On real solutions of the Dirac equation for a one-dimensional Majorana particle

We construct general solutions of the time-dependent Dirac equation in (1+1) dimensions with a Lorentz scalar potential, subject to the so-called Majorana condition, in the Majorana representation. In this situation, these solutions are real-valued and describe a one-dimensional Majorana single particle. We specifically obtain solutions for the following cases: a Majorana particle at rest inside a box, a free (i.e., in a penetrable box with the periodic boundary condition), in an impenetrable box with no potential (here we only have four boundary conditions), and in a linear potential. All these problems are treated in a very detailed and systematic way. In addition, we obtain and discuss various results related to real wave functions. Finally, we also wish to point out that, in choosing the Majorana representation, the solutions of the Dirac equation with a Lorentz scalar potential can be chosen to be real but do not need to be real. In fact, complex solutions for this equation can also be obtained. Thus, a Majorana particle cannot be described only with the Dirac equation in the Majorana representation without explicitly imposing the Majorana condition.


I. INTRODUCTION
In 1937, Majorana posed the question of whether a particle with a spin of 1/2 could be its own antiparticle [1]. Essentially, Majorana noted that if the gamma matrices present in the free Dirac equation were forced to satisfy the condition (iγ µ ) * = iγ µ (1 4 is the 4×4 identity matrix), then Eq. It is found that these results are still valid when a (real) Lorentz scalar potential is included into the free Dirac equation. In fact, no other Lorentz potential can be added; for example, in (1+1) dimensions, neither a Lorentz two-vector nor a Lorentz pseudoscalar potential can be added [2]. On a side note, the Klein-Fock-Gordon and Maxwell equations are also real equations; therefore, they can accommodate real solutions. These equations, however, can also have complex solutions.
The particle described by the real-valued solution of the Dirac equation is the Majorana particle, which would be, in fact, a massive fermion that is its own antiparticle (at least in (3+1) dimensions); hence, the particle must be electrically neutral [1,3]. In (1+1) dimensions, it is understood that this particle is the one-dimensional Majorana particle. To date, no elementary particle has been identified as a Majorana particle. Among the known spin-1/2 particles, only neutrinos could be of a Majorana nature [4]. However, a different type of Majorana fermion have recently emerged within condensed matter systems as exotic quasiparticle excitations that are their own antiparticles; for instance, see [4][5][6][7] and the references within.
Thus, in the Majorana representation, the equation that describes a Majorana particle is the Dirac equation together with the reality condition of the wave function Precisely, in the Majorana representation, Ψ * is the charge-conjugate wave function of Ψ, Ψ C [2,7]; therefore, the reality condition given by Eq. (2) expresses the invariance of Ψ under the charge-conjugation operation, i.e., The latter relation is what defines a Majorana particle in any representation [8] and is called the Majorana condition. Certainly, depending on the representation, the equation for the Majorana particle is a real or a complex system of coupled equations or a complex single equation for a single component of the Dirac wave function [2,7]. Moreover, it is important to remember that Ψ C has the same transformation properties as Ψ under proper Lorentz transformations [8]; hence, the Majorana condition is Lorentz covariant.
Traditionally, the Dirac wave function Ψ has been physically associated with the Dirac particle, and its charge-conjugate wave function Ψ C has been associated with its antiparticle.
This is because charge conjugation is essentially an operation that changes a particle into its antiparticle (and vice versa). However, in a single-particle theory, if Ψ describes the particle's state with positive energy, for example, in a given Lorentz scalar potential, then Ψ C describes the particle's state (not its antiparticle's state) with negative energy, in the same potential (and whose absence appears as the presence of a physical antiparticle with positive energy, according to the so-called Dirac hole theory). Apropos of this, in the Majorana representation, Ψ * can describe that particle's state with negative energy, but the Majorana condition continues to give us a real solution for the time-dependent Dirac equation. In any case, one can always invoke the hole theory to obtain a physical picture of the negative energy states (see, for example, pp. 131-134 in Ref. [9]). However, as is well known, the latter takes us out of the Dirac single-particle theory itself (recall the unobservable infinite Dirac sea of negative energy particles, the so-called "vacuum" state).
To recap, The Dirac single-particle two-component wave function Ψ = [ φ 1 φ 2 ] T is generally complexvalued (the symbol T represents the transpose of a matrix). The Dirac gamma matricesγ µ , with µ = 0, 1, satisfyγ µγν +γ νγµ = 2g µν1 2 , where g µν = diag(1, −1) (1 2 is the 2 × 2 identity matrix). In the Majorana representation, the Dirac gamma matrices satisfy (iγ µ ) * = iγ µ , and the operator acting on Ψ in Eq. (4) is real (the raised asterisk * denotes a complex conjugate). In the latter case, the solutions of the Dirac equation (4) can be chosen to be real, but clearly they do not need to be real. For the Majorana representation, we choose the following matrices:γ whereσ y andσ z are the Pauli matrices. Obviously, the matricesγ 0 ′ =σ y andγ 1 ′ = +iσ z , for example, can also be considered to be a Majorana representation (incidentally, the latter choice was used in Refs. [10,11]). These two Majorana representations are related bŷ where Ψ C , the charge-conjugate wave function, also satisfies Eq. (4) (remember that the scalar potential would be independent of the charge of the particle [2]), but this implies that and the matrixŜ C can be chosen to be unitary (up to a phase factor). In the Majorana representation, we haveŜ C =1 2 ; therefore, (iγ µ ) * = iγ µ (as expected), and also Ψ C = Ψ * (by virtue of Eq. (7)). Consequently, the Majorana condition takes the form i.e., the entire Dirac wave function must be real, and it becomes the Majorana wave function.
It is worth mentioning that the Majorana condition can also be written as Ψ = exp(iθ)Ψ C , where θ is an arbitrary phase. We could call this condition the generalized Majorana condition, and it remains covariant under a Lorentz transformation, as it must [8]. Certainly, if this version of the Majorana condition is explicitly used, every solution Ψ that describes a Majorana particle will depend on the phase θ. We can always choose θ = 0, but the freedom to choose a different value for the angle θ could be convenient. Lastly, it is important to note that this generalized Majorana condition also leads to real solutions in the Majorana representation. Moreover, this condition leads to the Majorana condition in the form given in Eq. (6) for a wave functionΨ that is physically indistinguishable from Ψ. In effect, if the generalized Majorana condition is verified, then the following relation is verified: The Dirac equation (4) in its Hamiltonian form is is the formally self-adjoint, or Hermitian, Hamiltonian operator, i.e.,ĥ =ĥ † (in the present paper, the symbol † denotes the Hermitian conjugate, or the adjoint, of a matrix and an operator). The Dirac matrices areα =γ 0γ1 andβ =γ 0 , andp = −i 1 2 ∂/∂x =p † is the formally self-adjoint, or Hermitian, Dirac momentum operator (which is, in the end, a 2 × 2 matrix). In any Majorana representation, we have that (iγ µ ) * = iγ µ ; therefore,α =α * and β = −β * . The latter implies thatĥ and as expected, the operator acting on Ψ in Eq. (9)  In the Majorana representation that we have chosen in the present article, we have that α =σ x andβ =σ y . By substituting these matrices into Eq. (10) and substituting the result obtained into Eq. (9), we obtain the following real system of two coupled equations for the functions φ 1 and φ 2 : (of course, you can cancel the imaginary number!). We can also prove that each component of Ψ satisfies an equation of the Klein-Fock-Gordon type, namely, The latter two equations are only slightly different, but in the free case (S = const), the equations are the same. In the latter situation, Eq. (13) is precisely the free Klein-Fock-Gordon equation with mass mc 2 + const.
Obviously, the system of equations (12) is real because it is precisely the Dirac equation in the Majorana representation. If we explicitly demand that the solutions φ 1 and φ 2 are real-valued, then Ψ = [ φ 1 φ 2 ] T describes a Majorana particle, and the Majorana condition (12) can also be obtained [10], but these do not describe a Majorana particle.

A. Majorana particle at rest inside a box
This case will allow us to present the essential ideas of the subject in an immediate and simple way. To describe a (one-dimensional) Majorana particle at rest inside a box with no potential (p = S = 0), the first thing we must do is solve the following Dirac equation: (see Eqs. (9) and (10)). In this case,ĥ is simply a 2 × 2 Hermitian constant matrix, i.e., h =ĥ † ; in fact,ĥ is also self-adjoint, because it is a bounded operator. It could be considered that this Dirac Hamiltonian operator acts on two-component column vectors Ψ that belong to the Hilbert space of the square integrable functions, where Ω is a region in R of width L (a box) that can be arbitrarily large. In general, Ψ is a function of x with values in C 2 (the two-dimensional complex linear space); hence, the Hilbert space could also be written as H = L 2 (Ω) ⊗ C 2 (a tensor product). The scalar . The eigenfunctions ofĥ are denoted by ψ Explicitly, these eigenfunctions can be written as follows and Moreover, they satisfy the following relations of normalization and orthogonality, as expected: but the following C 2 -orthogonality and normalization conditions for the two-component column vectors u(0) and v(0) (that are charge-conjugates of each other) must also be satisfied: The most general solution of Eq. (14) at t = 0, i.e., the initial state, can be written as where c p=0 (t = 0) are two arbitrary complex constants. But if Ψ(0) is required to be normalized, then the following constraint is verified: Because, in the end, we want to describe a Majorana particle at rest, we impose upon the wave function Ψ(0) the Majorana condition, i.e., Ψ(0) = Ψ C (0) = (Ψ(0)) * . The latter implies that c 0 (0)) * ; therefore, the most general initial state for the Majorana particle at rest can be written as follows where, henceforth, c.c. means a complex conjugate, i.e., the complex conjugate of the expression to the left of the symbol c.c..
The state of the system at any subsequent time t is simply whereÛ is the evolution operator. Note that, in addition to being unitary, this operator is real in this case. Thus, because Ψ(0) is real, then Ψ(t) will also be real (in the same way, if Ψ(0) would have been complex, then Ψ(t) would also be complex). It is worthwhile to mention that, becauseĥ is a matrix,Û(t) admits correctly the power series expansion in Eq. (24).
Precisely, by using the latter expansion when applyingÛ(t) to Ψ(0), we finally obtain from Eq. (23) the Majorana wave function at the instant t with c It can be demonstrated that the mean value ofĥ in the real normalized state Ψ(t) vanishes.
In effect, first of all, we have that where we have made use of Eq. (11). Second, just becauseĥ is a Hermitian matrix, the following expected result is verified: where in the penultimate step, we made use of the scalar product property χ, Φ = Φ, χ * .
Finally, from the results given in Eqs. (26) and (27), we obtain Clearly, the latter is a general result; in fact, the mean value of a formally self-adjoint operatorĤ =Ĥ † (not just a Hermitian matrix), which also satisfies the equalityĤ = −Ĥ * , in a real-valued state, always vanishes. This is simply because the former condition implies that Ĥ Ψ = Ĥ * Ψ , and the latter condition implies Ĥ Ψ∈R = − Ĥ * Ψ∈R ; thus, Ĥ Ψ∈R = 0. For instance, the Dirac momentum operatorp = −i 1 2 ∂/∂x satisfies the relationp = −p * , and because it is also a formally self-adjoint operator, we can obtain the result p Ψ∈R = 0. On the other hand, the Dirac or standard velocity operator,v ≡ cα, is a Hermitian matrix but satisfies the relationv = +v * (this is always so in the Majorana representation); therefore, the mean value v Ψ∈R does not have to be zero, although it is real-valued [13]. Likewise, the so-called classical velocity operator,v cl ≡ c 2pĥ−1 (that corresponds to the formula of classical relativistic mechanics that gives the velocity as a function of momentum and energy) [12] is a real and formally self-adjoint operator, i.e.,v cl =v * cl andv cl =v † cl ; therefore, its (real) expectation value in a real state, v cl Ψ∈R , does not have to be zero.
It is worth mentioning that in the present case (and approximately even when p ≈ 0, i.e., p ≪ mc), each component of Ψ(t) satisfies the equation of a harmonic oscillator with frequency ω ≡ mc 2 / , namely (see Eq. (13)), Solving these equations and obviously respecting the connections between φ 1 (t) and φ 2 (t) from the Dirac equation (14), we obtain which is simply Eq. (23); the latter is precisely the equation that gives us the wave function Ψ(t) from Ψ(0) via the evolution operator. Perhaps, Majorana knew of this simple result, which could have perfectly well been called a "Majorana oscillator" [14].

B. Free Majorana particle in a penetrable box
To describe a free (one-dimensional) Majorana particle (S = 0), we must first solve the following Dirac equation: (see Eqs. (9) and (10)). We place the particle inside a wide interval Ω ⊂ R (e.g., in a wide penetrable box of size L, with ends, for example, at x = 0 and x = L). The Hamiltonian is an unbounded Hermitian operator, i.e.,ĥ =ĥ † ; in fact,ĥ is also a self-adjoint operator because its domain, i.e., the set of Dirac wave functions in H = L 2 (Ω) ⊗ C 2 on whichĥ can act (≡ D(ĥ) ⊂ H), includes the periodic boundary condition, Ψ(L, t) = Ψ(0, t) (the condition that all these functions must satisfy and that we specifically choose to use in this section), and alsoĥΨ ∈ H (for example, see Ref. [2]). Precisely,ĥ satisfies the hermiticity condition (or, in this case, the self-adjointness condition) where Explicitly, these eigenfunctions can be written as follows and where p = 2πn L , n = 0, ±1, ±2, . . . , because we use the periodic boundary condition (it is understood that p = p n , but for simplicity, we do not place the subscript n on the letter p). Moreover, ψ (+) p and ψ (−) p satisfy the following expected relations of orthonormality and orthogonality: where δ p p ′ is the Kronecker delta. Again, the mutually charge-conjugate vectors, u(p) and v(p), must satisfy C 2 -orthogonality and normalization relations, namely, Thus, the free particle's states ψ p (x) are orthonormal eigenstates of the operatorsp andĥ, with eigenvalues p and E p and p and −E p , respectively. However, according to the Dirac hole theory (i.e., abandoning for a moment Dirac's theory as a single-particle theory), the state ψ (−) ±p (x) can be physically associated with the corresponding antiparticle, and it would have momentum ∓p and energy +E p . Certainly, the operatorp for a Dirac particle in a box is also self-adjoint when the boundary condition Ψ(L, t) = Ψ(0, t) is included in its own domain [13,15]. The states ψ then v cl → ∓c, i.e., the spectrum ofv cl is the interval [−c, +c]. Thus, this operator is certainly bounded and self-adjoint [12]. Notice finally that the periodic boundary condition did not provide non-trivial eigensolutions for the range of energies −mc 2 < E < +mc 2 .
The most general wave function at t = 0 can be written as where in the last expression we used the fact that p f p = p f −p . In addition, c provided that Ψ(x, 0) is a normalized wave function. Again, in a hole-theoretic description, the destruction (or removal from the "vacuum") of a negative-energy particle of momentum −p appears as the creation of its corresponding positive-energy antiparticle with momentum +p (a hole in the "vacuum"). We also write the sum in Eq. (39) in terms of ψ −p (0) ≡ d * p , and in the formalism of second quantization, they become time-independent operators, namely, b p →b p (the annihilation or absorption operator for the particle) and d * p →d † p (the creation operator for the antiparticle). In particle physics, it is customary to use these symbols.
Once again, because we are describing a Majorana particle, the initial state must obey the Majorana condition, Ψ(x, 0) = Ψ C (x, 0) = (Ψ(x, 0)) * . With the latter condition imposed upon Ψ(x, 0) (Eq. (39)), we obtain the result c Therefore, the most general initial wave function for the free Majorana particle in this penetrable box takes the form It is then clear that here we need only one type of constant, b p or d p . Abandoning the language of the single-particle theory itself, we could say that a Majorana particle is a superposition of a particle and its antiparticle, but this particle and antiparticle coincide, as expected.
The wave function Ψ(x, t) can be obtained by applying the evolution operatorÛ(t) = exp − i tĥ to the initial wave function Ψ(x, 0), but only formally. In fact, in this case, we cannot use the power series expansion of the exponential, becauseĥ is an unbounded operator, i.e., it cannot act on all the wave functions of the Hilbert space of the system.
However, the correct result is the same as that obtained following the usual prescription (for example, see Ref. [16]), namely, with +∞ p=−∞ c (+) Notice again thatÛ (t) is a real operator and therefore preserves the real character of the initial wave function. Additionally, notice that we would have to multiply the c.c. term in Eqs. (41) and (42) by exp(iθ) in the case of using the generalized Majorana condition. In the end, the real wave function in Eq. (42) can also be written as follows: where +∞ p=−∞ | b p | 2 = 1 2 . In the second quantized theory, the corresponding Majorana quantum field operator is Hermitian, i.e., Ψ(x, t) →Ψ(x, t) =Ψ † (x, t). This is because the wave function Ψ(x, t), i.e., the classical field, is real.
It is worth mentioning that the boundary condition we have used in this section, i.e., the periodic boundary condition, is just one of the infinite non-confining boundary conditions that a (one-dimensional) Majorana particle can support when it is inside a box, namely, with m 2 = 0 (and thus Eq. (44) is a non-confining family of boundary conditions) and (m 0 ) 2 + (m 2 ) 2 = 1. In addition, withĥ =ĥ † is satisfied) [2]. Therefore, if we impose Φ = χ = Ψ in Eq. (32), we obtain, as before, the expression Ψ,ĥΨ ≡ ĥ Ψ = ĥ Ψ, Ψ = Ψ,ĥΨ * = ĥ * Ψ . The latter result together with Eq. (26) (which is verified whenever Ψ is real-valued) leads to the result given in Eq. (28), i.e., ĥ Ψ = 0. Clearly, the boundary conditions in Eqs. (44) and (45) are real boundary conditions when the wave function describes a Majorana particle, i.e., when the wave function is real-valued. Lastly, the results obtained in this subsection obviously apply to those of subsection A if we only maintain the eigenvalue p = 0.

C. Free Majorana particle in an impenetrable box
To describe a free (one-dimensional) Majorana particle (S = 0) that is inside an impenetrable box of width L, we must first solve Eq. (31). In this case, the self-adjoint (Dirac) Hamiltonian operator has a domain that includes a general two-real-parameter family of confining boundary conditions, i.e., a two-real-parameter family at one end of the box, and another family with the same two parameters at the other end [17]. These boundary conditions cancel the probability current density at the ends of the box (which is why we call them confining boundary conditions) [2,17]. Likewise, these boundary conditions imposed upon Φ and χ in Eq. (32) lead to the vanishing of the boundary term present there (this is simply becauseĥ, with these boundary conditions within its domain, is a self-adjoint operator). As it was demonstrated in Ref. [2], from all the boundary conditions that confine a (one-dimensional) Dirac particle inside a box, only four of them can be used to confine a (one-dimensional) Majorana particle (due to the Majorana condition), namely, and  47)). Likewise, and These are two mixed boundary conditions. The authors of Ref. [11] also obtained the result that there are just four confining boundary conditions for a (one-dimensional) Majorana commonly used in that model, but in one dimension [2].
Once again, we use the following notation to identify the normalized eigenfunctions of for the eigenvalues E = +E q = +E −q = + (mc 2 ) 2 − (cq) 2 < mc 2 (positive-energies) and E = −E q > −mc 2 (negativeenergies), respectively. The latter eigenfunctions can be obtained from the former with the Naturally, for each of the four confining boundary conditions, we will have a different orthonormal system of eigenstates ofĥ. However, all these eigenstates will not be eigenstates of the Dirac momentum operatorp = −i 1 2 ∂/∂x; thus, the letter p in the eigenfunctions will just be a label. Moreover, the operatorp in an interval is self-adjoint just when Ψ(L, t) = M Ψ(0, t), whereM is a unitary matrix, but the confining boundary conditions in Eqs.
(46)-(49) are not included in that latter general boundary condition [13,15]. Thus, in the present problem, we do not have a completely (physically) acceptable momentum operator.
However, if we could instantly remove the walls of the box and immediately measure the momentum, that would put the particle in an eigenstate of the momentum operator, which would be, in this case, a self-adjoint operator with a continuous spectrum (see, for example, p. 269 in Ref. [18]). Explicitly, the eigenfunctions of the Hamiltonian can be written as follows. For For φ 1 (L, t) = φ 1 (0, t) = 0, and with energy E = +E q , and the corresponding negative-energy eigensolution, with energy E = −E q . The quantity q is the only positive solution of the equation where L > /mc.
and its solutions must satisfy the inequality p > 0. In this case, the eigensolution for p = 0 (⇒ E = ±mc 2 ) is just the trivial solution. Likewise, we do not have non-trivial eigensolutions in the range of energies −mc 2 < E < +mc 2 .
with the restriction that follows Eq. (65), which comes from the normalization of the initial state and that always remains equal to one. The correct result given by Eq. (66) can be obtained by formally applying the evolution operator to the wave function Ψ(x, 0). In fact, the four Hamiltonian operators considered here act in the manner presented in Eq. (31) but do not have equal domains because each of these has a different boundary condition.
Certainly, they are unbounded operators, by acting only on a subset of the Hilbert space.

D. A Majorana particle in a linear Lorentz scalar potential
In this case, we choose S = kx, where k > 0 is a constant. The Dirac equation we must solve is (see Eqs. (9) and (10)). We have a particle inside the infinite region Ω = R, i.e., the entire real line. In general, the Hamiltonian operator in the Hilbert space H = L 2 (Ω) ⊗ C 2 is essentially self-adjoint on the set of infinitely differentiable two-component functions with compact support in Ω, no matter how great the potential S is at infinity [19].
Specifically, it was shown in Ref. [20] that the spectrum ofĥ is purely discrete and unbounded above and below. Explicitly, the eigenfunctions can be written as follows and ψ (+) and ψ (−) where N = 1, 2, . . .. In Eqs. (69)-(71), a 0 and a N are (real) normalization constants, N ) C , as expected. As usual, these eigenvectors satisfy the relations of orthogonality, namely, The most general Dirac wave function at t = 0 can be written as and assuming that the eigenfunctions are normalized, the condition Ψ(x, 0) = 1 leads to the restriction where c Once again, the real general solution Ψ(x, t) is simply given by with the restriction that follows Eq. (75). Recently, for the problem addressed in this subsection, real particular solutions for a given N were obtained using the factorization method or the so-called supersymmetric procedure [21]. The gamma matrices used therein defined another Majorana representation, namely,γ 0 ′′ = −σ y andγ 1 ′′ = +iσ z . These matrices and the matrices used in the present article (γ 0 = +σ y andγ 1 = −iσ z ) are related byγ µ ′′ =σ xγ µσ x and [ φ ′′

III. CONCLUSIONS
A Majorana particle is its own antiparticle. Therefore, the condition that defines a particle of this type is given by Ψ = Ψ C (i.e., Ψ is invariant under charge conjugation). This system of coupled equations that can describe a Majorana particle as long as its solutions are real-valued. Clearly, the real general solutions obtained here are a superposition of real solutions for a given label or quantum number, but each of these real solutions is not a stationary solution but a superposition of positive and negative energy complex eigenstates (unless it is the real state corresponding to zero energy). Likewise, these real general solutions can perfectly be square integrable.
Because the equation describing the Majorana particle is the Dirac equation with a Lorentz scalar potential -although with the restriction imposed by the Majorana conditionthe study of the supersymmetric procedure in these circumstances turns out to be more important. In particular, this task could give us new solutions that describe the Majorana particle. A very good reference on this matter is Ref. [22].