S S symmetry Article Biquaternionic Dirac Equation Predicts Zero Mass for Majorana Fermions Avraham Nofech Department of Mathematics and Statistics, MacEwan University, Edmonton, AB T5J 4S2, Canada; [email protected] or [email protected] Received: 1 June 2020; Accepted: 3 July 2020; Published: 8 July 2020 Abstract: A biquaternionic version of the Dirac Equation is introduced, with a procedure for converting four-component spinors to elements of the Pauli algebra. In this version, mass appears as a coefficient between the 4-gradient of a spinor and its image under an outer automorphism of the Pauli algebra. The charge conjugation operator takes a particulary simple form in this formulation and switches the sign of the mass coefficient, so that for a solution invariant under charge conjugation the mass has to equal zero. The multiple of the charge conjugation operator by the imaginary unit turns out to be a complex Lorentz transformation. It commutes with the outer automorphism, while the charge conjugation operator itself anticommutes with it, providing a second more algebraic proof of the main theorem. Considering the Majorana equation, it is shown that non-zero mass of its solution is imaginary. Keywords: Dirac Equation; biquaternion; Majorana; mass; Pauli algebra; outer automorphism; neutrinoless double beta decay 1. Introduction It is well known that Majorana fermions can exist as composite particles [1], but the question whether they can exist as single elementary particles remains open since the work of Majorana [2,3]. There is ongoing work attempting to detect Majorana neutrinos using the neutrinoless double beta decay [4,5]. The aim of this article is to show that if a solution of the Dirac Equation coincides with its own image under the charge conjugation operator, then its mass must necessarily be zero. The very first step is to complete a four-component spinor with a second column, so that the four new scalar equations thus obtained will be exactly the four scalar equations of the conjugate Dirac Equation. Then the Dirac Equation would contain exactly the same eight scalar equations as its conjugate equation. This symmetry of the Dirac Equation exists only if mass is a real scalar. In the four rows by two columns spinor the bottom square becomes a quaternion and the top square becomes a product of a quaternion with an imaginary unit (3). The sum of these two squares thus becomes a biquaternion, and the difference its biquaternion conjugate. A biquaternionic form of the Dirac Equation is introduced, and shown to be equivalent to both the standard equation and its Hermitian conjugate (which both consist of the same eight scalar equations when completed with a second column). The biquaternionic form of the Dirac Equation (16) involves the so-called bar-star outer automorphism of the Pauli algebra (1). There is a simple expression for the conserved probability current in this formulation (29). A transformation of two-column spinors (10) results in a symmetry of the Dirac Equation that reverses the signs of mass, leaving everything else unchanged. This symmetry takes a particularly simple form in the biquaternionic formulation (31), where it reverses the sign of mass. From there it Symmetry 2020, 12, 1144; doi:10.3390/sym12071144 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1144 2 of 19 is an easy conclusion that Majorana solutions of DE that are transformed by this charge conjugation symmetry to themselves must have zero mass. When multiplied by the imaginary unit charge conjugation becomes a complex Lorentz transformation (2). This Lorentz transformation commutes with the bar-star automorphism while the charge conjugation operator itself anticommutes with the bar-star automorphism (12). This anticommutation is used in the second, more algebraic proof that the mass of a Majorana solution has to equal zero (13). For the Majorana equation it is shown that if mass of its solution is non-zero then it is imaginary. The main assumption used in this article is the validity of using the multiplicative structure of the biquaternionic spinor space; see also [6], which uses the multiplicative properties of the space-time algebra. 2. Notation and Terminology The letter P is used for the Pauli algebra Cl3,0(R) ' Cl1,2(R) ' M2(C) ' P. Since it is isomorphic to the algebra of biquaternions, these terms are used interchangeably. The upper case Greek letter Y is used for four-component Dirac spinors, the lower case y for Pauli algebra, or equivalently, biquaternionic spinors. The standard Dirac–Pauli representation is used throughout. The asterisk is used both for Hermitian conjugate and for complex conjugation. The space inversion is denoted by an overbar: k z = z0 + z z¯ = z0 − z z = zks here z 2 P is an arbitrary element of the Pauli algebra split into scalar and vector components. Note. The use of overbar here is different from the Dirac adjoint! In matrix form an element z 2 P and its parity conjugate z¯ look like: 2 3 2 3 z0 + z3 −iz2 + z1 z0 − z3 iz2 − z1 6 7 6 7 z = 4 5 z¯ = 4 5 iz2 + z1 z0 − z3 −iz2 − z1 z0 + z3 The product of an element with its parity conjugate is a scalar: zz¯ = zz¯ = det(z)I2. This is formal and applies also to the product of the four-gradient with its parity conjugate: 2 2 2 3 ¶0 − r 0 ¯ = − r + r = 6 7 = ¶¶ (¶0 )(¶0 ) 4 5 I2 2 2 0 ¶0 − r ¯ m where is the D’Alembertian, ¶ = ¶m, ¶ = ¶ . The algebra of quaternions is a subalgebra of Pauli algebra characterised by the condition: H = f h 2 P j h∗ = h¯ g Another way to characterise the subalgebra of quaternions is as those elements of Pauli algebra whose scalar component is real and whose vector components are purely imaginary: h 2 H , h = h0 + ih hm 2 R Symmetry 2020, 12, 1144 3 of 19 Still another way to characterise quaternions is that they are represented by complex matrices of the form: 2 3 2 3 c −d∗ a b∗ 6 7 6 7 u, v 2 H u = 4 5 iv = 4 5 (1) d c∗ b −a∗ For any quaternion its product with its parity conjugate is the square of its norm: 2 −1 q¯ q 2 H qq¯ = jjqjj I2 q = qq¯ The division is unambiguous because qq¯ is a scalar. Charge conjugation is denoted either as yc or as Cˆ y when using operator notation, (11) and (31). 2.1. The “Bar-Star" Outer Automorphism of the Pauli Algebra ∗ Since Hermitian conjugation z ! z and parity (space inversion) z = z0 + z ! z¯ = z0 − z are both anti-automorphisms, their composition is an automorphism. It is straightforward to check that the composition of Hermitian conjugation and of parity does not depend on their order. This can be summed up as: (xy)∗ = x∗ y∗ Proposition 1. “Bar-star" is an outer automorphism. Proof. The proof is by checking the determinants: ∗ ∗ ∗ ∗ z0 + z3 −iz2 + z1 z0 − z3 iz2 − z1 ∗ ∗ det z = = z2 − z2 − z2 − z2 6= det z = = z2 − z2 − z2 − z2 0 1 2 3 0 1 2 3 ∗ ∗ ∗ ∗ iz2 + z1 z0 − z3 −iz2 − z1 z0 + z3 Proposition 2. Quaternions are those elements of the Pauli algebra that are fixed by the bar-star automorphism. Proof. Since both Hermitian conjugation and parity inversion are involutions it immediately follows that h∗ = h¯ () h = h∗ When writing the bar-star automorphism as an operator the letter iota is used:i ˆy = y∗. 2.2. The Algebra of Biquaternions The algebra of biquaternions is the complexification of the quaternion algebra. The Pauli algebra is the same as the algebra of biquaternions because any of its elements can be written in the form: z 2 P z = u + iv = (u0 + iu) + i(v0 + iv) u, v 2 H z0 = u0 + iv0 z = u + iv Proposition 3. The bar-star automorphism is the conjugation of the biquaternion algebra. Proof. This follows from the action of the bar-star automorphism on both the real and imaginary quaternion components being the identity, so its only result is the switching the sign of the imaginary component. Symmetry 2020, 12, 1144 4 of 19 Note. “Real" and “Imaginary" parts of a biquaternion. Let z = au + ibv, u, v 2 H , jjujj = jjvjj = 1 a, b 2 R be an arbitrary biquaternion written using two unit quaternions. Let us call it “real" if b = 0 and “purely imaginary" if a = 0. Of course in a “real" quaternion only the time coordinate is real, but the three spatial coordinates are purely imaginary. It is the opposite for the purely imaginary quaternion component of a biquaternion. 3. Symmetries of the Dirac Equation For convenience, the four-component spinor is written with a standard choice of letters instead of numbered indices: 2 3 2 3 y1(x) a 6 ( )7 6 7 6y2 x 7 6b7 6 7 = 6 7 (2) 4y3(x)5 4c5 y4(x) d where the letters represent complex scalar functions. Then the spinor is completed with a second column as follows: 2 3 2 3 a a b∗ 2 3 2 3 y iv 6b7 6b −a∗ 7 upp 6 7 −! Y = 6 7 = 6 ··· 7 = 6···7 u, v 2 H (3) 6c7 6c −d∗7 4 5 4 5 4 5 4 5 y u d d c∗ low The spinor thus completed with the right column can be rewritten as a column of two Pauli algebra spinors, the upper and the lower. The lower spinor is a quaternion and the upper spinor is a product of a quaternion with the imaginary unit; see (1).