Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representat...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra / I.Yu. Krivsky, V.M. Simulik // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43101:1-15. — Бібліогр.: 26 назв. — англ. |
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irk-123456789-321222012-04-10T12:14:14Z Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra Krivsky, I.Yu. Simulik, V.M. We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found. Among these symmetries (together with the 32-dimensional pure matrix algebra of invariance) the new, physically meaningful, spin 1 Poincar ´e symmetry of equation under consideration is proved. In order to provide the corresponding proofs, a 64-dimensional extended real Clifford-Dirac algebra is put into consideration. На підставі симетрійного аналізу стандартного рівняння Дірака з ненульовою масою доведено, що це рівняння може описувати не лише ферміони зі спіном 1/2, але й бозони зі спіном 1. Знайдено нові бозонні симетрії рівняння Дірака як у представленні Фолді - Вотхойзена, так і у представленні Паулі - Дірака. Серед цих симетрій (поряд з 32-вимірною чисто матричною алгеброю інваріантності) доведено нову, фізично важливу симетрію Пуанкаре спіна 1 згаданого рівняння. Для виконання зазначених доведень введено в розгляд 64-вимірну розширену дійсну алгебру Кліффорда - Дірака. 2010 Article Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra / I.Yu. Krivsky, V.M. Simulik // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43101:1-15. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 11.30-z., 11.30.Cp., 11.30.j http://dspace.nbuv.gov.ua/handle/123456789/32122 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found. Among these symmetries (together with the 32-dimensional pure matrix algebra of invariance) the new, physically meaningful, spin 1 Poincar ´e symmetry of equation under consideration is proved. In order to provide the corresponding proofs, a 64-dimensional extended real Clifford-Dirac algebra is put into consideration. |
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Article |
| author |
Krivsky, I.Yu. Simulik, V.M. |
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Krivsky, I.Yu. Simulik, V.M. Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra Condensed Matter Physics |
| author_facet |
Krivsky, I.Yu. Simulik, V.M. |
| author_sort |
Krivsky, I.Yu. |
| title |
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra |
| title_short |
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra |
| title_full |
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra |
| title_fullStr |
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra |
| title_full_unstemmed |
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra |
| title_sort |
fermi-bose duality of the dirac equation and extended real clifford-dirac algebra |
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Інститут фізики конденсованих систем НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/32122 |
| citation_txt |
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra / I.Yu. Krivsky, V.M. Simulik // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43101:1-15. — Бібліогр.: 26 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT krivskyiyu fermibosedualityofthediracequationandextendedrealclifforddiracalgebra AT simulikvm fermibosedualityofthediracequationandextendedrealclifforddiracalgebra |
| first_indexed |
2025-07-03T12:39:23Z |
| last_indexed |
2025-07-03T12:39:23Z |
| _version_ |
1836629479677493248 |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 4, 43101: 1–15
http://www.icmp.lviv.ua/journal
Fermi-Bose duality of the Dirac equation and extended
real Clifford-Dirac algebra
I.Yu. Krivsky, V.M. Simulik∗
Institute of Electron Physics, National Academy of Sciences of Ukraine,
21 Universitetska Str., 88000 Uzhgorod, Ukraine
Received May 12, 2010, in final form June 29, 2010
We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that
this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic sym-
metries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found.
Among these symmetries (together with the 32-dimensional pure matrix algebra of invariance) the new, phys-
ically meaningful, spin 1 Poincaré symmetry of equation under consideration is proved. In order to provide the
corresponding proofs, a 64-dimensional extended real Clifford-Dirac algebra is put into consideration.
Key words: spinor field, symmetry, group-theoretical analysis, supersymmetry, Foldy-Wouthuysen
representation, Clifford-Dirac algebra
PACS: 11.30-z., 11.30.Cp., 11.30.j
1. Introduction
The main physically meaningful result of this paper is as follows. It has been shown that the
Dirac equation (with nonzero mass!) can equally well describe both spin 1/2 fermionic field and
spin 1 bosonic field. The Dirac’s well-known result (complemented by Foldy-Wouthuysen (FW)
analysis and the Bargman-Wigner classification of fields), which is presented in each handbook on
quantum mechanics, contains only one (fermionic) half of our consideration.
An interest in the problem of the relationship between the Dirac and Maxwell equations, which
describe a spin 1 field, emerged immediately after the creation of quantum mechanics [1–11]. The
bosonic spin 1 representation of the Poincaré group P , the vector (1/2,1/2) and the tensor-scalar
(1, 0) ⊗ (0, 0) representations of the Lorentz group L, as the groups of invariance of the massless
Dirac equation, were found in our papers [12–17]. Here P is the universal covering of the proper
ortochronous Poincaré group P↑
+ = T(4)×)L↑
+ and L = SL(2,C) is that of the Lorentz group L↑
+ =
SO(1,3), respectively. In this paper, using the conception of the relativistic invariance of the theory,
we consider only its invariance with respect to some representations of the proper ortochronous
Poincaré group (the group P without reflections, CPT, etc.)
This paper is directly related to our previous papers [12–17]. It is the next logical step, which
follows from considerations [12–17], where each symmetry of the massless Dirac equation is mapped
into the corresponding symmetry of the Maxwell equations in terms of field strengths (generated
by the gradient-type current), and vice versa. Let us recall that P-symmetries of a field equation
(its invariance with respect to a certain representation of the group P) have the principal physical
meaning: the Bargman-Wigner classification of the fields is based on the P-symmetries of the
corresponding field equations. The relationship between the massless Dirac equation and the above
mentioned Maxwell equations [12–17] means, in particular, that each equation is invariant not only
with respect to the fermionic P-representation (for the s=1/2 spin doublet), but also with respect
to the bosonic tensor-scalar P-representation (with s=1,0).
Now we are able to present the similar results for the general case when the mass in the Dirac
equation is nonzero.
∗E-mail: vsimulik@gmail.com, sim@iep.uzhgorod.ua
c© I.Yu. Krivsky, V.M. Simulik 43101-1
http://www.icmp.lviv.ua/journal
I.Yu. Krivsky, V.M. Simulik
First we seek for the additional symmetries of the FW equation [18]. Then one can easily
find the corresponding symmetries of a standard Dirac equation fulfilling the well-known FW
transformation [18]. Thus, we find a wide set of symmetries of the Dirac equation in both FW
and standard Pauli-Dirac (PD) representations. The 32-dimensional algebra A32 of invariance is
found. In the FW representation, the algebra A32 is the maximal algebra of invariance in the
class of pure matrix operators (without any derivatives over the space variables). The physically
meaningful spin 1 symmetries are found here as subalgebras of the A32 algebra. Finally, we have
found the new hidden Poincaré symmetry of the Dirac equation and fulfilled the Bargman-Wigner
analysis of corresponding P-representation. This analysis proved that this hidden symmetry is spin
1 Poincaré symmetry of the Dirac equation. On the basis of the latter assertions we come to a
conclusion that the Dirac equation may also describe the spin 1 field, i.e. not only fermions of spin
1/2, but also bosons of spin 1. Thus, hereinbelow the investigation of the spin 1 symmetries of the
Dirac equation with nonzero mass started in [19, 20] is being continued.
In section 2, a brief review of general assertions about Poincaré invariance and covariance of
the spinor field theory is given. The ordinary handbook consideration is rewritten in terms of the
so-called primary generators, which are associated with the real (not imaginary) parameters of the
groups of invariance. Several tiny and hidden assumptions of group-theoretical approach to the
spinor field theory are explained.
In section 3, in the canonical FW-representation, the extended real Clifford-Dirac algebra
(ERCD) is put into consideration.
In section 4, the 32-dimensional maximal pure matrix algebra A32 of invariance of the FW
equation is found.
In section 5, the spin 1 Lorentz-symmetries of the FW and Dirac equations are obtained.
In section 6, the spin 1 Poincaré-symmetry of the FW equation is proved.
In section 7, the spin 1 Poincaré-symmetry of the standard Dirac equation is put into conside-
ration.
In section 8, a brief remix of our previous results [12–17] for bosonic symmetries of the massless
Dirac equation is presented. The specific characteristics of the case m = 0 in comparison with
nonzero mass are briefly considered. The possibility, which is open here, of generalization of all
results [12–17] to the general case of nonzero mass is mentioned.
In section 9, the brief general conclusions are given.
We use the units ~ = c = 1; the summation over a twice repeated index is performed.
2. Notations, assumptions and definitions
Here we consider the Dirac equation
(iγµ∂µ −m)ψ (x) = 0; ∂µ ≡ ∂/∂xµ, µ = 0, 3 = 0, 1, 2, 3, ψ ∈ S(M(1, 3))× C4 ≡ S4,4, (1)
as an equation in the test function space S4,4 of the 4-component functions over the Minkowski
space M(1,3). We note that the complete set {ψ} ≡ Ψ of solutions of the equation (1) contains the
generalized solutions belonging to the space S4,4∗ ⊃ S4,4 of the Schwartz’s generalized functions,
{ψgen} ⊂ S4,4∗. The mathematically well-defined consideration of this fact demands (see, e.g.,
the book [21] on the axiomatic approach to the field theory) the functional representation of the
elements of ψgen ⊂ S4,4∗, which makes the consideration very complicated. Hence, we recall that
the test function space S4,4 is dense in S4,4∗. It means that any element ψgen ⊂ S4,4∗ can be
approximated (with arbitrary degree of accuracy) by an element ψ ∈ S4,4 from the corresponding
Cauchy sequence in the space S4,4. Therefore, here for the equation (1) we have restricted ourselves
to the supposition ψ ∈ S4,4. Such supposition is physically verified and essentially simplifies the
consideration without any loss of generality and mathematical correctness.
The invariance of equation (1) with respect to the hidden transformations will be considered.
We seek for the Bose, rather than Fermi, symmetries. In order to fulfill this programme let us briefly
recall the conventional relativistic invariance of equation (1). In the modern consideration, it is
the invariance with respect to the universal covering P ⊃ L=SL(2,C) of the proper ortochronous
43101-2
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
inhomogeneous Poincaré group P↑
+ ⊃ L↑
+ = SO(1,3). For our purposes, in the space S4,4 it is
convenient to rewrite the standard Fermi (spinor) representation of the group P in terms of the
primary P-generators (p ≡ (pµ), j ≡ (jµν )). By definition, these generators are associated with
the real parameters (a, ω) ≡ (aµ, ωµν = −ωνµ) with well-known physical meaning. The generators
(pµ, jµν) are the orts of the real P-algebra, i.e. the algebra over the real numbers (we use the
similar symbols for groups and their algebras). The necessity of using the primary P-generators
and the real P-algebra will be evident below in our search for additional (hidden) symmetries of
the equation (1).
In arbitrary representations, the primary P-generators (pµ, jµν), as the orts of the real Lie
P-algebra, satisfy the commutation relations
[pµ, pν ] = 0, [pµ, jρσ] = gµρpσ−gµσpρ , [jµν , jρσ] = −gµρjνσ−gρνjσµ−gνσjµρ−gσµjρν , (2)
and generate a P-representation, which is defined by an exponential series
(a, ω) ∈ P → F̂(a, ω) = exp(aµpµ +
1
2
ωµνjµν)
inf
= 1 + aµpµ +
1
2
ωµνjµν , (3)
where symbol
inf
= defines “infinitesimally, i.e. in the neighborhood of the unit element of the
group P”.
The primary generators (pµ, jµν) for the ordinary local spinor (Fermi) PF-representation in
S4,4 are the following Lie operators
pρ = ∂ρ ≡ ∂/∂xρ, jρσ = mρσ + sρσ (mρσ ≡ xρ∂σ − xσ∂ρ , sρσ ≡ 1
4
[γρ, γσ]). (4)
The operators pµ, jµν from (4) commute with the Diracian iγµ∂µ − m. Therefore, formulae (3)
with generators (4) define the local spinor (Fermi) PF-representation of the group P in the form
ψ(x) → ψ′(x) = [F̂(a, ω) ≡ F̂1(ω)F̂2(a, ω)]ψ(x)
inf
= (1 + aµpµ +
1
2
ωµνjµν)ψ(x), (5)
which is the P-group of invariance of equation (1). In formula (5) the following notations are used:
ψ, ψ′ ∈ S4,4, F̂1(ω) ≡ exp(
1
2
ωµνsµν)
inf
= (1 +
1
2
ωµνsµν) ∼ (
1
2
, 0)⊗
(
0,
1
2
)
, (6)
F̂2(a, ω)ψ(x) ≡ exp(aµpµ +
1
2
ωµνmµν)ψ(x) = ψ(Λ−1(x − a))
inf
= (1 + aµpµ +
1
2
ωµνmµν)ψ(x), Λ−1 ∈ P↑
+ . (7)
We pay attention to the following detail of mathematical correctness of the consideration. The
space S4,4 is the common domain of definitions and values both for the generators (pµ, jµν) (4)
and for all functions from them, which we use here (in particular, for the exponential series (3)
convergent in the space S4,4). Further, we mark that usually the fermionic P-transformations of
the field ψ are written in the form
ψ(x) → ψ′(x′) = F̂1(ω)ψ(x), x′ = Λx+ a, Λ ∈ P↑
+ . (8)
However, this form (contrary to formula (5)) does not manifestly demonstrate the mathematical
definition of the group of invariance of equation (1), which is given by the definition: PF is a
group of invariance of equation (1), if PFΨ = Ψ; or for arbitrary solution of equation (1): if from
ψ(x) ∈ Ψ results F̂(a, ω)ψ(x) ∈ Ψ ⊂ S4,4, where F̂(a, ω) is given by (5).
Let us further note that in conventional field-theoretical approach, instead of our primary Lie
generators (4), the following operators are used
pstandρ = ipρ ≡ i∂ρ, jstandρσ = ijρσ ≡ mstand
ρσ + sstandρσ ,
(mstand
ρσ ≡ ixρ∂σ − ixσ∂ρ, sstandρσ ≡ i
4
[γρ, γσ]). (9)
43101-3
I.Yu. Krivsky, V.M. Simulik
It is evident that these generators are associated with the pure imaginary parameters (−iaµ, −iωµν)
and with corresponding complex P-algebra. Below, as well as in our other publications on the
symmetries, primary generators (4) and customary operators (9) should not be confused.
We recall that both pure matrix operators sρσ and pure differential operators mρσ from (4)
satisfy the same commutation relations as [jµν , jρσ] in (2). However, the operators sρσ and mρσ
(contrary to their sum jρσ) are not the symmetry operators (operators of invariance) of equation (1)
(operator q̂ is called a symmetry operator or the operator of invariance of equation (1), if equality
q̂Ψ = Ψ is valid, where Ψ ≡ {ψ} ⊂ S4,4 is a complete set of solutions of equation (1), see e.g.
the corresponding definition in [22]). Therefore, both pure matrix L-representation F̂1(ω) (6) and
infinite-dimensional L-representation
F̂2(ω)ψ(x) ≡ exp(
1
2
ωµνmµν)ψ(x) = ψ(Λ−1(x))
inf
= (1 +
1
2
ωµνmµν)ψ(x) (10)
in S4,4 are not the groups of invariance of equation (1). As a consequence of these facts (see arbitrary
conventional consideration in relevant papers, handbooks and monographs), both matrix L-spin
sρσ and orbital angular momentum mρσ (differential L angular momentum) do not generate the
conserved in time integral constants of motion, i.e. being taken separately both spin and orbital
angular momenta of the field ψ are not conserved. Therefore, both pure matrix F̂1(ω) (6) and
infinite-dimensional F̂2(ω) representations in S4,4 of the Lorentz group L are not the groups of
invariance of the Dirac equation (1).
Besides the local PF-representation, the so-called induced (see, e.g. [23, 24]) PF-representation
for the field ψ is useful and meaningful. Mathematically this representation can be related to
the conception of special role of the time variable t ∈ (−∞,∞) ⊂M(1,3). Indeed, in the general
consideration of the Dirac equation in the axiomatic approach, it follows from equation (1) that
the Dirac field ψ satisfies identically the Klein-Gordon equation
(∂µ∂µ +m2 ≡ ∂20 −△+m2)ψ(x) = 0, ψ ∈ S4,4∗, (11)
which is the equation of hyperbolic type. Hence, the generalized solutions of the Dirac equation
(1) are the ordinary functions of the time variable x0 = t ∈ (−∞,∞) ⊂M(1,3) (they are the
generalized functions of the variables −→x ≡ (xℓ) ∈ R3 ⊂ M(1, 3) only). Therefore, due to a special
role of the time variable x0 = t ∈ (xµ) (in obvious analogy with nonrelativistic theory), in general
consideration one can use the quantum-mechanical rigged Hilbert space
S3,4 ⊂ H3,4 ⊂ S3,4∗; S3,4 ≡ S(R3)× C4, (12)
where
H3,4 ≡ L2(R
3)× C4 ≡
{
f : R3 → C4,
∫
d3x|f(t,−→x )|2 <∞
}
, (13)
is the quantum-mechanical Hilbert space of the 4-component functions over R3 ⊂ M(1, 3) (depend-
ing parametrically on x0 = t), which are the square modulus integrable over the Lebesgue measure
d3x in the space R3 ⊂ M(1, 3). Just the space R3 is interpreted as the coordinate spectrum of the
quantum-mechanical particles described by the field ψ.
In this concept, the Dirac equation in the Schrödinger form
i∂0ψ = Hψ ↔ (∂0 − p̃0)ψ = 0; H ≡ γ0(−iγk∂k +m), p̃0 = −iH, (14)
(which is completely equivalent to the equation (1)) is presented in its integral form as follows:
ψ(t) = u(t0, t)ψ(t0); u(t0, t) = exp[−iH(t− t0)]; ψ(t0), ψ(t) ⊂ S3,4∗. (15)
In formula (15), the unitary operator u(t0, t) (with arbitrary-fixed parameters t0, t ∈ (−∞,∞) ⊂
M(1, 3)) is the operator of automorphism in the rigged Hilbert space (12) (below we set t0 = 0).
Recall that the test function space S3,4 has a few wonderful features. It consists of the functions
being infinitely smooth (infinitely differentiable with respect to xℓ) rapidly decreasing at the infinity
43101-4
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
|−→x | → ∞ in arbitrary direction in R3 together with its derivatives of arbitrary orders. Further, it
contains the space of the finite functions with the same properties. Moreover, the space S3,4 is kernel
in the triple (12). The latter one means that this space is dense both in the H3,4 and S3,4∗ spaces.
Therefore, hereinbelow we restrict our consideration to the suggestion ψ ∈ S3,4 in equation (14).
Such restriction is both mathematically correct and technically appropriate (it does not require
the use of the functional form of the elements ψ ∈ S3,4∗). It is also physically motivated. Indeed, an
arbitrary measurement of a construction from ψgen ⊂ S3,4∗ by an equipment of an arbitrary degree
of accuracy can be successfully approximated (with directly fixing an arbitrarily precise degree of
accuracy) by the corresponding constructions from the prelimit functions ψ ∈ S3,4 ⊂ H3,4 ⊂ S3,4∗.
For definiteness here we use the Pauli-Dirac (PD) representation of the Clifford-Dirac (CD)
γ-matrices
γµ : γµγν + γνγµ = 2gµν , g = (gµν ) = diag g(+−−−), (16)
γ0 =
∣∣∣∣
1 0
0 −1
∣∣∣∣ , γk =
∣∣∣∣
0 σk
−σk 0
∣∣∣∣ ; σ1 =
∣∣∣∣
0 1
1 0
∣∣∣∣ ,
σ2 =
∣∣∣∣
0 −i
i 0
∣∣∣∣ , σ3 =
∣∣∣∣
1 0
0 −1
∣∣∣∣ , ℓ = 1, 2, 3. (17)
In this representation, the SU(2)-spin primary generators are given by the matrices sℓn ≡ 1
4
[γℓ, γn].
Two of them are quazidiagonal and the sz-matrix is diagonal:
−→s = (s23, s31, s12) ≡ (sℓ) = − i
2
∣∣∣∣
−→σ 0
0 −→σ
∣∣∣∣ → sz ≡ s3 = − i
2
∣∣∣∣∣∣∣∣
1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 −1
∣∣∣∣∣∣∣∣
. (18)
Therefore, the operator sz contains the projections of the spin −→s on the axis z of the quantum-
mechanical spin- 1
2
doublet of particles.
The primary P-generators of the PF-representation in the space S3,4 have the form of the
following anti-Hermitian operators
p̃0 = −iH ≡ −γ0(γℓ∂l + im), p̃k = pk = ∂k ,
j̃kl = jkl = mkl + skl , j̃ok = t∂k − 1
2
{xk, p̃0}, (19)
where {A,B} ≡ AB+BA and t ∈ (−∞,∞) is the arbitrary-fixed parameter (the primary generators
q̃ = (p̃µ, j̃µν) (19) coincide with the corresponding operators −iq̃stand, where q̃stand are given
by the formulae (126)–(129) in [24]). Using the Heisenberg commutation relations in the form
[xℓ, ∂j ] = δℓj and the SU(2)-relations for sℓ (18), it is easy to show that the operators (19) satisfy the
commutation relations of P-algebra in the manifestly covariant form (2). Furthermore, generators
(19) commute with the operator (∂0 − p̃0) of the equation (14). Therefore, as a consequence of
anti-Hermiticity of arbitrary operator from (19), the induced PF-representation
(a, ω) ∈ P → Ũ(a, ω) = exp
(
aµp̃µ +
1
2
ωµν j̃µν
)
(20)
in the space S3,4 is unitary and is the group of invariance of equation (14). Hence, in the definition
wµ ≡ 1
2
εµνρσ p̃ρj̃νσ → w0 = sℓ∂ℓ ≡ −→s stand · −→p stand, (21)
the main Casimir operators for the generators (19) are given by
p̃µp̃µ ≡ p̃20 − ∂2ℓ ≡ m2I4 , I4 ≡ diag(1, 1, 1, 1), (22)
W ≡ wµwµ = m2−→s 2 = m2 1
2
(
1
2
+ 1)I4 . (23)
43101-5
I.Yu. Krivsky, V.M. Simulik
According to the Bargman-Wigner classification, precisely this fact means that in equation (14)
(for which the induced PF-representation (20) is the group of invariance) the field ψ is the Fermi
field (the field of quantummechanical spin- 1
2
doublet of particles with the mass m).
The relativistic invariance of the spinor field theory with respect to the representation (20) has
some special features. It should be stressed once more that in the induced PF-representation (20)
the time t = x0 plays a special role in comparison to the role of space variables xℓ. Moreover,
we use in the definition (13) such P-non-covariant objects as the Lebesgue measures d3x (or d3k
in the momentum representation of the rigged Hilbert space (12) for the field states ψ ∈ S3,4).
Nevertheless, the theory of the spinor field ψ based on the induced PF-representation (20) is
obviously relativistic invariant. The proof is given in the text after formulae (19) and by the
Bargman-Wigner analysis of the Casimir operators (22), (23).
Now we are in position to give some comparison of the local and induced PF-representations
(5) and (20). It is easy to see that in the set of solutions Ψ = {ψ} of equations (14)=(1) the
local PF-representation (5) and the induced PF-representation (20) coincide. Moreover, the main
Casimir operators for the Lie P-generators (4) have the form
ploc 2 ≡ pµpµ ≡ ∂µ∂µ , W loc ≡ m2−→s 2 =
3
4
∂µ∂µ . (24)
Therefore, the eigenvalues of the main Casimir operators for the PF-representations (5) and (20)
coincide. Note, by the way, that it is easy to show that formulae (24) have the same form for
arbitrary Lie operators (4) (i.e., for arbitrary local P-representations), for which the Lorentz spins
sµν generate the L-representations (s, 0)⊗ (0, s).
Mark that the local generators (4) in S4,4 are the functions of 14 independent operators
xρ, ∂ρ, sρσ (the Lorentz spin operators sρσ are the independent orts of the CD-algebra). The
conception of Hermiticity or anti-Hermiticity in S4,4 is not inherent both in these 14 operators and
P-generators (4). Therefore, the concept of unitarity is not inherent in the local PF-representation
(5) as well. It means that the PF-representation (5) itself (and, similarly, its generators (4)) does
not contain the information as to “which quantum-mechanical particles are described by the filed
ψ from equation (1)?”.
Contrary to these facts, the generators (19) of induced PF-representation in the space S3,4
are the anti-Hermitian functions of (particularly) other 11 independent operators −→x = (xℓ), ∂ℓ,
skℓ, γ
0, m or are the functions of the standard Hermitian in S3,4 operators: −→x = (xℓ), −→p stand =
(−i∂ℓ),
−→s stand = (isℓn), γ
0, m. Hence, the induced PF-representation (20) is unitary in the quantum-
mechanical rigged Hilbert space (12). However, the restriction of the local PF-representation (5) on
the set Ψ ⊂ S3,4 of solutions of equation (1)=(14) coincides with the induced PF-representation (20).
Nevertheless, both local and induced PF-representations have some common physical short-
comings (namely this is the reason why we have started with the FW-representation of the spinor
field theory, where PF-representation is free from the above shortcomings). These shortcomings
are related to the PD-representation of the spinor field ψ. In order to explain these assertions, we
recall the general solution of equation (14)=(1) in the rigged Hilbert space (12):
ψ(x) =
1
(2π)
3
2
∫
d3k
[
ar(
−→
k )v−r (
−→
k )e−ikx + b∗r(
−→
k )v+r (
−→
k )eikx
]
, r = 1, 2. (25)
Here kx ≡ ω̃t−−→
k −→x , ω̃ ≡
√−→
k 2 +m2 and the 4-component spinors are given by
v−r (
−→
k ) = N
∣∣∣∣
(ω̃ +m)dr
(−→σ · −→k )dr
∣∣∣∣ , v+r (
−→
k ) = N
∣∣∣∣
(−→σ · −→k )dr
(ω̃ +m)dr
∣∣∣∣ ;
N ≡ 1√
2ω̃(ω̃ +m)
, d1 =
∣∣∣∣
1
0
∣∣∣∣ , d2 =
∣∣∣∣
0
1
∣∣∣∣ . (26)
We want to emphasize that, in general, the four independent amplitudes ar(
−→
k ), br(
−→
k ) are the gen-
eralized functions from the rigged Hilbert space in the momentum representation. Still, if ψ ∈ S3,4 ⊂
43101-6
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
S3,4∗, then ar(
−→
k ), br(
−→
k ) ∈ S̃3,4 ≡ S̃(R3
~k
)×C4 (where R3
~k
is the spectrum of the quantummechanical
operator −→p ) and the space S̃3,4 is a kernel in the rigged Hilbert space S̃3,4 ⊂ L2(R
3
~k
)×C4 ⊂ S̃3,4∗.
Taking into account that any partner from the triplet of spaces (12) is invariant with respect to the
3-dimensional Fourier-transformation, one can omit the “tilde” symbol over these spaces. There-
fore, exactly the fundamental solutions e∓ikxv∓r (
−→
k ) of the equations (14)=(1) are the essentially
generalized solutions (for them (ar(
−→
k ), br(
−→
k )) ∼ δ(
−→
k −−→
k ′) ∈ S(R3
~k
)∗).
It is well-known that the spinors v∓r (
−→
k ) (26) are called the spin states of the doublet. Neverthe-
less, they are not the eigenvectors of the operator sz = s3 from −→s (18). They are the eigenvectors
of the operator sDirac
z ≡ s3Dirac ⊂ −→s Dirac from nonlocal spin operator in the PD-representation for
the spinor field ψ, which is presented by
−→s Dirac = V −1−→s V = −→s − i[−→γ ×∇]
2ω̂
+
∇× [−→s ×∇]
ω̂(ω̂ +m)
(27)
and which is obtained from the −→s (18) by the FW-transformation V [18] (the prime operator
−→s Dirac (27) coincides with the corresponding spin −i−→s FW, where −→s FW is given in the table 1 in
[18]). Indeed, it is easy to see that the following equations are valid:
szv
∓
1 (
−→
k ) 6= 1
2
v∓1 (
−→
k ), szv
∓
2 (
−→
k ) 6= −1
2
v∓2 (
−→
k );
sDirac
z v∓1 (
−→
k ) =
1
2
v∓1 (
−→
k ), sDirac
z v∓2 (
−→
k ) = −1
2
v∓2 (
−→
k ), (28)
where sz = s3(18), sDirac
z = sDirac 3 (27) (the assertion (28) is our small addition to the consid-
eration [18]). Moreover, operator −→s (18) does not commute with both the prime Dirac operator
from equation (1) and the operator ∂0 − p̃0 (or H) from (14). Therefore, for the spinor field ψ
in the PD-representation the operator −→s (18) cannot be interpreted as the quantum-mechanical
spin operator for the spin 1
2
-doublet of particles and does not generate the spin conservation law
(even in the absence of interaction), which contradicts the experiment. Furthermore, as is known
from [18], the operator −→x = (xℓ) ∈ R3 ⊂ M(1, 3) in the PD-representation for the field ψ also
cannot be interpreted as the quantum-mechanical operator of 3-coordinate for the spin 1
2
-doublet
of particles. Nevertheless, operators −→s (18) and −→x are the important structure operators (which
possess the physically meaningful quantum-mechanical spectra) for the construction of generators
(4), (19) and corresponding local and induced PF-representations.
As shown in [18] the above shortcomings follow from the non-diagonality of the Hamiltonian
H = ip̃0 in (14). Therefore, in the PD-representation of the field ψ, the particle and antipar-
ticle states are mixed. The progressive way of moving forward was suggested in [18]. The FW-
representation of the spinor field theory is free from the above shortcomings, e.g., the coordinate
−→x = (xℓ) of the FW-spinor φ and spin −→s (18) in this representation have direct physical meaning of
corresponding quantum-mechanical observables of spin- 1
2
doublet of particles. It is important that
just the components of this spin −→s are the elements of the CD-algebra in the PD-representation.
The above arguments impel us to start with the FW-representation of the spinor field (rather than
with the PD-representation) and to consider the CD-algebra in the PD-representation just inside
the FW-representation of the spinor field.
Our search for the hidden symmetries of the Dirac equation is based on the method of extension
of the CD-algebra as the real one (based on the inclusion of the i and C orts into it). This appears
to be possible exactly owing to considering the P-generators in terms of the primary operators.
3. The Foldy-Wouthuysen representation and the extended re al
Clifford-Dirac algebra
In order to derive the assertions mentioned in the introduction, we essentially use two construc-
tive ideas (see Ansatz 1 and Ansatz 2 below).
43101-7
I.Yu. Krivsky, V.M. Simulik
Ansatz 1. The above physical arguments (see section 2) impelled us to start not with the
standard Dirac equation but with its FW representation [18]
(i∂0 − γ0ω̂)φ(x) = 0, φ = V ψ, (29)
V =
−i−→γ · ∇+ ω̂ +m√
2ω̂(ω̂ +m)
, V −1 = V (∇ → −∇), ω̂ ≡
√
−∆+m2, ∇ ≡ (∂ℓ). (30)
In the space (12), the operator V is unitary. The general solution of the FW-equation (29) (the
Dirac equation in the FW-representation) in the quantum-mechanical rigged Hilbert space (12) is
given by
φ(x) = V ψ(x) =
1
(2π)
3
2
∫
d3k
[
ar(
−→
k )D−
r e
−ikx + b∗r(
−→
k )D+
r e
ikx
]
, r = 1, 2, (31)
where the 4-dimensional Cartezian basis vectors in the space C4 have the form
D−
r ≡ V v−r (
−→
k ) =
∣∣∣∣
dr
0
∣∣∣∣ , D+
r ≡ V v+r (
−→
k ) =
∣∣∣∣
0
dr
∣∣∣∣ , (32)
and amplitudes ar(
−→
k ), br(
−→
k ) as well as two-component vectors dr are the same as in (25), (26).
Contrary to the basis (26), the vectors (32) are the eigenvectors of the quantum-mechanical spin
operator sz ∈ s3 from −→s (18). The fundamental solutions e∓ikxD∓
r of equation (29) are the rela-
tivistic quantummechanical de Broglie waves (with the determined values of the spin sz from (18))
for the spin- 1
2
doublet of particles.
Note that both the set of fundamental solutions {Ψ∓
~kr
} of equation (1)=(14) and {Φ∓
~kr
} of the
equation (29),
Ψ∓
~kr
(x) ≡ 1
(2π)3/2
e∓ikxv∓r (
−→
k ), Φ∓
~kr
(x) ≡ 1
(2π)3/2
e∓ikxD∓
r , (33)
are the common eigenvectors of the complete set of mutually commute independent operators in
S3,4, namely (−→p stand = (−i∂ℓ), s
Dirac
z ) for the set {Ψ∓
~kr
} and (−→p stand, sz) for the set {Φ∓
~kr
} (they
certainly satisfy one and the same orthonormality and completeness conditions). These facts present
the evident proof of an adequate physical meaning of the spin operators −→s Dirac (27) and −→s (18):
the first is the true spin for the spinor field ψ in PD-representation and the second is the true spin
for the spinor field φ in FW-representation.
Moreover, now the PF-generators of the unitary (induced) PF-representation for the field φ
(obtained from the operators (19) with the help of V -transformation (30)) are the functions of 11
directly experimentally observed operators (−→x = (xℓ), −→p stand = (−i∂ℓ),
−→s stand = (isℓn), γ
0, m)
or prime operators (−→x = (xℓ), ∇ = (∂ℓ),
−→s = (sℓ)(18), γ0, m):
p̌0 = −iγ0ω̂, p̌n = p̃n = ∂n ,
ǰln = j̃ln = xl∂n − xn∂l + sln , ǰ0k = x0∂k + iγ0
{
xkω̂ +
∂k
2ω̂
+
(−→s ×−→
∂ )k
ω̂ +m
}
(34)
(the standard form of the primary generators (34) see e.g. in the formulae (D-64)–(D-67) in [24]).
Now the unitary PF-representation for the field φ,
(a, ω) ∈ P → Ǔ(a, ω) = exp
(
aµp̌µ +
1
2
ωµν ǰµν
)
, (35)
(the PF-group of invariance of the equation (29)) is free from the above mentioned shortcomings
inherent in the local (3) and induced (20) PF-representations. The Casimir operators for the gener-
ators (34) are the same as in (22), (23). The theory of the spinor field φ in the FW-representation,
43101-8
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
which is recalled here, can be evidently generalized for the fields of any spin −→s (where (sℓ) ≡ −→s
are the arbitrary generators of an SU(2)-algebra representation).
Not a matter of special separate role of the time variable t = x0, the theory of the spinor
field φ in the FW-representation is the relativistic invariant (namely, P-invariant) in the sense of
two necessary aspects. (i) The dynamical aspect: the unitary PF-representation (35) is the group
of invariance of the equation (29) (the main conservation laws for the spinor field φ in the FW-
representation are the consequences of this assertion). (ii) The kinematical aspect: if the solution
(31) with arbitrary-fixed amplitudes ar(
−→
k ), br(
−→
k ) is a given state of the spinor field φ in the
arbitrary-fixed inertial frame of references (IFR) Σ, then for the observer in the (a, ω)-transformed
IFR Σ′ the solution φ′(x) = Ǔ(a, ω)φ(x), x ∈ M(1, 3), is the same state of the spinor field.
Some other details of physical arguments for our start with the FW-representation are as
follows. Just the components of the field φ of the FW equation (29) coincide with the quantum-
mechanical wave functions of the particle doublet. Furthermore, it is the FW-representation, where
the operators of directly experimentally observed quantities of this doublet are the corresponding
direct sums of the quantum-mechanical observables of single particles, which form the doublet.
Therefore, it is in the canonical (i.e. FW) representation where these operators have the status
of true observables of the particle doublet. In particular, the 16-dimensional Clifford-Dirac (CD)
algebra (see formulae (36) below) generated by the γµ-matrices (17) contains the generators of
the SU(2) group, which commute with the Hamiltonian γ0ω̂ of the FW equation (29). The latter
assertion means that the spin −→s (18) of the free particle doublet is conserved. Therefore, the
operator −→s (18) given in the set {φ} of solutions (31) of equation (29) has the status of true
spin. This is the reason to consider the prime CD algebra in the PD representation (17) as an
algebra in the set {φ} ⊂ (12). In order to avoid misunderstandings, note that in our consideration
the γµ-matrices in the FW representation have the form (17), rather than V γµV −1. Of course,
V −1(i∂0 − γ0ω̂)V = i∂0 −H with the γµ-matrices (17).
Finally, one has a simple technical reason to prefer the FW-representation. In the FW-repre-
sentation the operator of equation (29) contains only one diagonal γ0-matrix instead of four γµ-
matrices in equation (1)=(14). Hence, equation (29) and the FW-representation are much more
convenient in searching for the matrix symmetries (in comparison with the prime equation (1)=(14)
and the PD-representation for the spinor field). Note that this method and the advantages of the
FW-representation have been well-known since the appearance of publications [18, 24] (see e.g. the
table 1 in [18], where the physically important symmetry operator −→s Dirac (27) was found, or the
generators (126)–(129) in [24]). We recall them only for our purposes.
Ansatz 2. Here (see the formula (38) below) we introduce a 64-dimensional extended real
Clifford-Dirac algebra (ERCD algebra) as the algebra of 4 × 4 pure matrix operators in the set
{φ} ⊂ (12). We essentially apply it here as constructive mathematics for our purposes. For the
physical purposes, where the parameters of the relativistic groups are real, it is sufficient to consider
the 16-dimensional standard CD algebra in the complex space (12) as real algebra. It enables us,
along with the application of the imaginary unit “i” and the operator Ĉ of complex conjugation
in the set {φ} ⊂ (12), to extend this algebra to the ERCD algebra in the space (12). It is on the
basis of the ERCD algebra that we are able to find the maximal pure matrix algebra (without
the space-time derivatives ∂µ) of invariance of the FW equation and the corresponding algebra of
invariance of the Dirac equation with an arbitrary mass in the standard PD representation. The
ERCD algebra is the complete set of operators, the part of which generates the Pauli-Gursey-
Ibragimov (PGI) algebra. Let us recall that the 8-dimensional (for m=0) and 4-dimensional (for
a nonzero mass) PGI algebras of invariance of the Dirac equation are considered in detail, for
instance, in [13, 17] (these algebras were put into consideration in the original papers by W. Pauli,
F. Gursey and N. Ibragimov, see, e.g. [25, 26]).
Finally, it is useful to choose 16 independent (ind) generators – orts of standard CD algebra as
{indCD} ≡
{
I, αµ̂ν̂ = 2sµ̂ν̂ : sµ̌ν̌ ≡ 1
4
[γµ̌, γν̌ ] , sµ̌5 = −s5µ̌ ≡ 1
2
γµ̌ ;
γ4 ≡ γ0γ1γ2γ3, µ̂, ν̂ = 0, 5, µ̌, ν̌ = 0, 4
}
, (36)
43101-9
I.Yu. Krivsky, V.M. Simulik
where γ4 ≡ γ0γ1γ2γ3 = iγstand5 and the matrices sµ̂ν̂ are the prime generators of the SO(1,5)⊃SO(1,3)=
L↑
+ group (associated with the real parameters – the angles ωµ̂ν̂ of rotations in µ̂ν̂ planes of the
space M(1,5) ⊃ M(1,3)), these operators satisfy the following commutation relations
[sµ̂ν̂ , sρ̂σ̂] = −gµ̂ρ̂sν̂σ̂ − gρ̂ν̂sσ̂µ̂ − gν̂σ̂sµ̂ρ̂ − gσ̂µ̂sρ̂ν̂ ; µ̂, ν̂ = 0, 5,
(gµ̂ν̂ ) = diag(+1,−1,−1,−1,−1,−1). (37)
By complementing the orts (36) of the real CD algebra with the operators i, Ĉ and with all
possible products of orts (36) and operators i and Ĉ, we define the ERCD algebra as the linear
manifold spanned on the orts
{ERCD} =
{
indCD, i · indCD, Ĉ · indCD, iĈ · indCD
}
. (38)
Thus, the ERCD algebra generators are the compositions of the standard CD algebra generators
(36) and the generators of the PGI algebra [25, 26], i.e. it is the maximal set of independent matrices,
which can be constructed from the elements i, Ĉ, and (36).
All the physically meaningful symmetries of the FW and Dirac equations put into consideration
below are constructed using the elements of the ERCD algebra.
Note that all elements S3,4, H3,4, S3,4∗ of the rigged Hilbert space (12) are the spaces with
involution. It means that together with the element f the rigged Hilbert space also contains the
complex conjugated element f∗ = Ĉf . Namely this circumstance (together with the comprehension
of the main physical symmetry groups P ⊃ L, its subgroups and corresponding algebras as real
spaces, i.e., the spaces over the field of real numbers, to which the parameters of the symmetry
groups belong) makes it possible to consider and search for the hidden symmetries of the Dirac
equation in the extension of the matrix algebras as the real ones due to adding the operators i and
Ĉ to the orts of these algebras. In this context, the title ERCD algebra illustrates the fact that for
both algebras (for CD algebra (36) as well as for ERCD algebra (38)) the part of the orts pairs
commutes or anticommutes between each other and, furthermore, the square of every ort is equal
to +1 or to −1.
4. Maximal pure matrix algebra of invariance of the Foldy-Wo uthuysen
equation
Consider the 32-dimensional subalgebra A32 = SO(6)⊕ ε̂ · SO(6)⊕ ε̂ of the ERCD algebra. It
is easy to see that the part of ERCD algebra generators, namely the operators
sAB =
1
4
[γA, γB] = −sBA , A,B = 1, 6, γ5 ≡ γ1γ3Ĉ, γ6 ≡ iγ1γ3Ĉ, (39)
(here A,B = 1, 6 includes ǎb̌ = 1, 4 from (36)) and ε̂ = iγ0, satisfy the commutation relations
[sAB, sCD] = δACsBD + δCBsDA + δBDsAC + δDAsCB ,
[sAB, ε̂] = 0; A,B,C,D = 1, 6. (40)
Therefore, operators (39) generate a representation of the SO(6) ⊃ SO(3) group of space rotations
in the space R6 ⊂ M(1, 6) ⊃ M(1, 3). On this basis, together with the additional SO(6) Casimir
operator ε̂ ≡ iγ0 = −γ1γ2γ3γ4γ5γ6, we define the SO(6)⊕ ε̂ algebra and, finally, the A32 = SO(6)⊕
ε̂·SO(6)⊕ ε̂ algebra. The commutation relations for the ε̂·SO(6) generators s̃AB = ε̂sAB differ from
(40) by the common factor ε̂ = iγ0.The 32-dimensional subalgebra A32 = SO(6)⊕ε̂·SO(6)⊕ε̂
of the ERCD algebra in the rigged Hilbert space (12) is the maximal pure matrix
algebra (without the space-time derivatives ∂µ) of invariance of the FW-equation (29).
The proof of this assertion is carried out by straightforward calculations of (i) the corre-
sponding commutation relations (40) for elements of this algebra, (ii) the commutators between
43101-10
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
the elements of A32 and the operator (i∂0 − γ0ω̂) of the FW equation (29). The maximality of
A32 as the algebra of invariance of the equation (29) is the consequence of the maximality of the
dim(ERCD)=64 (in the class of pure matrix operators).
Note that antihermitian matrices {γA : γ1, γ2, γ3, γ4 ≡ γ0γ1γ2γ3, γ5 ≡ γ1γ3Ĉ, γ6 ≡ iγ1γ3Ĉ,
γ7 ≡ iγ0} satisfy the commutation relations γAγB + γBγA = −2δAB, A,B = 1, 7.
The explicit form of the elements of corresponding algebra A32 of invariance of the Dirac
equation in the PD-representation is found from the elements (39) and ε̂ with the help of the
FW-transformation (30): V −1(A32, ε̂)V . In PD-representation this algebra of invariance (of the
prime Dirac equation (1)) is given by the nonlocal operators.
5. Spin 1 Lorentz-symmetries of the Foldy-Wouthuysen and Di rac
equations
The FW-equation (29) is invariant with respect to the two different spin 1 representations of the
Lorentz group L (below sVµν are the generators of the irreducible vector (1/2,1/2) representation and
sTS
µν are the generators of the reducible tensor-scalar (1, 0)⊕(0, 0) representation of the SO(1,3)= L↑
+
algebra). The explicit forms of the corresponding pure matrix operators are given by
sTS
µν =
{
sTS
0k = sI0k + sII0k, sTS
mn = sImn + sIImn
}
, sVµν =
{
sV0k = −sI0k + sII0k, sVmn = sTS
mn
}
, (41)
where sI,IIµν are the following elements of A32 algebra:
sIµν =
{
sI0k =
i
2
γ4γk , sImk =
1
4
[γm , γk]
}
, γ4 ≡ γ0γ1γ2γ3 , (k,m = 1, 3), (42)
sIIµν =
{
sII01 =
i
2
γ2Ĉ, sII02 = −1
2
γ2Ĉ, sII03 =
1
2
γ0 ,
sII12 =
i
2
, sII31 =
i
2
γ2γ0Ĉ, sII23 =
1
2
γ2γ0Ĉ
}
. (43)
The sets sIµν (42) and sIIµν (43) determine in A32 the generators of two different versions of (1/2, 0)⊕
(0, 1/2) representation of the L-algebra. The validity of these assertions are evident after the
transition sBose
µν =WsV,TS
µν W−1 to the Bose-representation of the γ-matrices (γµBose =WγµW−1),
where the operator W is given by
W =
1√
2
∣∣∣∣∣∣∣∣
0 −1 0 Ĉ
0 i 0 iĈ
−1 0 Ĉ 0
−1 0 −Ĉ 0
∣∣∣∣∣∣∣∣
, W−1 =
1√
2
∣∣∣∣∣∣∣∣
0 0 −1 −1
−1 −i 0 0
0 0 Ĉ −Ĉ
Ĉ iĈ 0 0
∣∣∣∣∣∣∣∣
, WW−1 =W−1W = 1.
(44)
In such Bose-representation, the γµBose =WγµW−1 matrices contain not only the operator i, but
also the operator Ĉ of complex conjugation:
γ0Bose =
∣∣∣∣∣∣∣∣
0 i 0 0
−i 0 0 0
0 0 0 1
0 0 1 0
∣∣∣∣∣∣∣∣
, γ1Bose=
∣∣∣∣∣∣∣∣
0 0 0 Ĉ
0 0 iĈ 0
0 −iĈ 0 0
−Ĉ 0 0 0
∣∣∣∣∣∣∣∣
, γ2Bose=
∣∣∣∣∣∣∣∣
0 0 −iĈ 0
0 0 0 Ĉ
iĈ 0 0 0
0 −Ĉ 0 0
∣∣∣∣∣∣∣∣
,
γ3Bose =
∣∣∣∣∣∣∣∣
0 iĈ 0 0
−iĈ 0 0 0
0 0 0 Ĉ
0 0 −Ĉ 0
∣∣∣∣∣∣∣∣
, iBose=
∣∣∣∣∣∣∣∣
0 −1 0 0
1 0 0 0
0 0 0 i
0 0 i 0
∣∣∣∣∣∣∣∣
, ĈBose=
∣∣∣∣∣∣∣∣
Ĉ 0 0 0
0 −Ĉ 0 0
0 0 Ĉ 0
0 0 0 Ĉ
∣∣∣∣∣∣∣∣
, (45)
(of course, they satisfy the CD-relations (16)). Calculation of operator constructions (41) also
needs the explicit forms of operators iBose, ĈBose. Therefore, we present them in (45) too. Now,
43101-11
I.Yu. Krivsky, V.M. Simulik
together with γµBose-matrices, they are not so simple as iI4, ĈI4 in standard PD-representation
(I4 is 4× 4 unit matrix). Nevertheless, we note, the Lorentz spin matrices sBose
µν =WsV,TS
µν W−1 in
the Bose-representation do not contain the operator Ĉ and take the explicit forms well known for
the matrix representations (1, 0)⊗ (0, 0) and (1/2,1/2) of the group L ∼SO(1,3):
sV,TSBose
12 =
∣∣∣∣∣∣∣∣
0 −1 0 0
1 0 0 0
0 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣
, sV,TSBose
31 =
∣∣∣∣∣∣∣∣
0 0 1 0
0 0 0 0
−1 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣
, sV,TSBose
23 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 −1 0
0 1 0 0
0 0 0 0
∣∣∣∣∣∣∣∣
,
sTSBose
01 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 −i 0
0 i 0 0
0 0 0 0
∣∣∣∣∣∣∣∣
, sTSBose
02 =
∣∣∣∣∣∣∣∣
0 0 i 0
0 0 0 0
−i 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣
, sTSBose
03 =
∣∣∣∣∣∣∣∣
0 −i 0 0
i 0 0 0
0 0 0 0
0 0 0 0
∣∣∣∣∣∣∣∣
,
sVBose
01 =
∣∣∣∣∣∣∣∣
0 0 0 −1
0 0 0 0
0 0 0 0
−1 0 0 0
∣∣∣∣∣∣∣∣
, sVBose
02 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 0 −1
0 0 0 0
0 −1 0 0
∣∣∣∣∣∣∣∣
, sVBose
03 =
∣∣∣∣∣∣∣∣
0 0 0 0
0 0 0 0
0 0 0 −1
0 0 −1 0
∣∣∣∣∣∣∣∣
.
(46)
We refer to the transition φ→Wφ with W (44), which links the present fermionic and bosonic
multipletes, as a new natural form of the supersymmetry transformation. The corresponding sym-
metries of the Dirac equation (1) are obtained using FW-transformation (30).
6. Spin 1 Poincar é-symmetries of the Foldy-Wouthuysen equation
The FW equation (29) is invariant not only with respect to the well-known standard spin
1/2 PF-representation (35), but also with respect to the canonical-type spin 1 represen-
tation (Bose representation) of the Poincaré group P , i.e. with respect to the unitary (in
the set {φ} of solutions of equation (29)) PB-representation, which is determined by the primary
generators
p̂0 = p̌0 = −iγ0ω̂, p̂n = p̌n = ∂n , ĵln = xl∂n − xn∂l + sIln + sIIln ,
ĵ0k = x0∂k + iγ0
{
xkω̂ +
∂k
2ω̂
+
[(−→s I +−→s II)×−→
∂ ]k
ω̂ +m
}
, (47)
where ω̂ is given in (30), sIln and sIIln are given in (42), (43), respectively, and ~sI,II = (s23, s31, s12)
I,II.
The proof is performed by straightforward calculations of (i) the corresponding P-commutators
(2) between the generators (47), (ii) the commutators between generators (47) and operator (i∂0−
γ0ω̂), (iii) the Casimir operators of the Poincaré group for the generators (47). According to the
Bargman-Wigner classification of the P-covariant fields, just these facts (especially (iii)) visualize
the hidden Bose essence of the PB-representation, generated by the operators (47). For the PB-
representation the explicit form of main Casimir operators as follows:
p̂µp̂µ = m2, WB = wµwµ = m2(−→s TS)2, (48)
(compare with (22), (23)), where wµ ≡ 1
2
εµνρσ p̂ρĵνσ and, after diagonalization carried out with
the help of operator W (44),
(−→s TS)2 = (−→s TSBose)2 = −1(1 + 1)
∣∣∣∣
I 3 0
0 0
∣∣∣∣ , I3 ≡
∣∣∣∣∣∣
1 0 0
0 1 0
0 0 1
∣∣∣∣∣∣
. (49)
43101-12
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
7. Spin 1 symmetries of the Dirac equation with nonzero mass
It is easy to see that the prime Dirac equation (1) has all the above mentioned spin
1 symmetries of the FW equation . The corresponding explicit forms of the generators qPD
in the manifold {ψ} are obtained from the corresponding formulae (39), (41)–(43), (47) for the
FW generators qFW with the help of the FW operator V (30): qPD = V −1qFWV . As a meaningful
example, we present here the explicit form for the spin 1 generators of PB-symmetries of
the Dirac equation
p̂PD
0 = p̃0 = −iH, p̂PD
k = ∂k ,
ĵPD
kl = xk∂l − xl∂k + skl + ŝkl , ĵPD
0k = x0∂k − xkp̃0 + s0k +
εkℓnŝ0ℓ∂n
ω̂ +m
, (50)
where εkln is the Levi-Chivitta tensor, and the operators sµν , ŝµν = V −1sIIµνV have the form
sµν =
1
4
[γµ, γν ] , µ, ν = 0, 3,
ŝµν =
{
ŝ01 =
1
2
iγ2Ĉ, ŝ02 = −1
2
γ2Ĉ, ŝ03 = γ0
−→γ · −→p stand +m
2ω̂
=
H
2ω̂
,
ŝ12 =
i
2
, ŝ31 = − iH
2ω̂
γ2Ĉ, ŝ23 = −H
2ω̂
γ2Ĉ
}
(51)
(the part of the Lorentz spin operators from (51) is not a pure matrix because they depend on the
pseudodifferential operator ω̂ ≡
√
−∆+m2 well-defined in the space S3,4). Of course, the Casimir
operators for the PB-generators (50) have the same final form (48), (49) as for the generators
(47). The Dirac equation in the Bose-representation of the γ-matrices (γµBose =WγµW−1) is the
Maxwell-type equation for a massive tensor-scalar field.
Note that the generators (47) without the additional terms sIIln (43) and ~sII = (s23, s31, s12)
II
directly coincide with the well-known generators (34) of standard Fermi (spin 1/2) PF-symmetries
of the FW-equation (similar situation occurs for the generators (50) taken without the terms
including the operators ŝµν from (51) – they coincide with the operators (19) of the induced
PF-representation (20)). These well-known forms determine the Fermi-case, while the operators
suggested here are related to the Bose interpretation of equations (1), (29), which is found here
also to be possible. The only difference of our Fermi-case from the spin 1/2 generators in [18] is
that we use the prime form of generators related to the real parameters of the Poincaré group.
8. The case of zero mass, brief remix
Having analysed the arbitrary mass case presented here, the specific character of the zero mass
case [12–17] becomes evident. For m = 0 the analogues of the additional operators ŝµν (51) are
the local pure matrix Lie operators (see, e.g. formulae (12) in [13]). Therefore, for the m = 0 case,
the corresponding PB-generators are much simpler than those in (50) and contain no nonlocal
terms. Moreover, for the m = 0 case, there is no need to turn to the FW-representation and
ERCD algebra. In [12–17] all Bose symmetries of the massless Dirac equation were found in the
standard PD-representation of the spinor field ψ on the basis of the ordinary CD algebra and
well-known [25, 26] PGI operators. Thus, the search for the additional bosonic symmetries of the
massless Dirac equation is technically much easier. Therefore, our results [12–17] contain a lot of
additional meaningful information in comparison with the results presented here for the nonzero
mass. In [12–17], the full consideration of the Fermi-Bose duality of the massless Dirac equation is
presented. Furthermore, the Fermi-Bose duality of the Maxwell equations with the gradient-type
sources, which are the Bose partner of the massless Dirac equation, is investigated in detail. In our
further publications we will be in position to extend and generalize on the same level all our results
presented briefly here and to reproduce all the results of [12–17] for the case of nonzero mass. For
43101-13
I.Yu. Krivsky, V.M. Simulik
example, the unitary relationship between the fermionic amplitudes ar(
−→
k ), br(
−→
k ) of the solution
(25) and the bosonic amplitudes of the general solution of the Maxwell-type equation for a massive
tensor-scalar field will be presented (the analogue of the unitary relationship (31) in [15], or (40) in
[16], from the case m = 0). Hence, the subsequent steps of the analysis of the Fermi-Bose duality
of the Dirac equation with arbitrary mass will be presented.
9. Brief conclusions
The following four principal results have been proved and presented.
1. In the FW-representation for the Dirac equation the new mathematical object – the 64-
dimensional Extended real Clifford-Dirac (ERCD) algebra – is put into consideration. The
ERCD algebra is a pure matrix algebra, i.e. the algebra without any derivatives from the
space variables, transformations of reflections, inversions etc.
2. The 32-dimensional subalgebra A32 = SO(6) ⊕ iγ0 · SO(6) ⊕ iγ0 (iγ0 = ε̂) of the ERCD
algebra is proved to be new and maximal pure matrix algebra of invariance of the FW
equation. Its image V −1(A32)V is the algebra of invariance of the standard Dirac equation
in PD-representation.
3. It is shown that some subsets of generators from the A32 have the meaning of the well-known
fermionic SL(2,C)- and SU(2)-spins (see formulae (42), (43)), and the others (41) have the
meaning of new bosonic SL(2,C)- and SU(2)-spins. Therefore, as the symmetry operators,
they have a beneficial physical interpretation.
4. On the basis of the above (see item 3) spins, operators −→x and ∇ (together with well-known
fermionic PF-representation), a new (hidden) bosonic PB-representation of the proper or-
tochronous Poincaré group P , as the group of invariance of the Dirac and FW equations, has
been found. Using the Bargmann-Wigner analysis this new P-representation has been proved
to be the PB-representation for the Dirac field as the field of particles with the spin 1.
Therefore, here the assertion that “the Dirac equation can describe the bosons, not only the
fermions” is put into consideration as an interesting new possibility. The details and consequences
of this assertion will be presented in our further publication, where all results of our papers [12–17]
on the m = 0 case will be repeated for the general case of nonzero mass. Thus, we suggest here a
beginning of a natural approach to the supersymmetry of the Dirac equation, in which the Fermi
and Bose superpartners are linked by the transformation (44).
The possibilities of applying the ERCD algebra are much more extended than a few examples
with the Dirac and FW equations considered above. In general, the ERCD algebra can be applied
to any problem, in which the standard CD algebra can be used. Therefore, the introduction of the
ERCD algebra is a meaningful independent result in the field of mathematical physics.
43101-14
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
References
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Фермi-Бозе дуалiзм рiвняння Дiрака та розширена дiйсна
алгебра Клiффорда-Дiрака
I.Ю. Кривський, В.М. Симулик
Iнститут електронної фiзики, Нацiональна академiя наук України,
вул. Унiверситетська, 21, 88000 Ужгород
На основi симетрiйного аналiзу стандартного рiвняння Дiрака з ненульовою масою доведено, що
це рiвняння може описувати не лише фермiони зi спiном 1/2, але й бозони зi спiном 1. Знайдено
новi бозоннi симетрiї рiвняння Дiрака як у представленнi Фолдi-Вотхойзена, так i у представленнi
Паулi-Дiрака. Серед цих симетрiй (поряд з 32-вимiрною чисто матричною алгеброю iнварiантностi)
доведено нову, фiзично важливу симетрiю Пуанкаре спiна 1 згаданого рiвняння. Для виконання
зазначених доведень введено в розгляд 64-вимiрну розширену дiйсну алгебру Клiффорда-Дiрака.
Ключовi слова: спiнорне поле, симетрiя, теоретико-груповий аналiз, суперсиметрiя,
представлення Фолдi-Вотхойзена, алгебра Клiффорда-Дiрака
43101-15
Introduction
Notations, assumptions and definitions
The Foldy-Wouthuysen representation and the extended real Clifford-Dirac algebra
Maximal pure matrix algebra of invariance of the Foldy-Wouthuysen equation
Spin 1 Lorentz-symmetries of the Foldy-Wouthuysen and Dirac equations
Spin 1 Poincaré-symmetries of the Foldy-Wouthuysen equation
Spin 1 symmetries of the Dirac equation with nonzero mass
The case of zero mass, brief remix
Brief conclusions
|