Principles of Electrodynamics

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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.


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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. 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. 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:. 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.


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In time this problem was "fixed" by the technique of renormalization. However, Feynman himself remained unhappy about it, calling it a "dippy process". 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. 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. Mathematically, QED is an abelian gauge theory with the symmetry group U 1. The left-hand side is like the original Dirac equation , and the right-hand side is the interaction with the electromagnetic field.

Now, if we impose the Lorenz gauge condition. 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. 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] : 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.


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  7. 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.

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    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] : — 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] : — The predictive success of quantum electrodynamics largely rests on the use of perturbation theory, expressed in Feynman diagrams.

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    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 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".

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    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.

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    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. An argument by Freeman Dyson shows that the radius of convergence of the perturbation series in QED is zero. 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. The problem is essentially that QED appears to suffer from quantum triviality issues. From Wikipedia, the free encyclopedia. Relativistic quantum field theory of electromagnetism. Feynman diagram. Standard Model. Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism.

    Incomplete theories. Anderson P. Main articles: History of quantum mechanics and History of quantum field theory. 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. 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.

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    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. Main article: Self-energy. Physics portal. Princeton University Press. Dirac Proceedings of the Royal Society of London A. Fermi Reviews of Modern Physics. Bibcode : RvMP Physical Review. Bibcode : PhRv Weisskopf Oppenheimer Bethe Tomonaga Progress of Theoretical Physics.

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    There is one thing I recommend though: learn it backwards compared to Griffiths; i. Otherwise you might end up as confused as me, feeling lied to. I recommend this generally: try to learn the non-simplified cases first. This book is very short pages as compared to say Griffiths at pages!

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