Editing Tensor (section)

=== Spinors ===
{{Main|Spinor}}
When changing from one [[orthonormal basis]] (called a ''frame'') to another by a rotation, the components of a tensor transform by that same rotation.  This transformation does not depend on the path taken through the space of frames.  However, the space of frames is not [[simply connected]] (see [[orientation entanglement]] and [[plate trick]]): there are continuous paths in the space of frames with the same beginning and ending configurations that are not deformable one into the other.  It is possible to attach an additional discrete invariant to each frame that incorporates this path dependence, and which turns out (locally) to have values of ±1.<ref>{{cite book|title=The road to reality: a complete guide to the laws of our universe|url={{google books |plainurl=y |id=VWTNCwAAQBAJ|page=203}}|first=Roger|last=Penrose|author-link=Roger Penrose|publisher=Knopf|year=2005|pages=203–206}}</ref>  A [[spinor]] is an object that transforms like a tensor under rotations in the frame, apart from a possible sign that is determined by the value of this discrete invariant.<ref>{{cite book|first=E. |last=Meinrenken|title=Clifford Algebras and Lie Theory|chapter=The spin representation|series=Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics|volume=58|pages=49–85|doi=10.1007/978-3-642-36216-3_3|publisher=Springer |year=2013|isbn=978-3-642-36215-6}}</ref><ref>{{citation|first=S. H.|last=Dong |title=Wave Equations in Higher Dimensions|chapter=2. Special Orthogonal Group SO(''N'')|publisher=Springer|year=2011|pages=13–38}}</ref>

Spinors are elements of the [[spin representation]] of the rotation group, while tensors are elements of its [[tensor representation]]s. Other [[classical group]]s have tensor representations, and so also tensors that are compatible with the group, but all non-compact classical groups have infinite-dimensional unitary representations as well.