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The Infeld–Van der Waerden symbols, sometimes called simply Van der Waerden symbols, are an invariant symbol associated to the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
used in quantum field theory. They are named after
Leopold Infeld Leopold Infeld (20 August 1898 – 15 January 1968) was a Polish physicist who worked mainly in Poland and Canada (1938–1950). He was a Rockefeller fellow at Cambridge University (1933–1934) and a member of the Polish Academy of Sciences. Ea ...
and
Bartel Leendert van der Waerden Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics. Biography Education and early career Van der Waerden learned advanced mathematics at the University of Amster ...
. The Infeld–Van der Waerden symbols are index notation for Clifford multiplication of covectors on left handed
spinors In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
giving a right-handed spinors or vice versa, i.e. they are off diagonal blocks of
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
. The symbols are typically denoted in Van der Waerden notation as \sigma^m_\quad\text\quad\bar^. and so have one Lorentz index (m), one left-handed (undotted Greek), and one right-handed (dotted Greek) Weyl
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
index. They satisfy representation theory of the Lorentz group The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representati ...
or more properly its Lie algebra. Labeling irreducible representations by (j,\bar), the spinor and its complex conjugate representations are the left and right
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...
s :(\tfrac,0) and (0,\tfrac), while the tangent vectors live in the vector representation :(\tfrac,\tfrac). The tensor product of one left and right fundamental representation is the vector representation,(\tfrac,0)\otimes(0,\tfrac)=(\tfrac,\tfrac). A dual statement is that the tensor product of the vector, left, and right fundamental representations contains the
trivial representation In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ...
which is in fact generated by the construction of the Lie algebra representations through the Clifford algebra (see below) (\tfrac,0)\otimes(0,\tfrac)\otimes(\tfrac,\tfrac)=(0,0)\oplus\cdots .


Infeld–Van der Waerden symbols and representations of the Clifford algebra

Consider the space of positive Weyl spinors S of a Lorentzian vector space (T,g) with dual (T^\vee, g^\vee). Then the negative Weyl spinors can be identified with the vector space \bar S^\vee of complex conjugate dual spinors. The Weyl spinors implement "two halves of a Clifford algebra representation" i.e. they come with a multiplication by covectors implemented as maps :\sigma:T^\vee \to \mathrm(S, \bar S^\vee) and :\bar\sigma: T^\vee \to \mathrm(\bar S^\vee, S) which we will call Infeld–Van der Waerden maps. Note that in a natural way we can also think of the maps as a sesquilinear map associating a vector to a left and righthand spinor :\sigma \in T \otimes \bar S^\vee \otimes S^\vee \cong \mathrm(\bar S \otimes S, T) respectively \bar \sigma \in T \otimes S \otimes \bar S \cong \mathrm(S^\vee \otimes \bar S^\vee, T) . That the Infeld–Van der Waerden maps implement "two halves of a Clifford algebra representation" means that for covectors a,b \in T^\vee :\bar\sigma(a)\sigma(b) + \bar\sigma(b)\sigma(a) = 2g^\vee(a,b)1_ resp. :\sigma(a)\bar\sigma(b) + \sigma(b)\bar\sigma(a) = 2g^\vee(a,b)1_, so that if we define :\gamma = \begin0 & \bar \sigma \\\sigma & 0 \end:T^\vee \to \mathrm(S\oplus \bar S^\vee) then :\gamma(a)\gamma(b) + \gamma(b)\gamma(a) = 2g^\vee(a,b)1_. Therefore \gamma extends to a proper Clifford algebra representation \mathrm(T^\vee, g^\vee) \to \mathrm(S\oplus \bar S^\vee). The Infeld–Van der Waerden maps are real (or hermitian) in the sense that the complex conjugate dual maps :\sigma^\dagger(a): S \mathop\limits^ \bar S \mathop\limits^ S^\vee \mathop\limits^ \bar S^\vee coincides (for a real covector a) : :\sigma(a) = \sigma(\bar a)^\dagger. Likewise we have \bar\sigma(a) = \bar\sigma(\bar a)^\dagger. Now the Infeld the Infeld–Van der Waerden symbols are the components of the maps \bar\sigma and \sigma with respect to bases of T and S with induced bases on T^\vee and \bar S^\vee. Concretely, if T is the tangent space at a point O with local coordinates x^m (m = 0, \ldots, 3) so that \partial_m is a basis for T and dx^m is a basis for T^\vee, and s_\alpha (\alpha = 0,1 ) is a basis for S, s^\alpha is a dual basis for S^\vee with complex conjugate dual basis \bar s^ of \bar S^\vee, then : \sigma(dx^m)(s_\alpha) = \sigma^m_\bar s^ : \bar\sigma(dx^m)(\bar s^) = \bar\sigma^s_\beta Using local frames of the (co)tangent bundle and a Weyl spinor bundle, the construction carries over to a differentiable manifold with a spinor bundle.


Applications

The \sigma symbols are of fundamental importance for calculations in
quantum field theory in curved spacetime In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory treats spacetime as a fixed, classical background, while givi ...
, and in supersymmetry. In the presence of a
tetrad Tetrad ('group of 4') or tetrade may refer to: * Tetrad (area), an area 2 km x 2 km square * Tetrad (astronomy), four total lunar eclipses within two years * Tetrad (chromosomal formation) * Tetrad (general relativity), or frame field ** Tetra ...
e^\mu_m for "soldering" local Lorentz indices to tangent indices, the contracted version \sigma^\mu_ can also be thought of as a soldering form for building a tangent vector out of a pair of left and right Weyl spinors.


Conventions

In flat
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, A standard component representation is in terms of the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
, hence the \sigma notation. In an orthonormal basis with a standard spin frame, the conventional components are\begin \sigma^0_ \ &\dot\ \delta_ \,, \\ \sigma^i_ \ &\dot\ (\sigma^i)_ \,, \\ \bar^ \ &\dot\ \delta^ \,, \\ \bar^ \ &\dot\ -(\sigma^i)^ \,. \end Note that these are the blocks of the
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
in the Weyl Chiral basis convention. There are, however, many conventions.


References

{{DEFAULTSORT:Infeld-Van der Waerden symbols Mathematical physics Representation theory of Lie groups Spinors