Quantum electrodynamics

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.

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

History

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.

Feynman and Oppenheimer at Los Alamos
Feynman (center) and Oppenheimer (right) at Los Alamos.

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 a Nobel prize in physics in 1965 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.

Feynman's view of quantum electrodynamics

Introduction

Near the end of his life, Richard P. 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

  1. 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.
  2. If a process involves a number of independent sub-processes, then its probability amplitude is the product of the component probability amplitudes.

Basic constructions

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.

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.

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

Probability amplitudes

Feynmans QED probability amplitudes
Feynman replaces complex numbers with spinning arrows, which start at emission and end at detection of a particle. The sum of all resulting arrows represents the total probability of the event. In this diagram, light emitted by the source S bounces off a few segments of the mirror (in blue) before reaching the detector at P. The sum of all paths must be taken into account. The graph below depicts the total time spent to traverse each of the paths above.

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 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.

AdditionComplexes
Addition of probability amplitudes as complex numbers
MultiplicationComplexes
Multiplication of probability amplitudes as 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

Propagators

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 Klein–Gordon equation, 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.

Mass renormalization

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

Conclusions

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

Mathematics

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. electronpositron 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.

Equations of motion

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, .)

Interaction picture

This theory can be straightforwardly quantized by treating bosonic and fermionic sectors 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.

Feynman diagrams

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 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

Qed rules
Qed2e

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

Compton qed

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.

Renormalizability

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

Vacuum polarization

One-loop contribution to the vacuum polarization function

Electron self energy

One-loop contribution to the electron self-energy function

Vertex correction

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. It is important to note that 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 presently under very active research, are renormalizable theories.

Nonconvergence of series

An argument by Freeman Dyson shows that the radius of convergence of the perturbation series in QED is zero.[23] 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.[24] 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.

See also

References

  1. ^ a b c d e f g h i j k Feynman, Richard (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 978-0-691-12575-6.
  2. ^ 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.
  3. ^ E. Fermi (1932). "Quantum Theory of Radiation". Reviews of Modern Physics. 4: 87–132. Bibcode:1932RvMP....4...87F. doi:10.1103/RevModPhys.4.87.
  4. ^ 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.
  5. ^ V. F. Weisskopf (1939). "On the Self-Energy and the Electromagnetic Field of the Electron". Physical Review. 56: 72–85. Bibcode:1939PhRv...56...72W. doi:10.1103/PhysRev.56.72.
  6. ^ R. Oppenheimer (1930). "Note on the Theory of the Interaction of Field and Matter". Physical Review. 35 (5): 461–77. Bibcode:1930PhRv...35..461O. doi:10.1103/PhysRev.35.461.
  7. ^ Lamb, Willis; Retherford, Robert (1947). "Fine Structure of the Hydrogen Atom by a Microwave Method,". Physical Review. 72 (3): 241–43. Bibcode:1947PhRv...72..241L. doi:10.1103/PhysRev.72.241.
  8. ^ Foley, H.M.; Kusch, P. (1948). "On the Intrinsic Moment of the Electron". Physical Review. 73 (3): 412. Bibcode:1948PhRv...73..412F. doi:10.1103/PhysRev.73.412.
  9. ^ a b H. Bethe (1947). "The Electromagnetic Shift of Energy Levels". Physical Review. 72 (4): 339–41. Bibcode:1947PhRv...72..339B. doi:10.1103/PhysRev.72.339.
  10. ^ Schweber, Silvan (1994). "Chapter 5". QED and the Men Who Did it: Dyson, Feynman, Schwinger, and Tomonaga. Princeton University Press. p. 230. ISBN 978-0-691-03327-3.
  11. ^ S. Tomonaga (1946). "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields". Progress of Theoretical Physics. 1 (2): 27–42. Bibcode:1946PThPh...1...27T. doi:10.1143/PTP.1.27.
  12. ^ J. Schwinger (1948). "On Quantum-Electrodynamics and the Magnetic Moment of the Electron". Physical Review. 73 (4): 416–17. Bibcode:1948PhRv...73..416S. doi:10.1103/PhysRev.73.416.
  13. ^ J. Schwinger (1948). "Quantum Electrodynamics. I. A Covariant Formulation". Physical Review. 74 (10): 1439–61. Bibcode:1948PhRv...74.1439S. doi:10.1103/PhysRev.74.1439.
  14. ^ R. P. Feynman (1949). "Space–Time Approach to Quantum Electrodynamics". Physical Review. 76 (6): 769–89. Bibcode:1949PhRv...76..769F. doi:10.1103/PhysRev.76.769.
  15. ^ R. P. Feynman (1949). "The Theory of Positrons". Physical Review. 76 (6): 749–59. Bibcode:1949PhRv...76..749F. doi:10.1103/PhysRev.76.749.
  16. ^ R. P. Feynman (1950). "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction". Physical Review. 80 (3): 440–57. Bibcode:1950PhRv...80..440F. doi:10.1103/PhysRev.80.440.
  17. ^ a b F. Dyson (1949). "The Radiation Theories of Tomonaga, Schwinger, and Feynman". Physical Review. 75 (3): 486–502. Bibcode:1949PhRv...75..486D. doi:10.1103/PhysRev.75.486.
  18. ^ F. Dyson (1949). "The S Matrix in Quantum Electrodynamics". Physical Review. 75 (11): 1736–55. Bibcode:1949PhRv...75.1736D. doi:10.1103/PhysRev.75.1736.
  19. ^ "The Nobel Prize in Physics 1965". Nobel Foundation. Retrieved 2008-10-09.
  20. ^ Guralnik, G. S.; Hagen, C. R.; Kibble, T. W. B. (1964). "Global Conservation Laws and Massless Particles". Physical Review Letters. 13 (20): 585–87. Bibcode:1964PhRvL..13..585G. doi:10.1103/PhysRevLett.13.585.
  21. ^ Guralnik, G. S. (2009). "The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles". International Journal of Modern Physics A. 24 (14): 2601–27. arXiv:0907.3466. Bibcode:2009IJMPA..24.2601G. doi:10.1142/S0217751X09045431.
  22. ^ a b c d e f g Peskin, Michael; Schroeder, Daniel (1995). An introduction to quantum field theory (Reprint ed.). Westview Press. ISBN 978-0201503975.
  23. ^ Kinoshita, Toichiro (June 5, 1997). "Quantum Electrodynamics has Zero Radius of Convergence Summarized from Toichiro Kinoshita". Retrieved May 6, 2017.
  24. ^ Espriu and Tarrach (Apr 30, 1996). "Ambiguities in QED: Renormalons versus Triviality". Physics Letters B. 383: 482–486. arXiv:hep-ph/9604431. Bibcode:1996PhLB..383..482E. doi:10.1016/0370-2693(96)00779-4.

Further reading

Books

  • De Broglie, Louis (1925). Recherches sur la theorie des quanta [Research on quantum theory]. France: Wiley-Interscience.
  • Feynman, Richard Phillips (1998). Quantum Electrodynamics (New ed.). Westview Press. ISBN 978-0-201-36075-2.
  • Jauch, J.M.; Rohrlich, F. (1980). The Theory of Photons and Electrons. Springer-Verlag. ISBN 978-0-387-07295-1.
  • Greiner, Walter; Bromley, D.A.; Müller, Berndt (2000). Gauge Theory of Weak Interactions. Springer. ISBN 978-3-540-67672-0.
  • Kane, Gordon, L. (1993). 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). Photons and Atoms: Introduction to Quantum Electrodynamics. Wiley-Interscience. ISBN 978-0-471-18433-1.

Journals

  • Dudley, J.M.; Kwan, A.M. (1996). "Richard Feynman's popular lectures on quantum electrodynamics: The 1979 Robb Lectures at Auckland University". American Journal of Physics. 64 (6): 694–98. Bibcode:1996AmJPh..64..694D. doi:10.1119/1.18234.

External links

Anomalous magnetic dipole moment

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. (The magnetic moment, also called magnetic dipole moment, is a measure of the strength of a magnetic source.)

The "Dirac" magnetic moment, corresponding to tree-level Feynman diagrams (which can be thought of as the classical result), can be calculated from the Dirac equation. It is usually expressed in terms of the g-factor; the Dirac equation predicts g = 2. For particles such as the electron, this classical result differs from the observed value by a small fraction of a percent. The difference is the anomalous magnetic moment, denoted a and defined as

Cavity quantum electrodynamics

Cavity quantum electrodynamics (cavity QED) is the study of the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of light photons is significant. It could in principle be used to construct a quantum computer.

The case of a single 2-level atom in the cavity is mathematically described by the Jaynes-Cummings model, and undergoes vacuum Rabi oscillations , that is between an excited atom and n-1 photons, and a ground state atom and n photons.

If the cavity is in resonance with the atomic transition, a half-cycle of oscillation starting with no photons coherently swaps the atom qubit's state onto the cavity field's, , and can be repeated to swap it back again; this could be used as a single photon source (starting with an excited atom), or as an interface between an atom or trapped ion quantum computer and optical quantum communication.

Other interaction durations create entanglement between the atom and cavity field; for example, a quarter-cycle on resonance starting from gives the maximally entangled state (a Bell state) . This can in principle be used as a quantum computer, mathematically equivalent to a trapped ion quantum computer with cavity photons replacing phonons.

Circuit quantum electrodynamics

Circuit quantum electrodynamics (circuit QED) provides a means of studying the fundamental interaction between light and matter. As in the field of cavity quantum electrodynamics, a single photon within a single mode cavity coherently couples to a quantum object (atom). In contrast to cavity QED, the photon is stored in a one-dimensional on-chip resonator and the quantum object is no natural atom but an artificial one. These artificial atoms usually are mesoscopic devices which exhibit an atom-like energy spectrum.

The field of circuit QED is a prominent example for quantum information processing and a promising candidate for future quantum computation.

Euler–Heisenberg Lagrangian

In physics, the Euler–Heisenberg Lagrangian describes the non-linear dynamics of electromagnetic fields in vacuum. It was first obtained by Werner Heisenberg and Hans Heinrich Euler in 1936. By treating the vacuum as a medium, it predicts rates of quantum electrodynamics (QED) light interaction processes.

Gupta–Bleuler formalism

In quantum field theory, the Gupta–Bleuler formalism is a way of quantizing the electromagnetic field. The formulation is due to theoretical physicists Suraj N. Gupta and Konrad Bleuler.

History of quantum field theory

In particle physics, the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field in the late 1920s. Major advances in the theory were made in the 1940s and 1950s, and led to the introduction of renormalized quantum electrodynamics (QED). QED was so successful and accurately predictive that efforts were made to apply the same basic concepts for the other forces of nature. By the late 1970s, these efforts successfully utilized gauge theory in the strong nuclear force and weak nuclear force, producing the modern standard model of particle physics.

Efforts to describe gravity using the same techniques have, to date, failed. The study of quantum field theory is still flourishing, as are applications of its methods to many physical problems. It remains one of the most vital areas of theoretical physics today, providing a common language to several different branches of physics.

Julian Schwinger

Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize winning American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics (QED), in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order. Schwinger was a physics professor at several universities.

Schwinger is recognized as one of the greatest physicists of the twentieth century, responsible for much of modern quantum field theory, including a variational approach, and the equations of motion for quantum fields. He developed the first electroweak model, and the first example of confinement in 1+1 dimensions. He is responsible for the theory of multiple neutrinos, Schwinger terms, and the theory of the spin 3/2 field.

Lamb shift

In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2S1/2 and 2P1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which these states should have the same energy.

Interaction between vacuum energy fluctuations and the hydrogen electron in these different orbitals is the cause of the Lamb shift, as was shown subsequent to its discovery. The Lamb shift has since played a significant role through vacuum energy fluctuations in theoretical prediction of Hawking radiation from black holes.

This effect was first measured in 1947 in the Lamb–Retherford experiment on the hydrogen microwave spectrum and this measurement provided the stimulus for renormalization theory to handle the divergences. It was the harbinger of modern quantum electrodynamics developed by Julian Schwinger, Richard Feynman, Ernst Stueckelberg, Sin-Itiro Tomonaga and Freeman Dyson. Lamb won the Nobel Prize in Physics in 1955 for his discoveries related to the Lamb shift.

Positron

The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1 e, a spin of 1/2 (same as electron), and has the same mass as an electron. When a positron collides with an electron, annihilation occurs. If this collision occurs at low energies, it results in the production of two or more gamma ray photons.

Positrons can be created by positron emission radioactive decay (through weak interactions), or by pair production from a sufficiently energetic photon which is interacting with an atom in a material.

Positronium

Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom, specifically an onium. The system is unstable: the two particles annihilate each other to predominantly produce two or three gamma-rays, depending on the relative spin states. The orbit and energy levels of the two particles are similar to that of the hydrogen atom (which is a bound state of a proton and an electron). However, because of the reduced mass, the frequencies of the spectral lines are less than half of the corresponding hydrogen lines.

Precision tests of QED

Quantum electrodynamics (QED), a relativistic quantum field theory of electrodynamics, is among the most stringently tested theories in physics.

The most precise and specific tests of QED consist of measurements of the electromagnetic fine-structure constant, α, in various physical systems. Checking the consistency of such measurements tests the theory.

Tests of a theory are normally carried out by comparing experimental results to theoretical predictions. In QED, there is some subtlety in this comparison, because theoretical predictions require as input an extremely precise value of α, which can only be obtained from another precision QED experiment. Because of this, the comparisons between theory and experiment are usually quoted as independent determinations of α. QED is then confirmed to the extent that these measurements of α from different physical sources agree with each other.

The agreement found this way is to within ten parts in a billion (10−8), based on the comparison of the electron anomalous magnetic dipole moment and the Rydberg constant from atom recoil measurements as described below. This makes QED one of the most accurate physical theories constructed thus far.

Besides these independent measurements of the fine-structure constant, many other predictions of QED have been tested as well.

QED vacuum

The Quantum Electrodynamic Vacuum or QED vacuum is the field-theoretic vacuum of quantum electrodynamics. It is the lowest energy state (the ground state) of the electromagnetic field when the fields are quantized. When Planck's constant is hypothetically allowed to approach zero, QED vacuum is converted to classical vacuum, which is to say, the vacuum of classical electromagnetism.Another field-theoretic vacuum is the QCD vacuum of the Standard Model.

Vacuum state

In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. Zero-point field is sometimes used as a synonym for the vacuum state of an individual quantized field.

According to present-day understanding of what is called the vacuum state or the quantum vacuum, it is "by no means a simple empty space". According to quantum mechanics, the vacuum state is not truly empty but instead contains fleeting electromagnetic waves and particles that pop into and out of existence.The QED vacuum of quantum electrodynamics (or QED) was the first vacuum of quantum field theory to be developed. QED originated in the 1930s, and in the late 1940s and early 1950s it was reformulated by Feynman, Tomonaga and Schwinger, who jointly received the Nobel prize for this work in 1965. Today the electromagnetic interactions and the weak interactions are unified in the theory of the electroweak interaction.

The Standard Model is a generalization of the QED work to include all the known elementary particles and their interactions (except gravity). Quantum chromodynamics is the portion of the Standard Model that deals with strong interactions, and QCD vacuum is the vacuum of quantum chromodynamics. It is the object of study in the Large Hadron Collider and the Relativistic Heavy Ion Collider, and is related to the so-called vacuum structure of strong interactions.

Vertex function

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.

Ward–Takahashi identity

In quantum field theory, a Ward–Takahashi identity is an identity between correlation functions that follows from the global or gauge symmetries of the theory, and which remains valid after renormalization.

The Ward–Takahashi identity of quantum electrodynamics was originally used by John Clive Ward and Yasushi Takahashi to relate the wave function renormalization of the electron to its vertex renormalization factor F1(0), guaranteeing the cancellation of the ultraviolet divergence to all orders of perturbation theory. Later uses include the extension of the proof of Goldstone's theorem to all orders of perturbation theory.

The Ward–Takahashi identity is a quantum version of the classical Noether's theorem, and any symmetries in a quantum field theory can lead to an equation of motion for correlation functions. This generalized sense should be distinguished when reading literature, such as Michael Peskin and Daniel Schroeder's textbook, from the original sense of the Ward identity.

What Do You Care What Other People Think?

"What Do You Care What Other People Think?": Further Adventures of a Curious Character (1988) is the second of two books consisting of transcribed and edited, oral reminiscences from American physicist Richard Feynman. It follows Surely You're Joking, Mr. Feynman!. Richard Feynman received the Nobel Prize in Physics in 1965 for his contributions to the development of quantum electrodynamics. He is a theoretical physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics for which he proposed the parton model. For his contributions to the development of quantum electrodynamics, Feynman, jointly with Julian Schwinger and Shin'ichirō Tomonaga, was awarded the Nobel Prize in Physics in 1965.

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