symbol must be a tensor, and so must contain a factor of , where is the determinant of the metric tensor. Since this is negative, that factor is imaginary. Thus This matrix is given the special symbol , owing to its importance when one is considering improper transformations of space-time, that is, those that change the orientation of the basis vectors. In the standard representation, it is This matrix will also be found to anticommute with the other four Dirac matrices: It takes a leading role when questions of parity arise because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime. Comparison with related theories Pauli theory The necessity of introducing half-integer spin goes back experimentally to the results of the Stern–Gerlach experiment. A beam of atoms is run through a strong inhomogeneous magnetic field, which then splits into parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two; the ground state therefore could not be integer, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into three parts, corresponding to atoms with . The conclusion is that silver atoms have net intrinsic angular momentum of . Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so in SI units: (Note that bold faced characters imply Euclidean vectors in 3 dimensions, whereas the Minkowski four-vector can be defined as .) Here and represent the components of the electromagnetic four-potential in their standard SI units, and the three sigmas are the Pauli matrices. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field in SI units: This Hamiltonian is now a matrix, so the Schrödinger equation based on it must use a two-component wave function. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form: A second application of the Dirac operator will now reproduce the