It is of general theoretical interest to investigate the properties of superluminal matter wave equations for spin one-half particles. One can either enforce superluminal propagation by an explicit substitution of the real mass term for an imaginary mass, or one can use a matrix representation of the imaginary unit that multiplies the mass term. The latter leads to the tachyonic Dirac equation, while the equation obtained by the substitution m
The superluminal propagation of matter waves is a highly intriguing subject which is not without controversy. The subluminal (tardyonic) energy-momentum relation 
needs to be changed to the superluminal (tachyonic) dispersion relation
. Recently, it has been argued that the tachyonic Dirac equation [1,2] provides for a convenient framework for the description of tachyonic particles; in this equation, the mass is multiplied by a matrix representation of the imaginary unit. Here, starting from the Dirac Hamiltonian, we explore a Dirac equation where the mass is explicitly multiplied by the imaginary unit and we find certain fundamental relations for the corresponding spin-1/2 field theory. We also explore certain algebraic properties of modified Dirac theories with an imaginary mass term, and pertaining consequences for the eigenvalue spectrum of the imaginary-mass Dirac Hamiltonian. The tachyonic formulation [3-8] of a fundamental field theory is the only one compatible with Lorentz invariance, and therefore, compatible with special relativity. We exclusively use this concept in the following and avoid any breaking of Lorentz invariance.
According to the summary overview presented in Ref. [
. The best estimate for the neutrino mass square has been determined as negative in all experiments [10-16], but the result has been consistent with a vanishing neutrino mass within experimental error bars. In direct measurements of the neutrino velocity [17-19], the best estimate derived from experimental data has been superluminal (
), but again, consistent with the hypothesis 
within experimental error bars (see also Ref. [
, which (just like all other available experimental results) neither excludes subluminal nor superluminal propagation.
The neutrino is generally regarded as the most prominent candidate for a superluminal particle in the lowenergy domain [20,22-25]. However, the existence of conceivable superluminal particles in hitherto unexplored kinematic regions cannot be excluded, either; our study is of theoretical nature and not tied to a specific particle. It has recently been argued [1,2] that the tachyonic theory of spin-1/2 particles is easier to implement as compared to spinless particles, and we here continue this line of thought by analyzing a theory where the imaginary mass is used explicitly in the Dirac equation, rather than a matrix representation thereof. The latter has been used in Refs. [1,2,22,26,27]. We use natural units with
.
It is useful to recall that the subluminal (tardyonic) Dirac Hamiltonian 
reads
Here, 
is the momentum operator. We use the Dirac matrices in the standard Dirac representation (
, and
),
The Hamiltonian 
can be modified into a Hamiltonian describing superluminal (tachyonic) particles by the simple replacement 
(see Ref. [
Alternatively, one can choose a matrix representation of the imaginary unit, and write the tachyonic Dirac Hamiltonian [1,2,22,26,27] as
with 
and
Both 
and 
are pseudo-Hermitian, which implies that eigenvalues are either real or come in complexconjugate pairs, 
and
. Here, we also show that 
and 
fulfill additional, modified pseudo-Hermiticity conditions (“quasi-pseudo-Hermiticity”), which allow us to further conclude that if 
is a resonance eigenvalue, so is
, and thus, the eigenvalues either come in (real) pairs 
and
, or they occur in the rectangular complex configuration
, 
, 
, and
. The quantization of the imaginary-mass Dirac theory naturally leads to helicity-dependent anticommutators.
We proceed as follows. In Section 2, we derive a few algebraic properties of the Hamiltonians 
and 
which determine the general properties of their spectra. The field theory defined by the Hamiltonian 
is quantized in Section 3. In Section 4, we analyze the Hamiltonian 
which is obtained from (3) by the replacement
. The quantization of the imaginary-mass Dirac theory is shown to yield rather interesting insight into helicity-dependent anticommutators. Conclusions are reserved for Section 5.
It is useful to derive a few algebraic properties of 
and 
which determine the structure of the spectra of these Hamiltonians. We explicitly refer to the coordinatespace representations 
and
We use the following matrices,
These fulfill
, 
, and
. By elementary calculation, we infer that
where the superscript + denotes the Hermitian adjoint. The relations (9a) and (9c) imply the pseudo-Hermiticity of the Hamiltonians 
and
, respectively, in the sense of Refs. [29-38]. As shown in Refs. [1,29], for a pseudo-Hermitian Hamiltonian, if 
is a resonance eigenvalue, so is
. Indeed, it has been shown in Ref. [
has both real eigenvalues (corresponding to plane-wave solutions of positive and negative energy), and also resonances and anti-resonances whose resonance energies are manifestly complex. The resonances correspond to evanescent waves whose wavelength is too long to support superluminal propagation; these waves therefore decay exponentially.
In comparison to the structure of Equations (9a) and (9c), the relations (9b) and (9d) feature an additional minus sign. They correspond to additional “quasi-pseudoHermitian” properties of 
and
. These additional properties imply that if 
is a resonance eigenvalue, so is
. This can be shown as follows. Let 
be an eigenfunction of a general Hamiltonian 
with eigenvalue
. Then, because the spectrum of the Hermitian adjoint of an operator consists of the complex-conjugate eigenvalues, there exists a wavefunction 
with the property
from which we infer that
and so, in view of Equation (9b), we have 
. So, if E is a resonance eigenvalue of
, so is
, with a corresponding eigenvector
. The same property is implied for 
by Equation (9d). If E is real, then Equations (9b) and (9d) imply that energy eigenvalues come in pairs E and
, whereas if they are manifestly complex, then they exhibit a rectangular configuration (in the complex plane) consisting of E, 
, 
and
.
First, we observe that both 
and 
commute with the helicity operator,
where
The quantization of the tachyonic theory defined by the Hamiltonian 
has been discussed in Ref. [
. The corresponding covariant form the imaginary-mass Dirac equation reads as
Of course, it could be argued that the solutions of the imaginary-mass Dirac equation can be written down immediately, by simply replacing 
in the wellknown bispinor solutions of the ordinary Dirac equation, as given in Chapter 2 of Ref. [
where the latter term is the projector onto positive-energy states. Here, 
is the Feynman dagger. When replacing 
in the solution of the Dirac equation given in Equation (2.37) of Ref. [
(using the notation of Ref. [
for the spinors and 
for the Dirac adjoint bispinors. But then,
which is not equal to a compact projector form, as an elementary calculation shows. By contrast, compact formulas for sums over spins can be obtained in the helicity basis, as shown in the following.
For tachyonic particles, in analogy to the formalism developed in Ref. [
are given by
where 
and 
constitute the polar and azimuthal angles of the wave vector k, with
. We also recall the normalized positive-energy chirality and helicity eigenspinors of the massless Dirac equation as follows
,
Canonically, the subscripts 
of the u and v spinors correspond to the chirality (eigenvalue of
), which (in the massless limit) is equal to helicity for positive-energy eigenstates, and equal to the negative of the chirality for negative-energy eigenstates. This is because the positiveand negative-energy solutions are multiplied by 
and
, respectively [see Equation (22)]. Using the relation
we find
The massless limit 
is recovered as 
and
. The negativeenergy eigenstates of the imaginary-mass Dirac equation are given as
In the massless limit, the solutions 
and 
are recovered, 
and
. The states are normalized with respect to the condition
The positiveand negative-energy solutions of the imaginary-mass Dirac equation are thus given as
Here, 
is a solution for positive energy, and 
constitutes a solution for negative energy. All above formulas are valid for
, so that 
is real rather than complex. For
, one encounters resonances, which complete the spectrum. These are derived from Equations (19) and (20) by the identification
The Dirac adjoint is
. By an elementary calculation, one shows that
This can otherwise be seen as follows. One first realizes that the adjoint equation of 
reads as
, and so
Here, we have used the adjoint equation for the “first” 
and the original form of the imaginary-mass Dirac equation for the “second”
.
In analogy to Ref. [
and 
bispinors in the following normalization,
Under charge conjugation, the spinors transform as 
and
. In analogy with Ref. [
Note that the 
prescription selects the resonances (as analytic continuations of the positive-energy solutions) and antiresonances (as analytic continuations of the negative-energy solutions). This ensures that the waves are evanescent in their respective propagation direction in time. The second term in the sum in Equation (26) describes the absorption of a negative-energy tachyonic particle that propagates backward in time; this process is of course equivalent to the emission of a positive-energy antiparticle propagating forward in time by the Feinberg-Sudarshan reinterpretation principle, as explained in Ref. [
where 
creates antiparticles. For the imaginary-mass formalism, we postulate the same anticommutators as in Ref. [
The nonvanishing anticommutators read as follows,
The σ-dependent anticommutator implies that the norm of the right-handed helicity (positive chirality) neutrino one-particle state is negative, and that the righthanded helicity particle state has negative norm and can be excluded from the physical spectrum if one imposes a Gupta-Bleuler type condition (according to Chapter 3 of Ref. [
where 
is the helicity (for positive-energy states) and the negative of the helicity (for negative-energy states) and 
is the tensor product in bispinor space. The following two relations
are analogous to those found for the bispinor solutions of the tachyonic Dirac described in Ref. [
in these equations are due to the quantization conditions (28). Using Equation (31), we can derive the compact result,
where 
is the distribution encountered in Equations (3.55) and (3.56) of Ref. [
The equal-time anticommutator of the fields thus reads as
, with the full, unfiltered Dirac-δ function and 
as well as 
and the time
. Furthermore, with the help of Equations (27) and (31), one obtains the propagator 
(time-ordered product),
The chirality projectors are invariant under multiplication by
, in view of the relation 
.
For consistency reasons, the imaginary-mass Dirac propagator should be connected with a Green function,
where 
is the energy argument of the Green function and 
is the imaginary-mass Dirac Hamiltonian. In momentum space, we can replace 
by
. An elementary calculation then shows that
Introducing the 
prescription as before, we find that
Having determined the propagator, let us briefly comment on the non-invariance of the imaginary-mass Dirac Hamiltonian under time reversal. Indeed, time reversal exchanges the inand out-states of a process. In the calculation of a cross section, one has to square an invariant amplitude, which also exchanges inand out-states, and leads to the occurrence of a propagator of the form
which is obtained from (37) under the replacement
. In the time-reversed Hamiltonian, according to Ref. [
It is instructive to consider the Hamiltonian which is obtained from the imaginary-mass Dirac Hamiltonian in Equation (3) by the replacement
, which amounts to an inversion of the sign of the mass term,
A preliminary remark is in order. Within 
symmetric quantum mechanics [30-38], the one-dimensional quantum mechanical Hamiltonians 
and 
(with x being the coordinate) have been used as paradigmatic examples of an anharmonic (cubic) oscillator with imaginary coupling
. The Hamiltonians 
and 
have the same spectrum [31,33,37,38], and moreover, the eigenvalues can be shown to be analytic functions in the complex G plane where
, and the g plane has a branch cut along the negative real axis.
As is to be expected, the Hamiltonians 
and 
have the same spectrum, because 
fulfills the same algebraic relations (9a) and (9b) as
. Moreover, the plane-wave eigenstates of 
are solutions of the covariant equation
where
for positive-energy and negative-energy states, respectively. We find
The negative-energy eigenstates are given as
The states are normalized with respect to the condition
We normalize the 
and 
bispinors according to
and obtain the following two relations,
These are the analogues of Equations (31a) and (31b) and differ from Equations (31a) and (31b) by the replacement 
in the numerator. However, in the denominator no change takes place, because the denominator is obtained as
. The field operator is
with
, and with an obvious identification of the field operators according to Equation (27). The nonvanishing anticommutators read as follows,
These imply that the inversion of the mass term does not change the fact that again, right-handed particle and left-handed antiparticle states acquire a negative norm. It is very instructive to clarify by an explicit, detailed calculation that the inversion of the mass term does not change the pattern by which helicity components are suppressed for particle and antiparticle states.
In the current work, we investigate the relativistic (tachyonic) quantum theory defined by the Hamiltonian
, which is obtained from the ordinary Dirac Hamiltonian by the simple replacement
. In Section 2, we show that the Hamiltonian 
is pseudoHermitian and has an additional quasi-pseudo-Hermitian property given in Equation (9b). Eigenvalues come in a specific structure in the complex plane. Namely, if E is a resonance eigenvalue, so is
, 
, and
. This pattern is manifest in the spectrum calculated for the tachyonic Dirac Hamiltonian 
in Ref. [
calculated here. Plane-wave solutions of the imaginary-mass Dirac equation are given in Equations (19) and (20).
In Section 3, we complement recent work on the tachyonic Dirac Hamiltonian [
Obviously, the left-handedness of particle states and the right-handedness of antiparticles states imply that both the tachyonic Dirac equation as well as the imaginary-mass Dirac equation represent candidates for the description of neutrinos, if improved experimental techniques [10,18,19,41,42] finally allow us to decide if neutrinos propagate at superluminal or subluminal speeds, which would amount to deciding whether the neutrino mass square is positive or negative [9-16].
This work was supported by the NSF and by the National Institute of Standards and Technology (precision measurement grant). The author acknowledges helpful conversations with B. J. Wundt.
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Mass from Observation of the β Decay of Molecular Tritium,” Physical Review Letters, Vol. 67, No. 8, 1991, p. 957. doi:10.1103/PhysRevLett.67.957
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