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In mathematics , and there especially in differential geometry , a spinor is a vector in a smallest representation of a spin group . The spin group is isomorphic to a subset of a Clifford algebra . Every Clifford algebra is isomorphic to a partial algebra of a real, complex or quaternionic matrix algebra. This has a canonical representation using column vectors, the spinors.

In physics , a spinor is usually a vector of a 2-dimensional complex representation of the spin group , which belongs to the group of Lorentz transformations of the Minkowski space . The turning behavior is particularly important here.

History of spinors

In 1913, Élie Cartan classified the irreducible complex representations of simple Lie groups . In addition to the known tensor representations, he also found a new two-valued representation in the form of spinors (and said beforehand that these could build up the other representations), especially for linear representations of the rotating groups. His textbook on spinors appeared later. Their significance especially in physics but was only after the discovery of the Dirac equation by Paul Dirac recognized in 1928 (they allowed him an equation first order, the Dirac equation, as linearization of an equation 2nd order, the Klein-Gordon equation to win) . Paul Ehrenfest wondered why the representation in Dirac (with the relativistic covariant Dirac equation) was four-dimensional, in contrast to the Pauli equation by Wolfgang Pauli , in which he also introduced his Pauli matrices , which was previously established for spin within the framework of non-relativistic quantum mechanics two-dimensional. Ehrenfest coined the name Spinor for the new types in 1928 and commissioned Bartel Leendert van der Waerden to analyze them mathematically, a study that Van der Waerden published in 1929.

Dirac worked largely independently when he introduced the spinors, and in his own words also independently of Pauli in the use of Pauli matrices. Pauli himself received substantial support in the mathematical interpretation of his equation in 1927 by Pascual Jordan (who pointed out the connection with quaternions).

Dirac's work was within the Lorentz group, the connection with spinors in Euclidean space was established by Cartan in his book in 1938 and Richard Brauer and Hermann Weyl in an essay in 1935 (using Clifford algebras). Claude Chevalley continued the algebraic theory of spinors in the context of Clifford algebras in his textbook 1954.

They became particularly important in differential geometry through the Atiyah-Singer index theorem in the early 1960s.

Spinors of quantum physics

Structure of the spin group (1,3)

The spin group is a subset of the even part of Clifford algebra . The entire algebra - as a vector space has 16 dimensions - is of the four canonical basis vectors , , , of the 4-dimensional Minkowski space of a square shape (in this coordinate basis) generated. Accordingly, the products of different basis vectors anticommute; applies to their squares , ie , .

The (a vector space 8-dimensional) sub-algebra of the straight elements is generated by two-fold products which contain: , , . These also anticommute; their squares have the value 1 .

For example, a base of consists of the one element, the four elements described below and :

The missing twofold products (i.e. those that do not contain) form a "doubly even" subalgebra generated by even products of :

The squares of have the value -1 , and each of the is (possibly up to the sign) the product of the other two, so etc. The subalgebra generated by the is isomorphic to the algebra of the quaternions . With regard to the Pauli matrices we identify , , ; More details below.

The volume element is still missing among the basis vectors of the even sub-algebra

This commutes with the entire even sub-algebra, it is true .

Isomorphic matrix algebra

It is easy to see that the even sub-algebra is generating and that the odd part of the algebra is as obtaining. Overall:

  • and generate subalgebras isomorphic to the quaternions,
  • these sub-algebras commute with each other and
  • together span the entire algebra.

This gives the isomorphism


which restricted an isomorphism


In the following it is always , where is an imaginary unit of the quaternions. Then the isomorphism can be defined as follows:

As a result, with and

Own spinors

In quantum mechanics, intrinsic spinors represent the basis vectors that describe the spin state of a particle. For a single spin 1/2 particle they can be viewed as the eigenvectors of the Pauli matrices . They form a complete orthonormal system.

Representation in the quaternions, Majorana spinors

There is an isomorphism which assigns the mapping to a tensor product . This is a quaternionic one-dimensional or a real four-dimensional representation of the entire Clifford algebra. The latter is called the Majorana spinor representation , after Ettore Majorana .

Representation in complex numbers, Weyl spinors

We define a bijective map as . This mapping is real linear and complex right antilinear, i.e. H. . Be the coordinate map. That's how we define

, through ,

d. H. one element from becomes the figure that through

is given, assigned. It is z. B.


The matrix in this figure is the first Pauli matrix , analogously and .

Thus, a complex two-dimensional representation of the even sub-algebra and thus also of the group. This representation of is called Weyl-Spinor representation , named after Hermann Weyl (see also: Pauli matrices ).

There is a conjugate representation for this , where

Weyl, Dirac and Majorana spinors

A true representation is an embedding of the algebra in a matrix group, or generally in the endomorphism group of a vector space. Elements of the spin group are to be mapped onto orthogonal or unitary matrices.

The following lemma : If , are self-adjoint unitary mappings with and , then breaks down into isomorphic, mutually orthogonal subspaces and . The triple can be mapped isomorphically

is the identity on . The occurring tensor product can also be understood here as the Kronecker product of matrices .

Weyl spinors

A Weyl spinor representation , named after Hermann Weyl , is a smallest complex representation of . This is also the smallest complex representation of the even sub-algebra .

Suppose we had a complex representation of in a Hermitian vector space . The images are (for the sake of brevity we will omit this in the following ) unitary, self-adjoint images of themselves.

and meet the requirements of the lemma, so we can use an isomorphic representation

with and

pass over.

In order to restrict the shape of , we consider the product and find that due to the interchangeability rules


the following form necessarily results


Since the vector space is complex, we can split it into mutually orthogonal subspaces and on which how or acts. Both subspaces result in separate representations, the minimal ones are complex conjugate to one another, the matrices are the Pauli matrices already mentioned , because if so is

In the minimal case it is , or vice versa. So there are two conjugated Weyl spinor representations .

Application: see Weyl equation

Dirac spinors

In the quantum electrodynamics or Atiyah-Singer index theory is Dirac operator defined. The "how" is not important, just that a representation of the entire Clifford algebra is needed. The Dirac-Spinor representation , according to Paul Dirac , is the smallest complex representation of when used in 3 + 1 space-time dimensions . However, higher-dimensional Dirac spinors are also considered, for example in string theory .

Given such a complex representation, we can analyze the representation of the even sub-algebra as above. To also determine the odd part, let's consider the image of . It commutates with and anticommutates with . As above, we note that


You convince yourself that the subspaces and are swapped, so we can replace the representation with an even more factored one:

with the pictures of the generators

The minimal Dirac spinor representation is again the one with (and each isomorphic to it).

Dirac spinors in 3 + 1 dimensions are used within the framework of quantum electrodynamics for the mathematical description of fermions with spin 1/2. In the Standard Model of particle physics, these Dirac fermions include all fundamental fermions.

Majorana spinors

The Majorana spinor representation , according to Ettore Majorana , of both the spin group and the Clifford algebra is the smallest real representation of . We can take over the analysis from above up to the point where and are defined. Here we are now able to disassemble and , reversed both subspaces, however , so

with and

After multiplying out, we get for

with the pictures of the generators

In elementary particle physics, they are used to describe Majorana fermions , which, however, have not yet been observed.

Turning behavior

From the above, perhaps the most essential property of spinors for physics is not easy to recognize or to deduce:

  • For particles with an integer spin  (measured in units of Planck's reduced quantum of action ), so-called bosons , the wave function is multiplied by the factor  for a full rotation  by, i.e. H. it remains unchanged. 
  • On the other hand, for particles with half-integer spin, the fermions , a full rotation by  the factor -1 results in the wave function. I.e. With a full rotation, these particles change the sign of their quantum mechanical phase or they have to perform two full rotations in order to get back to their original state, similar to the hour hand of a clock.

Integer or half-integer values ​​of are the only possibilities for the expression of the spin.

Generalization in Mathematics

In mathematics, especially in differential geometry, a spinor is understood to be a (mostly smooth) section of the spinor bundle. The spinor bundle is a vector bundle that arises as follows: Starting from an oriented Riemannian manifold (M, g) , one forms bundle P of the ON repere. This consists of all oriented orthonormal bases point by point:

This is a main fiber bundle with structural group . A spin structure is then a pair (Q, f) consisting of a main fiber bundle Q with structure group spin n and an image that fulfills the following properties:

  1. , where and are the projections of the main fiber bundles and
  2. , where is the two-fold overlay image.

A spin structure does not exist for every manifold, if one exists, the manifold is called spin. The existence of a spin structure is equivalent to the disappearance of the second Stiefel-Whitney class .

Given a spin structure (Q, f) , the (complex) spinor bundle is constructed as follows: The irreducible representation of the (complex) Clifford algebra (which is unique when restricted to the spin group) is used (compare here ) and the spinor bundle is formed as an associated vector bundle


where the equivalence relation is given by .

Analogous constructions can also be carried out if the Riemannian metric is replaced by a pseudoriemannian. The spinors described above are spinors in the sense described here over the manifold with the pseudo-Euclidean metric . In this case, the spinor bundle is a trivial vector bundle.

See also

Individual evidence

  1. Cartan, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. Soc. Math. France, Vol. 41, 1913, pp. 53-96
  2. The history of spinors is presented, for example, in Marcel Berger , A panoramic view of Riemannian Geometry, Springer 2003, pp. 695f
  3. Cartan, The theory of spinors, Hermann 1966, Dover 1981, first published in French in 1938 as Leçons sur la théorie des spineurs by Hermann in two volumes
  4. Martina Schneider, Between Two Disciplines. BL van der Waerden and the development of quantum mechanics, Springer 2011, p. 122
  5. Van der Waerden, Göttinger Nachrichten, Spinoranalyse, Nachrichten Ges. Wiss. Göttingen, 1929, p. 100. The essay opens with the question asked by Ehrenfest.
  6. ^ Pauli, Letter to Jordan March 12, 1927, in Pauli, Briefwechsel, Volume 1, Springer 1979, p. 385
  7. Brauer, Weyl, Spinors in n dimensions, American Journal of Mathematics, Volume 37, 1935, pp. 425-449
  8. ^ Chevalley, The algebraic theory of spinors. New York, Columbia University Press 1954. Reprinted in Chevalley's Collected Works, Volume 2 (Springer 1996) with an afterword by Jean-Pierre Bourguignon .
  9. Berger, loc. cit. The construction of spinor bundles on Riemannian manifolds was folklore according to Berger in the 1950s and the year 1963 was outstanding in the history of spinors not only because of the introduction of the Atiyah-Singer index theorem, but also because of the formula for scalar curvature by André Lichnerowicz