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These symmetries are near-symmetries because each is broken in the present-day universe. However, the Standard Model predicts that the combination of the three (that is, the simultaneous application of all three transformations) must be a symmetry, called CPT symmetry. CP violation, the violation of the combination of ... |
A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the Standard Model. Supersymmetry is based on the idea that there is another physical symmetry beyond those already developed in the Standard Model, specifically a symmetry between bosons and fermions. Supersymmetry asserts t... |
The transformations describing physical symmetries typically form a mathematical group. Group theory is an important area of mathematics for physicists. |
Continuous symmetries are specified mathematically by "continuous groups" (called Lie groups). Many physical symmetries are isometries and are specified by symmetry groups. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere fo... |
Discrete groups describe discrete symmetries. For example, the symmetries of an equilateral triangle are characterized by the symmetric group S. |
A type of physical theory based on "local" symmetries is called a "gauge" theory and the symmetries natural to such a theory are called gauge symmetries. Gauge symmetries in the Standard Model, used to describe three of the fundamental interactions, are based on the SU(3) × SU(2) × U(1) group. (Roughly speaking, the sy... |
Also, the reduction by symmetry of the energy functional under the action by a group and spontaneous symmetry breaking of transformations of symmetric groups appear to elucidate topics in particle physics (for example, the unification of electromagnetism and the weak force in physical cosmology). |
The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. ... |
The following table summarizes some fundamental symmetries and the associated conserved quantity. |
Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various particle fields. The commutator of two of these infinitesimal transformations are equivalent to a third infinitesimal transformation of the same kind hence they form a Lie alg... |
A general coordinate transformation described as the general field formula_6 (also known as a diffeomorphism) has the infinitesimal effect on a scalar formula_7, spinor formula_8 or vector field formula_9 that can be expressed (using the index summation convention): |
Without gravity only the Poincaré symmetries are preserved which restricts formula_6 to be of the form: |
where M is an antisymmetric matrix (giving the Lorentz and rotational symmetries) and P is a general vector (giving the translational symmetries). Other symmetries affect multiple fields simultaneously. For example, local gauge transformations apply to both a vector and spinor field: |
where formula_17 are generators of a particular Lie group. So far the transformations on the right have only included fields of the same type. Supersymmetries are defined according to how the mix fields of "different" types. |
Another symmetry which is part of some theories of physics and not in others is scale invariance which involve Weyl transformations of the following kind: |
If the fields have this symmetry then it can be shown that the field theory is almost certainly conformally invariant also. This means that in the absence of gravity h(x) would restricted to the form: |
with D generating scale transformations and K generating special conformal transformations. For example, super-Yang–Mills theory has this symmetry while general relativity doesn't although other theories of gravity such as conformal gravity do. The 'action' of a field theory is an invariant under all the symmetries of ... |
In string theories, since a string can be decomposed into an infinite number of particle fields, the symmetries on the string world sheet is equivalent to special transformations which mix an infinite number of fields. |
There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise ... |
By definition of a symmetry of a quantum system, there is a group action on formula_7. For each formula_12, there is a corresponding transformation formula_13 of formula_7. More specifically, if formula_15 is some symmetry of the system (say, rotation about the x-axis by 12°), then the corresponding transformation form... |
Each map formula_13 preserves, by definition of symmetry, the ray product on formula_7 induced by the inner product on formula_1; according to Wigner's theorem, this transformation of formula_7 comes from a unitary or anti-unitary transformation formula_27 of formula_1. Note, however, that the formula_27 associated to ... |
where formula_34 will depend on formula_15 and formula_36. Thus, the map sending formula_15 to formula_27 is a "projective unitary representation" of formula_4, or possibly a mixture of unitary and anti-unitary, if formula_4 is disconnected. In practice, anti-unitary operators are always associated with time-reversal s... |
It is important physically that in general formula_41 does not have to be an ordinary representation of formula_4; it may not be possible to choose the phase factors in the definition of formula_27 to eliminate the phase factors in their composition law. An electron, for example, is a spin-one-half particle; its Hilber... |
For many interesting classes of groups formula_4, tells us that every projective unitary representation of formula_4 comes from an ordinary representation of the universal cover formula_51 of formula_4. Actually, if formula_1 is finite dimensional, then regardless of the group formula_4, every projective unitary repres... |
The requirement referred to above is that the Lie algebra formula_59 does not admit a nontrivial one-dimensional central extension. This is the case if and only if the second cohomology group of formula_59 is trivial. In this case, it may still be true that the group admits a central extension by a "discrete" group. Bu... |
Thus, in favorable cases, the quantum system will carry a unitary representation of the universal cover formula_51 of the symmetry group formula_4. This is desirable because formula_1 is much easier to work with than the non-vector space formula_7. If the representations of formula_51 can be classified, much more infor... |
In this case, to obtain an ordinary representation, one has to pass to the Heisenberg group, which is a nontrivial one-dimensional central extension of formula_75. |
The group of translations and Lorentz transformations form the Poincaré group, and this group should be a symmetry of a relativistic quantum system (neglecting general relativity effects, or in other words, in flat space). Representations of the Poincaré group are in many cases characterized by a nonnegative mass and a... |
While the spacetime symmetries in the Poincaré group are particularly easy to visualize and believe, there are also other types of symmetries, called internal symmetries. One example is color SU(3), an exact symmetry corresponding to the continuous interchange of the three quark colors. |
Many (but not all) symmetries or approximate symmetries form Lie groups. Rather than study the representation theory of these Lie groups, it is often preferable to study the closely related representation theory of the corresponding Lie algebras, which are usually simpler to compute. |
Now, representations of the Lie algebra correspond to representations of the universal cover of the original group. In the finite-dimensional case—and the infinite-dimensional case, provided that applies—irreducible projective representations of the original group correspond to ordinary unitary representations of the u... |
Thus, to summarize, the irreducible projective representations of SO(3) are in one-to-one correspondence with the irreducible ordinary representations of its Lie algebra so(3). The two-dimensional "spin 1/2" representation of the Lie algebra so(3), for example, does not correspond to an ordinary (single-valued) represe... |
Although the above symmetries are believed to be exact, other symmetries are only approximate. |
As an example of what an approximate symmetry means, suppose an experimentalist lived inside an infinite ferromagnet, with magnetization in some particular direction. The experimentalist in this situation would find not one but two distinct types of electrons: one with spin along the direction of the magnetization, wit... |
In general, an approximate symmetry arises when there are very strong interactions that obey that symmetry, along with weaker interactions that do not. In the electron example above, the two "types" of electrons behave identically under the strong and weak forces, but differently under the electromagnetic force. |
An example from the real world is isospin symmetry, an SU(2) group corresponding to the similarity between up quarks and down quarks. This is an approximate symmetry: While up and down quarks are identical in how they interact under the strong force, they have different masses and different electroweak interactions. Ma... |
and the laws of physics are "approximately" invariant under applying a determinant-1 unitary transformation to this space: |
For example, formula_85 would turn all up quarks in the universe into down quarks and vice versa. Some examples help clarify the possible effects of these transformations: |
In general, particles form isospin multiplets, which correspond to irreducible representations of the Lie algebra SU(2). Particles in an isospin multiplet have very similar but not identical masses, because the up and down quarks are very similar but not identical. |
Isospin symmetry can be generalized to flavour symmetry, an SU(3) group corresponding to the similarity between up quarks, down quarks, and strange quarks. This is, again, an approximate symmetry, violated by quark mass differences and electroweak interactions—in fact, it is a poorer approximation than isospin, because... |
Nevertheless, particles can indeed be neatly divided into groups that form irreducible representations of the Lie algebra SU(3), as first noted by Murray Gell-Mann and independently by Yuval Ne'eman. |
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by a: "T"a(p) = p + a. |
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation. Discrete translational symmetry is invariant under discrete translation. |
Analogously an operator "A" on functions is said to be translationally invariant with respect to a translation operator formula_1 if the result after applying "A" doesn't change if the argument function is translated. |
Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noether's theorem, space translational symmetry of a physical system is equivalent to the momentum conservation law. |
Translational symmetry of an object means that a particular translation does not change the object. For a given object, the translations for which this applies form a group, the symmetry group of the object, or, if the object has more kinds of symmetry, a subgroup of the symmetry group. |
Translational invariance implies that, at least in one direction, the object is infinite: for any given point p, the set of points with the same properties due to the translational symmetry form the infinite discrete set {p + "n"a | "n" ∈ Z} = p + Z a. Fundamental domains are e.g. H + [0, 1] a for any hyperplane H for ... |
In spaces with dimension higher than 1, there may be multiple translational symmetry. For each set of "k" independent translation vectors, the symmetry group is isomorphic with Z"k". |
In particular, the multiplicity may be equal to the dimension. This implies that the object is infinite in all directions. In this case, the set of all translations forms a lattice. Different bases of translation vectors generate the same lattice if and only if one is transformed into the other by a matrix of integer c... |
Alternatively, e.g. a rectangle may define the whole object, even if the translation vectors are not perpendicular, if it has two sides parallel to one translation vector, while the other translation vector starting at one side of the rectangle ends at the opposite side. |
For example, consider a tiling with equal rectangular tiles with an asymmetric pattern on them, all oriented the same, in rows, with for each row a shift of a fraction, not one half, of a tile, always the same, then we have only translational symmetry, wallpaper group "p"1 (the same applies without shift). With rotatio... |
In 2D there may be translational symmetry in one direction for vectors of any length. One line, not in the same direction, fully defines the whole object. Similarly, in 3D there may be translational symmetry in one or two directions for vectors of any length. One plane (cross-section) or line, respectively, fully defin... |
An example of translational symmetry in one direction in 2D nr. 1) is: |
Note: The example is not an example of rotational symmetry. |
(get the same by moving one line down and two positions to the right), and of translational symmetry in two directions in 2D (wallpaper group p1): |
(get the same by moving three positions to the right, or one line down and two positions to the right; consequently get also the same moving three lines down). |
In both cases there is neither mirror-image symmetry nor rotational symmetry. |
For a given translation of space we can consider the corresponding translation of objects. The objects with at least the corresponding translational symmetry are the fixed points of the latter, not to be confused with fixed points of the translation of space, which are non-existent. |
In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass cannot change, so quantity can neither be added nor be removed. Therefore, the ... |
The law implies that mass can neither be created nor destroyed, although it may be rearranged in space, or the entities associated with it may be changed in form. For example, in chemical reactions, the mass of the chemical components before the reaction is equal to the mass of the components after the reaction. Thus, ... |
The concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dynamics. Historically, mass conservation was demonstrated in chemical reactions independently by Mikhail Lomonosov and later rediscovered by Antoine Lavoisier in the late 18th century. The formulation of this law was... |
The conservation of mass only holds approximately and is considered part of a series of assumptions coming from classical mechanics. The law has to be modified to comply with the laws of quantum mechanics and special relativity under the principle of mass-energy equivalence, which states that energy and mass form one c... |
Mass is also not generally conserved in open systems. Such is the case when various forms of energy and matter are allowed into, or out of, the system. However, unless radioactivity or nuclear reactions are involved, the amount of energy escaping (or entering) such systems as heat, mechanical work, or electromagnetic r... |
For systems where large gravitational fields are involved, general relativity has to be taken into account, where mass-energy conservation becomes a more complex concept, subject to different definitions, and neither mass nor energy is as strictly and simply conserved as is the case in special relativity. |
The law of conservation of mass can only be formulated in classical mechanics when the energy scales associated to an isolated system are much smaller than formula_1, where formula_2 is the mass of a typical object in the system, measured in the frame of reference where the object is at rest, and formula_3 is the speed... |
The law can be formulated mathematically in the fields of fluid mechanics and continuum mechanics, where the conservation of mass is usually expressed using the continuity equation, given in differential form asformula_4where formula_5 is the density (mass per unit volume), formula_6 is the time, formula_7 is the diver... |
The interpretation of the continuity equation for mass is the following: For a given closed surface in the system, the change in time of the mass enclosed by the surface is equal to the mass that traverses the surface, positive if matter goes in and negative if matter goes out. For the whole isolated system, this condi... |
The continuity equation for the mass is part of Euler equations of fluid dynamics. Many other convection–diffusion equations describe the conservation and flow of mass and matter in a given system. |
In chemistry, the calculation of the amount of reactant and products in a chemical reaction, or stoichiometry, is founded on the principle of conservation of mass. The principle implies that during a chemical reaction the total mass of the reactants is equal to the total mass of the products. For example, in the follow... |
where one molecule of methane () and two oxygen molecules are converted into one molecule of carbon dioxide () and two of water (). The number of molecules as result from the reaction can be derived from the principle of conservation of mass, as initially four hydrogen atoms, 4 oxygen atoms and one carbon atom are pres... |
Many engineering problems are solved by following the mass distribution in time of a given system, this practice is known as mass balance. |
An important idea in ancient Greek philosophy was that "Nothing comes from nothing", so that what exists now has always existed: no new matter can come into existence where there was none before. An explicit statement of this, along with the further principle that nothing can pass away into nothing, is found in Empedoc... |
A further principle of conservation was stated by Epicurus around 3rd century BC, who, describing the nature of the Universe, wrote that "the totality of things was always such as it is now, and always will be". |
Jain philosophy, a non-creationist philosophy based on the teachings of Mahavira (6th century BC), states that the universe and its constituents such as matter cannot be destroyed or created. The Jain text Tattvarthasutra (2nd century AD) states that a substance is permanent, but its modes are characterised by creation... |
The conservation of mass was obscure for millennia because of the buoyancy effect of the Earth's atmosphere on the weight of gases. For example, a piece of wood weighs less after burning; this seemed to suggest that some of its mass disappears, or is transformed or lost. This was not disproved until careful experiments... |
Once understood, the conservation of mass was of great importance in progressing from alchemy to modern chemistry. Once early chemists realized that chemical substances never disappeared but were only transformed into other substances with the same weight, these scientists could for the first time embark on quantitativ... |
Following the pioneering work of Lavoisier, the exhaustive experiments of Jean Stas supported the consistency of this law in chemical reactions, even though they were carried out with other intentions. His research indicated that in certain reactions the loss or gain could not have been more than from 2 to 4 parts in 1... |
The law conservation of mass and the analogous law of conservation of energy were finally overruled by a more general principle known as the mass–energy equivalence. Special relativity also redefines the concept of mass and energy, which can be used interchangeably and are relative to the frame of reference. Several de... |
In special relativity, the conservation of mass does not apply if the system is open and energy escapes. However, it does continue to apply to totally closed (isolated) systems. If energy cannot escape a system, its mass cannot decrease. In relativity theory, so long as any type of energy is retained within a system, t... |
Also, mass must be differentiated from matter, since matter may "not" be perfectly conserved in isolated systems, even though mass is always conserved in such systems. However, matter is so nearly conserved in chemistry that violations of matter conservation were not measured until the nuclear age, and the assumption o... |
The mass associated with chemical amounts of energy is too small to measure. |
The change in mass of certain kinds of open systems where atoms or massive particles are not allowed to escape, but other types of energy (such as light or heat) are allowed to enter, escape or be merged, went unnoticed during the 19th century, because the change in mass associated with addition or loss of small quanti... |
Mass conservation remains correct if energy is not lost. |
The conservation of relativistic mass implies the viewpoint of a single observer (or the view from a single inertial frame) since changing inertial frames may result in a change of the total energy (relativistic energy) for systems, and this quantity determines the relativistic mass. |
The principle that the mass of a system of particles must be equal to the sum of their rest masses, even though true in classical physics, may be false in special relativity. The reason that rest masses cannot be simply added is that this does not take into account other forms of energy, such as kinetic and potential e... |
For moving massive particles in a system, examining the rest masses of the various particles also amounts to introducing many different inertial observation frames (which is prohibited if total system energy and momentum are to be conserved), and also when in the rest frame of one particle, this procedure ignores the m... |
The conservation of both relativistic and invariant mass applies even to systems of particles created by pair production, where energy for new particles may come from kinetic energy of other particles, or from one or more photons as part of a system that includes other particles besides a photon. Again, neither the rel... |
The mass-energy equivalence formula gives a different prediction in non-isolated systems, since if energy is allowed to escape a system, both relativistic mass and invariant mass will escape also. In this case, the mass-energy equivalence formula predicts that the "change" in mass of a system is associated with the "ch... |
In general relativity, the total invariant mass of photons in an expanding volume of space will decrease, due to the red shift of such an expansion. The conservation of both mass and energy therefore depends on various corrections made to energy in the theory, due to the changing gravitational potential energy of such ... |
The groundwater energy balance is the energy balance of a groundwater body in terms of incoming hydraulic energy associated with groundwater inflow into the body, energy associated with the outflow, energy conversion into heat due to friction of flow, and the resulting change of energy status and groundwater level. |
When multiplying the horizontal velocity of groundwater (dimension, for example, m3/day per m2 cross-sectional area) with the groundwater potential (dimension energy per m3 water, or "E"/m3) one obtains an energy flow (flux) in "E"/day per m2 cross-sectional area. |
Summation or integration of the energy flux in a vertical cross-section of unit width (say 1 m) from the lower flow boundary (the impermeable layer or base) up to the water table in an unconfined aquifer gives the energy flow "Fe" through the cross-section in "E"/day per m width of the aquifer. |
While flowing, the groundwater loses energy due to friction of flow, i.e. hydraulic energy is converted into heat. At the same time, energy may be added with the recharge of water coming into the aquifer through the water table. Thus one can make an hydraulic energy balance of a block of soil between two nearby cross-s... |
In mathematical terms this balance can be obtained by differentiating the cross-sectional integral of "Fe" in the direction of flow using the Leibniz rule, taking into account that the level of the water table may change in the direction of flow. |
The mathematics is simplified using the Dupuit–Forchheimer assumption. |
The hydraulic friction losses can be described in analogy to "Joule's law" in electricity (see Joule's law#Hydraulic equivalent), where the friction losses are proportional to the square value of the current (flow) and the electrical resistance of the material through which the current occurs. In groundwater hydraulics... |
The trial and error procedure is cumbersome and therefore computer programs may be developed to aid in the calculations. |
The energy balance of groundwater flow can be applied to flow of groundwater to subsurface drains. The computer program "EnDrain" compares the outcome of the traditional drain spacing equation, based on Darcy's law together with the continuity equation (i.e. conservation of mass), with the solution obtained by the ener... |
The Carter constant is a conserved quantity for motion around black holes in the general relativistic formulation of gravity. Carter's constant was derived for a spinning, charged black hole by Australian theoretical physicist Brandon Carter in 1968. Carter's constant along with the energy, axial angular momentum, and ... |
Carter noticed that the Hamiltonian for motion in Kerr spacetime was separable in Boyer–Lindquist coordinates, allowing the constants of such motion to be easily identified using Hamilton-Jacobi theory. The Carter constant can be written as follows: |
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