Electromagnetic field

From Wikipedia, the free encyclopedia

An electromagnetic field (also EMF or EM field) is a physical field produced by electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction. It is one of the four fundamental forces of nature (the others are gravitation, weak interaction and strong interaction).

The field can be viewed as the combination of an electric field and a magnetic field. The electric field is produced by stationary charges, and the magnetic field by moving charges (currents); these two are often described as the sources of the field. The way in which charges and currents interact with the electromagnetic field is described by Maxwell's equations and the Lorentz force law.
From a classical perspective in the history of electromagnetism, the electromagnetic field can be regarded as a smooth, continuous field, propagated in a wavelike manner; whereas from the perspective of quantum field theory, the field is seen as quantized, being composed of individual particles.^{[citation needed]}

Structure of the electromagnetic field

The electromagnetic field may be viewed in two distinct ways: a continuous structure or a discrete structure.

Continuous structure

Classically, electric and magnetic fields are thought of as being produced by smooth motions of charged objects. For example, oscillating charges produce electric and magnetic fields that may be viewed in a 'smooth', continuous, wavelike fashion. In this case, energy is viewed as being transferred continuously through the electromagnetic field between any two locations. For instance, the metal atoms in a radio transmitter appear to transfer energy continuously. This view is useful to a certain extent (radiation of low frequency), but problems are found at high frequencies (see ultraviolet catastrophe).

Discrete structure

The electromagnetic field may be thought of in a more 'coarse' way. Experiments reveal that in some circumstances electromagnetic energy transfer is better described as being carried in the form of packets called quanta (in this case, photons) with a fixed frequency. Planck's relation links the energy E of a photon to its frequency ν through the equation:^[1]

where h is Planck's constant, named in honor of Max Planck, and ν is the frequency of the photon . Although modern quantum optics tells us that there also is a semi-classical explanation of the photoelectric effect—the emission of electrons from metallic surfaces subjected to electromagnetic radiation—the photon was historically (although not strictly necessarily) used to explain certain observations. It is found that increasing the intensity of the incident radiation (so long as one remains in the linear regime) increases only the number of electrons ejected, and has almost no effect on the energy distribution of their ejection. Only the frequency of the radiation is relevant to the energy of the ejected electrons.

This quantum picture of the electromagnetic field (which treats it as analogous to harmonic oscillators) has proved very successful, giving rise to quantum electrodynamics, a quantum field theory describing the interaction of electromagnetic radiation with charged matter. It also gives rise to quantum optics, which is different from quantum electrodynamics in that the matter itself is modelled using quantum mechanics rather than quantum field theory.

Dynamics of the electromagnetic field

In the past, electrically charged objects were thought to produce two different, unrelated types of field associated with their charge property. An electric field is produced when the charge is stationary with respect to an observer measuring the properties of the charge, and a magnetic field (as well as an electric field) is produced when the charge moves (creating an electric current) with respect to this observer. Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole — the electromagnetic field. Recall that until 1831 electricity and magnetism had been viewed as unrelated phenomena. In 1831, Michael Faraday, one of the great thinkers of his time, made the seminal observation that time-varying magnetic fields could induce electric currents and then, in 1864, James Clerk Maxwell published his famous paper on a dynamical theory of the electromagnetic field. See Maxwell 1864 5, page 499; also David J. Griffiths (1999), Introduction to electrodynamics, third Edition, ed. Prentice Hall, pp. 559-562"(as quoted in Gabriela, 2009).

Once this electromagnetic field has been produced from a given charge distribution, other charged objects in this field will experience a force (in a similar way that planets experience a force in the gravitational field of the Sun). If these other charges and currents are comparable in size to the sources producing the above electromagnetic field, then a new net electromagnetic field will be produced. Thus, the electromagnetic field may be viewed as a dynamic entity that causes other charges and currents to move, and which is also affected by them. These interactions are described by Maxwell's equations and the Lorentz force law. (This discussion ignores the radiation reaction force.)

Electromagnetic field as a feedback loop

The behavior of the electromagnetic field can be resolved into four different parts of a loop:

• the electric and magnetic fields are generated by electric charges,
• the electric and magnetic fields interact with each other,
• the electric and magnetic fields produce forces on electric charges,
• the electric charges move in space.

A common misunderstanding is that (a) the quanta of the fields act in the same manner as (b) the charged particles that generate the fields. In our everyday world, charged particles, such as electrons, move slowly through matter with a drift velocity of a fraction of a centimeter (or inch) per second, but fields propagate at the speed of light - approximately 300 thousand kilometers (or 186 thousand miles) a second. The mundane speed difference between charged particles and field quanta is on the order of one to a million, more or less. Maxwell's equations relate (a) the presence and movement of charged particles with (b) the generation of fields. Those fields can then affect the force on, and can then move other slowly moving charged particles. Charged particles can move at relativistic speeds nearing field propagation speeds, but, as Einstein showed^{[citation needed]}, this requires enormous field energies, which are not present in our everyday experiences with electricity, magnetism, matter, and time and space.

The feedback loop can be summarized in a list, including phenomena belonging to each part of the loop:

• charged particles generate electric and magnetic fields
• the fields interact with each other
  • changing electric field acts like a current, generating 'vortex' of magnetic field
  • Faraday induction: changing magnetic field induces (negative) vortex of electric field
  • Lenz's law: negative feedback loop between electric and magnetic fields
• fields act upon particles
  • Lorentz force: force due to electromagnetic field
    • electric force: same direction as electric field
    • magnetic force: perpendicular both to magnetic field and to velocity of charge
• particles move
  • current is movement of particles
• particles generate more electric and magnetic fields; cycle repeats

Mathematical description

There are different mathematical ways of representing the electromagnetic field. The first one views the electric and magnetic fields as three-dimensional vector fields. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as E(x, y, z, t) (electric field) and B(x, y, z, t) (magnetic field).
If only the electric field (E) is non-zero, and is constant in time, the field is said to be an electrostatic field. Similarly, if only the magnetic field (B) is non-zero and is constant in time, the field is said to be a magnetostatic field. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using Maxwell's equations.^[2]

With the advent of special relativity, physical laws became susceptible to the formalism of tensors. Maxwell's equations can be written in tensor form, generally viewed by physicists as a more elegant means of expressing physical laws.

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed by Maxwell's equations. In the vector field formalism, these are:

(Gauss's law)

(Gauss's law for magnetism)

(Faraday's law)

(Ampère-Maxwell law)

where is the charge density, which can (and often does) depend on time and position, is the permittivity of free space, is the permeability of free space, and J is the current density vector, also a function of time and position. The units used above are the standard SI units. Inside a linear material, Maxwell's equations change by switching the permeability and permittivity of free space with the permeability and permittivity of the linear material in question. Inside other materials which possess more complex responses to electromagnetic fields, these terms are often represented by complex numbers, or tensors.

The Lorentz force law governs the interaction of the electromagnetic field with charged matter.
When a field travels across to different media, the properties of the field change according to the various boundary conditions. These equations are derived from Maxwell's equations. The tangential components of the electric and magnetic fields as they relate on the boundary of two media are as follows:^[3]

(current-free)

(charge-free)

The angle of refraction of an electric field between media is related to the permittivity of each medium:

The angle of refraction of a magnetic field between media is related to the permeability of each medium:

Properties of the field

Reciprocal behavior of electric and magnetic fields

The two Maxwell equations, Faraday's Law and the Ampère-Maxwell Law, illustrate a very practical feature of the electromagnetic field. Faraday's Law may be stated roughly as 'a changing magnetic field creates an electric field'. This is the principle behind the electric generator.

Ampere's Law roughly states that 'a changing electric field creates a magnetic field'. Thus, this law can be applied to generate a magnetic field and run an electric motor.

Light as an electromagnetic disturbance

Maxwell's equations take the form of an electromagnetic wave in a volume of space not containing charges or currents (free space) – that is, where and J are zero. Under these conditions, the electric and magnetic fields satisfy the electromagnetic wave equation:^[4]

James Clerk Maxwell was the first to obtain this relationship by his completion of Maxwell's equations with the addition of a displacement current term to Ampere's Circuital law.

Relation to and comparison with other physical fields

Being one of the four fundamental forces of nature, it is useful to compare the electromagnetic field with the gravitational, strong and weak fields. The word 'force' is sometimes replaced by 'interaction' because modern particle physics models electromagnetism as an exchange of particles known as gauge bosons.

Electromagnetic and gravitational fields

Sources of electromagnetic fields consist of two types of charge – positive and negative. This contrasts with the sources of the gravitational field, which are masses. Masses are sometimes described as gravitational charges, the important feature of them being that there are only positive masses and no negative masses. Further, gravity differs from electromagnetism in that positive masses attract other positive masses whereas same charges in electromagnetism repel each other.

The relative strengths and ranges of the four interactions and other information are tabulated below:

Theory	Interaction	mediator	Relative Magnitude	Behavior	Range
Chromodynamics	Strong interaction	gluon	1038	1	10−15 m
Electrodynamics	Electromagnetic interaction	photon	1036	1/r2	infinite
Flavordynamics	Weak interaction	W and Z bosons	1025	1/r5 to 1/r7	10−16 m
Geometrodynamics	Gravitation	graviton	100	1/r2	infinite

Applications

Static E and M fields and static EM fields

When an EM field (see electromagnetic tensor) is not varying in time, it may be seen as a purely electrical field or a purely magnetic field, or a mixture of both. However the general case of a static EM field with both electric and magnetic components present, is the case that appears to most observers. Observers who see only an electric or magnetic field component of a static EM field, have the other (electric or magnetic) component suppressed, due to the special case of the immobile state of the charges that produce the EM field in that case. In such cases the other component becomes manifest in other observer frames.
A consequence of this, is that any case that seems to consist of a "pure" static electric or magnetic field, can be converted to an EM field, with both E and M components present, by simply moving the observer into a frame of reference which is moving with regard to the frame in which only the “pure” electric or magnetic field appears. That is, a pure static electric field will show the familiar magnetic field associated with a current, in any frame of reference where the charge moves. Likewise, any new motion of a charge in a region that seemed previously to contain only a magnetic field, will show that that the space now contains an electric field as well, which will be found to produces an additional Lorentz force upon the moving charge.

Thus, electrostatics, as well as magnetism and magnetostatics, are now seen as studies of the static EM field when a particular frame has been selected to suppress the other type of field, and since an EM field with both electric and magnetic will appear in any other frame, these "simpler" effects are merely the observer's. The "applications" of all such non-time varying (static) fields are discussed in the main articles linked in this section.

Time-varying EM fields in Maxwell’s equations

An EM field that varies in time has two “causes” in Maxwell’s equations. One is charges and currents (so-called “sources”), and the other cause for an E or M field is a change in the other type of field (this last cause also appears in “free space” very far from currents and charges).
An electromagnetic field very far from currents and charges (sources) is called electromagnetic radiation (EMR) since it radiates from the charges and currents in the source, and has no "feedback" effect on them, and is also not affected directly by them in the present time (rather, it is indirectly produced by a sequences of changes in fields radiating out from them in the past). EMR consists of the radiations in the electromagnetic spectrum, including radio waves, microwave, infrared, visible light, ultraviolet light, X-rays, and gamma rays. The many commercial applications of these radiations are discussed in the named and linked articles.

A notable application of visible light is that this type of energy from the Sun powers all life on Earth that either makes or uses oxygen.

A changing electromagnetic field which is physically close to currents and charges (see near and far field for a definition of “close”) will have a dipole characteristic that is dominated by either a changing electric dipole, or a changing magnetic dipole. This type of dipole field near sources is called an electromagnetic near-field.

Changing electric dipole fields, as such, are used commercially as near-fields mainly as a source of dielectric heating. Otherwise, they appear parasitically around conductors which absorb EMR, and around antennas which have the purpose of generating EMR at greater distances.

Changing magnetic dipole fields (i.e., magnetic near-fields) are used commercially for many types of magnetic induction devices. These include motors and electrical transformers at low frequencies, and devices such as metal detectors and MRI scanner coils at higher frequencies. Sometimes these high-frequency magnetic fields change at radio frequencies without being far-field waves and thus radio waves; see RFID tags. See also near-field communication. Further uses of near-field EM effects commercially, may be found in the article on virtual photons, since at the quantum level, these fields are represented by these particles. Far-field effects (EMR) in the quantum picture of radiation, are represented by ordinary photons.

Health and safety

The potential health effects of the very low frequency EMFs surrounding power lines and electrical devices are the subject of on-going research and a significant amount of public debate. The US National Institute for Occupational Safety and Health (NIOSH) has issued some cautionary advisories but stresses that the data is currently too limited to draw good conclusions.^[5]

The potential effects of electromagnetic fields on human health vary widely depending on the frequency and intensity of the fields. For more information on the health effects due to specific parts of the electromagnetic spectrum, see the following articles:

• Static electric fields: see Electric shock
• Static magnetic fields: see MRI#Safety
• Extremely low frequency (ELF): see Power lines#Health concerns
• Radio frequency (RF): see Electromagnetic radiation and health
• Light: see Laser safety
• Ultraviolet (UV): see Sunburn
• Gamma rays: see Gamma ray
• Mobile telephony: see Mobile phone radiation and health

Introduction to gauge theory

From Wikipedia, the free encyclopedia

A gauge theory is a type of theory in physics. Modern physical theories, such as the theory of electromagnetism, describe the nature of reality in terms of fields, e.g., the electromagnetic field, the gravitational field, and fields for the electron and all other elementary particles. A general feature of these field theories is that the fundamental fields cannot be directly measured; however, there are observable quantities that can be measured experimentally, such as charges, energies, and velocities. 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;^[1][2] the lack of change in the measurable quantities, despite the field being transformed, is a property called 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.^[3] Under a gauge transformation in which a constant is added to V, no observable change occurs in E or B.

With the advent of quantum mechanics in the 1920s, 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 explaining all of the fundamental interactions except gravity.

History and importance

The earliest field theory having a gauge symmetry was Maxwell's formulation of electrodynamics in 1864. 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. Later Hermann Weyl, in an attempt to unify general relativity and electromagnetism, conjectured (incorrectly, as it turned out) that invariance under the change of scale or "gauge" (a term inspired by the various track gauges of railroads) might also be a local symmetry of general relativity. Although Weyl's choice of the gauge was incorrect, the name "gauge" stuck to the approach. 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. Gauge symmetry was generalized mathematically in 1954 by Chen Ning Yang and Robert Mills in an attempt to describe the strong nuclear forces. This idea, dubbed Yang-Mills, 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.

In classical physics

Electromagnetism

Historically, the first example of gauge symmetry to be discovered was classical electromagnetism. A static electric field can be described in terms of an electric potential (voltage) that is defined at every point in space, and in practical work it is conventional to take the Earth as a physical reference that defines the zero level of the potential, or ground. 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.^[4] If the potential is a solution to Maxwell's equations then, after this gauge transformation, the new potential is also a solution to Maxwell's equations and no experiment can distinguish between these two solutions. In other words the laws of physics governing electricity and magnetism (that is, Maxwell equations) are invariant under gauge transformation.^[5] That is, Maxwell's equations have a gauge symmetry.
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 violate conservation of charge, at least temporarily, 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. We could posit a rule stating that creating the charge required an input of energy E₁=qV₁ and destroying it released E₂=qV₂, which would seem natural since qV measures the extra energy stored in the electric field because of the existence of a charge at a certain point. (There may also be energy associated, e.g., with the rest mass of the particle, but that is not relevant to the present argument.) Conservation of energy would be satisfied, because the net energy released by creation and destruction of the particle, qV₂-qV₁, would be equal to the work done in moving the particle from 1 to 2, qV₂-qV₁. But although this scenario salvages conservation of energy, it violates gauge symmetry. Gauge symmetry requires that the laws of physics be invariant under the transformation , which implies that no experiment should be able to measure the absolute potential, without reference to some external standard such as an electrical ground. But the proposed rules E₁=qV₁ and E₂=qV₂ for the energies of creation and destruction would allow an experimenter to determine the absolute potential, simply by checking how much energy input was required in order to create the charge q at a particular point in space. The conclusion is that if gauge symmetry holds, and energy is conserved, then charge must be conserved.^[6]

The Cartesian coordinate grid on this square has been distorted by a coordinate transformation, so that there is a nonlinear relationship between the old (x,y) coordinates and the new ones. Einstein's equations of general relativity are still valid in the new coordinate system. Such changes of coordinate system are the gauge transformations of general relativity.

General relativity

As discussed above, the gauge transformations for classical (i.e., non-quantum mechanical) general relativity are arbitrary coordinate transformations.^[7] (Technically, the transformations must be invertible, and both the transformation and its inverse must be smooth, in the sense of being differentiable an arbitrary number of times.)

An example of a symmetry in a physical theory: translation invariance

Some global symmetries under changes of coordinate predate both general relativity and the concept of a gauge. For example, translation invariance was introduced in the era of Galileo, who eliminated the Aristotelian concept that various places in space, such as the earth and the heavens, obeyed different physical rules.

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—100 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 100 years. If the time when the older observer did the experiment was t, the time of the modern experiment is t+100 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.

In other words, if in the theory we change the time t to t+100 years (or indeed any other time shift) the theoretical predictions do not change.^[8]

Another example of a symmetry: the invariance of Einstein's field equation under arbitrary coordinate transformations

In Einstein's general relativity, coordinates like x, y, z, and t are not only "relative" in the global sense of translations like , rotations, etc., but become completely arbitrary, so that for example one can define an entirely new timelike coordinate according to some arbitrary rule such as , where has units of time, and yet Einstein's equations will have the same form.^[7][9]

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.

In quantum mechanics

Quantum electrodynamics

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.

Aharonov–Bohm experiment

Double-slit diffraction and interference pattern

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. The frequency of the electron wave is related to the kinetic energy of an individual electron particle via the quantum-mechanical relation E = hf. 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 was imposed on one side of the axis but not on the other, the results of the experiment would be affected. The part of the electron wave passing through that side oscillates at a different rate, since its energy has had −eV added to it, where −e is the charge of the electron and V the electrical potential. 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.

Schematic of double-slit experiment in which Aharonov–Bohm effect can be observed: electrons pass through two slits, interfering at an observation screen, with the interference pattern shifted when a magnetic field B is turned on in the cylindrical solenoid, marked in blue on the diagram.

Explanation with potentials

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.^[10] In this example, turning on the solenoid only causes a magnetic field B to exist within the solenoid. 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.

Gauge invariance: the results of the experiments are independent of the choice of the gauge for the potentials

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.^[11] If both clocks are sped up by the same amount, the phase relationship between them is unchanged, and the results of experiments are the same. Not only that, but it is not even necessary to change the speed of each clock by a fixed amount. We could change the angle of the hand on each clock by a varying amount θ, where θ could depend on both the position in space and on time. 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.

Summary

In summary, gauge symmetry attains its full importance in the context of quantum mechanics. In the application of quantum mechanics to electromagnetism, i.e., quantum electrodynamics, gauge symmetry applies to both electromagnetic waves and electron waves. These two gauge symmetries are in fact intimately related. If a gauge transformation θ is applied to the electron waves, for example, then one must also apply a corresponding transformation to the potentials that describe the electromagnetic waves.^[12] Gauge symmetry is required in order to make quantum electrodynamics a renormalizable theory, i.e., one in which the calculated predictions of all physically measurable quantities are finite.

Types of gauge symmetries

The description of the electrons in the subsection above as little clocks is in effect a statement of the mathematical rules according to which the phases of electrons are to be added and subtracted: they are to be treated as ordinary numbers, except that in the case where the result of the calculation falls outside the range of 0≤θ<360°, we force it to "wrap around" into the allowed range, which covers a circle. Another way of putting this is that a phase angle of, say, 5° is considered to be completely equivalent to an angle of 365°. 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). "Abelian" means that addition commutes, so that θ + φ = φ + θ. 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.

Gauge fixing of a twisted cylinder.

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. We could, however, draw an arbitrary curve along the cylinder, defined by some function θ(x), where x measures distance along the axis of the cylinder. Once this arbitrary choice (the choice of gauge) has been made, it becomes possible to detect it if someone later twists the cylinder.

In 1954, 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.

Gauge bosons

Surprisingly, gauge symmetry can give a deeper explanation for the existence of interactions, such as the electrical 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 other. 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 we observe that Mom's favorite vase is broken, we can only infer that the blame belongs to one twin or the other, but we cannot tell whether the blame is 100% Alice's and 0% Betty's, or vice versa. 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 a complete swaps of identity. For example, if we observe that a certain amount of energy exists in a certain location in space, there is no experiment that can tell us whether that energy is 100% A's and 0% B's, 0% A's and 100% B's, or 20% A's and 80% B's, or some other mixture. The fact that the symmetry is local means that we cannot even count on these proportions to remain fixed as the particles propagate through space. The details of how this is represented mathematically depend on technical issues relating to the spins of the particles, but for our present purposes we consider a spinless particle, for which it turns out that the mixing can be specified by some arbitrary choice of gauge θ(x), where an angle θ = 0° represents 100% A and 0% B, θ = 90° means 0% A and 100% B, and intermediate angles represent mixtures.

According to the principles of quantum mechanics, particles do not actually have trajectories through space. Motion can only be described in terms of waves, and the momentum p of an individual particle is related to its wavelength λ by p = h/λ. In terms of empirical measurements, the wavelength can only be determined by observing a change in the wave between one point in space and another nearby point (mathematically, by differentiation). A wave with a shorter wavelength oscillates more rapidly, and therefore changes more rapidly between nearby points. Now suppose that we arbitrarily fix a gauge at one point in space, by saying that the energy at that location is 20% A's and 80% B's. We then measure the two waves at some other, nearby point, in order to determine their wavelengths. But there are two entirely different reasons that the waves could have changed. They could have changed because they were oscillating with a certain wavelength, or they could have changed because the gauge function changed from a 20-80 mixture to, say, 21-79. If we ignore the second possibility, the resulting theory doesn't work; strange discrepancies in momentum will show up, violating the principle of conservation of momentum. Something in the theory must be changed.

Again there are technical issues relating to spin, but in several important cases, including electrically charged particles and particles interacting via nuclear forces, the solution to the problem is to impute physical reality to the gauge function θ(x). We say that if the function θ oscillates, it represents a new type of quantum-mechanical wave, and this new wave has its own momentum p = h/λ, which turns out to patch up the discrepancies that otherwise would have broken conservation of momentum. In the context of electromagnetism, the particles A and B would be charged particles such as electrons, and the quantum mechanical wave represented by θ would be the electromagnetic field. (Here we ignore the technical issues raised by the fact that electrons actually have spin 1/2, not spin zero. This oversimplification is the reason that the gauge field θ comes out to be a scalar, whereas the electromagnetic field is actually represented by a vector consisting of V and A.) The result is that we have an explanation for the presence of electromagnetic interactions: if we try to construct a gauge-symmetric theory of identical, non-interacting particles, the result is not self-consistent, and can only be repaired by adding electrical and magnetic fields that cause the particles to interact.

Although the function θ(x) describes a wave, the laws of quantum mechanics require that it also have particle properties. In the case of electromagnetism, the particle corresponding to electromagnetic waves is the photon. In general, such particles are called gauge bosons, where the term "boson" refers to a particle with integer spin. In the simplest versions of the theory, gauge bosons are massless, but it is also possible to construct versions in which they have mass, as is the case for the gauge bosons that transmit the nuclear decay forces.

Conservation law

From Wikipedia, the free encyclopedia

A conservation law is a system of homogeneous first-order quasilinear hyperbolic equation that (in one space dimension) can be put into the form:^[1]

where the dependent varible y is called the density of a conserved quantity, and A(y) is called the current jacobian, and the subscript notation for partial derivatives has been employed. The more general inhomogenous case:

is not a conservation equation but the general kind of balance equation describing a dissipative system. The dependent variable y is called a nonconserved quantity, and the inhomogeneous term s(y,x,t) is the a (quantity)-source, or dissipation. For example balance equations of this kind are the momentum and energy Navier-Stokes equations, or the entropy balance for a general isolated system.

In mathematical physics, conservation equations express a particular measurable property of an isolated physical system does not change as the system evolves. One particularly important result concerning conservation equations is Noether's theorem, which states that there is a one-to-one correspondence between each one of them and a differentiable symmetry in the system. For example, the conservation of energy follows from the time-invariance of physical systems, and the fact that physical systems behave the same regardless of how they are oriented in space gives rise to the conservation of angular momentum.

Differential forms

In the one-dimensional space a conservation equation is a first-order quasilinear hyperbolic equation that can be put into the advection form:

where the dependent variable y(x,t) is called the density of the conserved (scalar) quantity (c.q.(d.) = conserved quantity (density)) , and a(y) is called the current coefficient, usually corresponding to the partial derivative in the conserved quantity of a current density (c.d.) of the conserved quantity j(y):^[1]

In this case since the chain rule applies:

the conservation equation can be put into the current density form:

In a space with more than one dimension the former definition can be extended to an equation that can be put into the form:

where the conserved quantity is y(r,t), denotes the scalar product, ∇ is the nabla operator, here indicating a gradient, and a(y) is a vector of current coefficients, analogously corresponding to the divergence of a vector c.d. associated to the c.q. j(y):

This is the case for the continuity equation:

Here the conserved quantity is the mass, with density ρ(r,t) and current density ρu, identical to the momentum density, while u(r,t) is the flow velocity.

In the general case a conservation equation can be also a system of this kind of equations (a vector equation) in the form:^[1]

where y is called the conserved (vector) quantity, ∇ y is its gradient, 0 is the zero vector, and A(y) is called the Jacobian of the current density. In fact as in the former scalar case, also in the vector case A(y) usually corresponding to the Jacobian of a current density matrix J(y):

and the conservation equation can be put into the form:

For example this the case for Euler equations (fluid dynamics). In the simple incompressible case they are:

where:

• u is the flow velocity vector, with components in a N-dimensional space u₁, u₂ ... u_N,
• s is the specific pressure (pressure per unit density) giving the source term,

It can be shown that the conserved (vector) quantity and the c.d. matrix for these equations are respectively:

where denotes the tensor product.

Integral and weak forms

Conservation equations can be also expressed in integral form: the advantage of the latter is substantially that it requires less smoothness of the solution, which paves the way to weak form, extending the class of admissible solutions to include discontinuous solutions.^[2] By integrating in any space-time domain the current density form in 1-D space:

and by using Green's theorem, the integral form is:

In a similar fashion, for the scalar multidimensional space, the integral form is:

where the line integration is performed along the boundary of the domain, in an anticlock-wise manner.^[2]

Moreover, by defining a test function φ(r,t) continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition. In 1-D space it is:

Note that in the weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is sufficiently smooth to admit these derivatives.^[2]

Exact laws

A partial listing of physical conservation equations due to symmetry that are said to be exact laws, or more precisely have never been [proven to be] violated:

Conservation Law	Respective Noether symmetry invariance		Number of dimensions
Conservation of mass-energy	Time invariance	Lorentz invariance symmetry	1	translation about time axis
Conservation of linear momentum	Galilean invariance		3	translation about x,y,z position
Conservation of angular momentum	Rotation invariance		3	rotation about x,y,z axes
CPT symmetry (combining charge, parity and time conjugation)	Lorentz invariance		1+1+1	(charge inversion q→-q) + (position inversion r→-r) + (time inversion t→-t)
Conservation of electric charge	Gauge invariance		1⊗4	scalar field (1D) in 4D spacetime (x,y,z + time evolution)
Conservation of color charge	SU(3) Gauge invariance		3	r,g,b
Conservation of weak isospin	SU(2)L Gauge invariance		1	weak charge
Conservation of probability	Probability invariance		1⊗4	total probability always=1 in whole x,y,z space, during time evolution

Approximate laws

There are also approximate conservation laws. These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions.

• Conservation of mass (applies for non-relativistic speeds and when there are no nuclear reactions)
• Conservation of baryon number (See chiral anomaly)
• Conservation of lepton number (In the Standard Model)
• Conservation of flavor (violated by the weak interaction)
• Conservation of parity
• Invariance under Charge conjugation
• Invariance under time reversal
• CP symmetry, the combination of charge and parity conjugation (equivalent to time reversal if CPT holds)