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As an example, consider decreasing of the moment of inertia, e.g. when a figure skater is pulling in her/his hands, speeding up the circular motion. In terms of angular momentum conservation, we have, for angular momentum "L", moment of inertia "I" and angular velocity "ω": |
Using this, we see that the change requires an energy of: |
so that a decrease in the moment of inertia requires investing energy. |
This can be compared to the work done as calculated using Newton's laws. Each point in the rotating body is accelerating, at each point of time, with radial acceleration of: |
Let us observe a point of mass "m", whose position vector relative to the center of motion is parallel to the z-axis at a given point of time, and is at a distance "z". The centripetal force on this point, keeping the circular motion, is: |
Thus the work required for moving this point to a distance "dz" farther from the center of motion is: |
For a non-pointlike body one must integrate over this, with "m" replaced by the mass density per unit "z". This gives: |
which is exactly the energy required for keeping the angular momentum conserved. |
Note, that the above calculation can also be performed per mass, using kinematics only. Thus the phenomena of figure skater accelerating tangential velocity while pulling her/his hands in, can be understood as follows in layman's language: The skater's palms are not moving in a straight line, so they are constantly accelerating inwards, but do not gain additional speed because the accelerating is always done when their motion inwards is zero. However, this is different when pulling the palms closer to the body: The acceleration due to rotation now increases the speed; but because of the rotation, the increase in speed does not translate to a significant speed inwards, but to an increase of the rotation speed. |
In Lagrangian mechanics, angular momentum for rotation around a given axis, is the conjugate momentum of the generalized coordinate of the angle around the same axis. For example, formula_85, the angular momentum around the z axis, is: |
where formula_87 is the Lagrangian and formula_88 is the angle around the z axis. |
Note that formula_89, the time derivative of the angle, is the angular velocity formula_90. Ordinarily, the Lagrangian depends on the angular velocity through the kinetic energy: The latter can be written by separating the velocity to its radial and tangential part, with the tangential part at the x-y plane, around the z-axis, being equal to: |
Since the lagrangian is dependent upon the angles of the object only through the potential, we have: |
which is the torque on the i-th object. |
Suppose the system is invariant to rotations, so that the potential is independent of an overall rotation by the angle "θ"z (thus it may depend on the angles of objects only through their differences, in the form formula_93). We therefore get for the total angular momentum: |
And thus the angular momentum around the z-axis is conserved. |
This analysis can be repeated separately for each axis, giving conversation of the angular momentum vector. However, the angles around the three axes cannot be treated simultaneously as generalized coordinates, since they are not independent; in particular, two angles per point suffice to determine its position. While it is true that in the case of a rigid body, fully describing it requires, in addition to three translational degrees of freedom, also specification of three rotational degrees of freedom; however these cannot be defined as rotations around the Cartesian axes (see Euler angles). This caveat is reflected in quantum mechanics in the non-trivial commutation relations of the different components of the angular momentum operator. |
Equivalently, in Hamiltonian mechanics the Hamiltonian can be described as a function of the angular momentum. As before, the part of the kinetic energy related to rotation around the z-axis for the i-th object is: |
In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have more left than right, or vice versa. |
Such events are expected to be prohibited according to classical conservation laws, but we know there must be ways they can be broken, because we have evidence of charge–parity non-conservation ("CP violation"). It is possible that other imbalances have been caused by breaking of a "chiral law" of this kind. Many physicists suspect that the fact that the observable universe contains more matter than antimatter is caused by a chiral anomaly. Research into chiral symmetry breaking laws is a major endeavor in particle physics research at this time. |
The chiral anomaly originally referred to the anomalous decay rate of the neutral pion, as computed in the current algebra of the chiral model. These calculations suggested that the decay of the pion was suppressed, clearly contradicting experimental results. The nature of the anomalous calculations was first explained by Adler and Bell & Jackiw. This is now termed the Adler–Bell–Jackiw anomaly of quantum electrodynamics. This is a symmetry of classical electrodynamics that is violated by quantum corrections. |
At the time that the Adler–Bell–Jackiw anomaly was being explored in physics, there were related developments in differential geometry that appeared to involve the same kinds of expressions. These were not in any way related to quantum corrections of any sort, but rather were the exploration of the global structure of fiber bundles, and specifically, of the Dirac operators on spin structures having curvature forms resembling that of the electromagnetic tensor, both in four and three dimensions (the Chern–Simons theory). After considerable back and forth, it became clear that the structure of the anomaly could be described with bundles with a non-trivial homotopy group, or, in physics lingo, in terms of instantons. |
Instantons are a form of topological soliton; they are a solution to the "classical" field theory, having the property that they are stable and cannot decay (into plane waves, for example). Put differently: conventional field theory is built on the idea of a vacuum - roughly speaking, a flat empty space. Classically, this is the "trivial" solution; all fields vanish. However, one can also arrange the (classical) fields in such a way that they have a non-trivial global configuration. These non-trivial configurations are also candidates for the vacuum, for empty space; yet they are no longer flat or trivial; they contain a twist, the instanton. The quantum theory is able to interact with these configurations; when it does so, it manifests as the chiral anomaly. |
In mathematics, non-trivial configurations are found during the study of Dirac operators in their fully generalized setting, namely, on Riemannian manifolds in arbitrary dimensions. Mathematical tasks include finding and classifying structures and configurations. Famous results include the Atiyah–Singer index theorem for Dirac operators. Roughly speaking, the symmetries of Minkowski spacetime, Lorentz invariance, Laplacians, Dirac operators and the U(1)xSU(2)xSU(3) fiber bundles can be taken to be a special case of a far more general setting in differential geometry; the exploration of the various possibilities accounts for much of the excitement in theories such as string theory; the richness of possibilities accounts for a certain perception of lack of progress. |
Besides explaining the decay of the pion, it has a second very important role. The one loop amplitude includes a factor that counts the grand total number of leptons that can circulate in the loop. In order to get the correct decay width, one must have exactly three generations of quarks, and not four or more. In this way, it plays an important role in constraining the Standard model. It provides a direct physical prediction of the number of quarks that can exist in nature. |
Current day research is focused on similar phenomena in different settings, including non-trivial topological configurations of the electroweak theory, that is, the sphalerons. Other applications include the hypothetical non-conservation of baryon number in GUTS and other theories. |
In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this (global) chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved. The non-conservation happens in a process of tunneling from one vacuum to another. Such a process is called an instanton. |
In the case of a symmetry related to the conservation of a fermionic particle number, one may understand the creation of such particles as follows. The definition of a particle is different in the two vacuum states between which the tunneling occurs; therefore a state of no particles in one vacuum corresponds to a state with some particles in the other vacuum. In particular, there is a Dirac sea of fermions and, when such a tunneling happens, it causes the energy levels of the sea fermions to gradually shift upwards for the particles and downwards for the anti-particles, or vice versa. This means particles which once belonged to the Dirac sea become real (positive energy) particles and particle creation happens. |
Technically, in the path integral formulation, an anomalous symmetry is a symmetry of the action formula_3, but not of the measure and therefore "not" of the generating functional |
of the quantized theory ( is Planck's action-quantum divided by 2). The measure formula_5 consists of a part depending on the fermion field formula_6 and a part depending on its complex conjugate formula_7. The transformations of both parts under a chiral symmetry do not cancel in general. Note that if formula_8 is a Dirac fermion, then the chiral symmetry can be written as formula_9 where formula_10 is the chiral gamma matrix acting on formula_8. From the formula for formula_12 one also sees explicitly that in the classical limit, anomalies don't come into play, since in this limit only the extrema of formula_3 remain relevant. |
The anomaly is proportional to the instanton number of a gauge field to which the fermions are coupled. (Note that the gauge symmetry is always non-anomalous and is exactly respected, as is required for the theory to be consistent.) |
The chiral anomaly can be calculated exactly by one-loop Feynman diagrams, e.g. Steinberger's "triangle diagram", contributing to the pion decays, and formula_14. The amplitude for this process can be calculated directly from the change in the measure of the fermionic fields under the chiral transformation. |
Wess and Zumino developed a set of conditions on how the partition function ought to behave under gauge transformations called the Wess–Zumino consistency condition. |
Fujikawa derived this anomaly using the correspondence between functional determinants and the partition function using the Atiyah–Singer index theorem. See Fujikawa's method. |
The Standard Model of electroweak interactions has all the necessary ingredients for successful baryogenesis, although these interactions have never been observed and may be insufficient to explain the total baryon number of the observed universe if the initial baryon number of the universe at the time of the Big Bang is zero. Beyond the violation of charge conjugation formula_15 and CP violation formula_16 (charge+parity), baryonic charge violation appears through the Adler–Bell–Jackiw anomaly of the formula_17 group. |
Baryons are not conserved by the usual electroweak interactions due to quantum chiral anomaly. The classic electroweak Lagrangian conserves baryonic charge. Quarks always enter in bilinear combinations formula_18, so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current formula_19 is conserved: |
However, quantum corrections known as the sphaleron destroy this conservation law: instead of zero in the right hand side of this equation, there is a non-vanishing quantum term, |
where is a numerical constant vanishing for ℏ =0, |
and the gauge field strength formula_23 is given by the expression |
Electroweak sphalerons can only change the baryon and/or lepton number by 3 or multiples of 3 (collision of three baryons into three leptons/antileptons and vice versa). |
An important fact is that the anomalous current non-conservation is proportional to the total derivative of a vector operator, formula_25 (this is non-vanishing due to instanton configurations of the gauge field, which are pure gauge at the infinity), where the anomalous current formula_26 is |
which is the Hodge dual of the Chern–Simons 3-form. |
In the language of differential forms, to any self-dual curvature form formula_28 we may assign the abelian 4-form formula_29. Chern–Weil theory shows that this 4-form is locally "but not globally" exact, with potential given by the Chern-Simons 3-form locally: |
Again, this is true only on a single chart, and is false for the global form formula_31 unless the instanton number vanishes. |
To proceed further, we attach a "point at infinity" onto formula_32 to yield formula_33, and use the clutching construction to chart principal A-bundles, with one chart on the neighborhood of and a second on formula_34. The thickening around , where these charts intersect, is trivial, so their intersection is essentially formula_35. Thus instantons are classified by the third homotopy group formula_36, which for formula_37 is simply the third 3-sphere group formula_38. |
The divergence of the baryon number current is (ignoring numerical constants) |
Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (converted into kinetic energy) when the objects fall towards each other. Gravitational potential energy increases when two objects are brought further apart. |
For two pairwise interacting point particles, the gravitational potential energy formula_1 is given by |
where formula_3 and formula_4 are the masses of the two particles, formula_5 is the distance between them, and formula_6 is the gravitational constant. |
Close to the Earth's surface, the gravitational field is approximately constant, and the gravitational potential energy of an object reduces to |
where formula_4 is the object's mass, formula_9 is the gravity of Earth, and formula_10 is the height of the object's center of mass above a chosen reference level. |
In classical mechanics, two or more masses always have a gravitational potential. Conservation of energy requires that this gravitational field energy is always negative, so that it is zero when the objects are infinitely far apart. The gravitational potential energy is the potential energy an object has because it is within a gravitational field. |
The force between a point mass, formula_3, and another point mass, formula_4, is given by Newton's law of gravitation: |
To get the total work done by an external force to bring point mass formula_4 from infinity to the final distance formula_5 (for example the radius of Earth) of the two mass points, the force is integrated with respect to displacement: |
Because formula_18, the total work done on the object can be written as: |
In general relativity gravitational energy is extremely complex, and there is no single agreed upon definition of the concept. It is sometimes modelled via the Landau–Lifshitz pseudotensor that allows retention for the energy–momentum conservation laws of classical mechanics. Addition of the matter stress–energy tensor to the Landau–Lifshitz pseudotensor results in a combined matter plus gravitational energy pseudotensor that has a vanishing 4-divergence in all frames—ensuring the conservation law. Some people object to this derivation on the grounds that pseudotensors are inappropriate in general relativity, but the divergence of the combined matter plus gravitational energy pseudotensor is a tensor. |
In particle physics, lepton number (historically also called lepton charge) is a conserved quantum number representing the difference between the number of leptons and the number of antileptons in an elementary particle reaction. Lepton number is an additive quantum number, so its sum is preserved in interactions (as opposed to multiplicative quantum numbers such as parity, where the product is preserved instead). Mathematically, the lepton number formula_1 is defined by formula_2, where formula_3 is the number of leptons and formula_4 is the number of antileptons. |
Lepton number was introduced in 1953 to explain the absence of reactions such as formula_5 in the Cowan–Reines neutrino experiment, which instead observed formula_6. This process, inverse beta decay, conserves lepton number, as the incoming antineutrino has lepton number –1, while the outgoing positron (antielectron) also has lepton number –1. |
In addition to lepton number, lepton family numbers are defined as |
Prominent examples of lepton flavor conservation are the muon decays formula_7 and formula_8. In these, the creation of an electron is accompanied by the creation of an electron antineutrino, and the creation of a positron is accompanied by the creation of an electron neutrino. Likewise, a decaying negative muon results in the creation of a muon neutrino, while a decaying positive muon results in the creation of a muon antineutrino. |
Violations of the lepton number conservation laws. |
Lepton flavor is only approximately conserved, and is notably not conserved in neutrino oscillation. However, total lepton number is still conserved in the Standard Model. |
Numerous searches for physics beyond the Standard Model incorporate searches for lepton number or lepton flavor violation, such as the decays formula_9. Experiments such as MEGA and SINDRUM have searched for lepton number violation in muon decays to electrons; MEG set the current branching limit of order 10−13 and plans to lower to limit to 10−14 after 2016. Some theories beyond the Standard Model, such as supersymmetry, predict branching ratios of order 10−12 to 10−14. The Mu2e experiment, in construction as of 2017, has a planned sensitivity of order 10−17. |
Because the lepton number conservation law in fact is violated by chiral anomalies, there are problems applying this symmetry universally over all energy scales. However, the quantum number "B" − "L" is commonly conserved in Grand Unified Theory models. |
If neutrinos turn out to be Majorana fermions, neither the lepton number nor "B" − "L" would be conserved, e.g. in neutrinoless double beta decay. |
In mathematical physics, inversion transformations are a natural extension of Poincaré transformations to include all conformal one-to-one transformations on coordinate space-time. They are less studied in physics because unlike the rotations and translations of Poincaré symmetry an object cannot be physically transformed by the inversion symmetry. Some physical theories are invariant under this symmetry, in these cases it is what is known as a 'hidden symmetry'. Other hidden symmetries of physics include gauge symmetry and general covariance. |
In 1831 the mathematician Ludwig Immanuel Magnus began to publish on transformations of the plane generated by inversion in a circle of radius "R". His work initiated a large body of publications, now called inversive geometry. The most prominently named mathematician became August Ferdinand Möbius once he reduced the planar transformations to complex number arithmetic. In the company of physicists employing the inversion transformation early on was Lord Kelvin, and the association with him leads it to be called the Kelvin transform. |
In the following we shall use imaginary time (formula_1) so that space-time is Euclidean and the equations are simpler. The Poincaré transformations are given by the coordinate transformation on space-time parametrized by the 4-vectors "V" |
where formula_3 is an orthogonal matrix and formula_4 is a 4-vector. Applying this transformation twice on a 4-vector gives a third transformation of the same form. The basic invariant under this transformation is the space-time length given by the distance between two space-time points given by 4-vectors "x" and "y": |
These transformations are subgroups of general 1-1 conformal transformations on space-time. It is possible to extend these transformations to include all 1-1 conformal transformations on space-time |
We must also have an equivalent condition to the orthogonality condition of the Poincaré transformations: |
Because one can divide the top and bottom of the transformation by formula_8 we lose no generality by setting formula_9 to the unit matrix. We end up with |
Applying this transformation twice on a 4-vector gives a transformation of the same form. The new symmetry of 'inversion' is given by the 3-tensor formula_11 This symmetry becomes Poincaré symmetry if we set formula_12 When formula_13 the second condition requires that formula_3 is an orthogonal matrix. This transformation is 1-1 meaning that each point is mapped to a unique point only if we theoretically include the points at infinity. |
The invariants for this symmetry in 4 dimensions is unknown however it is known that the invariant requires a minimum of 4 space-time points. In one dimension, the invariant is the well known cross-ratio from Möbius transformations: |
Because the only invariants under this symmetry involve a minimum of 4 points, this symmetry cannot be a symmetry of point particle theory. Point particle theory relies on knowing the lengths of paths of particles through space-time (e.g., from formula_16 to formula_17). The symmetry can be a symmetry of a string theory in which the strings are uniquely determined by their endpoints. The propagator for this theory for a string starting at the endpoints formula_18 and ending at the endpoints formula_19 is a conformal function of the 4-dimensional invariant. A string field in endpoint-string theory is a function over the endpoints. |
In theoretical physics, an invariant is an observable of a physical system which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition. Invariants of a system are deeply tied to the symmetries imposed by its environment. |
Invariance is an important concept in modern theoretical physics, and many theories are expressed in terms of their symmetries and invariants. |
In classical and quantum mechanics, invariance of space under translation results in momentum being an invariant and the conservation of momentum, whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the conservation of energy. In general, by Noether's theorem, any invariance of a physical system under a continuous symmetry leads to a fundamental conservation law. |
In crystals, the electron density is periodic and invariant with respect to discrete translations by unit cell vectors. In very few materials, this symmetry can be broken due to enhanced electron correlations. |
Another examples of physical invariants are the speed of light, and charge and mass of a particle observed from two reference frames moving with respect to one another (invariance under a spacetime Lorentz transformation), and invariance of time and acceleration under a Galilean transformation between two such frames moving at low velocities. |
Quantities can be invariant under some common transformations but not under others. For example, the velocity of a particle is invariant when switching coordinate representations from rectangular to curvilinear coordinates, but is not invariant when transforming between frames of reference that are moving with respect to each other. Other quantities, like the speed of light, are always invariant. |
Physical laws are said to be invariant under transformations when their predictions remain unchanged. This generally means that the form of the law (e.g. the type of differential equations used to describe the law) is unchanged in transformations so that no additional or different solutions are obtained. |
Covariance and contravariance generalize the mathematical properties of invariance in tensor mathematics, and are frequently used in electromagnetism, special relativity, and general relativity. |
In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. |
A family of particular transformations may be "continuous" (such as rotation of a circle) or "discrete" (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see "Symmetry group"). |
These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as group representations and can, in addition, be exploited to simplify many problems. |
Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is known in mathematical terms as the Poincaré group, the symmetry group of special relativity. Another important example is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea in general relativity. |
Invariance is specified mathematically by transformations that leave some property (e.g. quantity) unchanged. This idea can apply to basic real-world observations. For example, temperature may be homogeneous throughout a room. Since the temperature does not depend on the position of an observer within the room, we say that the temperature is "invariant" under a shift in an observer's position within the room. |
Similarly, a uniform sphere rotated about its center will appear exactly as it did before the rotation. The sphere is said to exhibit spherical symmetry. A rotation about any axis of the sphere will preserve how the sphere "looks". |
The above ideas lead to the useful idea of "invariance" when discussing observed physical symmetry; this can be applied to symmetries in forces as well. |
For example, an electric field due to an electrically charged wire of infinite length is said to exhibit cylindrical symmetry, because the electric field strength at a given distance "r" from the wire will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius "r". Rotating the wire about its own axis does not change its position or charge density, hence it will preserve the field. The field strength at a rotated position is the same. This is not true in general for an arbitrary system of charges. |
In Newton's theory of mechanics, given two bodies, each with mass "m", starting at the origin and moving along the "x"-axis in opposite directions, one with speed "v"1 and the other with speed "v"2 the total kinetic energy of the system (as calculated from an observer at the origin) is and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the "y"-axis. |
The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if "v"1 and "v"2 are interchanged. |
Symmetries may be broadly classified as "global" or "local". A "global symmetry" is one that keeps a property invariant for a transformation that is applied simultaneously at all points of spacetime, whereas a "local symmetry" is one that keeps a property invariant when a possibly different symmetry transformation is applied at each point of spacetime; specifically a local symmetry transformation is parameterised by the spacetime co-ordinates, whereas a global symmetry is not. This implies that a global symmetry is also a local symmetry. Local symmetries play an important role in physics as they form the basis for gauge theories. |
The two examples of rotational symmetry described above – spherical and cylindrical – are each instances of continuous symmetry. These are characterised by invariance following a continuous change in the geometry of the system. For example, the wire may be rotated through any angle about its axis and the field strength will be the same on a given cylinder. Mathematically, continuous symmetries are described by transformations that change continuously as a function of their parameterization. An important subclass of continuous symmetries in physics are spacetime symmetries. |
Continuous "spacetime symmetries" are symmetries involving transformations of space and time. These may be further classified as "spatial symmetries", involving only the spatial geometry associated with a physical system; "temporal symmetries", involving only changes in time; or "spatio-temporal symmetries", involving changes in both space and time. |
Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold. The underlying local diffeomorphisms associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system. |
Some of the most important vector fields are Killing vector fields which are those spacetime symmetries that preserve the underlying metric structure of a manifold. In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of isometries. |
A "discrete symmetry" is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called "reflections" or "interchanges". |
The Standard Model of particle physics has three related natural near-symmetries. These state that the universe in which we live should be indistinguishable from one where a certain type of change is introduced. |
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