Spinor
A spinor is in mathematics, and there specifically in differential geometry, 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 sub - algebra of a real, complex or quaternionic matrix algebra. This has a canonical representation by column vectors, the spinors.
A spinor is in physics usually a vector of a 2- dimensional complex representation of the spin group, which belongs to the group of Lorentz transformations of Minkowski space. Important here is mainly the rotational behavior.
- 2.1 Weyl spinors
- 2.2 Dirac spinors
- 2.3 Majorana spinors
Spinors of quantum physics
Structure of the group Spin (1,3)
The group is a subset of the straight part of the Clifford algebra. The entire algebra - as vector space has 16 dimensions - is generated by the four canonical basis vectors, of the 4-dimensional Minkowski space with a square shape ( in this coordinate basis). Accordingly, the products of different basis vectors anticommute; applies to their squares, ie, .
The (as - vector space 8 -dimensional ) subalgebra of even elements is generated by two-fold products containing: ,, . This also anticommute; their squares have the value 1
A base consists of (for example ) from the unit element, and the four elements described below and:
The missing two-time products (ie those that do not contain ) form a "double straight" subalgebra which is generated by the products of straight:
The squares have the value -1, and each is (possibly up to sign ) the product of the other two, ie, etc. The sub- algebra generated by the algebra is isomorphic to the quaternions. With regard to the Pauli matrices we identify; More details below.
Among the basis vectors of the even subalgebra is missing the volume element
This commutes with the entire straight subalgebra, it is.
Isomorphic Matrix Algebra
It is easy to see that just generate subalgebra and that the odd part of the algebra than to receive. Overall, the following applies:
- And generate the quaternions isomorphic subalgebras respectively,
- These subalgebras commute and
- Join forces on the entire algebra.
This provides the isomorphism
The restricted isomorphism
Results.
It should be in the following always, with an imaginary unit quaternions is. Then the isomorphism can be defined as follows:
As a consequence, result with and
Representation in the quaternions, Majorana spinors
There is an isomorphism, which assigns a tensor imaging. This is a real four-dimensional or one-dimensional quaternionisch representation of the entire Clifford algebra. As the latter, it has the name Majorana spinor representation, according to Ettore Majorana.
Representation in the complex numbers, Weyl spinors
We define a bijective mapping as. This picture is real linear and complex antilinear right, that is. Be the coordinate mapping. Thus we define
That is, an element from the figure that by
Is optionally assigned. It is, for example,
The matrix of this picture is the first Pauli matrix, and similarly applies.
Thus, a complex two-dimensional representation of the straight sub- algebra and thus the group. This representation is called by Weyl spinor representation, named after Hermann Weyl (see also Pauli matrices ).
It is added to this is a conjugate representation,
Weyl, Dirac and Majorana spinors
A faithful representation is an embedding of the algebra into a matrix group, or generally in the Endomorphismengruppe a vector space. Thus elements of the spin- group should be mapped onto orthogonal or unitary matrices.
For this purpose the following lemma: Are self-adjoint unitary maps on with and so decomposes into isomorphic, mutually orthogonal subspaces and. The triples can be mapped to isomorphic to
Is the identity. The occurring tensor can also be interpreted as the Kronecker product of matrices here.
Weyl spinors
A Weyl spinor representation, named after Hermann Weyl, is a complex representation of the smallest. This is also the smallest complex representation of the even subalgebra.
Suppose we have available a complex representation of in a Hermitian vector space. The images are ( for brevity, we let the further away) unitary, self-adjoint representations of self.
And satisfy the conditions of the lemma, so we can be an isomorphic representation
Proceed.
To restrict the form of, we consider the product and find that due to the commutation relations
The following figure is compulsory
Since the vector space is complex, we can split it into mutually orthogonal subspaces and on which such or acts. Both subspaces arising separate presentations, each being the minimum to one another complex conjugate, the matrices are the Pauli matrices already mentioned, because if, it is
In the minimum case, or vice versa. So there are two conjugated Weyl spinor representations.
Application: see Weyl equation
Dirac spinors
In quantum electrodynamics or Atiyah-Singer index theory of the Dirac operator is defined. The "how " is not important, just that, a representation of the entire Clifford algebra is required. The Dirac spinor representation, according to Paul Dirac, is the smallest complex representation of.
If such a complex representation given, then we can analyze the above representation of the even subalgebra. To determine also the odd part, we consider the image of. It commutes with and antikommutiert with, as above, we note that
It is satisfied that the subspaces and swap, so we can replace the view through a further factorized:
The minimum Dirac spinor representation is again with (and any isomorphic to it).
Majorana spinors
The Majorana spinor representation, according to Ettore Majorana, both the spin group and the Clifford algebra is the smallest real representation of. We can analyze the top cover up to the location where and on are defined. Here we can now search disassemble and swapped both subspaces, however, is thus
After multiplying out, we obtain
Rotational behavior
From the above, for the physics perhaps most important property of the spinors is not easy to detect or to deduce:
- For particles of integral spin ( measured in units of the reduced Planck's constant ), so-called bosons, the wave function of one complete rotation about the factor is multiplied, that is, it remains unchanged.
- In contrast, results for particles with half-integer spin, fermions, with a full rotation around the factor -1 for the wave function. That these particles move in a full rotation, the sign of its quantum mechanical phase or they need to perform two full rotations, to get back to its initial 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, specifically in differential geometry, under a spinor is understood as a (usually smooth) section of the Spinorbündels. The spinor bundle is a vector bundle, which is formed as follows: Starting from an oriented Riemannian manifold (M, g) is one of the bundle P ON Repere. This is pointwise from all oriented orthonormal bases:
This is a main fiber bundle structure group. A spin structure is then a pair (Q, f) of a principal bundle Q with structure group spinning and a mapping which satisfies the following properties:
A spin structure does not exist at any manifold, there exists a so called the spin multiplicity. The existence of a spin structure is equivalent to the vanishing of the second Stiefel- Whitney class.
Given a spin structure (Q, f) we construct the (complex) spinor bundle as follows: One uses the irreducible (with restriction on the spin Group unique) representation of the (complex) Clifford algebra (see here ) and forms the spinor bundle as an associated vector bundle
Where the equivalence relation is given by.
Analogous constructions can be carried out even if the Riemannian metric is replaced by a pseudoriemannsche. The spinors described above are spinors in the sense described here on the manifold with the pseudo -Euclidean metric. The spinor bundle is a trivial vector bundle in this case.