Projective Geometry and $\cal PT$-Symmetric Dirac Hamiltonian

The $(3 + 1)$-dimensional (generalized) Dirac equation is shown to have the same form as the equation expressing the condition that a given point lies on a given line in 3-dimensional projective space. The resulting Hamiltonian with a $\gamma_5$ mass term is not Hermitian, but is invariant under the combined transformation of parity reflection $\cal P$ and time reversal $\cal T$. When the $\cal PT$ symmetry is unbroken, the energy spectrum of the free spin-$\frac {1}{2}$ theory is real, with an appropriately shifted mass.


I. INTRODUCTION AND SUMMARY
Conventional quantum mechanics requires the Hamiltonian H of any physical system be Hermitian (transpose + complex conjugation) so that the energy spectrum is real. But as shown in the seminal paper by Bender and Boettcher [1], there is an alternative formulation of quantum mechanics in which the requirement of Hermiticity is replaced by the condition of space-time PT reflection symmetry. (For a recent review, see Ref. [2].) If H has an unbroken PT symmetry, then the energy spectrum is real. Examples of PT -symmetric non-Hermitian quantum-mechanical Hamiltonians include the class of Hamiltonians with complex potentials: H = p 2 + x 2 (ix) ǫ with ǫ > 0. Incredibly the energy levels of these Hamiltonians turn out to be real and positive. [1] Now Hermiticity is an algebraic requirement whereas the condition of PT symmetry appears to be more geometric in nature. Thus one may wonder whether a purely geometric consideration can naturally lead to a Hamiltonian which is PT -symmetric rather than its Hermitian counterpart. In this note we provide one such example.
In the next section, we "derive" the (3 + 1)-dimensional Dirac equation from a consideration of the condition that a given point lies on a given line in 3-dimensional projective space. By associating the (homogeneous) coordinates of the point with the Dirac spinor components ψ(x, t), and the coordinates of the line with the four-momentum and two real mass parameters m 1 and m 2 of the Dirac particle, we are led to an equation taking on the form of a generalized Dirac equation with Hamiltonian density As noted in Ref. [3], the Hamiltonian H = dx H(x, t) associated with the above H is not Hermitian but is invariant under combined P and T reflection. For µ 2 ≡ m 2 1 − m 2 2 ≥ 0, it is equivalent to a Hermitian Hamiltonian for the conventional free fermion field theory with mass µ. Studies of spin-1 2 theories in the framework of projective geometry have been undertaken before. See, e.g., Ref. [4]. 1 But the idea that there may be a natural connection between the projective geometrical approach (perhaps also other geometrical approaches) and PT -symmetric Hamiltonians as pointed out in this note appears to be novel.

II. PROJECTIVE GEOMETRY AND PT -SYMMETRIC DIRAC EQUATION
It is convenient to use homogeneous coordinates to express the geometry in a projective space. [5] A point x ≡ (x, y, z) in three-dimensional Euclidean space can be expressed by the ratios of four coordinates (x 1 , x 2 , x 3 , x 4 ) which are called the homogenous coordinates of that point. One possible definition of ( with d being the distance of the point from the origin. Obviously, for any constant c, (cx 1 , cx 2 , cx 3 , cx 4 ) and (x 1 , x 2 , x 3 , x 4 ) represent the same point x.
Consider the line through two points (a 1 , a 2 , a 3 , a 4 ) and to lie on that line, the following determinant has to vanish, for any (r 1 , r 2 , r 3 , r 4 ), This gives, for any r 1 , With the aid of the Plucker coordinates of the line defined by Eq.
(3) can be written as related to that of Kummer's 16 6 configuration. All these authors, explicitly or implicitly, put one of the two masses, viz., m 2 in (1), to be zero by hand. In this note, we "derive" the generalized Dirac equation from the projective geometrical approach in a relatively simple way and point out that there is no need to put m 2 = 0 and perhaps it is even natural to keep both masses m 1 and m 2 .
Similarly, for any r 2 , r 3 , r 4 , the following equations respectively must hold Note that the Plucker line coordinates are not independent; the identical relation that connects them can be found by expanding the determinant where 0 is a 2×2 zero matrix, 1 is a 2×2 unit matrix, and σ are the three 2×2 Pauli matrices.
Let us further write the six Plucker coordinates (under the so-called Klein transformation) in terms of p µ with µ running over 0, 1, 2, 3 (to be interpreted as the four-momentum of the Dirac particle) and m 1 and m 2 (to be interpreted as two real mass parameters) as follows: Then we can rewrite Eqs. (5) and (6) as the generalized Dirac equation in energy-momentum space! In coordinate space, we get The above choice (10) of p ij in terms of p µ , m 1 and m 2 is dictated by the representation of the Dirac matrices we have adopted. A different representation would result in a different choice. To wit, if we use the Weyl or chiral representation for the Dirac matrices we have to choose the Plucker coordinates according to p 34 = +m 1 − m 2 , p 12 = +m 1 + m 2 , to yield (11) or (12).
Associated with the generalized Dirac equation (12) is the Hamiltonian density for the free Dirac particle given in (1). Following Bender et al. [3], one can check that the Hamiltonian H is not Hermitian because the m 2 term changes sign under Hermitian conjugation. However

H is invariant under combined P and T reflection given by
Pψ(x, t)P = γ 0 ψ(−x, t), and T ψ(x, t)T = C −1 γ 5 ψ(x, −t), where C is the charge-conjugation matrix, defined by C −1 γ µ C = −γ T µ . Therefore, the projective geometrical approach yields (at least in this particular example) a PT -symmetric Hamiltonian rather than a Hermitian Hamiltonian. [6] By iterating (12), one obtains Thus, the physical mass that propagates under this equation is real for µ 2 ≥ 0, i.e., Plucker coordinates which, in general, can naturally accommodate two types of masses (in addition to the four energy-momentum) for the spin-1 2 particles. Finally we note that (17) in the form of (−p µ p µ + m 2 1 − m 2 2 )ψ = 0 is simply a reflection of the relation (8) among the Plucker coordinates when they are written in terms of p µ , m 1 and m 2 given by either (10) or (14).