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supersymmetry
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(Definition)
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Definition 0.1 Supersymmetry or Poincaré, (extended) quantum symmetry is usually defined as an extension of ordinary spacetime symmetries obtained by adjoining $N$ spinorial generators $Q$ whose anticommutator yields a translation generator: $\left\{Q ,Q \right\} = \left\{P\right\}$ .
As further explained in ref. [1]:
``This (super) symmetry...(of the superspace)... can be realized on ordinary fields (that are defined as certain functions of physical spacetime(s)) by transformations that mix bosons and fermions. Such realizations suffice to study supersymmetry (one can write invariant actions, etc.)
but are as cumbersome and inconvenient as doing vector calculus component by component. A compact alternative to this `component field' approach is given by the superspace-superfield approach", which is defined next.
Definition 0.2 Quantum superspace, or superspacetimes, can be defined as an extension(s) of ordinary spacetime(s) to include additional anticommuting coordinates, for example, in the form of $N$ two-component Weyl spinors $\theta$ .
Definition 0.3 (Quantum) superfields $\Psi(x , \theta)$ are functions defined over such superspaces, or superspacetimes. Taylor series expansions of the superfield functions can be then performed with respect to the anticommuting coordinates $\theta$ ; this Taylor series has only a finite number of terms and the
series expansion coefficients obtained in this manner are the ordinary `component fields' specified above.
Remarks: Supersymmetry is expected to be manifested, or observable, in such superspaces, that is, the supersymmetry algebras are represented by translations and rotations involving both the spacetime and the anticommuting coordinates. Then, the transformations of the `component fields' can be computed from the Taylor expansion of the translated and rotated superfields. Especially important are those transformations that mix boson and fermion symmetries; further details are found in ref. [2].
- 1
- J.S. Gates, Jr, et al. ``Superspace''., arxiv-hep-th/0108200 preprint (1983).
- 2
- ``Preprint of 1,001 Lessons in Supersymmetry.'' on line PDF.
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"supersymmetry" is owned by bci1. [ full author list (2) ]
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Cross-references: Taylor expansion, rotations, algebras, coefficients, series, terms, number, finite, Taylor series, superfields, spinors, coordinates, compact, component, Calculus, vector, actions, invariant, transformations, functions, fields, superspace, translation, generators, extension, symmetry, Poincaré
There are 15 references to this entry.
This is version 19 of supersymmetry, born on 2008-07-30, modified 2008-10-20.
Object id is 10893, canonical name is SupersymmetryOrSupersymmetries.
Accessed 1876 times total.
Classification:
| AMS MSC: | 81R15 (Quantum theory :: Groups and algebras in quantum theory :: Operator algebra methods) | | | 81R50 (Quantum theory :: Groups and algebras in quantum theory :: Quantum groups and related algebraic methods) | | | 81Q60 (Quantum theory :: General mathematical topics and methods in quantum theory :: Supersymmetric quantum mechanics) | | | 55-02 (Algebraic topology :: Research exposition ) | | | 55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory) |
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Pending Errata and Addenda
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