In physics, and specifically in quantum field theory, a bispinor is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons.
Bispinors transform in a certain "spinorial" fashion under the action of the Lorentz group, which describes the symmetries of Minkowski spacetime.
They occur in the relativistic spin-1/2 wave function solutions to the Dirac equation.
Each of the two component spinors transform differently under the two distinct complex-conjugate spin-1/2 representations of the Lorentz group.
This pairing is of fundamental importance, as it allows the represented particle to have a mass, carry a charge, and represent the flow of charge as a current, and perhaps most importantly, to carry angular momentum.
More precisely, the mass is a Casimir invariant of the Lorentz group (an eigenstate of the energy), while the vector combination carries momentum and current, being covariant under the action of the Lorentz group.
The angular momentum is carried by the Poynting vector, suitably constructed for the spin field.
By contrast, the article below concentrates primarily on the Weyl, or chiral representation, is less focused on the Dirac equation, and more focused on the geometric structure, including the geometry of the Lorentz group.
Bispinors are elements of a 4-dimensional complex vector space (1/2, 0) ⊕ (0, 1/2) representation of the Lorentz group.
, a matrix, is an element of the spinor representation (for spin 1/2) of the Lorentz group.
is the boost parameter which is the rapidity multiplied by the normalized direction of the velocity, and
A bilinear form of bispinors can be reduced to five irreducible (under the Lorentz group) objects: where
For reference, the commutation relations of so(3,1) are with the spacetime metric η = diag(−1, 1, 1, 1).
The Dirac matrices satisfy where { , } is the anticommutator, I4 is a 4×4 unit matrix, and ημν is the spacetime metric with signature (+,−,−,−).
They act on the subspace Vγ the γμ span in the passive sense, according to In (C2), the second equality follows from property (D1) of the Clifford algebra.
Now define an action of so(3,1) on the σμν, and the linear subspace Vσ ⊂ Cl4(C) they span in Cl4(C) ≈ MnC, given by The last equality in (C4), which follows from (C2) and the property (D1) of the gamma matrices, shows that the σμν constitute a representation of so(3,1) since the commutation relations in (C4) are exactly those of so(3,1).
The Lie algebra of so(3,1) is thus embedded in Cl4(C) by π as the real subspace of Cl4(C) spanned by the σμν.
Now introduce any 4-dimensional complex vector space U where the γμ act by matrix multiplication.
The elements of U, when endowed with the transformation rule given by S, are called bispinors or simply spinors.
To include space parity inversion in this formalism, one sets as representative for P = diag(1, −1, −1, −1).
This space corresponds to the Clifford algebra itself so that all linear operators on U are elements of the latter.
The Dirac matrices are a set of four 4×4 matrices forming the Dirac algebra, and are used to intertwine the spin direction with the local reference frame (the local coordinate frame of spacetime), as well as to define charge (C-symmetry), parity and time reversal operators.
After choosing the signature, there are many ways of constructing a representation in the 4×4 matrices, and many are in common use.
In order to make this example as general as possible we will not specify a representation until the final step.
As with the example of the Pauli algebra discussed above, the spin direction is defined by a unit vector in 3 dimensions, (a, b, c).
They form a complete set of commuting operators for the Dirac algebra.
So we can apply it to a spinor with the value 1 in one of its components, and 0 in the others, which gives a column of the matrix.
Continuing the example, we put (a, b, c) = (0, 0, 1) and have and so our desired projection operator is The 4×4 gamma matrices used in the Weyl representation are for k = 1, 2, 3 and where
The first and third columns give the same result: More generally, for electrons and positrons with spin oriented in the (a, b, c) direction, the projection operator is where the upper signs are for the electron and the lower signs are for the positron.
The representation of the resulting spinor in the Dirac basis can be obtained using the rule given in the bispinor article.