Degenerate Solutions to the Massless Dirac and Weyl Equations and a Proposed Method for Controlling the Quantum State of Weyl Particles

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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.

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

 iam + −= 0 (1) satisfying the conditions  † = 0 and  T  2 0 where   are the standard Dirac matrices and  =+    0123 i , 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 aqA= 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 uc uihx yexp,,, 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 uuvv,,, 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]

  cossin −  22

ii eesincos 22 uu== (3a)   cossin  22  ii eesincos −  22

  sincos  22

ii −eecossin 22 vv== (3b)   −sincos  22  ii eecossin  22

The 4-potentials corresponding to the above solutions are

hhhh (aaaa0123,,,,,, ) =  (4) txyz 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

bas=+ (5) where

TTT0  1  200   2  3 (0123,,,1,,,    ) =−−TTT222  (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  

ssqq BA= = sinsincos − i zy (7b)  ssssqqqq +−++−sincoscossincossin jk  zxyx 

Here U b= q 0 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

shshssh222 E = −−+−−+−−qqqqqqq222 ijk  (8a) xx  tyy  ttzzt 

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 EkEWWWWW1122,,,, are real constants, the electromagnetic fields (8) take the form

222hhh E = −+++2 qqqijk EB, B = WW (10) x  ty  tz t where

EWWWWWWW=E1cos k( z + t) + 1  i + E 2 cos  k( z + t) + 2  j (11a)

BijW=+Ek WWWWWW2211 +−+coscos z tEk + ( z) t ( ) (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

1 S =EB 4 WW (12) 1 = − EkztEkzt2222coscos ( +++++) ( )  k 4  WWWWWW1122    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. (2a) and h E= z − t − ( ) in Eq. (2b), they take the familiar form

=+− c ucuiEzt exp p ( 12) ( ) (13a)

 =c v + c vexp − iE z − t a ( 12) ( ) (13b) describing free massless particles (2a) or antiparticles (2b) with energy E moving along the +−z d i r e c t i o n . However, the solutions (2) are much more general, and can describe the state of particles (or antiparticles) in a wide variety of different electromagnetic fields, through an appropriate choice of the arbitrary real function h For example, setting qq hHtxHtyEzt= −−+− ( ) ( ) ( ) (14) 2212 in Eq. (3a), the electromagnetic fields (8), in the case that sq = 0, take the form

dHdH E ==12i + j, B 0 (15) dtdt describing an arbitrary combination of time-dependent electric fields along the x- and y-axes. If, e.g.,

EE0102 H111222( txtHtyt) =+=+ sin,(  sin ) ( ) ( ) (16) 12 where E01,,,,, 1  1E 02  2  2 are real constants, the electromagnetic fields (15) become

E =E01cos( 1 t +  1)i+ E 02 cos(  2 t +  2 ) j, B = 0 (17) corresponding to an arbitrary combination of oscillating electric fields along the x- and y-axes. Thus, the spinor

 E01 q E 02 q p =(cucu1 + 2)exp i x sin( 1 t +  1) + y sin (  2 t +  2 ) + Ezt( − ) (18) 2212

6 describes a massless Dirac particle moving along the +−z direction in an arbitrary combination of oscillating electric fields along the x- and y-axes. Further, the state of a particle described by the above spinor will not be affected in the presence of the wide variety of electromagnetic fields given by Eq. (8).

3. General forms of solutions to the Weyl equation and a proposed method to fully control the quantum state of Weyl particles using appropriate electromagnetic fields

In this section we shall provide some general forms of solutions to the Weyl equation, calculating also the corresponding electromagnetic fields. Based on these results, we will propose a method for fully controlling the quantum state of Weyl particles using appropriate electromagnetic fields.

T In the case that c1 = 0 or c2 = 0,   =2 0, and consequently, according to Theorem 5.4 in [10], the spinors (2) can be written as =(, ) or  = −(, ) , where  is solution to the Weyl equation, either in the form

 ia+= 0 (19a) corresponding to particles with spin parallel to their direction of motion (positive helicity), or in the form

00 iiaa  −220  +00 − = (19b) corresponding to particles with spin antiparallel to their direction of motion (negative  helicity). Here,  are the standard Pauli matrices and aqA= , where q is the electric charge of the particle and A the electromagnetic 4-potential, as in the case of the Dirac equation. It should be mentioned that equations (19a), (19b) can also describe antiparticles with negative and positive helicity respectively. Specifically, all spinors of the form

 cos 2 1 = exp,ih ,( , x y z t) (20) i  e sin  2 are solutions to the Weyl equation (19a) for the real 4-potential (4). Also, all spinors of the form

 −sin  2  2 = expih( x , y , z , t) (21) i  e cos 2

7 are solutions to the Weyl equation (19b) for the same 4-potential. Further, according to Theorem 3.1 in [10], the spinors (20) will also be solutions to the Weyl equation (19a) for the 4-potentials

b1=+ a  1 s  1 (22) where

†1†2†3 111111    (01112131,,,1,,, ) =−−−††† 111111    (23) =−−−(1,sincos,sinsin,cos ) and s is an arbitrary real function of the spatial coordinates and time. Similarly, the spinors (21) will also be solutions to the Weyl equation (19b) for the 4-potentials

b a222 s=+ (24) where

† 1† 2† 3 222222       (02122232,,,1,,, ) = ††† 2  22  22 2 (25) =−−−(1, sincos , sinsin , cos )

It is clear that for the spinors (20), (21), aa12= and 12= , consequently bb12=

Further, the 4-potentials bb12, are the same with the 4-potentials (5), corresponding to massless Dirac particles. Thus, the state of a Weyl particle described by the spinors (20) or (21) will not be affected under the influence of the wide variety of electromagnetic fields given by equations (7). Further, all the physical properties of massless Dirac particles, described by spinors of the form (2), regarding their electromagnetic interactions, will not change in the case of Weyl particles described by spinors (20) or (21). This is expected since Weyl particles are in fact massless Dirac particles with either positive or negative helicity.

Another interesting remark is that the spinors resulting from multiplying the spinors (2), or (20), (21) with an arbitrary real function dw( ) , where

w=++− xyxsin zt cossin sincos (26) are also solutions to the Dirac and Weyl equations for the same 4-potentials. However, in this case, the 4-vector probability current, defined as

†  † 0 1 j  =  (27) † 0 2  † 0 3 

8 in the case of Dirac particles, and in a similar way in the case of Weyl particles, becomes proportional to d 2 . Consequently, the continuity equation [11]

 = j 0 (28) is not generally satisfied, except if d is constant. For example, setting dw=−e xp(  2 ) (29) where  is a real positive constant, we obtain that j  → 0 as

w → (30)

This practically means that particles described by those spinors behave in an unusual way, existing only at a specific “region” of space and time, and disappearing as w tends to infinity. Obviously, the nature of these particles and the way that they could be incorporated in the current framework of Physics, is an interesting open question.

Another particularly important remark regarding Weyl particles, is that the solutions (20), (21) can be generalized, allowing the angles (, ) to be functions of time. In this case, the spinors (20), (21) become

 (t) cos 2 1 = exp,,ih ,( x y z t) (31a) it  (t) e ( ) sin  2

 (t) −sin  2  2 = exp,,ih ,( x y z t) (31b) it  (t) e ( ) cos  2 and it can be shown that they are solutions to the Weyl equations (19a), (19b) for the 4-potentials

h d  h11 d   h d   h d  (a0, a 1 , a 2 , a 3 ) = + , + sin , − cos , − (32a) t dt  x22 dt  y dt  z dt and

h d  h11 d   h d   h d  (a0, a 1  , a 2 , a 3 ) = + , − sin , + cos , + (32b) t dt  x22 dt  y dt  z dt respectively. Further, according to Theorem 3.1 in [10], the spinors (31) will also be solutions to the Weyl equations (19a), (19b) for the 4-potentials

b= a  +  s, b  = a  +  s (33) respectively, where

(0,  1 ,  2 ,  3 ) =( 1, − sin (t) cos ( t) , − sin ( t) sin ( t) , − cos  ( t)) (34)

In both cases s is an arbitrary real function of the spatial coordinates and time.

This result is extremely important because it provides the opportunity to fully control the quantum state of a Weyl particle applying the appropriate electromagnetic fields. Specifically, setting

hx( y,,,sincossinsincos z tEtttxttytzt) =++−0 ( ) ( ) ( ) ( ) ( ) ( ) (35)

the spinors (31) describe a particle with time varying energy Et0 ( ) and propagation direction, defined by the angles (tt), ( ). The electromagnetic fields corresponding to the above choice of h x( y z, ,t , ) are

2 1 dE0 dddd  Ei(tEE) =−+−−+4sin cossin4 sincos4 cos002  2qdtdtdtdt dt  2 1 dddddE0  +−+++−+sin4 cos4sincos4EE00 sin 2 j 2qdt dtdtdtdt  2 1 dE0 dd +−++4cos4 sin E0 2 k 2qdtdtdt (36a)

in the case of the 4-potentials (32a) corresponding to the Weyl equation (19a) and

1 dE d d2  d  d   0  Ei(t) = −4sin cos  + sin  4 E00 sin  +2 + cos  − 4 E cos  2q dt dt dt dt dt 2 1 d d dE0  d  d  +sin − 4EE00 cos  − 4sin  − cos  4 sin  + 2 j 2q dt dt dt dt dt 2 1 dE0 dd + −4cos + 4E0 sin − 2 k 2q dt dt dt (36b) in the case of the 4-potentials (32b) corresponding to the Weyl equation (19b). In both cases the magnetic field is zero. Further, the state of the particle won’t change if the following electromagnetic fields

1  ssdd  Eris ( ,sincoscoscossinsints) =−+−  qtxdtdt  1 ssdd  +−++sinsincossinsincos sj (37a) qtydtdt  1 ssd  +−−sinsins k qtzdt

11ssss  Brijs ( ,sinsincossincoscost) =−+−+  qzyqzx  (37b) 1 ss +−sincossin k qyx corresponding to the 4-potential (s s, , s , s123   ) are added to the fields given by equations (36). Equations (36), (37) describe the family of electromagnetic fields that should be used in order to fully control the state of a Weyl particle, regarding both its energy Et0 ( ) and its propagation direction ((tt), ( )) as functions of time. Obviously, this result is expected to play a particularly important role regarding the practical applications of materials supporting Weyl particles [2, 5-9].

4. A discussion on the transition from degenerate to non-degenerate solutions as the particles acquire mass

In this section we shall discuss the transition from degenerate to non-degenerate solutions as the particles acquire mass, showing that, in an approximate sense, our results can also be extended to the case of massive particles, provided that the rest energy of the particles is much smaller than their total energy and the magnitude of the electromagnetic fields corresponding to the exact degenerate solutions is sufficiently small.

In the case of massive particles, the eigenvectors (3) describing the spin state of the particle become

       cos   − sin    22      

ii       eesin    cos    22    uu==    (38a) pp       cos    sin    E++ m22    E m        ppii    eesin   − cos    E++ m22    E m   

 pp  sincos EmEm++22  ppii − eecossin EmEm++22 vv== (38b)    −sincos  22  ii  eecossin  22 where p =−Em22 is the modulus of the momentum of the particle in natural units ( ==c 1). Substituting the above eigenvectors into the spinors (2) it is found that they are still solutions to the massless Dirac equation for the 4-potentials (4). However, these spinors are no longer degenerate since  †  0. Specifically, it is found that  † =2c1ee2 + c 2 + in the case of particles, and pp( 1 2 ) ( )  † = − 2eecc122+ + e m= E aa ( 12) ( ) in the case of antiparticles, where . However, in the case that the rest energy of the particle is much smaller than its total energy, or mE in natural units, e → 0 , and consequently the spinors (2) with the “massive” eigenvectors (38) can be considered as nearly degenerate, since † † →pp 0 and →aa 0.

Further, the spinors (2) are no longer solutions to the massless Dirac equation for the 4-potentials (5). Indeed, substituting the spinors (2) with the spin eigenvectors given by equations (38) into the massless Dirac equation for the 4-potentials (5), it takes the form

1− e i+= b 1− s * (39a) ppp   1+ e  in the case of particles, and

1− e i  +b = − 1− s * (39b) a a a  1+ e 

* * in the case of antiparticles. Here  p , a are the unperturbed spinors for particles and antiparticles respectively. In the case that the rest energy of the particles is much smaller than their total energy (e 1), the above equations take the simpler form

* ie p +b  p = s  p (40a)

 * ie a +b  a = − s  a (40b)

This result implies that as the ratio of the rest energy to the total energy of the particles increases, the function s should be restricted to smaller values, suppressing the effects of degeneracy. On the other hand, as the ratio of the rest energy to the total energy of the particles decreases, the function s is allowed to take larger values, and the degeneracy becomes more evident. Finally, as the ratio of the rest energy to the total energy of the particles tends to zero, there is no restriction on the values of the function and the theory of degeneracy becomes fully applicable.

5. Conclusions In conclusion, we have found general forms of degenerate solutions to the massless Dirac and Weyl equations, and calculated the electromagnetic fields corresponding to these solutions. Specifically, we have shown that the state of massless Dirac - described by degenerate spinors - and Weyl particles is not affected by the presence of a spatially constant, but with arbitrary time-dependence, electric field, applied along the direction of motion of the particles, or a plane electromagnetic wave, e.g. a laser beam, propagating in a direction opposite to the direction of motion of the particles. Further, based on a special class of solutions to the Weyl equation, we have proposed a method for fully controlling the quantum state of Weyl particles using appropriate electromagnetic fields. Obviously, these results are expected to play a significant role regarding the practical applications of materials supporting massless Dirac and/or Weyl particles. Finally, we have studied the transition from degenerate to non-degenerate solutions as the particles acquire mass, showing that the concept of degeneracy is still valid, in an approximate sense, provided that the rest energy of the particles is much smaller than their total energy and the magnitude of the electromagnetic fields corresponding to the exact degenerate solutions is sufficiently small.

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