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Wick's Theorem
Wick's theorem
Wick's theorem
is a method of reducing high-order derivatives to a combinatorics problem.[1] It is named after Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to sums of products of pairs of these operators
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Quantum Chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a non-abelian gauge theory, with symmetry group SU(3). The QCD analog of electric charge is a property called color. Gluons are the force carrier of the theory, like photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics. A large body of experimental evidence for QCD has been gathered over the years. QCD exhibits two main properties: Color
Color
confinement, plasmas. This is a consequence of the constant force between two color charges as they are separated: In order to increase the separation between two quarks within a hadron, ever-increasing amounts of energy are required
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Parity (physics)
In quantum mechanics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it is also often described by the simultaneous flip in the sign of all three spatial coordinates (a point reflection): P : ( x y z ) ↦ ( − x − y − z ) . displaystyle mathbf P : begin pmatrix x\y\zend pmatrix mapsto begin pmatrix -x\-y\-zend pmatrix . It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. A parity transformation on something achiral, on the other hand, can be viewed as an identity transformation
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Regularization (physics)
In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in order to make them finite by the introduction of a suitable parameter called regulator. The regulator, also known as a "cutoff", models our lack of knowledge about physics at unobserved scales (e.g. scales of small size or large energy levels). It compensates for (and requires) the possibility that "new physics" may be discovered at those scales which the present theory is unable to model, while enabling the current theory to give accurate predictions as an "effective theory" within its intended scale of use. It is distinct from renormalization, another technique to control infinities without assuming new physics, by adjusting for self-interaction feedback. Regularization was for many decades controversial even amongst its inventors, as it combines physical and epistemological claims into the same equations
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Klein–Gordon Equation
The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second order in space and time and manifestly Lorentz covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation.[1] Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pi mesons are unstable and also experience the strong interaction (with unknown interaction term in the Hamiltonian[2]), the practical utility is limited. The equation can be put into the form of a Schrödinger equation
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Topological Charge
In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are topological invariants associated with topological defects or soliton-type solutions of some set of differential equations modeling a physical system, as the solitons themselves owe their stability to topological considerations. The specific "topological considerations" are usually due to the appearance of the fundamental group or a higher-dimensional homotopy group in the description of the problem, quite often because the boundary, on which the boundary conditions are specified, has a non-trivial homotopy group that is preserved by the differential equations
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Noether Charge
Noether's (first)[1] theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether
Emmy Noether
in 1915 and published in 1918.[2], although a special case was proven by E. Cosserat & F. Cosserat in 1909.[3] The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action. Noether's theorem
Noether's theorem
is used in theoretical physics and the calculus of variations. A generalization of the formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g
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Explicit Symmetry Breaking
In theoretical physics, explicit symmetry breaking is the breaking of a symmetry of a theory by terms in its defining equations of motion (most typically, to the Lagrangian or the Hamiltonian) that do not respect the symmetry. Usually this term is used in situations where these symmetry-breaking terms are small, so that the symmetry is approximately respected by the theory. An example is the spectral line splitting in the Zeeman effect, due to a magnetic interaction perturbation in the Hamiltonian of the atoms involved. Explicit symmetry breaking differs from spontaneous symmetry breaking. In the latter, the defining equations respect the symmetry but the ground state (vacuum) of the theory breaks it.[1] Explicit symmetry breaking is also associated with electromagnetic radiation
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Gauge Symmetry (mathematics)
In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory. A gauge symmetry of a Lagrangian L displaystyle L is defined as a differential operator on some vector bundle E displaystyle E taking its values in the linear space of (variational or exact) symmetries of L displaystyle L
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Poincaré Symmetry
The Poincaré group, named after Henri Poincaré
Henri Poincaré
(1906),[1] was first defined by Minkowski (1908) as the group of Minkowski spacetime isometries.[2][3] It is a ten-generator non-abelian Lie group
Lie group
of fundamental importance in physics.Contents1 Overview 2 Poincaré symmetry 3 Poincaré group 4 Poincaré algebra 5 Other dimensions 6 Super-Poincaré algebra 7 See also 8 Notes 9 ReferencesOverview[edit] A Minkowski spacetime isometry has the property that the interval between events is left invariant. For example, if everything was postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stop-watch you carried with you would be the same. Or if everything was shifted five kilometres to the west, or turned 60 degrees to the right, you would also see no change in the interval
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Lorentz Symmetry
In relativistic physics, Lorentz symmetry, named for Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers that are moving with respect to one another within an inertial frame. It has also been described as "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".[1] Lorentz covariance, a related concept, is a property of the underlying spacetime manifold. Lorentz covariance
Lorentz covariance
has two distinct, but closely related meanings:A physical quantity is said to be Lorentz covariant
Lorentz covariant
if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, four-tensors, and spinors
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Rotation Symmetry
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks the same.Contents1 Formal treatment1.1 Discrete rotational symmetry 1.2 Examples 1.3 Multiple symmetry axes through the same point 1.4 Rotational symmetry
Rotational symmetry
with respect to any angle 1.5 Rotational symmetry
Rotational symmetry
with translational symmetry2 See also 3 References 4 External linksFormal treatment[edit] See also: Rotational invariance Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation
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Time Translation Symmetry
Time
Time
translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time
Time
translation symmetry is the hypothesis that the laws of physics are unchanged, (i.e. invariant) under such a transformation. Time
Time
translation symmetry is a rigorous way to formulate the idea that the laws of physics are the same throughout history. Time
Time
translation symmetry is closely connected via the Noether theorem, to conservation of energy.[1] In mathematics, the set of all time translations on a given system form a Lie group. There are many symmetries in nature besides time translation, such as spacial translation or rotational symmetries
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T-symmetry
T-symmetry
T-symmetry
or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal: T : t ↦ − t . displaystyle T:tmapsto -t. Although in restricted contexts one may find this symmetry, the observable universe itself does not show symmetry under time reversal, primarily due to the second law of thermodynamics
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Space Translation Symmetry
In geometry, a translation "slides" a thing by a: Ta(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 T δ displaystyle T_ delta if the result after applying A doesn't change if the argument function is translated
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Charge Conjugation
Charge conjugation is a transformation that switches all particles with their corresponding antiparticles, and thus changes the sign of all charges: not only electric charge but also the charges relevant to other forces. In physics, C-symmetry means the symmetry of physical laws under a charge-conjugation transformation. Electromagnetism, gravity and the strong interaction all obey C-symmetry, but weak interactions violate C-symmetry.Contents1 Charge reversal in electroweak theory 2 Combination of charge and parity reversal 3 Charge definition 4 See also 5 ReferencesCharge reversal in electroweak theory[edit] The laws of electromagnetism (both classical and quantum) are invariant under this transformation: if each charge q were to be replaced with a charge −q, and thus the directions of the electric and magnetic fields were reversed, the dynamics would preserve the same form
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