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A spinor visualized as a vector pointing along the Möbius band, exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°. [a] In geometry and physics, spinors (pronounced "spinner" IPA / s p ɪ n ər /) are elements of a complex vector space that can be associated with Euclidean ...
The conventional definition of the spin quantum number is s = n / 2 , where n can be any non-negative integer. Hence the allowed values of s are 0, 1 / 2 , 1, 3 / 2 , 2, etc. The value of s for an elementary particle depends only on the type of particle and cannot be altered in any known way (in contrast to the spin ...
Often, the first example of spinors that a student of physics encounters are the 2×1 spinors used in Pauli's theory of electron spin. The Pauli matrices are a vector of three 2×2 matrices that are used as spin operators. Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain ...
The definition of a "vector" in physics (including both polar vectors and pseudovectors) is more specific than the mathematical definition of "vector" (namely, any element of an abstract vector space). Under the physics definition, a "vector" is required to have components that "transform" in a certain way under a proper rotation: In particular ...
A free vector is a vector quantity having an undefined support or region of application; it can be freely translated with no consequences; a displacement vector is a prototypical example of free vector. Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of metric.
In particle physics the spin–statistics theorem implies that the wavefunction of an uncharged fermion is a section of the associated vector bundle to the spin lift of an SO(N) bundle E. Therefore, the choice of spin structure is part of the data needed to define the wavefunction, and one often needs to sum over these choices in the partition ...
This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold. In general, there is no restriction: the spin connection may also contain torsion.
A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector. The prototypical example of a pseudoscalar is the scalar triple product, which can be written as the scalar product between one of the vectors in the triple product and the cross product between the two other vectors, where the latter is a ...