Quantum field theory |
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History |

In particle physics, **quantum electrodynamics** (**QED**) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction.

- History
- Feynman's view of quantum electrodynamics
- Introduction
- Basic constructions
- Probability amplitudes
- Propagators
- Mass renormalization
- Conclusions
- Mathematical formulation
- Equations of motion
- Interaction picture
- Feynman diagrams
- Nonperturbative phenomena
- Renormalizability
- Nonconvergence of series
- See also
- References
- Further reading
- Books
- Journals
- External links

In technical terms, QED can be described as a perturbation theory of the electromagnetic quantum vacuum. Richard Feynman called it "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the electron and the Lamb shift of the energy levels of hydrogen.^{ [1] }^{:Ch1}

The first formulation of a quantum theory describing radiation and matter interaction is attributed to British scientist Paul Dirac, who (during the 1920s) was able to compute the coefficient of spontaneous emission of an atom.^{ [2] }

Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. In the following years, with contributions from Wolfgang Pauli, Eugene Wigner, Pascual Jordan, Werner Heisenberg and an elegant formulation of quantum electrodynamics due to Enrico Fermi,^{ [3] } physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles. However, further studies by Felix Bloch with Arnold Nordsieck,^{ [4] } and Victor Weisskopf,^{ [5] } in 1937 and 1939, revealed that such computations were reliable only at a first order of perturbation theory, a problem already pointed out by Robert Oppenheimer.^{ [6] } At higher orders in the series infinities emerged, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself. With no solution for this problem known at the time, it appeared that a fundamental incompatibility existed between special relativity and quantum mechanics.

Difficulties with the theory increased through the end of the 1940s. Improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom,^{ [7] } now known as the Lamb shift and magnetic moment of the electron.^{ [8] } These experiments exposed discrepancies which the theory was unable to explain.

A first indication of a possible way out was given by Hans Bethe in 1947,^{ [9] } after attending the Shelter Island Conference.^{ [10] } While he was traveling by train from the conference to Schenectady he made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by Lamb and Retherford.^{ [9] } Despite the limitations of the computation, agreement was excellent. The idea was simply to attach infinities to corrections of mass and charge that were actually fixed to a finite value by experiments. In this way, the infinities get absorbed in those constants and yield a finite result in good agreement with experiments. This procedure was named renormalization.

Based on Bethe's intuition and fundamental papers on the subject by Shin'ichirō Tomonaga,^{ [11] } Julian Schwinger,^{ [12] }^{ [13] } Richard Feynman ^{ [14] }^{ [15] }^{ [16] } and Freeman Dyson,^{ [17] }^{ [18] } it was finally possible to get fully covariant formulations that were finite at any order in a perturbation series of quantum electrodynamics. Shin'ichirō Tomonaga, Julian Schwinger and Richard Feynman were jointly awarded with the 1965 Nobel Prize in Physics for their work in this area.^{ [19] } Their contributions, and those of Freeman Dyson, were about covariant and gauge-invariant formulations of quantum electrodynamics that allow computations of observables at any order of perturbation theory. Feynman's mathematical technique, based on his diagrams, initially seemed very different from the field-theoretic, operator-based approach of Schwinger and Tomonaga, but Freeman Dyson later showed that the two approaches were equivalent.^{ [17] } Renormalization, the need to attach a physical meaning at certain divergences appearing in the theory through integrals, has subsequently become one of the fundamental aspects of quantum field theory and has come to be seen as a criterion for a theory's general acceptability. Even though renormalization works very well in practice, Feynman was never entirely comfortable with its mathematical validity, even referring to renormalization as a "shell game" and "hocus pocus".^{ [1] }^{:128}

QED has served as the model and template for all subsequent quantum field theories. One such subsequent theory is quantum chromodynamics, which began in the early 1960s and attained its present form in the 1970s work by H. David Politzer, Sidney Coleman, David Gross and Frank Wilczek. Building on the pioneering work of Schwinger, Gerald Guralnik, Dick Hagen, and Tom Kibble,^{ [20] }^{ [21] } Peter Higgs, Jeffrey Goldstone, and others, Sheldon Lee Glashow, Steven Weinberg and Abdus Salam independently showed how the weak nuclear force and quantum electrodynamics could be merged into a single electroweak force.

Near the end of his life, Richard Feynman gave a series of lectures on QED intended for the lay public. These lectures were transcribed and published as Feynman (1985), * QED: The Strange Theory of Light and Matter *,^{ [1] } a classic non-mathematical exposition of QED from the point of view articulated below.

The key components of Feynman's presentation of QED are three basic actions.^{ [1] }^{:85}

- A photon goes from one place and time to another place and time.
- An electron goes from one place and time to another place and time.
- An electron emits or absorbs a photon at a certain place and time.

These actions are represented in the form of visual shorthand by the three basic elements of Feynman diagrams: a wavy line for the photon, a straight line for the electron and a junction of two straight lines and a wavy one for a vertex representing emission or absorption of a photon by an electron. These can all be seen in the adjacent diagram.

As well as the visual shorthand for the actions Feynman introduces another kind of shorthand for the numerical quantities called probability amplitudes. The probability is the square of the absolute value of total probability amplitude, . If a photon moves from one place and time to another place and time , the associated quantity is written in Feynman's shorthand as . The similar quantity for an electron moving from to is written . The quantity that tells us about the probability amplitude for the emission or absorption of a photon he calls *j*. This is related to, but not the same as, the measured electron charge *e*.^{ [1] }^{:91}

QED is based on the assumption that complex interactions of many electrons and photons can be represented by fitting together a suitable collection of the above three building blocks and then using the probability amplitudes to calculate the probability of any such complex interaction. It turns out that the basic idea of QED can be communicated while assuming that the square of the total of the probability amplitudes mentioned above (*P*(*A* to *B*), *E*(*C* to *D*) and *j*) acts just like our everyday probability (a simplification made in Feynman's book). Later on, this will be corrected to include specifically quantum-style mathematics, following Feynman.

The basic rules of probability amplitudes that will be used are:^{ [1] }^{:93}

- If an event can happen in a variety of different ways, then its probability amplitude is the
**sum**of the probability amplitudes of the possible ways. - If a process involves a number of independent sub-processes, then its probability amplitude is the
**product**of the component probability amplitudes.

Suppose, we start with one electron at a certain place and time (this place and time being given the arbitrary label *A*) and a photon at another place and time (given the label *B*). A typical question from a physical standpoint is: "What is the probability of finding an electron at *C* (another place and a later time) and a photon at *D* (yet another place and time)?". The simplest process to achieve this end is for the electron to move from *A* to *C* (an elementary action) and for the photon to move from *B* to *D* (another elementary action). From a knowledge of the probability amplitudes of each of these sub-processes – *E*(*A* to *C*) and *P*(*B* to *D*) – we would expect to calculate the probability amplitude of both happening together by multiplying them, using rule b) above. This gives a simple estimated overall probability amplitude, which is squared to give an estimated probability.^{[ citation needed ]}

But there are other ways in which the end result could come about. The electron might move to a place and time *E*, where it absorbs the photon; then move on before emitting another photon at *F*; then move on to *C*, where it is detected, while the new photon moves on to *D*. The probability of this complex process can again be calculated by knowing the probability amplitudes of each of the individual actions: three electron actions, two photon actions and two vertexes – one emission and one absorption. We would expect to find the total probability amplitude by multiplying the probability amplitudes of each of the actions, for any chosen positions of *E* and *F*. We then, using rule a) above, have to add up all these probability amplitudes for all the alternatives for *E* and *F*. (This is not elementary in practice and involves integration.) But there is another possibility, which is that the electron first moves to *G*, where it emits a photon, which goes on to *D*, while the electron moves on to *H*, where it absorbs the first photon, before moving on to *C*. Again, we can calculate the probability amplitude of these possibilities (for all points *G* and *H*). We then have a better estimation for the total probability amplitude by adding the probability amplitudes of these two possibilities to our original simple estimate. Incidentally, the name given to this process of a photon interacting with an electron in this way is Compton scattering.^{[ citation needed ]}

There is an *infinite number* of other intermediate processes in which more and more photons are absorbed and/or emitted. For each of these possibilities, there is a Feynman diagram describing it. This implies a complex computation for the resulting probability amplitudes, but provided it is the case that the more complicated the diagram, the less it contributes to the result, it is only a matter of time and effort to find as accurate an answer as one wants to the original question. This is the basic approach of QED. To calculate the probability of *any* interactive process between electrons and photons, it is a matter of first noting, with Feynman diagrams, all the possible ways in which the process can be constructed from the three basic elements. Each diagram involves some calculation involving definite rules to find the associated probability amplitude.

That basic scaffolding remains when one moves to a quantum description, but some conceptual changes are needed. One is that whereas we might expect in our everyday life that there would be some constraints on the points to which a particle can move, that is *not* true in full quantum electrodynamics. There is a possibility of an electron at *A*, or a photon at *B*, moving as a basic action to *any other place and time in the universe*. That includes places that could only be reached at speeds greater than that of light and also *earlier times*. (An electron moving backwards in time can be viewed as a positron moving forward in time.)^{ [1] }^{:89, 98–99}

Quantum mechanics introduces an important change in the way probabilities are computed. Probabilities are still represented by the usual real numbers we use for probabilities in our everyday world, but probabilities are computed as the square modulus of probability amplitudes, which are complex numbers.

Feynman avoids exposing the reader to the mathematics of complex numbers by using a simple but accurate representation of them as arrows on a piece of paper or screen. (These must not be confused with the arrows of Feynman diagrams, which are simplified representations in two dimensions of a relationship between points in three dimensions of space and one of time.) The amplitude arrows are fundamental to the description of the world given by quantum theory. They are related to our everyday ideas of probability by the simple rule that the probability of an event is the *square* of the length of the corresponding amplitude arrow. So, for a given process, if two probability amplitudes, **v** and **w**, are involved, the probability of the process will be given either by

or

The rules as regards adding or multiplying, however, are the same as above. But where you would expect to add or multiply probabilities, instead you add or multiply probability amplitudes that now are complex numbers.

Addition and multiplication are common operations in the theory of complex numbers and are given in the figures. The sum is found as follows. Let the start of the second arrow be at the end of the first. The sum is then a third arrow that goes directly from the beginning of the first to the end of the second. The product of two arrows is an arrow whose length is the product of the two lengths. The direction of the product is found by adding the angles that each of the two have been turned through relative to a reference direction: that gives the angle that the product is turned relative to the reference direction.

That change, from probabilities to probability amplitudes, complicates the mathematics without changing the basic approach. But that change is still not quite enough because it fails to take into account the fact that both photons and electrons can be polarized, which is to say that their orientations in space and time have to be taken into account. Therefore, *P*(*A* to *B*) consists of 16 complex numbers, or probability amplitude arrows.^{ [1] }^{:120–121} There are also some minor changes to do with the quantity *j*, which may have to be rotated by a multiple of 90° for some polarizations, which is only of interest for the detailed bookkeeping.

Associated with the fact that the electron can be polarized is another small necessary detail, which is connected with the fact that an electron is a fermion and obeys Fermi–Dirac statistics. The basic rule is that if we have the probability amplitude for a given complex process involving more than one electron, then when we include (as we always must) the complementary Feynman diagram in which we exchange two electron events, the resulting amplitude is the reverse – the negative – of the first. The simplest case would be two electrons starting at *A* and *B* ending at *C* and *D*. The amplitude would be calculated as the "difference", *E*(*A* to *D*) × *E*(*B* to *C*) − *E*(*A* to *C*) × *E*(*B* to *D*), where we would expect, from our everyday idea of probabilities, that it would be a sum.^{ [1] }^{:112–113}

Finally, one has to compute *P*(*A* to *B*) and *E*(*C* to *D*) corresponding to the probability amplitudes for the photon and the electron respectively. These are essentially the solutions of the Dirac equation, which describe the behavior of the electron's probability amplitude and the Maxwell's equations, which describes the behavior of the photon's probability amplitude. These are called Feynman propagators. The translation to a notation commonly used in the standard literature is as follows:

where a shorthand symbol such as stands for the four real numbers that give the time and position in three dimensions of the point labeled *A*.

A problem arose historically which held up progress for twenty years: although we start with the assumption of three basic "simple" actions, the rules of the game say that if we want to calculate the probability amplitude for an electron to get from *A* to *B*, we must take into account *all* the possible ways: all possible Feynman diagrams with those endpoints. Thus there will be a way in which the electron travels to *C*, emits a photon there and then absorbs it again at *D* before moving on to *B*. Or it could do this kind of thing twice, or more. In short, we have a fractal-like situation in which if we look closely at a line, it breaks up into a collection of "simple" lines, each of which, if looked at closely, are in turn composed of "simple" lines, and so on *ad infinitum*. This is a challenging situation to handle. If adding that detail only altered things slightly, then it would not have been too bad, but disaster struck when it was found that the simple correction mentioned above led to *infinite* probability amplitudes. In time this problem was "fixed" by the technique of renormalization. However, Feynman himself remained unhappy about it, calling it a "dippy process".^{ [1] }^{:128}

Within the above framework physicists were then able to calculate to a high degree of accuracy some of the properties of electrons, such as the anomalous magnetic dipole moment. However, as Feynman points out, it fails to explain why particles such as the electron have the masses they do. "There is no theory that adequately explains these numbers. We use the numbers in all our theories, but we don't understand them – what they are, or where they come from. I believe that from a fundamental point of view, this is a very interesting and serious problem."^{ [1] }^{:152}

Mathematically, QED is an abelian gauge theory with the symmetry group U(1). The gauge field, which mediates the interaction between the charged spin-1/2 fields, is the electromagnetic field. The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field is given in natural units by the real part of^{ [22] }^{:78}

where

- are Dirac matrices;
- a bispinor field of spin-1/2 particles (e.g. electron–positron field);
- , called "psi-bar", is sometimes referred to as the Dirac adjoint;
- is the gauge covariant derivative;
*e*is the coupling constant, equal to the electric charge of the bispinor field;*m*is the mass of the electron or positron;- is the covariant four-potential of the electromagnetic field generated by the electron itself;
- is the external field imposed by external source;
- is the electromagnetic field tensor.

Substituting the definition of *D* into the Lagrangian gives

From this Lagrangian, the equations of motion for the *ψ* and *A* fields can be obtained.

- Using the field-theoretic Euler–Lagrange equation for
*ψ*,

**(2)**

The derivatives of the Lagrangian concerning *ψ* are

Inserting these into (** 2 **) results in

with Hermitian conjugate

Bringing the middle term to the right-hand side yields

The left-hand side is like the original Dirac equation, and the right-hand side is the interaction with the electromagnetic field.

- Using the Euler–Lagrange equation for the
*A*field,

**(3)**

the derivatives this time are

Substituting back into (** 3 **) leads to

Now, if we impose the Lorenz gauge condition

the equations reduce to

which is a wave equation for the four-potential, the QED version of the classical Maxwell equations in the Lorenz gauge. (The square represents the D'Alembert operator, .)

This theory can be straightforwardly quantized by treating bosonic and fermionic sectors^{[ clarification needed ]} as free. This permits us to build a set of asymptotic states that can be used to start computation of the probability amplitudes for different processes. In order to do so, we have to compute an evolution operator, which for a given initial state will give a final state in such a way to have^{ [22] }^{:5}

This technique is also known as the S-matrix. The evolution operator is obtained in the interaction picture, where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:^{ [22] }^{:123}

and so, one has^{ [22] }^{:86}

where *T* is the time-ordering operator. This evolution operator only has meaning as a series, and what we get here is a perturbation series with the fine-structure constant as the development parameter. This series is called the Dyson series.

Despite the conceptual clarity of this Feynman approach to QED, almost no early textbooks follow him in their presentation. When performing calculations, it is much easier to work with the Fourier transforms of the propagators. Experimental tests of quantum electrodynamics are typically scattering experiments. In scattering theory, particles' momenta rather than their positions are considered, and it is convenient to think of particles as being created or annihilated when they interact. Feynman diagrams then *look* the same, but the lines have different interpretations. The electron line represents an electron with a given energy and momentum, with a similar interpretation of the photon line. A vertex diagram represents the annihilation of one electron and the creation of another together with the absorption or creation of a photon, each having specified energies and momenta.

Using Wick's theorem on the terms of the Dyson series, all the terms of the S-matrix for quantum electrodynamics can be computed through the technique of Feynman diagrams. In this case, rules for drawing are the following^{ [22] }^{:801–802}

To these rules we must add a further one for closed loops that implies an integration on momenta , since these internal ("virtual") particles are not constrained to any specific energy–momentum, even that usually required by special relativity (see Propagator for details).

From them, computations of probability amplitudes are straightforwardly given. An example is Compton scattering, with an electron and a photon undergoing elastic scattering. Feynman diagrams are in this case^{ [22] }^{:158–159}

and so we are able to get the corresponding amplitude at the first order of a perturbation series for the S-matrix:

from which we can compute the cross section for this scattering.

The predictive success of quantum electrodynamics largely rests on the use of perturbation theory, expressed in Feynman diagrams. However, quantum electrodynamics also leads to predictions beyond perturbation theory. In the presence of very strong electric fields, it predicts that electrons and positrons will be spontaneously produced, so causing the decay of the field. This process, called the Schwinger effect,^{ [23] } cannot be understood in terms of any finite number of Feynman diagrams and hence is described as nonperturbative. Mathematically, it can be derived by a semiclassical approximation to the path integral of quantum electrodynamics.

Higher-order terms can be straightforwardly computed for the evolution operator, but these terms display diagrams containing the following simpler ones^{ [22] }^{:ch 10}

- One-loop contribution to the vacuum polarization function
- One-loop contribution to the electron self-energy function
- One-loop contribution to the vertex function

that, being closed loops, imply the presence of diverging integrals having no mathematical meaning. To overcome this difficulty, a technique called renormalization has been devised, producing finite results in very close agreement with experiments. A criterion for the theory being meaningful after renormalization is that the number of diverging diagrams is finite. In this case, the theory is said to be "renormalizable". The reason for this is that to get observables renormalized, one needs a finite number of constants to maintain the predictive value of the theory untouched. This is exactly the case of quantum electrodynamics displaying just three diverging diagrams. This procedure gives observables in very close agreement with experiment as seen e.g. for electron gyromagnetic ratio.

Renormalizability has become an essential criterion for a quantum field theory to be considered as a viable one. All the theories describing fundamental interactions, except gravitation, whose quantum counterpart is only conjectural and presently under very active research, are renormalizable theories.

An argument by Freeman Dyson shows that the radius of convergence of the perturbation series in QED is zero.^{ [24] } The basic argument goes as follows: if the coupling constant were negative, this would be equivalent to the Coulomb force constant being negative. This would "reverse" the electromagnetic interaction so that *like* charges would *attract* and *unlike* charges would *repel*. This would render the vacuum unstable against decay into a cluster of electrons on one side of the universe and a cluster of positrons on the other side of the universe. Because the theory is "sick" for any negative value of the coupling constant, the series does not converge but are at best an asymptotic series.

From a modern perspective, we say that QED is not well defined as a quantum field theory to arbitrarily high energy.^{ [25] } The coupling constant runs to infinity at finite energy, signalling a Landau pole. The problem is essentially that QED appears to suffer from quantum triviality issues. This is one of the motivations for embedding QED within a Grand Unified Theory.

- Abraham–Lorentz force
- Anomalous magnetic moment
- Bhabha scattering
- Cavity quantum electrodynamics
- Circuit quantum electrodynamics
- Compton scattering
- Euler–Heisenberg Lagrangian
- Gupta–Bleuler formalism
- Lamb shift
- Landau pole
- Moeller scattering
- Non-relativistic quantum electrodynamics
- Photon polarization
- Positronium
- Precision tests of QED
- QED vacuum
*QED: The Strange Theory of Light and Matter*- Quantization of the electromagnetic field
- Scalar electrodynamics
- Schrödinger equation
- Schwinger model
- Schwinger–Dyson equation
- Vacuum polarization
- Vertex function
- Wheeler–Feynman absorber theory

In theoretical physics, a **Feynman diagram** is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced the diagrams in 1948. The interaction of subatomic particles can be complex and difficult to understand; Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula. According to David Kaiser, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations. Feynman diagrams have revolutionized nearly every aspect of theoretical physics." While the diagrams are applied primarily to quantum field theory, they can also be used in other fields, such as solid-state theory. Frank Wilczek wrote that the calculations which won him the 2004 Nobel Prize in Physics "would have been literally unthinkable without Feynman diagrams, as would [Wilczek's] calculations that established a route to production and observation of the Higgs particle."

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

**Spontaneous emission** is the process in which a quantum mechanical system transits from an excited energy state to a lower energy state and emits a quantized amount of energy in the form of a photon. Spontaneous emission is ultimately responsible for most of the light we see all around us; it is so ubiquitous that there are many names given to what is essentially the same process. If atoms are excited by some means other than heating, the spontaneous emission is called luminescence. For example, fireflies are luminescent. And there are different forms of luminescence depending on how excited atoms are produced. If the excitation is affected by the absorption of radiation the spontaneous emission is called fluorescence. Sometimes molecules have a metastable level and continue to fluoresce long after the exciting radiation is turned off; this is called phosphorescence. Figurines that glow in the dark are phosphorescent. Lasers start via spontaneous emission, then during continuous operation work by stimulated emission.

In particle physics, the **Dirac equation** is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

**Renormalization** is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their **self-interactions**. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.

In physics, the **Schwinger model**, named after Julian Schwinger, is the model describing 1+1D *Lorentzian* quantum electrodynamics which includes Electrons, coupled to Photons.

In theoretical physics, a **chiral anomaly** is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have more left than right, or vice versa.

In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, **relativistic wave equations** predict the behavior of particles at high energies and velocities comparable to the speed of light. In the context of quantum field theory (QFT), the equations determine the dynamics of quantum fields. The solutions to the equations, universally denoted as ψ or Ψ, are referred to as "wave functions" in the context of RQM, and "fields" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a wave equation or are generated from a Lagrangian density and the field-theoretic Euler–Lagrange equations.

In quantum electrodynamics, the **vertex function** describes the coupling between a photon and an electron beyond the leading order of perturbation theory. In particular, it is the one particle irreducible correlation function involving the fermion , the antifermion , and the vector potential **A**.

In theoretical physics, the **Rarita–Schwinger equation** is the relativistic field equation of spin-3/2 fermions. It is similar to the Dirac equation for spin-1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinger in 1941.

In the physics of gauge theories, **gauge fixing** denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.

This article describes the mathematics of the **Standard Model** of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3) × SU(2) × U(1). The theory is commonly viewed as containing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson.

In quantum electrodynamics, the **anomalous magnetic moment** of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle.

In mathematical physics, **spacetime algebra** (STA) is a name for the Clifford algebra Cl_{1,3}(**R**), or equivalently the geometric algebra G(M^{4}). According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime.

The **Bethe–Salpeter equation** describes the bound states of a two-body (particles) quantum field theoretical system in a relativistically covariant formalism. The equation was actually first published in 1950 at the end of a paper by Yoichiro Nambu, but without derivation.

In physics, **relativistic quantum mechanics** (**RQM**) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light *c*, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. *Non-relativistic quantum mechanics* refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. *Relativistic quantum mechanics* (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them also work with special relativity.

In physics, a **gauge theory** is a type of field theory in which the Lagrangian does not change under local transformations from certain Lie groups.

In quantum field theory, and especially in quantum electrodynamics, the interacting theory leads to infinite quantities that have to be absorbed in a renormalization procedure, in order to be able to predict measurable quantities. The renormalization scheme can depend on the type of particles that are being considered. For particles that can travel asymptotically large distances, or for low energy processes, the **on-shell scheme**, also known as the physical scheme, is appropriate. If these conditions are not fulfilled, one can turn to other schemes, like the minimal subtraction scheme.

In quantum field theory, and in the significant subfields of quantum electrodynamics (QED) and quantum chromodynamics (QCD), the **two-body Dirac equations (TBDE)** of constraint dynamics provide a three-dimensional yet manifestly covariant reformulation of the Bethe–Salpeter equation for two spin-1/2 particles. Such a reformulation is necessary since without it, as shown by Nakanishi, the Bethe–Salpeter equation possesses negative-norm solutions arising from the presence of an essentially relativistic degree of freedom, the relative time. These "ghost" states have spoiled the naive interpretation of the Bethe–Salpeter equation as a quantum mechanical wave equation. The two-body Dirac equations of constraint dynamics rectify this flaw. The forms of these equations can not only be derived from quantum field theory they can also be derived purely in the context of Dirac's constraint dynamics and relativistic mechanics and quantum mechanics. Their structures, unlike the more familiar two-body Dirac equation of Breit, which is a single equation, are that of two simultaneous quantum relativistic wave equations. A single two-body Dirac equation similar to the Breit equation can be derived from the TBDE. Unlike the Breit equation, it is manifestly covariant and free from the types of singularities that prevent a strictly nonperturbative treatment of the Breit equation.

In quantum field theory, the **nonlinear Dirac equation** is a model of self-interacting Dirac fermions. This model is widely considered in quantum physics as a toy model of self-interacting electrons.

- 1 2 3 4 5 6 7 8 9 10 11 Feynman, Richard (1985).
*QED: The Strange Theory of Light and Matter*. Princeton University Press. ISBN 978-0-691-12575-6. - ↑ P. A. M. Dirac (1927). "The Quantum Theory of the Emission and Absorption of Radiation".
*Proceedings of the Royal Society of London A*.**114**(767): 243–65. Bibcode:1927RSPSA.114..243D. doi: 10.1098/rspa.1927.0039 . - ↑ E. Fermi (1932). "Quantum Theory of Radiation".
*Reviews of Modern Physics*.**4**(1): 87–132. Bibcode:1932RvMP....4...87F. doi:10.1103/RevModPhys.4.87. - ↑ Bloch, F.; Nordsieck, A. (1937). "Note on the Radiation Field of the Electron".
*Physical Review*.**52**(2): 54–59. Bibcode:1937PhRv...52...54B. doi:10.1103/PhysRev.52.54. - ↑ V. F. Weisskopf (1939). "On the Self-Energy and the Electromagnetic Field of the Electron".
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*Modern Elementary Particle Physics*. Westview Press. ISBN 978-0-201-62460-1. - Miller, Arthur I. (1995).
*Early Quantum Electrodynamics: A Sourcebook*. Cambridge University Press. ISBN 978-0-521-56891-3. - Milonni, Peter W. (1994).
*The Quantum Vacuum: An Introduction to Quantum Electrodynamics*. Boston: Academic Press. ISBN 0124980805. LCCN 93029780. OCLC 422797902. - Schweber, Silvan S. (1994).
*QED and the Men Who Made It*. Princeton University Press. ISBN 978-0-691-03327-3. - Schwinger, Julian (1958).
*Selected Papers on Quantum Electrodynamics*. Dover Publications. ISBN 978-0-486-60444-2. - Tannoudji-Cohen, Claude; Dupont-Roc, Jacques; Grynberg, Gilbert (1997).
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- Feynman's Nobel Prize lecture describing the evolution of QED and his role in it
- Feynman's New Zealand lectures on QED for non-physicists
- http://qed.wikina.org/ – Animations demonstrating QED

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