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...[[linear combination]] of ''products of the bilinears'' of the individual spinors. It is named after Swiss physicist [[Markus Fierz]]. The Fierz identities a The signs in the table correspond to the case of ''commuting spinors'', otherwise, as is the case of fermions in physics, ''all coefficients ch ...

...vated by the need to form well-behaved, measurable quantities out of Dirac spinors, replacing the usual role of the [[Hermitian adjoint]]. == Spinors under Lorentz transformations == ...

==Spinors== Put it more explicitly, the Dirac spinors in the helicity basis for a fermion is ...

{{Short description|Matrices used for Lorentz group spinors}} ...iplication]] of covectors on left handed [[spinors]] giving a right-handed spinors or vice versa, i.e. they are off diagonal blocks of [[gamma matrices]]. The ...

*''Twistor and Killing spinors on [[Riemannian manifold]]s'', with Thomas Friedrich, Ralf Grunewald, and I <ref name=twist>Review of ''Twistor and Killing spinors on Riemannian manifolds'': N. J. Hitchin, {{MR|1084369}}</ref> ...

...equation]] without complex numbers, and its elements are called Majorana [[spinors]]. ...ation|first=Pierre|last=Deligne|authorlink=Pierre Deligne|chapter=Notes on spinors |title= Quantum Fields and Strings: A Course for Mathematicians |editor-fir ...

...first2=J. | title=Spontaneously broken supergauge symmetries and goldstone spinors | journal=Physics Letters B | publisher=Elsevier BV | volume=51 | issue=5 | ...

...a particle. Strictly speaking, they are not vectors at all, but in fact [[spinors]]. For a single spin 1/2 particle, they can be defined as the [[eigenvecto ...ritten as a [[linear combination]] of the [[basis (linear algebra)|basis]] spinors. ...

...sions]].<ref>{{Citation | last1=Cartan | first1=Élie | title=The theory of spinors | origyear=1938 | url=https://books.google.com/books?isbn=0486640701 |page= ...

...can all be expressed directly in terms of the [[Ricci spinors]] and [[Weyl spinors]], using [[Newman–Penrose formalism]]; see the link below. ...

...ns to the [[Dirac equation]] for spinor fields are often called ''harmonic spinors''.<ref>{{SpringerEOM|id=Spinor_structure&oldid=33893 |title=Spinor structur ...l=https://books.google.com/books?id=FzcbAQAAIAAJ |title=An Introduction to Spinors and Geometry with Applications in Physics |last2=Tucker |first2=Robin W. |d ...

{{Short description|Mathematical notation for tensors and spinors}} ...="Penrose1984">{{Cite book |last=Roger Penrose and Wolfgang Rindler |title=Spinors and Space-Time, Volume 1: Two-Spinor Calculus and Relativistic Fields |publ ...

...geometry]]. For example, effectively all [[Riemannian manifold]]s can have spinors and [[spin connection]]s built upon them, via the [[Clifford algebra]].<ref ...ion. The chiral spinors may be written as linear combinations of the Dirac spinors presented below; thus, nothing is lost or gained, other than a change in pe ...

...an der Waerden notation]], distinguishes left-handed and right-handed Weyl spinors from each other. Generators of other spin, such spin-3/2 or higher, are dis ...tion operator]] rather than Pauli matrices used for the two-component Weyl spinors.<ref>{{cite book|last=Năstase|first=H.|author-link=|date=2015|title=Introd ...

...presentation|representation]] of the action of the [[Clifford algebra]] on spinors. ...and its complex conjugate contains a top-dimensional component. Such pure spinors determine generalized almost complex structures. ...

...same decomposition holds for the compact form as well: the product of two spinors in [[SU(2)|su(2)]] being a vector in the [[rotation group]] [[O(3)]] and a ...

==Spinors== ...

...one always counts the number of ''real'' irreducible representations. The spinors with spins less than 3/2 that exist in each number of dimensions will be cl == A classification of spinors == ...

...tilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensions, notably in string theory and supergra {{Cite journal | last1=Petitjean | first1=Michel | title=Chirality of Dirac spinors revisited | journal=Symmetry | year=2020 | volume=12 | issue=4 | pages=616 ...

* {{cite journal | last=Bär | first=Christian | title=Real Killing spinors and holonomy | journal=[[Communications in Mathematical Physics]] | publish ...