field Theory to GR, would mediate the gravitational force between particles. Unlike the bosons mediating the other forces of nature, which are spin 1, gravitons in the simplest case would be bosons of spin 2, coupled to the Einstein mass-energy tensor. This difference in spin greatly complicates the renormalization of quantum gravity.

Consequently, alternative theories of quantum gravity are an active area of research among physicists, including popular theoretical attempts such as string theory and loop quantum gravity. None of these alternative approaches yet has experimental confirmation.

## References

- ↑ Bjorken and Drell.
*Quantum Field Theory*Chapter 3; McGraw Hill Inc. (1980) - ↑ http://theory.caltech.edu/people/jhs/strings/str114.html
- ↑ Pierre Ramond.
*Field Theory: A Modern Primer*, Chapter VIII; Benjamin-Cummings Inc. (1981) - ↑ G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom,
*New Determination of the Fine Structure Constant from the Electron g Value and QED,*Phys. Rev. Lett. 97, 030802 (2006), Erratum, Phys. Rev. Lett. 99, 039902 (2007). - ↑ B. Odom, D. Hanneke, B. D'Urso, and G. Gabrielse,
*New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron,*Phys. Rev. Lett. 97, 030801 (2006). - ↑ S. Dürr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S. D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K. K. Szabo, and G. Vulvert.
*Ab Initio Determination of Light Hadron Masses.**Science*21 Vol. 322. no. 5905 (Nov. 2008), pp. 1224 - 1227. DOI: 10.1126/science.1163233. - ↑ C. T. Davies
*et al.**High-Precision Lattice QCD Confronts Experiment*. Phys. Rev. Lett. 92, 22001 (2004). DOI: 10.1103/PhysRevLett.92.022001. - ↑
- ↑ Chris Quigg.
*Gauge Theories of the Strong, Weak, and Electromagnetic Interactions*; Benjamin/Cummings Co. (1983). - ↑ Chris Quigg.
*Gauge Theories of the Strong, Weak, and Electromagnetic Interactions*p.16; Benjamin/Cummings Co. (1983). - ↑ Chris Quigg.
*Gauge Theories of the Strong, Weak, and Electromagnetic Interactions*Chapter 6; Benjamin/Cummings Co. (1983). - ↑ Chris Quigg.
*Gauge Theories of the Strong, Weak, and Electromagnetic Interactions*Chapter 8; Benjamin/Cummings Co. (1983). - ↑ Chris Quigg.
*Gauge Theories of the Strong, Weak, and Electromagnetic Interactions*p.271; Benjamin/Cummings Co. (1983). - ↑ Bjorken and Drell.
*Quantum Field Theory*Chapter 6; McGraw Hill Inc. (1980) - ↑ Chris Quigg.
*Gauge Theories of the Strong, Weak, and Electromagnetic Interactions*Chapter 6; Benjamin/Cummings Co. (1983). - ↑ Chris Quigg.
*Gauge Theories of the Strong, Weak, and Electromagnetic Interactions*Chapter 8; Benjamin/Cummings Co. (1983). - ↑ Bjorken and Drell.
*Quantum Field Theory*Chapter 6; McGraw Hill Inc. (1980) - ↑ G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom,
*New Determination of the Fine Structure Constant from the Electron g Value and QED,*Phys. Rev. Lett. 97, 030802 (2006), Erratum, Phys. Rev. Lett. 99, 039902 (2007). - ↑ S. Dürr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S. D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K. K. Szabo, and G. Vulvert.
*Ab Initio Determination of Light Hadron Masses.**Science*21 Vol. 322. no. 5905 (Nov. 2008), pp. 1224 - 1227. DOI: 10.1126/science.1163233. - ↑ C. T. Davies
*et al.**High-Precision Lattice QCD Confronts Experiment*. Phys. Rev. Lett. 92, 22001 (2004). DOI: 10.1103/PhysRevLett.92.022001. - ↑ G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom,
*New Determination of the Fine Structure Constant from the Electron g Value and QED,*Phys. Rev. Lett. 97, 030802 (2006), Erratum, Phys. Rev. Lett. 99, 039902 (2007). - ↑ B. Odom, D. Hanneke, B. D'Urso, and G. Gabrielse,
*New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron,*Phys. Rev. Lett. 97, 030801 (2006). - ↑ Chris Quigg.
*Gauge Theories of the Strong, Weak, and Electromagnetic Interactions*Chapter 6; Benjamin/Cummings Co. (1983). - ↑ http://nobelprize.org/nobel_prizes/physics/laureates/2004/wilczek-lecture.pdf (p. 102)
- ↑ S. Dürr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S. D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K. K. Szabo, and G. Vulvert.
*Ab Initio Determination of Light Hadron Masses.*and mesons*Science*21 Vol. 322. no. 5905 (Nov. 2008), pp. 1224 - 1227. DOI: 10.1126/science.1163233. - ↑ C. T. Davies
*et al.**High-Precision Lattice QCD Confronts Experiment*. Phys. Rev. Lett. 92, 22001 (2004). DOI: 10.1103/PhysRevLett.92.022001.