# Quantum Field Theory of the Electromagnetic Force

The history of physics from ancient times to the modern day, focusing on light and matter. The first quantum field theory to be developed was known as quantum electrodynamics (QED). QED combines quantum mechanics with special relativity in order to describe how the electromagnetic force is carried by photons.

Last updated on 5th June 2017 by Dr Helen Klus

## 1. The electromagnetic force ↑

Quantum electrodynamics (QED) was the first quantum field theory to be discovered. QED describes the electromagnetic field, which is related to the electromagnetic force, the force that conveys electric charge. All charged particles, including protons and electrons, interact via the electromagnetic force.

The electromagnetic force is about 1039 (10,000, billion, billion, billion, billion) times stronger than the force of gravity, but it doesn't affect large objects very often. This is because most things have charges that cancel each other out, making them neutral, whereas gravity always affects everything with mass[1].

## 2. Quantum electrodynamics (QED) ↑

### 2.1 Paul Dirac and antimatter ↑

British physicist Paul Dirac coined the term 'quantum electrodynamics' in 1927, when he provided a quantum theory of the electromagnetic field that explained how an atom can decay to a lower, and therefore less energetic, state and still follow the laws of energy conservation[2].

Dirac showed that the atom does this by emitting the excess energy in the form of a photon. He did this by treating the electromagnetic field as if it is a gas made up of photons, which act as harmonic oscillators.

Within a year, Dirac published his relativistic theory of the electron, which combined quantum mechanics with German-Swiss-American physicist Albert Einstein's theory of special relativity[3]. This showed that the electron has a spin of +1/2 or -1/2 and predicted, not only the existence of the antielectron - an electron with a negative energy, and therefore an opposite spin and charge - but that all particles have corresponding antimatter partners.

Matter and antimatter annihilate each other upon contact, and the total rest mass of the particles is converted to kinetic energy, in accordance with special relativity.

In 1932, within four years of Dirac's prediction, American physicist Carl David Anderson discovered antielectrons, which he named positrons, from tracks produced by cosmic rays inside of a cloud chamber[4].

That same year, British physicist Patrick Blackett and Italian physicist Giuseppe "Beppo" Occhialini showed that photons can produce positrons and electrons in pairs if they are energetic enough[5]. They did this by improving the efficiency of cloud chamber records by linking the chamber to Geiger counters that trigger a camera when a particle arrives.

### 2.2 Early quantum electrodynamics ↑

Other physicists, including Austrian physicist Wolfgang Pauli[6], Hungarian-American physicist Eugene Wigner[7], German physicists Pascual Jordan[8], Max Born, and Werner Heisenberg[9], and Italian physicist Enrico Fermi[10], helped extend Dirac's idea to form the basis for modern QED theory.

While photons can be thought of as both particles and waves, QED treats photons as particles that 'carry' the electromagnetic force. Charged particles interact by emitting and absorbing photons. Photons do not experience the electromagnetic force themselves and so they do not interact with each other, but the effects of electromagnetism are produced by the energy and momentum they carry[11].

Feynman diagram showing how an electron changes trajectory when it emits a photon.
Image credit: Helen Klus/CC-NC-SA.

Feynman diagram showing one electron emitting a photon, and a second electron absorbing it. This exchange of energy is reflected in the electron's new trajectories. Image credit: Helen Klus/CC-NC-SA.

The photons that carry force are known as 'virtual' particles. Virtual particles are created the instant a particle emits or absorbs a photon, the total energy and momentum of the system is the same before and after, but Heisenberg's uncertainty principle allows 'extra' energy to exist in the form of a particle for a very brief period of time[12].

Each virtual particle can be thought of as a harmonic oscillator, where the strength of the field is given by the displacement from its rest position.

Virtual particles exist for such a short period of time that they are essentially invisible, and can only be detected by the effect they have on the particle that emits or absorbs them. Force carrying photons are therefore different from photons produced by other means, like in nuclear fusion, which could potentially exist forever[13].

The possible ways in which charged particles can interact by exchanging virtual photons are represented by Feynman diagrams. These were devised by American physicist Richard Feynman in the 1940s and 1950s[14a].

Feynman diagrams show a plot of time and space with straight lines used to depict fermions, like electrons, and wavy lines to depict bosons, like virtual photons. Antiparticles are represented as normal particles that are moving backwards in time.

### 2.3 Problems with early quantum electrodynamics ↑

By 1939 American physicist Robert Oppenheimer[15], Swiss physicist Felix Bloch and American physicist Arnold Nordsieck[16], and Austrian-American physicist Victor Weisskopf[17], had all shown that this version of QED couldn't be entirely correct. This is because it led to the prediction that the energy, mass, and charge of a single electron are all infinite, which clearly does not match observations.

American physicists Willis Lamb and Robert Retherford had found another problem with QED in 1947[18]. Lamb and Retherford measured hydrogen lines in the microwave spectrum in order to study the difference in energy between the l=0 and l=1 (S and P) quantum states. It was predicted that the two states should have equal energies, but a magnetic field could induce an energy difference between them.

Lamb and Retherford measured this difference and then calculated what the difference would be if there was no magnetic field. To their surprise, it was not zero. The two states did not have equal energies after all. This difference is known as the Lamb shift, and it couldn't currently be explained with QED.

### 2.4 Renormalisation ↑

At the 1947 Shelter Island Conference on Quantum Mechanics, which took place in Long Island, New York, over 20 physicists - including Lamb, Oppenheimer, Feynman, German-American physicist Hans Bethe, and American physicists Julian Schwinger and David Bohm - discussed how they could solve these problems[19].

On the train ride home, Bethe realised that the infinite values could be removed in a process known as renormalisation, where the infinities cancel out leaving just the measured values[20]. This theory was developed by Schwinger[21][22], Feynman[23][14b], and Japanese physicist Sin'ichirō Tomonaga[24][25], in the late 1940s. American physicist Freeman Dyson later showed that all of these approaches are equivalent[26].

The problem of Lamb shift was solved with the realisation that different corrections were needed for S and P states as they differ in their average distance from the nucleus.

## 3. Zero-point energy and the Casimir effect ↑

Quantum field theories state that all fundamental fields must be quantised at each point in space. This means that virtual particles are constantly coming into and out of existence almost everywhere. The temporary change of energy at a point in spacetime is known as a quantum fluctuation. The excess energy is known as zero-point energy, or vacuum energy[27]. This excess energy should add to the energy density of the universe.

If spacetime is infinitely divisible, then it should produce an infinite amount of energy, yet this does not seem to be the case. We will probably not understand how vacuum energy affects the energy density of the universe until we have developed a quantum field theory of gravity.

In 1948, Dutch physicists Hendrik Casimir and Dirk Polder discovered the Casimir effect, which demonstrates measurable forces possibly arising from vacuum energy[28]. Casimir and Polder showed that if two uncharged metal plates are placed close enough together in a vacuum, and are then pushed together slightly, then they will start to attract each other.

This is because the vacuum energy between the plates contains contributions from all whole wavelengths that fit in the gap between the plates. As they are pushed together, more wavelengths are excluded, and the radiation pressure between the plates decreases, pulling the plates together.

Illustration of the Casimir effect. Image credit: Emok/CC-SA.

This effect becomes dominant if the plates are less than a micrometre (one-thousandth of a millimetre) apart and was first demonstrated by physicist Steve Lamoreaux in 1997[29].

In 1961, Russian physicists Igor Ekhiel'evich Dzyaloshinskii, Evgeny Lifshitz, and Lev Pitaevskii, predicted that if the medium between the two plates is not a vacuum, then some materials can be made to repel each other via the Casimir effect[30]. This was shown experimentally in 2009[31].

Some argue that the Casimir effect does not provide evidence for vacuum energy, as it can also be explained in terms of relativistic van der Waals forces[32]. These are the forces between neutral atoms, which were given a quantum description by German physicist Fritz London in 1930[33].

Quantum van der Waals forces occur because the negative charge of the electrons in an atom, and the positive charge of the nuclei, are not always in the same place, relative to each other. The fluctuation of charge can result in attractive forces between atoms, in this case, the atoms that make up the metal plates.

In 1981, Russian physicists Viatcheslav Mukhanov and Gennady Chibisov showed that quantum fluctuations were present during the inflationary epoch of the early universe[34]. These fluctuations expanded during inflation, and this can explain the asymmetry in spacetime that led objects to becoming gravitationally bound, creating structure in the universe.

### 3.1 Black holes and Hawking radiation ↑

Virtual particles are usually created with an antimatter partner, and they annihilate each other almost instantly. A virtual particle can become 'real', however, if it's removed from its anti-partner and it gains the required amount of energy from an outside source. In 1974, British physicist Stephen Hawking showed that this is what happens at the edge of black holes[35].

To an observer on either side, the constant production of particles would make it seem as if the black hole was emitting radiation, and so this effect is known as Hawking radiation. Hawking showed that black holes will start to evaporate, and eventually disappear when they contain more Hawking radiation than matter and energy.

### 3.2 The black hole information paradox ↑

Black holes only have mass, and sometimes charge and angular momentum, but they retain no information about the matter that formed them. If black holes existed forever then this information would be thought of as existing within the black hole. If they evaporate by emitting Hawking radiation, then this information appears to be lost forever.

In quantum mechanics, information loss violates unitarity, which is another way of saying that it violates the conservation of probability, because it makes the sum of probabilities of all possible quantum outcomes different from 1. This may mean breaking the laws of energy conservation, and is known as the black hole information paradox[36][37].

### 3.3 Boltzmann brains ↑

It's possible that a vast amount of vacuum energy can be produced as long as it persists for a short enough time. In 2006, Canadian physicist Don Page popularised the highly speculative idea that every possible object could be created and, if we accept a material theory of the mind, then this includes conscious minds[38][39].

These are known as Boltzmann brains after Austrian physicist Ludwig Boltzmann, who first suggested that macroscopic objects could arise as the result of random interactions in the 1890s[40].

Page argues that if the universe continues to evolve in the way that we predict, then there will be a point when there are no more biological minds, and it is suggested that in this time, Boltzmann brains will outnumber all other observers who have ever existed.

This implies that it's more likely that you are a Boltzmann brain than the product of a biological mind, and so it is unlikely that the external world exists in the way that you think. Few Boltzmann brains will become 'real' and so, if they do exist, the past you remember is more likely to be an illusion than to have actually happened.

## 4. References ↑

1. Stenger, V. J., 2011, 'The Fallacy of Fine-Tuning: Why the Universe Is Not Designed for Us', Prometheus Books.

2. Dirac, P. A. M., 1927, 'The quantum theory of the emission and absorption of radiation', Proceedings of the Royal Society of London, Series A, 114, pp.243-265.

3. Dirac, P. A. M., 1928, 'The quantum theory of the electron', Proceedings of the Royal Society of London, Series A, 117, pp.610-624.

4. Anderson, C. D., 1933, 'The positive electron', Physical Review, 43, pp.491-498.

5. Blackett, P. M. S. and Occhialini, G., 1932, 'Photography of penetrating corpuscular radiation', Nature, 130, pp.363-363.

6. Pauli, W., 1927, 'Zur Quantenmechanik des magnetischen Elektrons' ('On the quantum mechanics of magnetic electrons'), Zeitschrift für Physik A Hadrons and Nuclei, 43, pp.601-623.

7. Weisskopf, V. and Wigner, E., 1930, 'Berechnung der natürlichen linienbreite auf grund der diracschen lichttheorie' ('Calculation of the natural line width on the basis of Dirac's theory of light'), Zeitschrift für Physik, 63, pp.54-73.

8. Jordan, P., 1929, 'Der gegenwärtige Stand der Quantenelektrodynamik' ('The present state of quantum electrodynamics'), Physikalische Zeitschrift, 30, pp.700-713.

9. Born, M., Heisenberg, W., and Jordan, P., 1926, 'Zur quantenmechanik II' ('On quantum mechanics II'), Zeitschrift für Physik, 35, pp.557-615.

10. Fermi, E., 1932, 'Quantum theory of radiation', Reviews of modern physics, 4, pp.87-132.

11. Miller, A. I., 1995, 'Early Quantum Electrodynamics: A Sourcebook', Cambridge University Press.

12. Heisenberg, W., 1925, 'Quantum-theoretical re-interpretation of kinematic and mechanical relations', Zeitschrift für Physik, 33, pp.879-893.

13. Hey, A. J. G. and Walters, P., 2003, 'The New Quantum Universe', Cambridge University Press.

14. (a, b) Feynman, R. P., 1948, 'A relativistic cut-off for classical electrodynamics', Physical Review, 74, pp.939-947.

15. Oppenheimer, J. R., 1930, 'Note on the theory of the interaction of field and matter', Physical Review, 35, pp.461-477.

16. Bloch, F. and Nordsieck, A., 1937, 'Note on the radiation field of the electron', Physical Review, 52, pp.54.

17. Weisskopf, V. F., 1939, 'On the self-energy and the electromagnetic field of the electron', Physical Review, 56, pp.72.

18. Lamb Jr, W. E. and Retherford, R. C., 1947, 'Fine structure of the hydrogen atom by a microwave method', Physical Review, 72, pp.241-243.

19. American Physical Society, 'Shelter Island Conference', last accessed 01-06-17.

20. Bethe, H. A., 1947, 'The electromagnetic shift of energy levels', Physical Review, 72, pp.339-341.

21. Schwinger, J., 1948, 'On quantum-electrodynamics and the magnetic moment of the electron', Physical Review, 73, pp.416-418.

22. Schwinger, J., 1948, 'Quantum electrodynamics. I. A covariant formulation', Physical Review, 74, pp.1439.

23. Feynman, R. P., 1948, 'Space-time approach to non-relativistic quantum mechanics', Reviews of Modern Physics, 20, pp.367-387.

24. Tomonaga, S. I., 1946, 'On the Effect of the Field Reactions on the Interaction of Mesotrons and Nuclear Particles', Progress of theoretical physics, 1, pp.83-101.

25. Koba, Z., Tani, T., and Tomonaga, S. I., 1947, 'On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields', Progress of Theoretical Physics, 2, pp.101-116.

26. Dyson, F. J., 1949, 'The radiation theories of Tomonaga, Schwinger, and Feynman', Physical Review, 75, pp.486.

27. Coughlan, G. D., Dodd, J. E., and Gripaios, B. M., 2006, 'The Ideas of Particle Physics: An Introduction for Scientists', Cambridge University Press.

28. Casimir, H. B. G. and Polder, D., 1948, 'The influence of retardation on the London-van der Waals forces', Physical Review, 73, pp.360.

29. Lamoreaux, S. K., 1997, 'Demonstration of the Casimir force in the 0.6 to 6 micrometre range', Physical Review Letters, 78, pp.5-8.

30. Dzyaloshinskii, I. E. E., Lifshitz, E. M. and Pitaevskii, L. P., 1961, 'The general theory of van der Waals' forces', Physics-Uspekhi, 4, pp.153-176.

31. Munday, J. N., Capasso, F. and Parsegian, V. A., 2009, 'Measured long-range repulsive Casimir–Lifshitz forces', Nature, 457, pp.170-173.

32. Jaffe, R. L., 2005, 'Casimir effect and the quantum vacuum', Physical Review D, 72, pp.021301-021309.

33. London, F., 1930, 'Zur theorie und systematik der molekularkräfte' ('On the theory and system of molecular forces'), Zeitschrift für Physik, 63, pp.245-279.

34. Mukhanov, V. F. and Chibisov, G. V., 1981, 'Quantum fluctuations and a nonsingular universe', JETP Letters, 33, pp.532-535.

35. Hawking, S. W., 1975, 'Particle creation by black holes', Communications in mathematical physics, 43, pp.199-220.

36. Hawking, S. W., 1976, 'Breakdown of predictability in gravitational collapse', Physical Review D, 14, pp.2460.

37. Maldacena, J., 1999, 'The large-N limit of superconformal field theories and supergravity', International journal of theoretical physics, 38, pp.1113-1133.

38. Page, D. N., 2006, 'The Lifetime of the Universe', Journal of the Korean Physical Society, 49, pp.711-714.

39. Page, D. N., 2008, 'Return of the Boltzmann brains', Physical Review D, 78, pp.063536.

40. Boltzmann, L., 1897, 'On Zermelo's Paper 'On the Mechanical Explanation of Irreversible Processes'', Annalen der Physik, 60, pp.392-398.

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### Light & Matter

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1. Atoms and Waves

2. Reflection, Refraction, and Diffraction

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7. Spectral Lines and Spectroscopy

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