Degenerate solutions to the massless Dirac and Weyl equations and a proposed method for controlling the quantum state of Weyl particles Georgios N. Tsigaridas1,*, Aristides I. Kechriniotis2, Christos A. Tsonos2 and Konstantinos K. Delibasis3 1Department of Physics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, GR-15780 Zografou Athens, Greece 2Department of Physics, University of Thessaly, GR-35100 Lamia, Greece 3Department of Computer Science and Biomedical Informatics, University of Thessaly, GR-35131 Lamia, Greece *Corresponding Author. E-mail: [email protected] Abstract In a recent work we have shown that all solutions to the Weyl equation and a special class of solutions to the Dirac equation are degenerate, in the sense that they remain unaltered under the influence of a wide variety of different electromagnetic fields. In the present article our previous work is significantly extended, providing a wide class of degenerate solutions to the Dirac equation for massless particles. The electromagnetic fields corresponding to these solutions are calculated, giving also some examples regarding both spatially constant electromagnetic fields and electromagnetic waves. Further, some general forms of solutions to the Weyl equation are presented and the corresponding electromagnetic fields are calculated. Based on these results, a method for fully controlling the quantum state of Weyl particles through appropriate electromagnetic fields is proposed. Finally, the transition from degenerate to non-degenerate solutions as the particles acquire mass is discussed. Keywords: Dirac equation, Weyl equation, Degenerate solutions, Massless particles, Electromagnetic 4-potentials, Electromagnetic fields, Electromagnetic waves, Nearly degenerate solutions 1. Introduction In recent years it has been shown that in certain materials as graphene sheets, Weyl semimetals, etc., ordinary charged particles can collectively behave as massless [1-6], offering a wide range of opportunities for novel technological applications [7-9]. Therefore, an in-depth understanding of the fundamental properties of these particles, especially regarding their interactions with electromagnetic fields is highly desirable in order to efficiently utilize these materials in novel nanoelectronic and nanophotonic components and devices. 1 In the present work we utilize a particularly important result published in a recent article of our group [10], in order to study thoroughly the electromagnetic interactions of these exotic massless particles. In more detail, in [10] we had shown that all solutions to the Weyl equation and a special class of solutions to the Dirac equation, especially regarding massless particles, are degenerate, in the sense that they remain unaltered in the presence of a wide variety of different electromagnetic fields. In this article our previous results are extended considerably, providing, in section 2, a general class of degenerate solutions to the Dirac equation for massless particles. The electromagnetic fields corresponding to these solutions are calculated, and some examples regarding both spatially constant electromagnetic fields and electromagnetic waves are discussed. Specifically, we have shown that the state of massless Dirac - described by degenerate spinors - and Weyl particles is not affected under the influence of spatially constant, but with arbitrary time-dependence, electric fields, applied along the direction of motion of the particles. We have also shown that the state of these particles remains unaltered in the presence of a plane electromagnetic wave, e.g. a laser beam, propagating in a direction opposite to the direction of motion of the particles. A schematic diagram of the main concept of this article is shown in figure 1. Figure 1: The main concept of this work. Further, in section 3, we present a general form of solutions to the Weyl equation, which could be utilized to fully control the quantum state of Weyl particles using appropriate electromagnetic fields. Obviously, these results are expected to play an important role regarding the practical applications of materials supporting massless charged particles [1-9]. Finally, in section 4, we study the transition from degenerate to non-degenerate solutions as the particles acquire mass. It is shown that, as far as the rest energy of the 2 particles is much smaller than their total energy, the concept of the degeneracy can also be extended, in an approximate sense, to massive particles, provided that the magnitude of the electromagnetic fields corresponding to the exact degenerate solutions is sufficiently small. 2. General form of degenerate solutions to the Dirac equation for massless particles and the corresponding electromagnetic fields In this section we shall provide a general form of degenerate solutions to the Dirac equation for massless particles. Further, we will calculate the electromagnetic fields corresponding to these solutions, providing also some examples, regarding both spatially constant electromagnetic fields and electromagnetic waves In a recent article [10] we have shown that all solutions to the Dirac equation i + a − m = 0 (1) satisfying the conditions † = 0 and T 2 0 where are the standard Dirac matrices and =+ 0i 1 2 3 , are degenerate corresponding to an infinite number of electromagnetic 4-potentials which are explicitly calculated in Theorem 5.4. Here m is the mass of the particle and a= qA where q is the electric charge of the particle and A the electromagnetic 4-potential. It should also be noted that Eq. (1) is written in natural units, where ==c 1. In our effort to find general forms of degenerate solutions to the Dirac equation, we have found that all spinors of the form =c u + c uexp ih x , y , z , t p ( 12) ( ) (2a) =c v + c vexp ih x , y , z , t a ( 12) ( ) (2b) are degenerate solutions to the Dirac equation corresponding to massless particles (2a) or antiparticles (2b) propagating along a direction in space defined by the angles (, ) in spherical coordinates. Here, cc12, are arbitrary complex constants, h is an arbitrary real functions of the spatial coordinates and time, and u,,, u v v are the eigenvectors describing the spin state of the particle (uu, ) or antiparticle (vv, ) . In the case of massless particles, they are given by the formulae [11] 3 cos − sin 22 ii eesin cos 22 uu== (3a) cos sin 22 ii eesin − cos 22 sin cos 22 ii −eecos sin 22 vv== (3b) −sin cos 22 ii eecos sin 22 The 4-potentials corresponding to the above solutions are hhhh (a0,,,,,, a 1 a 2 a 3 ) = (4) t x y z Further, according to Theorem 5.4 in [10], the spinors (3) will also be solutions to the Dirac equation for an infinite number of 4-potentials, given by the formula b=+ a s (5) where TTT0 1 2 0 0 2 3 (0, 1 , 2 , 3 ) = 1, −TTT2 , − 2 , 2 (6) =(1, − sin cos , − sin sin , − cos ) and s is an arbitrary real function of the spatial coordinates and time. The electromagnetic fields (in Gaussian units) corresponding to the above 4-potetials are [12] A s s 2 h E = −U − = sin cosq−− q 2 q i t t x x t (7a) s s 22 h s s h +sin sin q − q − 2 qjk + cos q − q − 2 q t y y tt t z z 4 ssqq BA= = sin sin − cos i zy (7b) sq s q s q s q + −sin cos + cos jk + sin cos − sin z x y x Here U= b0 q is the electric potential, A=(1 q)( b1 i + b 2 j + b 3 k) is the magnetic vector potential, where hq = h q and sq = s q . Also, in the above equations the speed of light has been set equal to unity, since we are working in the natural system of units, where ==c 1. Since the choice of the coordinate system is arbitrary, the direction of motion of the particles can be set to correspond to the +−z direction without loss of generality. In this case ==0 , and equations (7) are simplified, taking the form s 2 h s 2 h s s 2 h E = −q −2 qi + − q − 2 q j + q − q − 2 q k (8a) x x t y y t t z z t ss B = −+qqij (8b) yx Thus, the state of the particle (or antiparticle) will remain unchanged under a wide variety of electromagnetic fields. For example, supposing that the arbitrary function sq depends only on time, equations (8) imply that the state of the particle will not change in the presence of a spatially constant, but with arbitrary time-dependence electric field, applied along the direction of motion of the particle. Further, if the arbitrary function sq is given by the formula sEq= − W1cos kzt W( +) + W 1 xE − W 2 cos kzt W( +) + W 2 y (9) where EWWWWW1,,,, k 1 E 2 2 are real constants, the electromagnetic fields (8) take the form 2h 2 h 2 h E = −2 qij + q + q k + E, B = B WW (10) x t y t z t where EWWWWWWW=E1cos k( z + t) + 1 i + E 2 cos k( z + t) + 2 j (11a) BWWWWWWW=E2cos k( z + t) + 2 i − E 1 cos k( z + t) + 1 j (11b) Τhe electromagnetic fields (11) correspond to an electromagnetic wave, e.g. a laser beam, of arbitrary polarization, propagating along the −−z direction with Poynting vector 5 1 S =EB 4 WW (12) 1 = − E2cos 2 k( z+ t) + + E 2 cos 2 k( z + t) + k 4 WWWWWW1 1 2 2 Thus, the state of massless Dirac particles described by spinors of the general form (2), will not be affected in the presence of a plane electromagnetic wave, e.g. a laser beam, of arbitrary polarization, propagating in a direction opposite to the direction of motion of the particles. Another important remark is that, setting h=− E( z t) in Eq.