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Wess–Zumino Model
In theoretical physics, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory with linearly realised supersymmetry. In 1974, Julius Wess and Bruno Zumino studied, using modern terminology, dynamics of a single chiral superfield (composed of a complex scalar and a spinor fermion) whose cubic superpotential leads to a renormalizable theory. The treatment in this article largely follows that of Figueroa-O'Farrill's lectures on supersymmetry, and to some extent of Tong. The model is an important model in supersymmetric quantum field theory. It is arguably the simplest supersymmetric field theory in four dimensions, and is ungauged. The Wess–Zumino action Preliminary treatment Spacetime and matter content In a preliminary treatment, the theory is defined on flat spacetime (Minkowski space). For this article, the metric has ''mostly plus'' signature. The matter content is a real scalar field S, a real pseudosc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally 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.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numbers rather than real numbers. The ordinary space dimensions correspond to bosonic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom. The word "superspace" was first used by John Wheeler in an unrelated sense to describe the configuration space of general relativity; for example, this usage may be seen in his 1973 textbook ''Gravitation''. Informal discussion There are several similar, but not equivalent, definitions of superspace that have been used, and continue to be used in the mathematical and physics literature. One such usage is as a synonym for super Minkowski space. In this case, one takes ordinary Minkowski space, and extends it with anti-commuting fermionic degrees of freedom, taken to be anti- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
R-symmetry
In theoretical physics, the R-symmetry is the symmetry transforming different supercharges in a theory with supersymmetry into each other. In the simplest case of the ''N''=1 supersymmetry, such an R-symmetry is isomorphic to a global U(1) group or its discrete subgroup (for the Z2 subgroup it is called R-parity). For extended supersymmetry In theoretical physics, extended supersymmetry is supersymmetry whose infinitesimal generators Q_i^\alpha carry not only a spinor index \alpha, but also an additional index i=1,2 \dots \mathcal where \mathcal is integer (such as 2 or 4). Extended ..., the R-symmetry group becomes a global U(N) non-abelian group. In a model that is classically invariant under both ''N''=1 supersymmetry and conformal transformations, the closure of the superconformal algebra (at least on-shell) needs the introduction of a further bosonic generator that is associated to the R-symmetry. References * Supersymmetry {{quantum-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Conformal Transformation
In projective geometry, a special conformal transformation is a linear fractional transformation that is ''not'' an affine transformation. Thus the generation of a special conformal transformation involves use of multiplicative inversion, which is the generator of linear fractional transformations that is not affine. In mathematical physics, certain conformal maps known as spherical wave transformations are special conformal transformations. Vector presentation A special conformal transformation can be written : x'^\mu = \frac = \frac(x^\mu-b^\mu x^2)\,. It is a composition of an inversion (''x''''μ'' → ''x''''μ''/x2 = ''y''''μ''), a translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ... (''y''''μ'' → ''y''''μ'' − ''b''''μ'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superconformal Algebra
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group (in two Euclidean dimensions, the Lie superalgebra does not generate any Lie supergroup). Superconformal algebra in dimension greater than 2 The conformal group of the (p+q)-dimensional space \mathbb^ is SO(p+1,q+1) and its Lie algebra is \mathfrak(p+1,q+1). The superconformal algebra is a Lie superalgebra containing the bosonic factor \mathfrak(p+1,q+1) and whose odd generators transform in spinor representations of \mathfrak(p+1,q+1). Given Kač's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of p and q. A (possibly incomplete) list is * \mathfrak^*(2N, 2,2) in 3+0D thanks to \mathfrak(2,2)\simeq\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is called parity transformation. Invariance under parity transformation by a Dirac fermion is called chiral symmetry. Chirality and helicity The helicity of a particle is positive (“right-handed”) if the direction of its spin is the same as the direction of its motion. It is negative (“left-handed”) if the directions of spin and motion are opposite. So a standard clock, with its spin vector defined by the rotation of its hands, has left-handed helicity if tossed with its face directed forwards. Mathematically, ''helicity'' is the sign of the projection of the spin vector onto the momentum vector: “left” is negative, “right” is positive. The chira ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berezin Integral
In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variables (elements of the exterior algebra). It is not an integral in the Lebesgue sense; the word "integral" is used because the Berezin integral has properties analogous to the Lebesgue integral and because it extends the path integral in physics, where it is used as a sum over histories for fermions. Definition Let \Lambda^n be the exterior algebra of polynomials in anticommuting elements \theta_,\dots,\theta_ over the field of complex numbers. (The ordering of the generators \theta_1,\dots,\theta_n is fixed and defines the orientation of the exterior algebra.) One variable The ''Berezin integral'' over the sole Grassmann variable \theta = \theta_1 is defined to be a linear functional : \int f(\theta)+bg(\theta)\, d\theta = a\int f(\theta) \, d\theta + b\int g(\theta) \, d\theta, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Spinor
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions. None of the elementary particles in the Standard Model are Weyl fermions. Previous to the confirmation of the neutrino oscillations, it was considered possible that the neutrino might be a Weyl fermion (it is now expected to be either a Dirac or a Majorana fermion). In condensed matter physics, some materials can display quasiparticles that behave as Weyl fermions, leading to the notion of Weyl semimetals. Mathematically, any Dirac fermion can be decomposed as two Weyl fermions of opposite chirality coupled by the mass term. History The Dirac equation, was published in 1928 by Paul Dirac, first describing spin-½ particles in the framework ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Statistics Theorem
Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally biased portrayal of something Spin, spinning or spinnin may also refer to: Physics and mathematics * Spin, the rotation of an object around a central axis * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin group, a particular double cover of the special orthogonal group SO(''n'') * Spin tensor, a tensor quantity for describing spinning motion in special relativity and general relativity * Spin (aerodynamics), autorotation of an aerodynamically stalled aeroplane * SPIN bibliographic database, an indexing and abstracting service focusing on physics research Textile arts * Spinning (polymers), a process for creating polymer fibres * Spinning (textiles), the creation of yarn or thread by twisting fibe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Number
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed. Informal discussion Grassmann numbers are generated by anti-commuting elements or objects. The idea of anti-commuting objects arises in multiple areas of mathematics: they are typically seen in differential geometry, where the differential forms are anti-commuting. Differential forms are normally defined in terms of derivatives on a manifold; however, one can contemplate the situation where one "forgets" or "ignores" the existence of any underlying manifold, and "forgets" or "ignores" that the forms were defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super Minkowski Space
In mathematics and physics, super Minkowski space or Minkowski superspace is a supersymmetric extension of Minkowski space, sometimes used as the base manifold (or rather, supermanifold) for superfields. It is acted on by the super Poincaré algebra. Construction Abstract construction Abstractly, super Minkowski space is the space of (right) cosets within the Super Poincaré group of Lorentz group, that is, :\text \cong \frac. This is analogous to the way ordinary Minkowski spacetime can be identified with the (right) cosets within the Poincaré group of the Lorentz group, that is, :\text \cong \frac. The coset space is naturally affine, and the nilpotent, anti-commuting behavior of the fermionic directions arises naturally from the Clifford algebra associated with the Lorentz group. Direct sum construction For this section, the dimension of the Minkowski space under consideration is d = 4. Super Minkowski space can be concretely realized as the direct sum of Minkowsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supercharge
In theoretical physics, a supercharge is a generator of supersymmetry transformations. It is an example of the general notion of a charge in physics. Supercharge, denoted by the symbol Q, is an operator which transforms bosons into fermions, and vice versa. Since the supercharge operator changes a particle with spin one-half to a particle with spin one or zero, the supercharge itself is a spinor that carries one half unit of spin. Depending on the context, supercharges may also be called ''Grassmann variables'' or ''Grassmann directions''; they are generators of the exterior algebra of anti-commuting numbers, the Grassmann numbers. All these various usages are essentially synonymous; they refer to the \mathbb_2 grading between bosons and fermions, or equivalently, the grading between '' c-numbers'' and ''a-numbers''. Calling it a charge emphasizes the notion of a symmetry at work. Commutation Supercharge is described by the Super-Poincaré algebra. Supercharge commutes with th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
