Spin and pseudospin symmetries and the equivalent spectra of relativistic spin-1/2
and spin-0 particles
P. Alberto
Physics Department and Center for Computational Physics,
University of Coimbra, P-3004-516 Coimbra, Portugal
A. S. de Castro
Departamento de F́ısica e Qúımica, Universidade Estadual Paulista, 12516-410 Guaratinguetá, SP, Brazil
M. Malheiro
Departamento de F́ısica, Instituto Tecnológico de Aeronáutica,
CTA, 12228-900, São José dos Campos, SP, Brazil
and Instituto de F́ısica, Universidade Federal Fluminense, 24210-340 Niterói, Brazil
(Dated: November 4, 2018)
We show that the conditions which originate the spin and pseudospin symmetries in the Dirac
equation are the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0
particles in the presence of vector and scalar potentials. The conclusions do not depend on the
particular shapes of the potentials and can be important in different fields of physics. When both
scalar and vector potentials are spherical, these conditions for isospectrality imply that the spin-
orbit and Darwin terms of either the upper component or the lower component of the Dirac spinor
vanish, making it equivalent, as far as energy is concerned, to a spin-0 state. In this case, besides
energy, a scalar particle will also have the same orbital angular momentum as the (conserved) orbital
angular momentum of either the upper or lower component of the corresponding spin-1/2 particle.
We point out a few possible applications of this result.
PACS numbers: 11.30.-j,03.65.Pm
When describing some strong interacting systems it is often useful, because of simplicity, to approximate the
behavior of relativistic spin-1/2 particles by scalar spin-0 particles obeying the Klein-Gordon equation. An example
is the case of relativistic quark models used for studying quark-hadron duality because of the added complexity of
structure functions of Dirac particles as compared to scalar ones. It turns out that some results (e.g., the onset of
scaling in some structure functions) almost do not depend on the spin structure of the particle [1]. In this work we
will give another example of an observable, the energy, whose value may not depend on the spinor structure of the
particle, i.e., whether one has a spin-1/2 or a spin-0 particle. We will show that when a Dirac particle is subjected
to scalar and vector potentials of equal magnitude, it will have exactly the same energy spectrum as a scalar particle
of the same mass under the same potentials. As we will see, this happens because the spin-orbit and Darwin terms
in the second-order equation for either the upper or lower spinor component vanish when the scalar and vector
potentials have equal magnitude. It is not uncommon to find physical systems in which strong interacting relativistic
particles are subject to Lorentz scalar potentials (or position-dependent effective masses) that are of the same order
of magnitude of potentials which couple to the energy (time components of Lorentz four-vectors). For instance, the
scalar and vector (hereafter meaning time-component of a four-vector potential) nuclear mean-field potentials have
opposite signs but similar magnitudes, whereas relativistic models of mesons with a heavy and a light quark, like D-
or B-mesons, explain the observed small spin-orbit splitting by having vector and scalar potentials with the same sign
and similar strengths [2].
It is well-known that all the components of the free Dirac spinor, i.e., the solution of the free Dirac equation, satisfy
the free Klein-Gordon equation. Indeed, from the free Dirac equation
(i~γµ∂µ −mc)Ψ = 0 (1)
one gets
(−i~γν∂ν −mc)(i~γ
µ∂µ −mc)Ψ = (~
2∂µ∂µ +m
2c2)Ψ = 0 , (2)
where use has been made of the relation γµγν∂µ∂ν = ∂µ∂
µ. In a similar way, for the time-independent free Dirac
equation we would have
(cα · p+ βmc2)ψ = (−i~cα · ∇+ βmc2)ψ = Eψ , (3)
http://arxiv.org/abs/0704.0353v1
where, as usual, ψ(r) = Ψ(r, t) exp (i E t/~), α = γ0γ and β = γ0. Then, by left multiplying Eq. (3) by cα ·p+βmc2,
one gets the time-independent free Klein-Gordon equation
(c2p2 +m2c4)ψ = (−~2c2∇2 +m2c2)ψ = E2ψ , (4)
where the relation {β,α} = 0 was used. This all means that the free four-component Dirac spinor, and of course all
of its components, satisfy the Klein-Gordon equation. This is not surprising, because, after all, both free spin-1/2 and
spin-0 particles obey the same relativistic dispersion relation, E2 = p2c2 +m2c4, in spite of having different spinor
structures and thus different wave functions. Since there is no spin-dependent interaction, one expects both to have
the same energy spectrum.
We consider now the case of a spin-1/2 particle subject to a Lorentz scalar potential Vs plus a vector potential Vv.
The time-independent Dirac equation is given by
[cα · p+ β(mc2 + Vs)]ψ = (E − Vv)ψ (5)
It is convenient to define the four-spinors ψ± = P±ψ = [(I ± β)/2]ψ such that
, (6)
where φ and χ are respectively the upper and lower two-component spinors. Using the properties and anti-
commutation relations of the matrices β and α we can apply the projectors P± to the Dirac equation (5) and
decompose it into two coupled equations for ψ+ and ψ−:
cα · pψ− + (mc
2 + Vs)ψ+ = (E − Vv)ψ+ (7)
cα · pψ+ − (mc
2 + Vs)ψ− = (E − Vv)ψ− . (8)
Applying the operator cα · p on the left of these equations and using them to write ψ+ and ψ− in terms of α · pψ−
and α · pψ+ respectively, we finally get second-order equations for ψ+ and ψ−:
c2p2 ψ+ + c
[α · p∆]α · pψ+
E −∆+mc2
= (E −∆+mc2)(E − Σ−mc2)ψ+ (9)
c2p2 ψ− + c
[α · pΣ]α · pψ−
E − Σ−mc2
= (E −∆+mc2)(E − Σ−mc2)ψ− (10)
where the square brackets [ ] mean that the operator α · p only acts on the potential in front of it and we defined
Σ = Vv + Vs and ∆ = Vv − Vs. The second term in these equations can be further elaborated noting that the Dirac
αi matrices satisfy the relation αiαj = δij +
iǫijkSk where Sk, k = 1, 2, 3, are the spin operator components. The
second-order equations read now
c2 p2 ψ+ + c
[p∆] · pψ+ +
[p∆]× p · S ψ+
E −∆+mc2
= (E −∆+mc2)(E − Σ−mc2)ψ+ (11)
c2 p2 ψ− + c
[pΣ] · pψ− +
[pΣ]× p · S ψ−
E − Σ−mc2
= (E −∆+mc2)(E − Σ−mc2)ψ−. (12)
Now, if p∆ = 0, meaning that ∆ is constant or zero (if ∆ goes to zero at infinity, the two conditions are equivalent),
then the second term in eq. (11) disappears and we have
c2 p2ψ+ = (E −∆+mc
2)(E − Σ−mc2)ψ+ = [(E − Vv)
2 − (mc2 + Vs)
2]ψ+ , (13)
which is precisely the time-independent Klein-Gordon equation for a scalar potential Vs plus a vector potential Vv[14].
Since the second-order equation determines the eigenvalues for the spin-1/2 particle, this means that when p∆ = 0,
a spin-1/2 and a spin-0 particle with the same mass and subject to the same potentials Vs and Vv will have the same
energy spectrum, including both bound and scattering states. This last sufficient condition for isospectrality can be
relaxed to demand that just the combination mc2+Vs be the same for both particles, allowing them to have different
masses. This is so because this weaker condition does not change the gradient of ∆ and Σ and therefore the condition
p∆ = 0 will still hold. On the other hand, if the scalar and vector potentials are such that pΣ = 0, we would obtain a
Klein-Gordon equation for ψ−, and again the spectrum for spin-0 and spin-1/2 particles would be the same, provided
they are subjected to the same vector potential and mc2 + Vs is the same for both particles. If both Vs and Vv are
central potentials, i.e., only depend on the radial coordinate, then the numerators of the second terms in equations
(11) and (12) read
[p∆] · pψ+ +
[p∆]× p · S ψ+ = −~
∆′L · S ψ+ (14)
[pΣ] · pψ− +
[pΣ]× p · S ψ− = −~
Σ′L · S ψ− , (15)
where ∆′ and Σ′ are the derivatives with respect to r of the radial potentials ∆(r) and Σ(r), and L = r × p is the
orbital angular momentum operator. From these equations ones sees that these terms, which set apart the Dirac
second-order equations for the upper and lower components of the Dirac spinor from the Klein-Gordon equation and
thus are the origin of the different spectra for spin-1/2 and spin-0 particles, are composed of a derivative term, related
to the Darwin term which appears in the Foldy-Wouthuysen expansion, and a L · S spin-orbit term. If ∆′ = 0
(Σ′ = 0), then there is no spin-orbit term for the upper (lower) component of the Dirac spinor. In turn, since
the second-order equation determines the energy eigenvalues, this means that the orbital angular momentum of the
respective component is a good quantum number of the Dirac spinor. This can be a bit surprising, since one knows
that in general the orbital quantum number is not a good quantum number for a Dirac particle, since L2 does not
commute with a Dirac Hamiltonian with radial potentials. The reason why this does not happen in these cases was
reported in Refs. [3, 4], and we now review it in a slight different fashion. Let us consider in more detail the case of
spherical potentials such that ∆′ = 0. One knows that a spinor that is a solution of a Dirac equation with spherically
symmetric potentials can be generally written as
ψjm(r) =
gj l(r)
Yj lm(r̂)
j l̃ m
. (16)
where Yj lm are the spinor spherical harmonics. These result from the coupling of spherical harmonics and two-
dimensional Pauli spinors χms , Yj lm =
〈 l ml ; 1/2ms | j m 〉Ylmlχms , where 〈 l ml ; 1/2ms | j m 〉 is a
Clebsch-Gordan coefficient and l̃ = l ± 1, the plus and minus signs being related to whether one has aligned or
anti-aligned spin, i.e., j = l ± 1/2. The spinor spherical harmonics for the lower component satisfy the relation
j l̃m
= −σ · r̂Yj lm. The fact that the upper and lower components have different orbital angular momenta is related
to the fact, mentioned before, that L2 does not commute with the Dirac Hamiltonian
H = cα · p+ β(Vs +mc
2) + Vv = cα · p+ βmc
2 +ΣP+ +∆P− , (17)
where P± are the projectors defined above. However, when ∆
′ = 0, there is an extra SU(2) symmetry of H (so-called
“spin symmetry”) as first shown by Bell and Ruegg [5]. When we have spherical potentials, Ginocchio showed that
there is an additional SU(2) symmetry (for a recent review see [4]). The generators of this last symmetry are
L = LP+ +
α · pLα · pP− =
0 Up LUp
, (18)
where Up = σ · p/(
p2) is the helicity operator. One can check that L commutes with the Dirac Hamiltonian,
[H,L] = [cα · p,LP+ +
α · pLα · pP−] + [∆,
α · pLα · p] + [Σ,L]
= [∆,
α · pLα · p ] = 0 , (19)
where the last equality comes from the fact that ∆′ = 0. The Casimir L2 operator is given by L2 = L2P+ +
α · pP−. Applying this operator to the spinor ψjm (16), we get
2ψjm = L
α · pL2 α · pψ−
= ~2l(l+ 1)ψ+
α · p cL2 ψ+jm
E −∆+mc2
= ~2l(l + 1)ψ+
+ ~2l(l + 1)ψ−
= ~2l(l + 1)ψjm , (20)
where ψ±jm = P±ψjm and we used the relation, valid when ∆
′ = 0, ψ+jm = (E −∆ +mc
α · p
ψ−jm. From (20) we
see that ψjm is indeed an eigenstate of L
2. Thus the orbital quantum number of the upper component l is a good
quantum number of the system when the spherical potentials Vs(r) and Vv(r) are such that Vv(r) = Vs(r)+C∆, where
C∆ is an arbitrary constant. Also, according to we have said before, there is a state of a spin-0 particle subjected to
these same spherical potentials (or, at least, with a scalar potential such that the sum Vs +mc
2 is the same) that
has the same energy and the same orbital angular momentum as ψjm. In addition, the wave function of this scalar
particle would be proportional to the spatial part of the wave function of the upper component.
Note that the generator of the “spin symmetry” S is given by a similar expression as (18) just replacing L by ~/2σ
[4, 5], meaning that S2 ≡ S2 = 3/4 ~2I so that spin is also a good quantum number, as would be expected. Actually,
one can show that the total angular momentum operator J can be written as L + S, so that l, ml (eigenvalue of
Lz), s = 1/2, ms (eigenvalue of Sz) are good quantum numbers. Then, of course, j and m = ml +ms are also good
quantum numbers, but only in a trivial way, because there is no longer spin-orbit coupling. Therefore, in the spinor
(16) one could just replace the spinor spherical harmonic Yj lm by Yl mlχms and Yj l̃m by −σ · r̂ Yl mlχms . Note that if
∆ is a nonrelativistic potential, ∆ ≪ mc2 and ∆′ ≪ m2c4/(~c), i.e., it is slowly varying over a Compton wavelength.
In this case, the spin-orbit term will also get suppressed. In fact, the derivative of the ∆ potential is the origin of
the well-known relativistic spin-orbit effect which appears as a relativistic correction term in atomic physics or in the
v/c Foldy-Wouthuysen expansion (only the derivative of Vv appears because usually no Lorentz scalar potential Vs is
considered, and therefore ∆ = Vv).
When Σ′ = 0, or Vv(r) = −Vs(r) + CΣ, with CΣ an arbitrary constant, there is again a SU(2) symmetry, usually
called pseudospin symmetry ([5, 6]) which is relevant for describing the single-particle level structure of several nuclei.
This symmetry has a dynamical character and cannot be fully realized in nuclei because in Relativistic Mean-field
Theories the Σ potential is the only binding potential for nucleons [7, 8]. For harmonic oscillator potentials this is
no longer the case, since ∆, acting as an effective mass going to infinity, can bind Dirac particles [9, 10], even when
Σ = 0. As before, in the special case of spherical potentials, there is another SU(2) symmetry whose generators are
α · pLα · pP+ +LP− =
Up LUp 0
. (21)
In the same way as before, applying L̃
to ψjm, we would find that L̃
ψjm = ~
2 l̃(l̃ + 1)ψjm, that is, this time it
is the orbital quantum number of the lower component l̃ which is a good quantum number of the system and can
be used to classify energy levels. Again, provided the vector and scalar potentials are adequately related, there
would be a corresponding state of a spin-0 particle with the same energy and same orbital angular momentum l̃,
and, furthermore, its wave function would be proportional to the spatial part of the wave function of the lower
component. As before, the pseudospin symmetry generator S̃ can be obtained from L̃ by replacing L by ~/2σ. The
good quantum numbers of the system would be, besides l̃, m
, s̃ ≡ s = 1/2 and ms̃. Again, J = L̃ + S̃. It is
interesting that, as has been noted by Ginocchio [9], the generators of spin and pseudospin symmetries are related
through a γ5 transformation since S̃ = γ5Sγ5 and L̃ = γ5Lγ5. This property was used in a recent work to relate
spin symmetric and pseudospin symmetric spectra of harmonic oscillator potentials [11]. There it was shown that
for massless particles (or ultrarelativistic particles) the spin- and pseudo-spin spectra of Dirac particles are the same.
In addition, this means that spin-symmetric massless eigenstates of γ5 would be also pseudo-spin symmetric and
vice-versa. Since in this case ∆ = Σ = 0, or Vv = Vs = 0, this is, of course, just another way of stating the well-known
fact that free massless Dirac particles have good chirality.
Naturally, for free spin-1/2 particles described by spherical waves, both l and l̃ are good quantum numbers, which
just reflects the fact that one can have free spherical waves with any orbital angular momentum for the upper or
lower component and still have the same energy, as long as their linear momentum magnitude is the same, or, put in
another way, the energy of a free spin-1/2 particle cannot depend on its direction of motion.
In summary, we showed that when a relativistic spin-1/2 particle is subject to vector and scalar potentials such
that Vv = ±Vs + C±, where C± are constants, its energy spectrum does not depend on their spinorial structure,
being identical to the spectrum of a spin-0 particle which has no spinorial structure. This amounts to say that if
the potentials have these configurations there is no spin-orbit coupling and Darwin term. If the scalar and vector
potentials are spherical, one can classify the energy levels according to the orbital angular momentum quantum
number of either the upper or the lower component of the Dirac spinor. This would then correspond to having a
spin-0 particle with orbital angular momentum l or l̃, respectively. This spectral identity can of course happen only
with potentials which do not involve the spinorial structure of the Dirac equation in an intrinsic way. For instance, a
tensor potential of the form iβσµν (∂µAν − ∂νAµ) does not have an analog in the Klein-Gordon equation, so that one
could not have a spin-0 particle with the same spectrum as a spin-1/2 particle with such a potential. This is the case
of the so-called Dirac oscillator [12] (see [10] for a complete reference list), in which the Dirac equation contains a
potential of the form iβσ0imωri = imωβα · r. Another important potential, the electromagnetic vector potential A,
which is the spatial part of the electromagnetic four-vector potential, can be added via the minimal coupling scheme
to both the Dirac and the Klein-Gordon equations. Since α · (p− eA)α · (p− eA) = (p− eA)2 + 2e~∇×A · S, the
spectra of spin-0 and spin-1/2 particles cannot be identical as long as there is a magnetic field present, even though
the condition Vv = ±Vs +C± is fulfilled. It is important also to remark that, since for an electromagnetic interaction
Vv is the time-component of the electromagnetic four-vector potential, this last condition is gauge invariant in the
present case, in which we are dealing with stationary states, i.e, time-independent potentials. So, in the absence of a
external magnetic field (allowing, for instance, an electromagnetic vector potential A which is constant or a gradient
of a scalar function), a spin-0 and spin-1/2 particle subject to the same electromagnetic potential Vv and a Lorentz
scalar potential fulfilling the above relation would have the same spectrum.
The remark made above about the similarity of spin-0 and spin-1/2 wave functions can be relevant for calculations
in which the observables do not depend on the spin structure of the particle, like some structure functions. One such
calculation was made by Paris [13] in a massless confined Dirac particle, in which Vv = Vs. It would be interesting
to see how a Klein-Gordon particle would behave under the same potentials. More generally, this spectral identity
can also have experimental implications in different fields of physics, since, should such an identity be found, it would
signal the presence of a Lorentz scalar field having a similar magnitude as that of a time-component of a Lorentz
vector field, or at least differing just by a constant.
Acknowledgments
We acknowledge financial support from CNPQ, FAPESP and FCT (POCTI) scientific program.
[1] S. Jeschonnek and J. W. Van Orden, Phys. Rev. D 69, 054006 (2004).
[2] P. R. Page, T. Goldman, and J. N. Ginocchio, Phys. Rev. Lett. 86, 204.
[3] J. N. Ginocchio and A. Leviatan, Phys. Lett. B425, 1 (1998).
[4] J. N. Ginocchio, Phys. Rep. 414 165 (2005).
[5] J. S. Bell and H. Ruegg, Nucl. Phys. B98, 151 (1975).
[6] J. N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997).
[7] P. Alberto, M. Fiolhais, M. Malheiro, A. Delfino, and M. Chiapparini, Phys. Rev. Lett. 86, 5015 (2001).
[8] P. Alberto, M. Fiolhais, M. Malheiro, A. Delfino, and M. Chiapparini, Phys. Rev. C 65, 034307 (2002).
[9] J. N. Ginocchio, Phys. Rev. Lett. 95, 252501 (2005).
[10] R. Lisboa, M. Malheiro, A. S. de Castro, P. Alberto, and M. Fiolhais, Phys. Rev. C 69, 024319 (2004).
[11] A. S. de Castro, P. Alberto, R. Lisboa, and M. Malheiro, Phys. Rev. C 73, 054309 (2006).
[12] D. Itô, K. Mori, and E. Carriere, Nuovo Cimento A 51, 1119 (1967); M. Moshinsky and A. Szczepaniak, J. Phys. A 22,
L817 (1989).
[13] M. W. Paris, Phys. Rev. C 68, 025201 (2003).
[14] There are some authors who introduce a scalar potential Vs in the Klein-Gordon equation by making the replacement
m2c4 → m2c4 +V2
. Here we introduce it, as most authors do, as an effective mass m∗ 2 = (m+Vs/c
2)2, since it is the way
that it is introduced in the Dirac equation. The two potentials are related by V2
= (mc2 + Vs)
−m2c4.
	Acknowledgments
	References

  We show that the conditions which originate the spin and pseudospin
symmetries in the Dirac equation are the same that produce equivalent energy
spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector
and scalar potentials. The conclusions do not depend on the particular shapes
of the potentials and can be important in different fields of physics. When
both scalar and vector potentials are spherical, these conditions for
isospectrality imply that the spin-orbit and Darwin terms of either the upper
component or the lower component of the Dirac spinor vanish, making it
equivalent, as far as energy is concerned, to a spin-0 state. In this case,
besides energy, a scalar particle will also have the same orbital angular
momentum as the (conserved) orbital angular momentum of either the upper or
lower component of the corresponding spin-1/2 particle. We point out a few
possible applications of this result.

Spin and pseudospin symmetries and the equivalent spectra of relativistic spin-1/2
and spin-0 particles
P. Alberto
Physics Department and Center for Computational Physics,
University of Coimbra, P-3004-516 Coimbra, Portugal
A. S. de Castro
Departamento de F́ısica e Qúımica, Universidade Estadual Paulista, 12516-410 Guaratinguetá, SP, Brazil
M. Malheiro
Departamento de F́ısica, Instituto Tecnológico de Aeronáutica,
CTA, 12228-900, São José dos Campos, SP, Brazil
and Instituto de F́ısica, Universidade Federal Fluminense, 24210-340 Niterói, Brazil
(Dated: November 4, 2018)
We show that the conditions which originate the spin and pseudospin symmetries in the Dirac
equation are the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0
particles in the presence of vector and scalar potentials. The conclusions do not depend on the
particular shapes of the potentials and can be important in different fields of physics. When both
scalar and vector potentials are spherical, these conditions for isospectrality imply that the spin-
orbit and Darwin terms of either the upper component or the lower component of the Dirac spinor
vanish, making it equivalent, as far as energy is concerned, to a spin-0 state. In this case, besides
energy, a scalar particle will also have the same orbital angular momentum as the (conserved) orbital
angular momentum of either the upper or lower component of the corresponding spin-1/2 particle.
We point out a few possible applications of this result.
PACS numbers: 11.30.-j,03.65.Pm
When describing some strong interacting systems it is often useful, because of simplicity, to approximate the
behavior of relativistic spin-1/2 particles by scalar spin-0 particles obeying the Klein-Gordon equation. An example
is the case of relativistic quark models used for studying quark-hadron duality because of the added complexity of
structure functions of Dirac particles as compared to scalar ones. It turns out that some results (e.g., the onset of
scaling in some structure functions) almost do not depend on the spin structure of the particle [1]. In this work we
will give another example of an observable, the energy, whose value may not depend on the spinor structure of the
particle, i.e., whether one has a spin-1/2 or a spin-0 particle. We will show that when a Dirac particle is subjected
to scalar and vector potentials of equal magnitude, it will have exactly the same energy spectrum as a scalar particle
of the same mass under the same potentials. As we will see, this happens because the spin-orbit and Darwin terms
in the second-order equation for either the upper or lower spinor component vanish when the scalar and vector
potentials have equal magnitude. It is not uncommon to find physical systems in which strong interacting relativistic
particles are subject to Lorentz scalar potentials (or position-dependent effective masses) that are of the same order
of magnitude of potentials which couple to the energy (time components of Lorentz four-vectors). For instance, the
scalar and vector (hereafter meaning time-component of a four-vector potential) nuclear mean-field potentials have
opposite signs but similar magnitudes, whereas relativistic models of mesons with a heavy and a light quark, like D-
or B-mesons, explain the observed small spin-orbit splitting by having vector and scalar potentials with the same sign
and similar strengths [2].
It is well-known that all the components of the free Dirac spinor, i.e., the solution of the free Dirac equation, satisfy
the free Klein-Gordon equation. Indeed, from the free Dirac equation
(i~γµ∂µ −mc)Ψ = 0 (1)
one gets
(−i~γν∂ν −mc)(i~γ
µ∂µ −mc)Ψ = (~
2∂µ∂µ +m
2c2)Ψ = 0 , (2)
where use has been made of the relation γµγν∂µ∂ν = ∂µ∂
µ. In a similar way, for the time-independent free Dirac
equation we would have
(cα · p+ βmc2)ψ = (−i~cα · ∇+ βmc2)ψ = Eψ , (3)
http://arxiv.org/abs/0704.0353v1
where, as usual, ψ(r) = Ψ(r, t) exp (i E t/~), α = γ0γ and β = γ0. Then, by left multiplying Eq. (3) by cα ·p+βmc2,
one gets the time-independent free Klein-Gordon equation
(c2p2 +m2c4)ψ = (−~2c2∇2 +m2c2)ψ = E2ψ , (4)
where the relation {β,α} = 0 was used. This all means that the free four-component Dirac spinor, and of course all
of its components, satisfy the Klein-Gordon equation. This is not surprising, because, after all, both free spin-1/2 and
spin-0 particles obey the same relativistic dispersion relation, E2 = p2c2 +m2c4, in spite of having different spinor
structures and thus different wave functions. Since there is no spin-dependent interaction, one expects both to have
the same energy spectrum.
We consider now the case of a spin-1/2 particle subject to a Lorentz scalar potential Vs plus a vector potential Vv.
The time-independent Dirac equation is given by
[cα · p+ β(mc2 + Vs)]ψ = (E − Vv)ψ (5)
It is convenient to define the four-spinors ψ± = P±ψ = [(I ± β)/2]ψ such that
, (6)
where φ and χ are respectively the upper and lower two-component spinors. Using the properties and anti-
commutation relations of the matrices β and α we can apply the projectors P± to the Dirac equation (5) and
decompose it into two coupled equations for ψ+ and ψ−:
cα · pψ− + (mc
2 + Vs)ψ+ = (E − Vv)ψ+ (7)
cα · pψ+ − (mc
2 + Vs)ψ− = (E − Vv)ψ− . (8)
Applying the operator cα · p on the left of these equations and using them to write ψ+ and ψ− in terms of α · pψ−
and α · pψ+ respectively, we finally get second-order equations for ψ+ and ψ−:
c2p2 ψ+ + c
[α · p∆]α · pψ+
E −∆+mc2
= (E −∆+mc2)(E − Σ−mc2)ψ+ (9)
c2p2 ψ− + c
[α · pΣ]α · pψ−
E − Σ−mc2
= (E −∆+mc2)(E − Σ−mc2)ψ− (10)
where the square brackets [ ] mean that the operator α · p only acts on the potential in front of it and we defined
Σ = Vv + Vs and ∆ = Vv − Vs. The second term in these equations can be further elaborated noting that the Dirac
αi matrices satisfy the relation αiαj = δij +
iǫijkSk where Sk, k = 1, 2, 3, are the spin operator components. The
second-order equations read now
c2 p2 ψ+ + c
[p∆] · pψ+ +
[p∆]× p · S ψ+
E −∆+mc2
= (E −∆+mc2)(E − Σ−mc2)ψ+ (11)
c2 p2 ψ− + c
[pΣ] · pψ− +
[pΣ]× p · S ψ−
E − Σ−mc2
= (E −∆+mc2)(E − Σ−mc2)ψ−. (12)
Now, if p∆ = 0, meaning that ∆ is constant or zero (if ∆ goes to zero at infinity, the two conditions are equivalent),
then the second term in eq. (11) disappears and we have
c2 p2ψ+ = (E −∆+mc
2)(E − Σ−mc2)ψ+ = [(E − Vv)
2 − (mc2 + Vs)
2]ψ+ , (13)
which is precisely the time-independent Klein-Gordon equation for a scalar potential Vs plus a vector potential Vv[14].
Since the second-order equation determines the eigenvalues for the spin-1/2 particle, this means that when p∆ = 0,
a spin-1/2 and a spin-0 particle with the same mass and subject to the same potentials Vs and Vv will have the same
energy spectrum, including both bound and scattering states. This last sufficient condition for isospectrality can be
relaxed to demand that just the combination mc2+Vs be the same for both particles, allowing them to have different
masses. This is so because this weaker condition does not change the gradient of ∆ and Σ and therefore the condition
p∆ = 0 will still hold. On the other hand, if the scalar and vector potentials are such that pΣ = 0, we would obtain a
Klein-Gordon equation for ψ−, and again the spectrum for spin-0 and spin-1/2 particles would be the same, provided
they are subjected to the same vector potential and mc2 + Vs is the same for both particles. If both Vs and Vv are
central potentials, i.e., only depend on the radial coordinate, then the numerators of the second terms in equations
(11) and (12) read
[p∆] · pψ+ +
[p∆]× p · S ψ+ = −~
∆′L · S ψ+ (14)
[pΣ] · pψ− +
[pΣ]× p · S ψ− = −~
Σ′L · S ψ− , (15)
where ∆′ and Σ′ are the derivatives with respect to r of the radial potentials ∆(r) and Σ(r), and L = r × p is the
orbital angular momentum operator. From these equations ones sees that these terms, which set apart the Dirac
second-order equations for the upper and lower components of the Dirac spinor from the Klein-Gordon equation and
thus are the origin of the different spectra for spin-1/2 and spin-0 particles, are composed of a derivative term, related
to the Darwin term which appears in the Foldy-Wouthuysen expansion, and a L · S spin-orbit term. If ∆′ = 0
(Σ′ = 0), then there is no spin-orbit term for the upper (lower) component of the Dirac spinor. In turn, since
the second-order equation determines the energy eigenvalues, this means that the orbital angular momentum of the
respective component is a good quantum number of the Dirac spinor. This can be a bit surprising, since one knows
that in general the orbital quantum number is not a good quantum number for a Dirac particle, since L2 does not
commute with a Dirac Hamiltonian with radial potentials. The reason why this does not happen in these cases was
reported in Refs. [3, 4], and we now review it in a slight different fashion. Let us consider in more detail the case of
spherical potentials such that ∆′ = 0. One knows that a spinor that is a solution of a Dirac equation with spherically
symmetric potentials can be generally written as
ψjm(r) =
gj l(r)
Yj lm(r̂)
j l̃ m
. (16)
where Yj lm are the spinor spherical harmonics. These result from the coupling of spherical harmonics and two-
dimensional Pauli spinors χms , Yj lm =
〈 l ml ; 1/2ms | j m 〉Ylmlχms , where 〈 l ml ; 1/2ms | j m 〉 is a
Clebsch-Gordan coefficient and l̃ = l ± 1, the plus and minus signs being related to whether one has aligned or
anti-aligned spin, i.e., j = l ± 1/2. The spinor spherical harmonics for the lower component satisfy the relation
j l̃m
= −σ · r̂Yj lm. The fact that the upper and lower components have different orbital angular momenta is related
to the fact, mentioned before, that L2 does not commute with the Dirac Hamiltonian
H = cα · p+ β(Vs +mc
2) + Vv = cα · p+ βmc
2 +ΣP+ +∆P− , (17)
where P± are the projectors defined above. However, when ∆
′ = 0, there is an extra SU(2) symmetry of H (so-called
“spin symmetry”) as first shown by Bell and Ruegg [5]. When we have spherical potentials, Ginocchio showed that
there is an additional SU(2) symmetry (for a recent review see [4]). The generators of this last symmetry are
L = LP+ +
α · pLα · pP− =
0 Up LUp
, (18)
where Up = σ · p/(
p2) is the helicity operator. One can check that L commutes with the Dirac Hamiltonian,
[H,L] = [cα · p,LP+ +
α · pLα · pP−] + [∆,
α · pLα · p] + [Σ,L]
= [∆,
α · pLα · p ] = 0 , (19)
where the last equality comes from the fact that ∆′ = 0. The Casimir L2 operator is given by L2 = L2P+ +
α · pP−. Applying this operator to the spinor ψjm (16), we get
2ψjm = L
α · pL2 α · pψ−
= ~2l(l+ 1)ψ+
α · p cL2 ψ+jm
E −∆+mc2
= ~2l(l + 1)ψ+
+ ~2l(l + 1)ψ−
= ~2l(l + 1)ψjm , (20)
where ψ±jm = P±ψjm and we used the relation, valid when ∆
′ = 0, ψ+jm = (E −∆ +mc
α · p
ψ−jm. From (20) we
see that ψjm is indeed an eigenstate of L
2. Thus the orbital quantum number of the upper component l is a good
quantum number of the system when the spherical potentials Vs(r) and Vv(r) are such that Vv(r) = Vs(r)+C∆, where
C∆ is an arbitrary constant. Also, according to we have said before, there is a state of a spin-0 particle subjected to
these same spherical potentials (or, at least, with a scalar potential such that the sum Vs +mc
2 is the same) that
has the same energy and the same orbital angular momentum as ψjm. In addition, the wave function of this scalar
particle would be proportional to the spatial part of the wave function of the upper component.
Note that the generator of the “spin symmetry” S is given by a similar expression as (18) just replacing L by ~/2σ
[4, 5], meaning that S2 ≡ S2 = 3/4 ~2I so that spin is also a good quantum number, as would be expected. Actually,
one can show that the total angular momentum operator J can be written as L + S, so that l, ml (eigenvalue of
Lz), s = 1/2, ms (eigenvalue of Sz) are good quantum numbers. Then, of course, j and m = ml +ms are also good
quantum numbers, but only in a trivial way, because there is no longer spin-orbit coupling. Therefore, in the spinor
(16) one could just replace the spinor spherical harmonic Yj lm by Yl mlχms and Yj l̃m by −σ · r̂ Yl mlχms . Note that if
∆ is a nonrelativistic potential, ∆ ≪ mc2 and ∆′ ≪ m2c4/(~c), i.e., it is slowly varying over a Compton wavelength.
In this case, the spin-orbit term will also get suppressed. In fact, the derivative of the ∆ potential is the origin of
the well-known relativistic spin-orbit effect which appears as a relativistic correction term in atomic physics or in the
v/c Foldy-Wouthuysen expansion (only the derivative of Vv appears because usually no Lorentz scalar potential Vs is
considered, and therefore ∆ = Vv).
When Σ′ = 0, or Vv(r) = −Vs(r) + CΣ, with CΣ an arbitrary constant, there is again a SU(2) symmetry, usually
called pseudospin symmetry ([5, 6]) which is relevant for describing the single-particle level structure of several nuclei.
This symmetry has a dynamical character and cannot be fully realized in nuclei because in Relativistic Mean-field
Theories the Σ potential is the only binding potential for nucleons [7, 8]. For harmonic oscillator potentials this is
no longer the case, since ∆, acting as an effective mass going to infinity, can bind Dirac particles [9, 10], even when
Σ = 0. As before, in the special case of spherical potentials, there is another SU(2) symmetry whose generators are
α · pLα · pP+ +LP− =
Up LUp 0
. (21)
In the same way as before, applying L̃
to ψjm, we would find that L̃
ψjm = ~
2 l̃(l̃ + 1)ψjm, that is, this time it
is the orbital quantum number of the lower component l̃ which is a good quantum number of the system and can
be used to classify energy levels. Again, provided the vector and scalar potentials are adequately related, there
would be a corresponding state of a spin-0 particle with the same energy and same orbital angular momentum l̃,
and, furthermore, its wave function would be proportional to the spatial part of the wave function of the lower
component. As before, the pseudospin symmetry generator S̃ can be obtained from L̃ by replacing L by ~/2σ. The
good quantum numbers of the system would be, besides l̃, m
, s̃ ≡ s = 1/2 and ms̃. Again, J = L̃ + S̃. It is
interesting that, as has been noted by Ginocchio [9], the generators of spin and pseudospin symmetries are related
through a γ5 transformation since S̃ = γ5Sγ5 and L̃ = γ5Lγ5. This property was used in a recent work to relate
spin symmetric and pseudospin symmetric spectra of harmonic oscillator potentials [11]. There it was shown that
for massless particles (or ultrarelativistic particles) the spin- and pseudo-spin spectra of Dirac particles are the same.
In addition, this means that spin-symmetric massless eigenstates of γ5 would be also pseudo-spin symmetric and
vice-versa. Since in this case ∆ = Σ = 0, or Vv = Vs = 0, this is, of course, just another way of stating the well-known
fact that free massless Dirac particles have good chirality.
Naturally, for free spin-1/2 particles described by spherical waves, both l and l̃ are good quantum numbers, which
just reflects the fact that one can have free spherical waves with any orbital angular momentum for the upper or
lower component and still have the same energy, as long as their linear momentum magnitude is the same, or, put in
another way, the energy of a free spin-1/2 particle cannot depend on its direction of motion.
In summary, we showed that when a relativistic spin-1/2 particle is subject to vector and scalar potentials such
that Vv = ±Vs + C±, where C± are constants, its energy spectrum does not depend on their spinorial structure,
being identical to the spectrum of a spin-0 particle which has no spinorial structure. This amounts to say that if
the potentials have these configurations there is no spin-orbit coupling and Darwin term. If the scalar and vector
potentials are spherical, one can classify the energy levels according to the orbital angular momentum quantum
number of either the upper or the lower component of the Dirac spinor. This would then correspond to having a
spin-0 particle with orbital angular momentum l or l̃, respectively. This spectral identity can of course happen only
with potentials which do not involve the spinorial structure of the Dirac equation in an intrinsic way. For instance, a
tensor potential of the form iβσµν (∂µAν − ∂νAµ) does not have an analog in the Klein-Gordon equation, so that one
could not have a spin-0 particle with the same spectrum as a spin-1/2 particle with such a potential. This is the case
of the so-called Dirac oscillator [12] (see [10] for a complete reference list), in which the Dirac equation contains a
potential of the form iβσ0imωri = imωβα · r. Another important potential, the electromagnetic vector potential A,
which is the spatial part of the electromagnetic four-vector potential, can be added via the minimal coupling scheme
to both the Dirac and the Klein-Gordon equations. Since α · (p− eA)α · (p− eA) = (p− eA)2 + 2e~∇×A · S, the
spectra of spin-0 and spin-1/2 particles cannot be identical as long as there is a magnetic field present, even though
the condition Vv = ±Vs +C± is fulfilled. It is important also to remark that, since for an electromagnetic interaction
Vv is the time-component of the electromagnetic four-vector potential, this last condition is gauge invariant in the
present case, in which we are dealing with stationary states, i.e, time-independent potentials. So, in the absence of a
external magnetic field (allowing, for instance, an electromagnetic vector potential A which is constant or a gradient
of a scalar function), a spin-0 and spin-1/2 particle subject to the same electromagnetic potential Vv and a Lorentz
scalar potential fulfilling the above relation would have the same spectrum.
The remark made above about the similarity of spin-0 and spin-1/2 wave functions can be relevant for calculations
in which the observables do not depend on the spin structure of the particle, like some structure functions. One such
calculation was made by Paris [13] in a massless confined Dirac particle, in which Vv = Vs. It would be interesting
to see how a Klein-Gordon particle would behave under the same potentials. More generally, this spectral identity
can also have experimental implications in different fields of physics, since, should such an identity be found, it would
signal the presence of a Lorentz scalar field having a similar magnitude as that of a time-component of a Lorentz
vector field, or at least differing just by a constant.
Acknowledgments
We acknowledge financial support from CNPQ, FAPESP and FCT (POCTI) scientific program.
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[14] There are some authors who introduce a scalar potential Vs in the Klein-Gordon equation by making the replacement
m2c4 → m2c4 +V2
. Here we introduce it, as most authors do, as an effective mass m∗ 2 = (m+Vs/c
2)2, since it is the way
that it is introduced in the Dirac equation. The two potentials are related by V2
= (mc2 + Vs)
−m2c4.
	Acknowledgments
	References
