Spinor

In geometry and physics, spinors are elements of a vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight rotation. However, when a sequence of such small rotations is composed to form an overall final rotation, the resulting spinor transformation depends on which sequence of small rotations was used: unlike vectors and tensors, a spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360°; this property characterizes spinors. It is possible to associate a similar notion of spinor to Minkowski space in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles. Spinors are characterized by the specific way.

They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved. There are two topologically distinguishable classes of paths through rotations that result in the same overall rotation, as famously illustrated by the belt trick puzzle; these two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class, it doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class. In mathematical terms, spinors are described by a double-valued projective representation of the rotation group SO. Although spinors can be defined purely as elements of a representation space of the spin group, they are defined as elements of a vector space that carries a linear representation of the Clifford algebra.

The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, in applications the Clifford algebra is the easiest to work with. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations; the spinors are the column vectors. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what constitutes a "column vector", involves the choice of basis and gamma matrices in an essential way; as a representation of the spin group, this realization of spinors as column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.

What characterizes spinors and distinguishes them from geometric vectors and other tensors is subtle. Consider applying a rotation to the coordinates of a system. No object in the system itself has moved, only the coordinates have, so there will always be a compensating change in those coordinate values when applied to any object of the system. Geometrical vectors, for example, have components that will undergo the same rotation as the coordinates. More broadly, any tensor associated with the system has coordinate descriptions that adjust to compensate for changes to the coordinate system itself. Spinors do not appear at this level of the description of a physical system, when one is concerned only with the properties of a single isolated rotation of the coordinates. Rather, spinors appear when we imagine that instead of a single rotation, the coordinate system is rotated between some initial and final configuration. For any of the familiar and intuitive quantities associated with the system, the transformation law does not depend on the precise details of how the coordinates arrived at their final configuration.

Spinors, on the other hand, are constructed in such a way that makes them sensitive to how the gradual rotation of the coordinates arrived there: they exhibit path-dependence. It turns out that, for any final configuration of the coordinates, there are two inequivalent gradual rotations of the coordinate system that result in this same configuration; this ambiguity is called the homotopy class of the gradual rotation. The belt trick puzzle famously demonstrates two different rotations, one through an angle of 2π and the other through an angle of 4π, having the same final configurations but different classes. Spinors exhibit a sign-reversal that genuinely depends on this homotopy class; this distinguishes them from other tensors, none of which can feel the class. Spinors can be exhibited as concrete objects using a choice of Cartesian coordinates. In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to the thre

Physics

Physics is the natural science that studies matter, its motion, behavior through space and time, that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, its main goal is to understand how the universe behaves. Physics is one of the oldest academic disciplines and, through its inclusion of astronomy the oldest. Over much of the past two millennia, chemistry and certain branches of mathematics, were a part of natural philosophy, but during the scientific revolution in the 17th century these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, the boundaries of physics which are not rigidly defined. New ideas in physics explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics enable advances in new technologies.

For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have transformed modern-day society, such as television, domestic appliances, nuclear weapons. Astronomy is one of the oldest natural sciences. Early civilizations dating back to beyond 3000 BCE, such as the Sumerians, ancient Egyptians, the Indus Valley Civilization, had a predictive knowledge and a basic understanding of the motions of the Sun and stars; the stars and planets were worshipped, believed to represent gods. While the explanations for the observed positions of the stars were unscientific and lacking in evidence, these early observations laid the foundation for astronomy, as the stars were found to traverse great circles across the sky, which however did not explain the positions of the planets. According to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, all Western efforts in the exact sciences are descended from late Babylonian astronomy.

Egyptian astronomers left monuments showing knowledge of the constellations and the motions of the celestial bodies, while Greek poet Homer wrote of various celestial objects in his Iliad and Odyssey. Natural philosophy has its origins in Greece during the Archaic period, when pre-Socratic philosophers like Thales rejected non-naturalistic explanations for natural phenomena and proclaimed that every event had a natural cause, they proposed ideas verified by reason and observation, many of their hypotheses proved successful in experiment. The Western Roman Empire fell in the fifth century, this resulted in a decline in intellectual pursuits in the western part of Europe. By contrast, the Eastern Roman Empire resisted the attacks from the barbarians, continued to advance various fields of learning, including physics. In the sixth century Isidore of Miletus created an important compilation of Archimedes' works that are copied in the Archimedes Palimpsest. In sixth century Europe John Philoponus, a Byzantine scholar, questioned Aristotle's teaching of physics and noting its flaws.

He introduced the theory of impetus. Aristotle's physics was not scrutinized until John Philoponus appeared, unlike Aristotle who based his physics on verbal argument, Philoponus relied on observation. On Aristotle's physics John Philoponus wrote: “But this is erroneous, our view may be corroborated by actual observation more than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a small one, and so, if the difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other”John Philoponus' criticism of Aristotelian principles of physics served as an inspiration for Galileo Galilei ten centuries during the Scientific Revolution.

Galileo cited Philoponus in his works when arguing that Aristotelian physics was flawed. In the 1300s Jean Buridan, a teacher in the faculty of arts at the University of Paris, developed the concept of impetus, it was a step toward the modern ideas of momentum. Islamic scholarship inherited Aristotelian physics from the Greeks and during the Islamic Golden Age developed it further placing emphasis on observation and a priori reasoning, developing early forms of the scientific method; the most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn al-Haytham, in which he conclusively disproved the ancient Greek idea about vision, but came up with a new theory. In the book, he presented a study of the phenomenon of the camera obscura (his thousand-year-old

Fermion

In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. These particles obey the Pauli exclusion principle. Fermions include all quarks and leptons, as well as all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons. A fermion can be an elementary particle, such as the electron, or it can be a composite particle, such as the proton. According to the spin-statistics theorem in any reasonable relativistic quantum field theory, particles with integer spin are bosons, while particles with half-integer spin are fermions. In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is referred to as the spin statistics relation is in fact a spin statistics-quantum number relation; as a consequence of the Pauli exclusion principle, only one fermion can occupy a particular quantum state at any given time. If multiple fermions have the same spatial probability distribution at least one property of each fermion, such as its spin, must be different.

Fermions are associated with matter, whereas bosons are force carrier particles, although in the current state of particle physics the distinction between the two concepts is unclear. Weakly interacting fermions can display bosonic behavior under extreme conditions. At low temperature fermions show superfluidity for uncharged particles and superconductivity for charged particles. Composite fermions, such as protons and neutrons, are the key building blocks of everyday matter; the name fermion was coined by English theoretical physicist Paul Dirac from the surname of Italian physicist Enrico Fermi. The Standard Model recognizes two types of elementary fermions: leptons. In all, the model distinguishes 24 different fermions. There are six quarks, six leptons, along with the corresponding antiparticle of each of these. Mathematically, fermions come in three types: Weyl fermions, Dirac fermions, Majorana fermions. Most Standard Model fermions are believed to be Dirac fermions, although it is unknown at this time whether the neutrinos are Dirac or Majorana fermions.

Dirac fermions can be treated as a combination of two Weyl fermions. In July 2015, Weyl fermions have been experimentally realized in Weyl semimetals. Composite particles can be fermions depending on their constituents. More because of the relation between spin and statistics, a particle containing an odd number of fermions is itself a fermion, it will have half-integer spin. Examples include the following: A baryon, such as the proton or neutron, contains three fermionic quarks and thus it is a fermion; the nucleus of a carbon-13 atom is therefore a fermion. The atom helium-3 is made of two protons, one neutron, two electrons, therefore it is a fermion; the number of bosons within a composite particle made up of simple particles bound with a potential has no effect on whether it is a boson or a fermion. Fermionic or bosonic behavior of a composite particle is only seen at large distances. At proximity, where spatial structure begins to be important, a composite particle behaves according to its constituent makeup.

Fermions can exhibit bosonic behavior. This is the origin of superconductivity and the superfluidity of helium-3: in superconducting materials, electrons interact through the exchange of phonons, forming Cooper pairs, while in helium-3, Cooper pairs are formed via spin fluctuations; the quasiparticles of the fractional quantum Hall effect are known as composite fermions, which are electrons with an number of quantized vortices attached to them. In a quantum field theory, there can be field configurations of bosons which are topologically twisted; these are coherent states which behave like a particle, they can be fermionic if all the constituent particles are bosons. This was discovered by Tony Skyrme in the early 1960s, so fermions made of bosons are named skyrmions after him. Skyrme's original example involved fields which take values on a three-dimensional sphere, the original nonlinear sigma model which describes the large distance behavior of pions. In Skyrme's model, reproduced in the large N or string approximation to quantum chromodynamics, the proton and neutron are fermionic topological solitons of the pion field.

Whereas Skyrme's example involved pion physics, there is a much more familiar example in quantum electrodynamics with a magnetic monopole. A bosonic monopole with the smallest possible magnetic charge and a bosonic version of the electron will form a fermionic dyon; the analogy between the Skyrme field and the Higgs field of the electroweak sector has been used to postulate that all fermions are skyrmions. This could explain why all known fermions have baryon or lepton quantum numbers and provide a physical mechanism for the Pauli exclusion principle

Theoretical physics

Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize and predict natural phenomena. This is in contrast to experimental physics; the advancement of science depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations. For example, while developing special relativity, Albert Einstein was concerned with the Lorentz transformation which left Maxwell's equations invariant, but was uninterested in the Michelson–Morley experiment on Earth's drift through a luminiferous aether. Conversely, Einstein was awarded the Nobel Prize for explaining the photoelectric effect an experimental result lacking a theoretical formulation. A physical theory is a model of physical events, it is judged by the extent. The quality of a physical theory is judged on its ability to make new predictions which can be verified by new observations.

A physical theory differs from a mathematical theorem in that while both are based on some form of axioms, judgment of mathematical applicability is not based on agreement with any experimental results. A physical theory differs from a mathematical theory, in the sense that the word "theory" has a different meaning in mathematical terms. A physical theory involves one or more relationships between various measurable quantities. Archimedes realized that a ship floats by displacing its mass of water, Pythagoras understood the relation between the length of a vibrating string and the musical tone it produces. Other examples include entropy as a measure of the uncertainty regarding the positions and motions of unseen particles and the quantum mechanical idea that energy are not continuously variable. Theoretical physics consists of several different approaches. In this regard, theoretical particle physics forms a good example. For instance: "phenomenologists" might employ empirical formulas to agree with experimental results without deep physical understanding.

"Modelers" appear much like phenomenologists, but try to model speculative theories that have certain desirable features, or apply the techniques of mathematical modeling to physics problems. Some attempt to create approximate theories, called effective theories, because developed theories may be regarded as unsolvable or too complicated. Other theorists may try to unify, reinterpret or generalise extant theories, or create new ones altogether. Sometimes the vision provided by pure mathematical systems can provide clues to how a physical system might be modeled. Theoretical problems that need computational investigation are the concern of computational physics. Theoretical advances may consist in setting aside old, incorrect paradigms or may be an alternative model that provides answers that are more accurate or that can be more applied. In the latter case, a correspondence principle will be required to recover the known result. Sometimes though, advances may proceed along different paths. For example, an correct theory may need some conceptual or factual revisions.

However, an exception to all the above is the wave–particle duality, a theory combining aspects of different, opposing models via the Bohr complementarity principle. Physical theories become accepted if they are able to make correct predictions and no incorrect ones; the theory should have, at least as a secondary objective, a certain economy and elegance, a notion sometimes called "Occam's razor" after the 13th-century English philosopher William of Occam, in which the simpler of two theories that describe the same matter just as adequately is preferred. They are more to be accepted if they connect a wide range of phenomena. Testing the consequences of a theory is part of the scientific method. Physical theories can be grouped into three categories: mainstream theories, proposed theories and fringe theories. Theoretical physics began at least 2,300 years ago, under the Pre-socratic philosophy, continued by Plato and Aristotle, whose views held sway for a millennium. During the rise of medieval universities, the only acknowledged intellectual disciplines were the seven liberal arts of the Trivium like grammar and rhetoric and of the Quadrivium like arithmetic, geometry and astronomy.

During the Middle Ages and Renaissance, the concept of experimental science, the counterpoint to theory, began with scholars such as Ibn al-Haytham and Francis Bacon. As the Scientific Revolution gathered pace, the concepts of matter, space and causality began to acquire the form we know today, other sciences spun off from the rubric of natural philosophy, thus began the modern era of theory with the Copernican paradigm shift in astronomy, soon followed by Johannes Kepler's expressions for planetary orbits, which summarized the meticulous observations of Tycho Brahe.

Boson

In quantum mechanics, a boson is a particle that follows Bose–Einstein statistics. Bosons make up one of the two classes of the other being fermions; the name boson was coined by Paul Dirac to commemorate the contribution of Indian physicist and professor of physics at University of Calcutta and at University of Dhaka, Satyendra Nath Bose in developing, with Albert Einstein, Bose–Einstein statistics—which theorizes the characteristics of elementary particles. Examples of bosons include fundamental particles such as photons, W and Z bosons, the discovered Higgs boson, the hypothetical graviton of quantum gravity; some composite particles are bosons, such as mesons and stable nuclei of mass number such as deuterium, helium-4, or lead-208. An important characteristic of bosons is that their statistics do not restrict the number of them that occupy the same quantum state; this property is exemplified by helium-4. Unlike bosons, two identical fermions cannot occupy the same quantum space. Whereas the elementary particles that make up matter are fermions, the elementary bosons are force carriers that function as the'glue' holding matter together.

This property holds for all particles with integer spin as a consequence of the spin–statistics theorem. When a gas of Bose particles is cooled down to temperatures close to absolute zero the kinetic energy of the particles decreases to a negligible amount, they condense into the lowest energy level state; this state is called a Bose-Einstein condensate. It is believed. Bosons may be composite, like mesons. While most bosons are composite particles, in the Standard Model of Particle Physics there are five bosons which are elementary: the Standard Model requires one scalar boson H0 Higgs bosonthe four vector bosons that are the gauge bosons for the Standard Model:γ Photon g Gluons Z Neutral weak boson W± Charged weak bosons There may be a sixth tensor boson, the graviton, that would be the force-carrier for gravity, it remains a hypothetical elementary particle since all attempts so far to incorporate gravitation into the Standard Model have failed. If the graviton does exist, it must be a boson, could conceivably be a gauge boson.

Composite bosons, such as helium nuclei, are important in superfluidity and other applications of Bose–Einstein condensates. Bosons differ from fermions. Two or more identical fermions cannot occupy the same quantum state. Since bosons with the same energy can occupy the same place in space, bosons are force carrier particles, including composite bosons such as mesons. Fermions are associated with matter. Bosons are particles which obey Bose–Einstein statistics: When one swaps two bosons, the wavefunction of the system is unchanged. Fermions, on the other hand, obey Fermi–Dirac statistics and the Pauli exclusion principle: Two fermions cannot occupy the same quantum state, accounting for the "rigidity" or "stiffness" of matter which includes fermions, thus fermions are sometimes said to be the constituents of matter, while bosons are said to be the particles that transmit interactions, or the constituents of radiation. The quantum fields of bosons are bosonic fields; the properties of lasers and masers, superfluid helium-4 and Bose–Einstein condensates are all consequences of statistics of bosons.

Another result is that the spectrum of a photon gas in thermal equilibrium is a Planck spectrum, one example of, black-body radiation. Interactions between elementary particles are called fundamental interactions; the fundamental interactions of virtual bosons with real particles result. All known elementary and composite particles are bosons or fermions, depending on their spin: Particles with half-integer spin are fermions. In the framework of nonrelativistic quantum mechanics, this is a purely empirical observation. In relativistic quantum field theory, the spin–statistics theorem shows that half-integer spin particles cannot be bosons and integer spin particles cannot be fermions. In large systems, the difference between bosonic and fermionic statistics is only apparent at large densities — when their wave functions overlap. At low densities, both types of statistics are well approximated by Maxwell–Boltzmann statistics, described by classical mechanics. All observed elementary particles bosons.

The observed elementary bosons are all gauge bosons: photons, W and Z bosons, except the Higgs boson, a scalar boson. Photons are the force carriers of the electromagnetic field. W and Z bosons are the force carriers. Gluons are the fundamental force carriers underlying the strong force. Higgs bosons give Z bosons mass via the Higgs mechanism, their existence was confirmed by CERN on 14 March 2013. Many approaches to quantum gravity postulate a force carrier for gravity, the graviton, a boson of spin plus or minus two. Composite particles can be b

Spin (physics)

In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles, atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum; the orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution and appears when there is periodic structure to its wavefunction as the angle varies. The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. In some ways, spin is like a vector quantity. All elementary particles of a given kind have the same magnitude of spin angular momentum, indicated by assigning the particle a spin quantum number; the SI unit of spin is the or, just as with classical angular momentum. In practice, spin is given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant ħ, which has the same units of angular momentum, although this is not the full computation of this value.

The "spin quantum number" is called "spin", leaving its meaning as the unitless "spin quantum number" to be inferred from context. When combined with the spin-statistics theorem, the spin of electrons results in the Pauli exclusion principle, which in turn underlies the periodic table of chemical elements. Wolfgang Pauli in 1924 was the first to propose a doubling of electron states due to a two-valued non-classical "hidden rotation". In 1925, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested the simple physical interpretation of a particle spinning around its own axis, in the spirit of the old quantum theory of Bohr and Sommerfeld. Ralph Kronig anticipated the Uhlenbeck-Goudsmit model in discussion with Hendrik Kramers several months earlier in Copenhagen, but did not publish; the mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it; as the name suggests, spin was conceived as the rotation of a particle around some axis.

This picture is correct so far as spin obeys the same mathematical laws as quantized angular momenta do. On the other hand, spin has some peculiar properties that distinguish it from orbital angular momenta: Spin quantum numbers may take half-integer values. Although the direction of its spin can be changed, an elementary particle cannot be made to spin faster or slower; the spin of a charged particle is associated with a magnetic dipole moment with a g-factor differing from 1. This could only occur classically if the internal charge of the particle were distributed differently from its mass; the conventional definition of the spin quantum number, s, is s = n/2, where n can be any non-negative integer. Hence the allowed values of s are 1/2, 1, 3/2, 2, etc.. The value of s for an elementary particle depends only on the type of particle, cannot be altered in any known way; the spin angular momentum, S, of any physical system is quantized. The allowed values of S are S = ℏ s = h 4 π n, where h is the Planck constant and ℏ = h/2π is the reduced Planck constant.

In contrast, orbital angular momentum can only take on integer values of s. Those particles with half-integer spins, such as 1/2, 3/2, 5/2, are known as fermions, while those particles with integer spins, such as 0, 1, 2, are known as bosons; the two families of particles obey different rules and broadly have different roles in the world around us. A key distinction between the two families is. In contrast, bosons obey the rules of Bose–Einstein statistics and have no such restriction, so they may "bunch together" if in identical states. Composite particles can have spins different from their component particles. For example, a helium atom in the ground state has spin 0 and behaves like a boson though the quarks and electrons which make it up are all fermions; this has some profound consequences: Quarks and leptons, which make up what is classically known as matter, are all fermions with spin 1/2. The common idea that "matter takes up space" comes from the Pauli exclusion principle acting on these particles to prevent the fermions that make up matter from being in the same quantum state.

Further compaction would require electrons to occupy the same energy states, therefore a kind of pressure acts to resist the fermions being overly close. Elementary fermions with other spins are not known to exist. Elementary particles which are thought of as carrying forces are all bosons with spin 1, they include the photon which carries the electromagnetic force, the gluon, the W and Z bosons. The ability of bosons to occupy the same quantu

Supersymmetry

In particle physics, supersymmetry is a principle that proposes a relationship between two basic classes of elementary particles: bosons, which have an integer-valued spin, fermions, which have a half-integer spin. A type of spacetime symmetry, supersymmetry is a possible candidate for undiscovered particle physics, seen as an elegant solution to many current problems in particle physics if confirmed correct, which could resolve various areas where current theories are believed to be incomplete. A supersymmetrical extension to the Standard Model would resolve major hierarchy problems within gauge theory, by guaranteeing that quadratic divergences of all orders will cancel out in perturbation theory. In supersymmetry, each particle from one group would have an associated particle in the other, known as its superpartner, the spin of which differs by a half-integer; these superpartners would be undiscovered particles. For example, there would be a particle called a "selectron", a bosonic partner of the electron.

In the simplest supersymmetry theories, with "unbroken" supersymmetry, each pair of superpartners would share the same mass and internal quantum numbers besides spin. Since we expect to find these "superpartners" using present-day equipment, if supersymmetry exists it consists of a spontaneously broken symmetry allowing superpartners to differ in mass. Spontaneously-broken supersymmetry could solve many mysterious problems in particle physics including the hierarchy problem. There is no evidence at this time to show whether or not supersymmetry is correct, or what other extensions to current models might be more accurate. In part this is because it is only since around 2010 that particle accelerators designed to study physics beyond the Standard Model have become operational, because it is not yet known where to look nor the energies required for a successful search; the main reasons for supersymmetry being supported by physicists is that the current theories are known to be incomplete and their limitations are well established, supersymmetry would be an attractive solution to some of the major concerns.

Direct confirmation would entail production of superpartners in collider experiments, such as the Large Hadron Collider. The first runs of the LHC found no previously-unknown particles other than the Higgs boson, suspected to exist as part of the Standard Model, therefore no evidence for supersymmetry. Indirect methods include the search for a permanent electric dipole moment in the known Standard Model particles, which can arise when the Standard Model particle interacts with the supersymmetric particles; the current best constraint on the electron electric dipole moment put it to be smaller than 10−28 e·cm, equivalent to a sensitivity to new physics at the TeV scale and matching that of the current best particle colliders. A permanent EDM in any fundamental particle points towards time-reversal violating physics, therefore CP-symmetry violation via the CPT theorem; such EDM experiments are much more scalable than conventional particle accelerators and offer a practical alternative to detecting physics beyond the standard model as accelerator experiments become costly and complicated to maintain.

These findings disappointed many physicists, who believed that supersymmetry were by far the most promising theories for "new" physics, had hoped for signs of unexpected results from these runs. Former enthusiastic supporter Mikhail Shifman went as far as urging the theoretical community to search for new ideas and accept that supersymmetry was a failed theory; however it has been argued that this "naturalness" crisis was premature, because various calculations were too optimistic about the limits of masses which would allow a supersymmetry based solution. There are numerous phenomenological motivations for supersymmetry close to the electroweak scale, as well as technical motivations for supersymmetry at any scale. Supersymmetry close to the electroweak scale ameliorates the hierarchy problem that afflicts the Standard Model. In the Standard Model, the electroweak scale receives enormous Planck-scale quantum corrections; the observed hierarchy between the electroweak scale and the Planck scale must be achieved with extraordinary fine tuning.

In a supersymmetric theory, on the other hand, Planck-scale quantum corrections cancel between partners and superpartners. The hierarchy between the electroweak scale and the Planck scale is achieved in a natural manner, without miraculous fine-tuning; the idea that the gauge symmetry groups unify at high-energy is called Grand unification theory. In the Standard Model, the weak and electromagnetic couplings fail to unify at high energy. In a supersymmetry theory, the running of the gauge couplings are modified, precise high-energy unification of the gauge couplings is achieved; the modified running provides a natural mechanism for radiative electroweak symmetry breaking. TeV-scale supersymmetry provides a candidate dark matter particle at a mass scale consistent with thermal relic abundance calculations. Supersymmetry is motivated by solutions to several theoretical problems, for providing many desirable mathematical properties, for ensuring sensible behavior at high energies. Supersymmetric quantum field theory is much easier to analyze, as many more problems become mathematically tractable.

When supersymmetry is imposed as a local symmetry, Einstein's theory of general relativity is included automatically, the result is said to be a theory of supergravity. It is a necessary feature of