Bled Workshops in Physics Vol. 14, No. 2 A Proceedings to the 16th Workshop What Comes Beyond ... (p. 212) Bled, Slovenia, July 14-21, 2013 15 Massless and Massive Representations in the Spinor Technique T. Troha, D. Lukman and N.S. Mankoc Borstnik University of Ljubljana, Department of Physics, Jadranska 19,1000 Ljubljana, Abstract. This contribution uses the technique [1] for representing spinors and the definition of the discrete symmetries [3] to illustrate on a toy model [2] properties of massless and massive solutions of spinors. It might help to solve the problem about representations of Dirac, Weyl and Majorana, presented in the ref. [4] in this proceedings. Povzetek. Prispevek uporablja tehniko [1] predstavitve spinorjev in definicijo diskretnih simetrij [3] za ilustracijo lastnosti brezmasnih in masivnih resitev spinorjev v preprostem modelu. Prispeva lahko k resitvi problema upodobitev Diracovih, Weylovih in Majoraninih spinorjev predstavljenega v prispevku [4] v tem zborniku. 15.1 Introduction We study in a toy model defined in d = (5 + 1), presented in the refs. [2], massless and massive positive and negative energy solutions of the equations of motion, and look for, by taking into account the definition of the discrete symmetry operators in the second quantized picture (Cn, Pn and Tn, presented in the paper [3]) the antiparticle states to the particle ones. We present the representations in the spinor technique [1]. In this toy model the M5+1 manifold is assumed to break into M3+1 x an almost S2 sphere due to the zweibein in d = (5,6). We first study massless solutions in d = (3 + 1) assuming that the extra dimensions bring no contribution to the masses in d = (3 +1). We correspondingly solve the Weyl equation in d = (5 + 1) and present the representations and comment on particle and antiparticle states. Requiring that there is only one massless of a particular handedness and mass protected solution in d = (3 + 1), what is achieved by a particular choice of the spin connection fields on this (almost) S2 sphere, what consequently forces the rest of solutions to be massive, we comment on the corresponding particle and antiparticle states. In these two cases the spin in d = (5,6) is a conserved quantity. Assuming nonzero vacuum expectation values of the spin connection fields [13], the gauge fields of S56 with indices d = (5,6), which manifest as scalars in d = (3 +1) and carry the U(1) charge S56, all the spinors become massive and no charge is the conserved quantity any longer. The Weyl equation in d = (5 + 1) manifests Slovenia d = (3 + 1) as the Dirac equation for massive states. We present representations and comment on particle and antiparticle states also in this case. 15.2 Massless solutions Let us look for the solutions of the Weyl equations yapa^ = 0 in d = (5 + 1) for a particular choice of the coordinate system: pa = (p0,0,0, |p3|, 0,0). Then the Weyl equations read (-2iS03p0 = p3)^. (15.1) In Table I, taken from the paper [3], the solutions of Eq. 15.1 are presented, using the technique of the refs. [1]. We found for the basic states 03 12 56 = (+i)(+)(+) |vac >f 03 12 56 ^2 = (+i)[-][-] |vac >fam, 03 12 56 ^3 = [-i][-]( + ) |vac >fam, 03 12 56 = [-i] (+) [-] |vac >fam , (15.2) where |vac >fam is defined so that there are 2d-1 family members (this is, however, not a second quantized vacuum). All the basic states are eigenstates of the Cartan subalgebra (of the Lorentz transformation Lie algebra), for which we take: S03, S12, S56, with the eigenvalues, which can be read from Eq. (15.2) if taking 2 ab ab times the numbers ±i or ±1 in the parentheses of nilpotents (k) and projectors [k]: 56 ab 56 ab Sab (k)= k (k), Sab [k]= | [k]. The first two positive energy solutions ( , i = (1,2)) and the last two negative energy solutions (^^ , i = (3,4)) correspond to p3 = |p3|. These all are the solutions of the Weyl equations (r ,3+" ip°i = w) (153) for the choice p = (p1 ,p2,p3) (in our case is (0,0,p3)), presented in all text books. Here S = (S23, S31, S12), Sab = J(yayb -ybya),and F((d-1)+1) (in usual notation is for d = (3 + 1) named y5) determines handedness for fermions in any d. For d = (5 + 1) r(5+1' = naYa in ascending order, equal also to F(3+1' (-2S56). For 214 T. Troha, D. Lukman and N.S. Mankoc Borstnik i n c8 positive energy state p° |p°| p3 |p3 03 12 56 0 0 3 3 (-i) (-) l (-) e-i|pV+i|pV -1 -1 03 12 56 n n 3 3 (-i) [-] l [-] e-i 1 P0 1 x0+i 1 p3 1x3 -1 -1 03 12 56 0 0 3 3 [-i] [-] l (-) e-i|p°|x°-i|p3|x3 -1 03 12 56 0 0 3 3 [-i] (-) l [-] e-i|p°|x°-i|p3|x3 -1 negative energy state p° |p°| p3 |p3 03 12 56 ° ° 3 3 (-i) (-) l (-) e1 |p | x -i |p | x 03 12 56 ° ° 3 3 (-i) [-] l [-] ei|p°|x°-i|p3|x3 03 12 56 0 0 3 3 [-i] [-] l (-) ei|P0 | x0+i | p3 |x3 -1 03 12 56 0 0 3 3 [-i] (-) l [-] ei|P0|x0+i|P3|x3 -1 -2iS03) 2iS ) -(3+1) + + -(3+1) + + 2p 3 S 12 Table 15.1. Four positive energy states and four negative energy states, the solutions of Eq. (15.1), half have ^ positive and half negative. pa = (p0, 0, 0,p3, 0, 0), F(5+1) = -1, r((d-1 ) + 1) defines the handedness in d-dimensional space-time, S56 defines the charge in 3 12 ab ab d = (3 + 1), 2ppS| defines the helicity. Nilpotents (k) and projectors [k] operate on the vacuum state | vac >fam not written in the table. Table is taken from [3]. 56 S 56 S the choice pa = (p1 ,p2,p3,0,0) the solutions read p0 = 1 p0l, / 03 12 56 p1 i ip2 03 12 56 \ ° ° = -S56i < (ff T ifg)w56a > causes on the tree level the mass m of spinors: < p0± >= -iS562m. Solutions of the Weyl equation in d = (5 + 1 ) manifest in d = (3 + 1 ) as massive states with the mass 2m =< w56+ >=< w56- >. Let us make a choice of the coordinate system so that pa = (p0,0,0,0,0,0). One obtains two positive and two negative energy solutions 03 12 56 03 12 56 0 <°m = N((+i)(+)(+) —i [-i](+)[—]) e-imx , 03 12 56 03 12 56 0 = N([-i][-](+) -i (+i)[-][-]) e-imx , 03 12 56 03 12 56 0 « = N((+i)(+)(+) +i [-i](+)[-]) eimx , 03 12 56 03 12 56 0 ^nm9 = N([-i][-](+) +i (+i)[-][-]) eimx , (15.6) with m2 = (p0)2, m = 2 < (ff T ifg)w56a > 1. (To obtain true masses of spinors one must take into account loop corrections in all orders, to which also the dynamical scalar and vector gauge fields contribute.) In this discussion only one family is assumed. Let us present massive positive and negative energy solutions, the ones which coincide with vectors ( °-ff

a}+ = 0, {Pn ,YaPa}- = 0, {Tn ,YaPa}+ = 0, {Cn Pn ,YaPa}+ = 0, {Cn Pn Tn,YaPa}- = 0. (15.9) In even dimensional spaces namely neither Cn nor Pd-1 transforms states within the same Weyl representation, it is only Cn P_N-1, which does this and it is correspondingly a good symmetry, keeping the states within one Weyl representation. To obtain an antiparticle state to a chosen particle state above the Dirac sea we must accordingly apply on a particle state the operator Cn Pd-1 and then empty the obtained negative energy state. Emptying the corresponding negative energy state and putting it on the top of the Dirac sea determines the antiparticle state to the starting particle state. In the ref. [3] the second quantized charge conjugation operator Cn is defined as follows: First one applies on the particle state with positive energy put on the top of the Dirac sea, the operator Cn , which makes a choice of the corresponding negative energy state. Then by emptying this negative energy state in the Dirac sea one creates an antiparticle with the positive energy and all the properties of the starting single particle state above the Dirac sea, that is with the same d-momentum and all the spin degrees of freedom the same, except the S03 value, as the starting single particle state 2. We make now a statement. It is the operator emptying : = ^ Ya Kr(3+1), (15.10) yaea operating on the negative energy state, which empties the negative energy state creating the antiparticle state above the Dirac sea. Let us check on the massless states of Eq. (15.4) what we claimed: First Cn Pn applied on (p) (this state is put on the top of the Dirac sea) transforms this state into the negative energy state (P), then ^yae3 Ya K r(3+1' applied on ^neg1 (p) transforms this state into the positive energy antiparticle / 03 12 56 - 2 03 12 56 \ 0 0 state^5°s(-p) = N2 ( [-i] (+) | [-] + JP-c+Pr (+i) [-] I HJ e-i(|p |x , (put on the top of the Dirac sea), up to phase factors, what can easily be checked. All these states, particle and antiparticle ones, are the solutions of the massless Weyl equation in d = (3 + 1) (with pa = (p0,p1 ,p2,p3,0,0). The two massless particle/antiparticle pairs are therefore (^P°s(p), ^p°s(-p)) and (^°s(p), <°s(-p)). For pa = (p0,0,0, p3, 0,0) we find that the two particle/antiparticle pairs are correspondingly and and and ^P°s), presented in (Table 15.1). 2 S03 is involved in the boost (contributing in d = (3 + 1), together with the spin, to handedness) and does not determine the (ordinary) spin. 218 T. Troha, D. Lukman and N.S. Mankoc Borštnik One checks this by applying the operator Cn Pd-1 on the state , transforming this state into the state ^es, while emptying this negative state generates 4 ' Similarly CN Pd ^pos into ^J^8, while emptying this negative state generates If only one massless solution, let say the right handed one with respect to d = (3 +1), is allowed, as in the case presented in the ref. [2], then the only allowed particle/antiparticle pair is (^p°s, ^4°s). Before discussing the discrete symmetries of the massive states presented in (56) (56) (56) Eq. (15.5) let us pay attention that K (+) = — (+), since (+) = 1 (y5 + iy6) and we make a choice of y0,y1 real, y2 imaginary, y3 real, y5 imaginary, y6 real, and alternating real and imaginary ones we end up in even dimensional spaces with real yd. K makes complex conjugation, transforming i into —i. Let us look at discrete symmetries of the massive solutions (P = 0), i = 1,2. We see that Cn Pn applied on (Eq. (15.6)), put on the top of the Dirac sea, transforms this state into the negative energy state ^m. Emptying this negative energy state, that is applying ^yae3 ya K r(3+1) on ^Jm, makes the antiparticle state on the top of the Dirac sea. This state does not distinguish from the starting particle state. Massive states have no conserved charge and correspondingly are the particle and antiparticle solutions of the massive Weyl equation for pm = (m, 0,0,0) indistinguishable. For the general momentumpm = (p0,p1,p2,p3) the state ^Jm (P) inEq. (15.7) follows if we apply the discrete operator Cn Pn on the state ^4m(p). Emptying the state ^Jm (P) leads to the antiparticle state above the Dirac sea, which is