Majorana Fermions in a Box
Majorana fermion dynamics may arise at the edge of Kitaev wires or superconductors. Alternatively, it can be engineered by using trapped ions or ultracold atoms in an optical lattice as quantum simulators. This motivates the theoretical study of Majorana fermions confined to a finite volume, whose boundary conditions are characterized by self-adjoint extension parameters. While the boundary conditions for Dirac fermions in -d are characterized by a 1-parameter family, , of self-adjoint extensions, for Majorana fermions is restricted to . Based on this result, we compute the frequency spectrum of Majorana fermions confined to a 1-d interval. The boundary conditions for Dirac fermions confined to a 3-d region of space are characterized by a 4-parameter family of self-adjoint extensions, which is reduced to two distinct 1-parameter families for Majorana fermions. We also consider the problems related to the quantum mechanical interpretation of the Majorana equation as a single-particle equation. Furthermore, the equation is related to a relativistic Schrödinger equation that does not suffer from these problems.
Majorana fermions  result from Dirac fermions  by imposing a reality condition on the Dirac spinor . As a result, Majorana fermions are neutral and are their own antiparticles. In the minimal version of the standard model of particle physics, neutrinos are electrically neutral left-handed Weyl fermions  charged under the electroweak gauge symmetry. In this case, no renormalizable neutrino mass terms exist, and thus, in this minimal theoretical framework, neutrinos are massless particles. Since the observation of neutrino oscillations, it is known that neutrinos indeed must have a small non-zero mass. When one extends the standard model by introducing additional right-handed neutrino fields, one can construct gauge invariant Dirac mass terms which involve the Higgs field and give rise to non-zero neutrino masses via the Higgs mechanism of electroweak symmetry breaking. Gauge invariance then requires that the right-handed neutrino fields are neutral under all gauge interactions. This in turn implies that one can also construct gauge invariant renormalizable Majorana mass terms which do not involve the Higgs field and thus give rise to neutrino masses, unrelated to the energy scale of electroweak symmetry breaking. Since the right-handed component does not participate in the electroweak or strong gauge interactions, Majorana neutrinos are extremely weakly interacting. In particular, like any neutrino they easily penetrate even dense materials and can thus not be confined in any container. Still, in some extensions of the standard model with extra spatial dimensions, neutrinos may be confined to finite regions of the extra-dimensional space.
The confinement of Majorana neutrinos in finite regions of space is a more important issue in condensed matter physics. In particular, Majorana fermions, which may emerge as edge modes of Kitaev wires  or of superconductors , have been discussed in the context of topological quantum computation [7, 8, 9, 10, 11, 12]. Majorana fermions may also arise in engineered systems, such as ultracold atoms in optical lattices or ion traps [13, 14, 15]. We take these systems as a motivation to investigate the Majorana equation, restricted to a finite region in space, using the theory of self-adjoint extensions [16, 17]. In previous work, we have analyzed the Schrödinger, Pauli, and Dirac equations in a similar manner [18, 19]. For example, the perfectly reflecting wall of a box that confines nonrelativistic Schrödinger particles without spin is characterized by a single self-adjoint extension parameter. The most general boundary condition for relativistic Dirac fermions (which generalizes the boundary conditions of the MIT bag model [20, 21, 22]) is characterized by a 4-parameter family of self-adjoint extension parameters . As we will show, imposing the Majorana reality condition on the corresponding Dirac spinor restricts the admissible values of the self-adjoint extension parameters. We then study the Majorana equation both in and in dimensions, with confining spatial boundary conditions.
The rest of this paper is organized as follows. In Section 2 we investigate the Majorana equation in dimensions, review its symmetries, and relate it to a relativistic Schrödinger-type equation with a consistent quantum mechanical single-particle interpretation. In Section 3 we study the self-adjoint extension parameters that characterize a perfectly reflecting boundary. The Majorana equation is then solved for a particle confined to a finite interval. In Section 4 we extend these investigations to dimensions by reviewing the Majorana equation and its symmetries, and by again constructing an equivalent relativistic Schrödinger-type equation. In Section 5 we construct a family of self-adjoint extensions for -d Majorana fermions, confined to a finite region of space. Finally, Section 6 contains our conclusions.
2 Majorana Fermions in Dimensions
In this section we investigate the Majorana equation in dimensions. In particular, we review its symmetry properties and investigate some problems related to its quantum mechanical interpretation as a single-particle equation.
2.1 The Majorana equation in dimensions
Let us first consider the Dirac equation in dimensions
Here is the fermion mass, is the velocity of light, and we have put . A consistent choice of the -matrices is provided in the Dirac basis
where the space-time metric is given by . Alternatively, we can use a Majorana basis
in which the -matrices have purely imaginary entries. The Dirac and the Majorana basis are related by the unitary transformation
In the Majorana basis, the Dirac equation is consistent with imposing the reality condition . In the Dirac basis, the Majorana condition takes the form
Introducing the 2-component Dirac equation reduces to the 1-component Majorana equation
Here we have used . Unlike for the Schrödinger or Dirac equation, the right-hand side of the Majorana equation involves both and . As a consequence, it can not be interpreted as an ordinary quantum mechanical Hamiltonian acting on a wave function . In any case, a quantum mechanical single-particle interpretation is problematical already for the Dirac equation. Putting this caveat aside, one can still use the Dirac Hamiltonian as well as other quantum mechanical operators of the Dirac theory, acting on constrained Majorana wave functions, to define expectation values for Majorana fermions. For the expectation value of the energy one then obtains
In the last step we have used the Majorana equation. As we will see in the next subsection, the total “probability” is indeed conserved. As a consequence, the energy expectation value of a Majorana fermion state, evaluated with the Dirac Hamiltonian, always vanishes. The same is true for the momentum operator
Here we have used partial integration and we have assumed that the wave function vanishes at spatial infinity. The expectation values of energy and momentum vanish because a Majorana fermion is an equal weight superposition of positive and negative energy and momentum states. As a consequence, the solutions of the Majorana equation do not include stationary energy eigenstates with a unique (positive or negative) energy.
2.2 Conserved “probability” current
The Majorana equation is not invariant against multiplication of by an arbitrary phase, but only against a change of sign. As a result, fermion number is conserved only modulo 2. Interestingly, the Majorana equation still inherits the conserved current of the Dirac equation,
which, after imposing the Majorana condition eq.(2.1), takes the form
Indeed, by using the Majorana equation (2.1), we obtain
Although, just like for the Dirac equation, a quantum mechanical single-particle interpretation of the Majorana equation is problematical, and despite the fact that Majorana fermion number is conserved only modulo 2, the continuity equation implies that the total “probability”
2.3 Lorentz invariance
Let us consider a Lorentz boost
Under Lorentz boosts a Dirac spinor transforms as
For a Majorana spinor this implies
It is straightforward to show that the Majorana equation is indeed invariant under this transformation.
2.4 Parity, time-reversal, and charge conjugation
Let us now consider the discrete symmetries P, T, and C for Majorana fermions in one spatial dimension. For a Dirac fermion, the parity transformation P takes the form
This is inconsistent with the Majorana condition . However, combining the Dirac parity operation with a phase multiplication by (which alone is not a symmetry of the Majorana equation) we obtain the Majorana parity transformation
which indeed leaves the Majorana equation invariant
As one would expect, under P the probability and current densities transform as
For a Majorana fermion, we define time-reversal as
which again leaves the Majorana equation invariant
Under time-reversal the probability and current densities transform as
Finally, let us consider charge conjugation C, which for a Dirac fermion takes the form
As it should, this implies that a Majorana fermion is C-invariant
2.5 Propagation of wave packets
By inserting the plane wave ansatz
into the Majorana equation (2.1) one obtains
such that the most general wave packet solution of the Majorana equation is given by
The normalization condition, inherited from the Dirac equation, then takes the form
We have seen that the expectation values of energy and momentum vanish because a Majorana fermion is its own antiparticle. Let us now calculate the expectation value of the velocity operator
which takes the form
and hence is time-independent. It is straightforward but somewhat tedious to calculate the expectation value of the position operator and one obtains
The oscillatory contribution to involving is reminiscent of “Zitterbewegung”. This term is not present for the propagation of wave packets following the nonrelativistic free particle Schrödinger equation for which .
2.6 Relation of the Majorana equation to a relativistic
As we discussed before, it is well known that a quantum mechanical single-particle interpretation of the Dirac or Majorana equation is problematical. The right-hand side of the Majorana equation cannot even be viewed as a quantum mechanical Hamiltonian acting on a wave function, because it involves both and . Let us map to a Schrödinger-type wave function
Remarkably, obeys a relativistic Schrödinger equation with only positive energy states. In particular, the equation for has a consistent quantum mechanical single-particle interpretation, with playing the role of the Hamiltonian. In the context of point-particle relativistic quantum mechanics it is no problem that this Hamiltonian is nonlocal (i.e. is contains derivatives of arbitrary order).
Interestingly, while the Majorana equation allows only a sign change of , the relativistic Schrödinger equation allows global phase changes
which give rise to a nonlocal conserved probability current that was constructed in . This current is not directly related to the conserved local Majorana current of eq.(2.10). One can invert the relation between and to obtain
The simple symmetry of eq.(2.34) then turns into the complicated nonlocal transformation
Similarly, the simple Lorentz transformation for a Majorana spinor of eq.(2.15) turns into a complicated nonlocal transformation rule for , which is not very illuminating in the present context but may be interesting to study in more details in the framework of relativistic quantum mechanics of free particles (in contrast to quantum field theory) .
The Schrödinger-type wave function inherits its P and T symmetry properties from the Majorana “wave function”
The introduction of and its corresponding relativistic Schrödinger equation may provide a consistent quantum mechanical single-particle interpretation of the Majorana equation. Based on this, one could evaluate new expectation values. For example, when evaluated with (rather than with the Dirac spinor that obeys the Majorana condition), one would obtain without any additional contribution from “Zitterbewegung”, such as the one in eq.(2.5). While this is interesting, it is not the subject of the current paper. Here we stay with the original Majorana equation by imposing the Majorana condition on a Dirac spinor, and accept the problems of its quantum mechanical interpretation as a single-particle equation.
3 Majorana Fermions Confined to an Interval
In this section we investigate Majorana fermions in a 1-dimensional box. In particular, we study the self-adjoint extension parameters that characterize a perfectly reflecting boundary condition and we solve the Majorana equation for a particle confined to an interval.
3.1 Perfectly Reflecting Walls for Majorana Fermions
It is well known to the experts, but only rarely emphasized in quantum mechanics textbooks, that a quantum mechanical wave function need not necessarily vanish at a perfectly reflecting wall [26, 27, 28, 18]. In fact, the most general perfectly reflecting Robin boundary condition is characterized by a self-adjoint extension parameter and takes the form . The standard textbook boundary condition then corresponds to the special case . The general Robin boundary condition ensures that the nonrelativistic probability current vanishes at the boundary. This implies that no probability is leaking out of the box. More than this is not required for a consistent unitary quantum mechanical evolution.
Let us begin by studying the -d Dirac equation on the positive -axis with a perfectly reflecting boundary at . In order to investigate the Hermiticity of the Dirac Hamiltonian, we consider
which leads to the Hermiticity condition
We now introduce the self-adjoint extension condition
which reduces eq.(3.2) to
In order for to be self-adjoint, the domains of and must coincide, i.e. . To achieve this, one must request
i.e. must be purely imaginary. Hence, for Dirac fermions in 1-d there is a 1-parameter family of self-adjoint extensions that characterizes a perfectly reflecting wall. The self-adjointness condition eq.(3.3) implies
Hence, as in the nonrelativistic case, the current vanishes at the perfectly reflecting wall.
Majorana fermions obey the additional constraint , such that
Hence, Majorana fermions admit only two discrete self-adjoint extensions, no longer a continuous 1-parameter family.
3.2 Majorana fermion in a 1-d box
Let us consider a 1-d box endowed with perfectly reflecting boundary conditions. For Majorana fermions this means
In order to maintain parity symmetry, we demand that the parity transformed field also obeys the boundary condition
We now make the ansatz
with . Imposing the boundary conditions of eq.(3.8) then implies
If we choose parity-violating boundary conditions with , this implies
which is equivalent to the nonrelativistic momentum quantization condition for the standard box boundary condition . On the other hand, using parity-symmetric boundary conditions with , one obtains the quantization condition
Let us first consider the massless limit , , such that
This solution also applies to massive fermions in the high-energy limit . In the nonrelativistic limit, on the other hand, we obtain
In the low-energy limit this again leads to
It should be noted that the discrete -values resulting from the quantization conditions as well as the corresponding discrete frequencies do not yield stationary energy eigenstates. This is because the solution of eq.(3.10) is again a superposition of states with positive and negative energy .
In the parity-respecting case (), the probability density corresponding to the wave function of eq.(3.10) takes the form
and the normalization factor is given by
In the parity-violating case (), the probability density is given by a more complicated expression, which we don’t display here explicitly. The corresponding probability density is illustrated in Fig.2.
4 Majorana Fermions in Dimensions
In this section we extend our previous considerations from to dimensions. We again consider the Majorana equation and its symmetries as well as a mapping to a relativistic Schrödinger equation.
4.1 The Majorana equation in dimensions
We start out with the Dirac equation in dimensions
For the -matrices we choose the Dirac basis
where are the Pauli matrices and we use the space-time metric . Next, we consider the Majorana basis
in which the matrices again have purely imaginary entries. In this basis, the Majorana condition takes the simple form
The Dirac and the Majorana basis are now related by the unitary transformation
In the Dirac basis, the Majorana condition reads