A gauge theory is a type of theory in physics. The word gauge means a measurement , a thickness, an in-between distance as in railroad tracks , or a resulting number of units per certain parameter a number of loops in an inch of fabric or a number of lead balls in a pound of ammunition. A general feature of these field theories is that the fundamental fields cannot be directly measured; however, some associated quantities can be measured, such as charges, energies, and velocities. For example, say you cannot measure the diameter of a lead ball, but you can determine how many lead balls, which are equal in every way, are required to make a pound. Using the number of balls, the elemental mass of lead, and the formula for calculating the volume of a sphere from its diameter, one could indirectly determine the diameter of a single lead ball.
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A gauge theory is a type of theory in physics. The word gauge means a measurement , a thickness, an in-between distance as in railroad tracks , or a resulting number of units per certain parameter a number of loops in an inch of fabric or a number of lead balls in a pound of ammunition.
A general feature of these field theories is that the fundamental fields cannot be directly measured; however, some associated quantities can be measured, such as charges, energies, and velocities. For example, say you cannot measure the diameter of a lead ball, but you can determine how many lead balls, which are equal in every way, are required to make a pound.
Using the number of balls, the elemental mass of lead, and the formula for calculating the volume of a sphere from its diameter, one could indirectly determine the diameter of a single lead ball.
In field theories, different configurations of the unobservable fields can result in identical observable quantities. A transformation from one such field configuration to another is called a gauge transformation ;   the lack of change in the measurable quantities, despite the field being transformed, is a property called gauge invariance. For example, if you could measure the color of lead balls and discover that when you change the color, you still fit the same number of balls in a pound, the property of "color" would show gauge invariance.
Since any kind of invariance under a field transformation is considered a symmetry , gauge invariance is sometimes called gauge symmetry. Generally, any theory that has the property of gauge invariance is considered a gauge theory.
For example, in electromagnetism the electric and magnetic fields, E and B are observable, while the potentials V "voltage" and A the vector potential are not. With the advent of quantum mechanics in the s, and with successive advances in quantum field theory , the importance of gauge transformations has steadily grown. Gauge theories constrain the laws of physics, because all the changes induced by a gauge transformation have to cancel each other out when written in terms of observable quantities.
Over the course of the 20th century, physicists gradually realized that all forces fundamental interactions arise from the constraints imposed by local gauge symmetries , in which case the transformations vary from point to point in space and time.
Perturbative quantum field theory usually employed for scattering theory describes forces in terms of force-mediating particles called gauge bosons. The nature of these particles is determined by the nature of the gauge transformations. The culmination of these efforts is the Standard Model , a quantum field theory that accurately predicts all of the fundamental interactions except gravity. The earliest field theory having a gauge symmetry was Maxwell 's formulation, in —65, of electrodynamics " A Dynamical Theory of the Electromagnetic Field ".
The importance of this symmetry remained unnoticed in the earliest formulations. Similarly unnoticed, Hilbert had derived Einstein's equations of general relativity by postulating a symmetry under any change of coordinates. After the development of quantum mechanics , Weyl, Fock and London modified their gauge choice by replacing the scale factor with a change of wave phase , and applying it successfully to electromagnetism. This idea, dubbed Yang—Mills theory , later found application in the quantum field theory of the weak force , and its unification with electromagnetism in the electroweak theory.
The importance of gauge theories for physics stems from their tremendous success in providing a unified framework to describe the quantum-mechanical behavior of electromagnetism , the weak force and the strong force. This gauge theory, known as the Standard Model , accurately describes experimental predictions regarding three of the four fundamental forces of nature.
Historically, the first example of gauge symmetry to be discovered was classical electromagnetism. But only differences in potential are physically measurable, which is the reason that a voltmeter must have two probes, and can only report the voltage difference between them.
Thus one could choose to define all voltage differences relative to some other standard, rather than the Earth, resulting in the addition of a constant offset. In other words, the laws of physics governing electricity and magnetism that is, Maxwell equations are invariant under gauge transformation.
Generalizing from static electricity to electromagnetism, we have a second potential, the magnetic vector potential A , which can also undergo gauge transformations. These transformations may be local. That is, rather than adding a constant onto V , one can add a function that takes on different values at different points in space and time.
If A is also changed in certain corresponding ways, then the same E and B fields result. The detailed mathematical relationship between the fields E and B and the potentials V and A is given in the article Gauge fixing , along with the precise statement of the nature of the gauge transformation.
The relevant point here is that the fields remain the same under the gauge transformation, and therefore Maxwell's equations are still satisfied. Gauge symmetry is closely related to charge conservation. Suppose that there existed some process by which one could briefly violate conservation of charge by creating a charge q at a certain point in space, 1, moving it to some other point 2, and then destroying it. We might imagine that this process was consistent with conservation of energy.
Outside of the interval during which the particle exists, conservation of energy would be satisfied, because the net energy released by creation and destruction of the particle, qV 2 - qV 1 , would be equal to the work done in moving the particle from 1 to 2, qV 2 - qV 1. But although this scenario salvages conservation of energy, it violates gauge symmetry. The conclusion is that if gauge symmetry holds, and energy is conserved, then charge must be conserved.
As discussed above, the gauge transformations for classical i. Some global symmetries under changes of coordinate predate both general relativity and the concept of a gauge. For example, Galileo and Newton introduced the notion of translation invariance [ when?
Suppose, for example, that one observer examines the properties of a hydrogen atom on Earth, the other—on the Moon or any other place in the universe , the observer will find that their hydrogen atoms exhibit completely identical properties. Again, if one observer had examined a hydrogen atom today and the other— years ago or any other time in the past or in the future , the two experiments would again produce completely identical results.
The invariance of the properties of a hydrogen atom with respect to the time and place where these properties were investigated is called translation invariance. Recalling our two observers from different ages: the time in their experiments is shifted by years. Both observers discover the same laws of physics. Because light from hydrogen atoms in distant galaxies may reach the earth after having traveled across space for billions of years, in effect one can do such observations covering periods of time almost all the way back to the Big Bang , and they show that the laws of physics have always been the same.
Invariance of the form of an equation under an arbitrary coordinate transformation is customarily referred to as general covariance , and equations with this property are referred to as written in the covariant form. General covariance is a special case of gauge invariance. Maxwell's equations can also be expressed in a generally covariant form, which is as invariant under general coordinate transformation as Einstein's field equation.
Until the advent of quantum mechanics, the only well known example of gauge symmetry was in electromagnetism, and the general significance of the concept was not fully understood. For example, it was not clear whether it was the fields E and B or the potentials V and A that were the fundamental quantities; if the former, then the gauge transformations could be considered as nothing more than a mathematical trick.
In quantum mechanics, a particle such as an electron is also described as a wave. For example, if the double-slit experiment is performed with electrons, then a wave-like interference pattern is observed.
The electron has the highest probability of being detected at locations where the parts of the wave passing through the two slits are in phase with one another, resulting in constructive interference. If there are no electric or magnetic fields present in this experiment, then the electron's energy is constant, and, for example, there will be a high probability of detecting the electron along the central axis of the experiment, where by symmetry the two parts of the wave are in phase.
But now suppose that the electrons in the experiment are subject to electric or magnetic fields. For example, if an electric field were imposed on one side of the axis but not on the other, the results of the experiment would be affected.
The results of the experiment will be different, because phase relationships between the two parts of the electron wave have changed, and therefore the locations of constructive and destructive interference will be shifted to one side or the other.
It is the electric potential that occurs here, not the electric field, and this is a manifestation of the fact that it is the potentials and not the fields that are of fundamental significance in quantum mechanics. It is even possible to have cases in which an experiment's results differ when the potentials are changed, even if no charged particle is ever exposed to a different field.
One such example is the Aharonov—Bohm effect , shown in the figure. But the solenoid has been positioned so that the electron cannot possibly pass through its interior. If one believed that the fields were the fundamental quantities, then one would expect that the results of the experiment would be unchanged. In reality, the results are different, because turning on the solenoid changed the vector potential A in the region that the electrons do pass through. Now that it has been established that it is the potentials V and A that are fundamental, and not the fields E and B , we can see that the gauge transformations, which change V and A , have real physical significance, rather than being merely mathematical artifacts.
Note that in these experiments, the only quantity that affects the result is the difference in phase between the two parts of the electron wave. Suppose we imagine the two parts of the electron wave as tiny clocks, each with a single hand that sweeps around in a circle, keeping track of its own phase. Although this cartoon ignores some technical details, it retains the physical phenomena that are important here.
Not only that, but it is not even necessary to change the speed of each clock by a fixed amount. This would have no effect on the result of the experiment, since the final observation of the location of the electron occurs at a single place and time, so that the phase shift in each electron's "clock" would be the same, and the two effects would cancel out.
This is another example of a gauge transformation: it is local, and it does not change the results of experiments. In summary, gauge symmetry attains its full importance in the context of quantum mechanics. In the application of quantum mechanics to electromagnetism, i. These two gauge symmetries are in fact intimately related. Experiments have verified this testable statement about the interference patterns formed by electron waves. Except for the "wrap-around" property, the algebraic properties of this mathematical structure are exactly the same as those of the ordinary real numbers.
In mathematical terminology, electron phases form an Abelian group under addition, called the circle group or U 1. Group means that addition associates and has an identity element , namely "0". Also, for every phase there exists an inverse such that the sum of a phase and its inverse is 0. Other examples of abelian groups are the integers under addition, 0, and negation, and the nonzero fractions under product, 1, and reciprocal.
As a way of visualizing the choice of a gauge, consider whether it is possible to tell if a cylinder has been twisted. If the cylinder has no bumps, marks, or scratches on it, we cannot tell.
Once this arbitrary choice the choice of gauge has been made, it becomes possible to detect it if someone later twists the cylinder. In , Chen Ning Yang and Robert Mills proposed to generalize these ideas to noncommutative groups.
A noncommutative gauge group can describe a field that, unlike the electromagnetic field, interacts with itself. For example, general relativity states that gravitational fields have energy, and special relativity concludes that energy is equivalent to mass.
Hence a gravitational field induces a further gravitational field. The nuclear forces also have this self-interacting property. Surprisingly, gauge symmetry can give a deeper explanation for the existence of interactions, such as the electric and nuclear interactions. This arises from a type of gauge symmetry relating to the fact that all particles of a given type are experimentally indistinguishable from one another. Imagine that Alice and Betty are identical twins, labeled at birth by bracelets reading A and B.
Because the girls are identical, nobody would be able to tell if they had been switched at birth; the labels A and B are arbitrary, and can be interchanged. Such a permanent interchanging of their identities is like a global gauge symmetry.
There is also a corresponding local gauge symmetry, which describes the fact that from one moment to the next, Alice and Betty could swap roles while nobody was looking, and nobody would be able to tell. If Alice and Betty are in fact quantum-mechanical particles rather than people, then they also have wave properties, including the property of superposition , which allows waves to be added, subtracted, and mixed arbitrarily. It follows that we are not even restricted to complete swaps of identity.
Réduction des symétries de jauges : une nouvelle approche géométrique
Toggle navigation. Have you forgotten your login? Discrete-time quantum walks and gauge theories. Pablo Arnault 1 Details. Pablo Arnault 1 AuthorId : Author. Hide details. Abstract : A quantum Q computer QC , i.
Introduction to gauge theory
Abstract: The Bargmann-Wigner formalism is adapted to spherical surfaces embedded in three to eleven dimensions. This is demonstrated to generate wave equations in spherical space for a variety of antisymmetric tensor fields. Some of these equations are gauge invariant for particular values of the parameters characterizing them. For spheres embedded in three, four, and five dimensions, this gauge invariance can be generalized so as to become non-Abelian.
The Bargmann-Wigner equations in spherical space