Bled Workshops in Physics
Vol. 14, No. 2 JLV
Proceedings to the 16th Workshop What Comes Beyond ... (p. 199) Bled, Slovenia, July 14-21, 2013
13 Dirac and Higher-Spin Equations of Negative Energies
V.V. Dvoeglazov
Unidad Académica de Física Universidad Autónoma de Zacatecas*
Abstract. It is easy to check that both algebraic equation det(p —m) = 0 and det(p + m) = 0 for 4-spinors u— and v— have solutions with po = ±Ep = p2 + m2. The same is true for higher-spin equations. Meanwhile, every book considers the p0 = Ep only for both u— and v— spinors of the (1/2,0) © (0,1/2)) representation, thus applying the Dirac-Feynman-Stueckelberg procedure for elimination of negative-energy solutions. Recent works of Ziino (and, independently, of several others) show that the Fock space can be doubled. We reconsider this possibility on the quantum field level for both s = 1/2 and higher spins particles.
Povzetek. Zlahka preverimo, da imata algebrajski enačbi det(p — m) = 0 in det(p + m) = 0 za 4-spinorja u— in v— rešitvi za po = ±Ep = ±\/p2 + m2. Enako velja za enačbe za spinorje z višjimi spini. Vseeno učbeniki obravnavajo samo p0 = Ep za oba spinorja u— in v— upodobitve (1/2, 0) © (0,1/2)), torej uporabijo postopek Diraca, Feynmana in Stueckel-berga za izločitev resitev z negativnimi energijami. Nedavni članki Ziina (in, neodvisno, nekaterih drugih) kazejo, da lahko Fockov prostor podvojimo. Ponovno obravnavamo to možnost na nivoju kvantnih polj, tako za delce s spinom s = 1/2 kot za tiste z visjimi spini.
The Dirac equation is:
[iy^ - m]¥(x) = 0.	(13.1)
At least, 3 methods of its derivation exist [1-3]:
•	the Dirac one (the Hamiltonian should be linear in 3/3xT, and be compatible with Ep — p2c2 = m2c4);
•	the Sakurai one (based on the equation (Ep — a ■ p)(Ep + a ■ = m2^);
•	the Ryder one (the relation between 2-spinors at rest is (0) = (0)).
The y^ are the Clifford algebra matrices
Y^YV + YVY^ = .	(13.2)
Usually, everybody uses the following definition of the field operator [4]:
^(x) = X ^[uh(p)ah(p)e-ip x + vh(p)bh(p)e+ip x], (13.3)
( )3 h ^ p
' d3p i
e-mail: valeri@fisica.uaz.edu.mx
as given ab initio. After introducing exp^ip^x1) the 4-spinors ( u— and v— ) satisfy the momentum-space equations: (p — m)uh(p) = 0 and (p + m)vh(p) = 0, respectively; the h is the polarization index. It is easy to prove from the characteristic equations det(p ^ m) = (pO — p2 — m2)2 = 0 that the solutions should satisfy the energy-momentum relation p0 = ±Ep = p2 + m2.
The general scheme of construction of the field operator has been presented in [5]. In the case of the (1/2,0) © (0,1/2) representation we have:
¥(x) =
1
(2n)3
d4p 6(p2 - m2)e-ipx¥(p) =
1
(2n)-1
L
h
r d4p
d4pô(p2 - Ep)e ip xUh(po,pWpo,p) =
(13.4)
(2n)3 1
(2n)3
^[6(po - Ep) + 6(po + Ep)][6(po) + 9(-po)]e-ip-^uh(p)ah(p)
L
h J
^[6(po - Ep) + 6(po + Ep)]
2Ep
6(po)Uh(p)ah(p)e-ip-
+ 6(po)uh(-p)ah(-p)e+ipx
(2n)-
L
h
rdEp e(po)
2Ep
Uh(p)ah(p)|po=Ep e-i<EPt-px)
+ Uh(-p)ah(-p)|p0=E„ e+i(Ept-px)
During the calculations above we had to represent 1 = 0(po) + 0(—po) in order to get positive- and negative-frequency parts.1 Moreover, during these calculations we did not yet assumed, which equation this field operator (namely, the u(p) spinor) satisfies, with negative- or positive- mass?
In general we should transform uh(—p) to the v(p). The procedure is the following one [7]. In the Dirac case we should assume the following relation in the field operator:
X vh(p)b[(p) = X uh(—p)M—p).	(13.5)
h h
We know that [3]
u^(p)uA(p) = +mS^A ,	(13.6)
u^(p)uA(—p)= 0,	(13.7)
v|i(p)vA(p) = —mS^A ,	(13.8)
v |(p)ua(p)= 0,	(13.9)
but we need A|A(p) = v|(p)uA(—p). By direct calculations, we find
—mb|(p) = X A|a(p)qa(—■p).	(13.10)
See [6] for some discussion.
13 Dirac and Higher-Spin Equations of Negative Energies 201 Hence, A^a = — im(a • n) and
bJi(p) = • n)^AQA(—p).
(13.11)
Multiplying (13.5) by up.(—p) we obtain
-p) = —iH(CT • n)^AbA(p).
(13.12)
The equations are self-consistent.2
However, other ways of thinking are possible. First of all to mention, we have, in fact, uh(Ep, p) and uh(—Ep, p) originally, which satisfy the equations:3
[Ep (±y°) — y • p — m] Uh(±Ep, p) = 0 .
(13.14)
Due to the properties U^y°U = —Y0, U^y1 U = +Y1 with the unitary matrix
0 1 = y°Y5 in the Weyl basis,4 we have
U =
1 0
[EpY0 — Y • p — m] UV(—Ep, p) = 0 .
(13.15)
Thus, unless the unitary transformations do not change the physical content, we have that the negative-energy spinors y5Y°u- (see (13.15)) satisfy the accustomed "positive-energy" Dirac equation. Their explicite forms y5Y°u- are different from the textbook "positive-energy" Dirac spinors. They are the following ones:5
U (p) =
N
v/2m(—Ep + m)
U (p) =
N
v/2m(—Ep + m)
/—p+ + m\
—Pr
p- — m \ —pr
( —pi \
-p- + m
—pi \ p+ — m ,
(13.16)
(13.17)
2	In the (1,0) © (0,1 ) representation the similar procedure leads to somewhat different situation:
a^(p) = [1 — 2(S • n)2]^AQa(—p).	(13.13)
This signifies that in order to construct the Sankaranarayanan-Good field operator (which was used by Ahluwalia, Johnson and Goldman [Phys. Lett. B (1993)], it satisfies [Y(aV3(a3v — (i3/E3t) m2]V(x) = 0, we need additional postulates. For instance, one can try to construct the left- and the right-hand side of the field operator separately each other [6].
3	Remember that, as before, we can always make the substitution p —> —p in any of the integrands of (13.4).
4	The properties of the U— matrix are opposite to those of P^y°P = +Y°, P^YlP = —Yl with the usual P = y°, thus giving [—EpY° + Y • p — m] Puh(—Ep, p) = — [p + m] v?(Ep, p) = 0. While, the relations of the spinors vh(Ep, p) = YsUh(Ep, p) are well-known, it seems that the relations of the v— spinors of the positive energy to u— spinors of the negative energy are frequently forgotten, v?(Ep, p) = Y°uh(— Ep, p).
5	We use tildes because we do not yet know their polarization properties.
a

Ep = \Jp2 + m2 > 0, p0 = ±Ep, p± = E ±pz, pr,i = px ± ipy. Their normalization is to -2N2.
What about the v(p) = y0u- transformed with the y0 matrix? Are they equal to vh(p) = Y5uh(p)? The answer is NO. Obviously, they also do not have well-known forms of the usual v— spinors in the Weyl basis differing by phase factor and in the sign at the mass term (!)
Next, one can prove that the matrix
P = ei0Y° = eie( 0J)	(13.18)
can be used in the parity operator as well as in the original Weyl basis. The parity-transformed function V(t, —x) = P¥(t, x) must satisfy
[iYx — m]¥'(t, —x) = 0,	(13.19)
with 9^ = (9/9t, —Vi). This is possible when P-1 y0P = Y0 and P-1 YiP = —Yi. The matrix (13.18) satisfies these requirements, as in the textbook case. However, if we would take the phase factor to be zero we obtain that while uh (p) have the eigenvalue +1, but
PRU(p) = PRy5Y0u(—Ep, p) = —■U(p), PRU(p) = PRy5Y0u(—Ep, p) = —U(p).
(13.20)
Perhaps, one should choose the phase factor 6 = n. Thus, we again confirmed that the relative (particle-antiparticle) intrinsic parity has physical significance only.
Similar formulations have been presented by [8], and [9]. The group-theoretical basis for such doubling has been given in the papers by Gelfand, Tsetlin and Soko-lik [10], who first presented the theory in the 2-dimensional representation of the inversion group in 1956 (later called as "the Bargmann-Wightman-Wigner-type quantum field theory" in 1993).
M. Markov wrote long ago two Dirac equations with the opposite signs at the mass term [8].
[iY x3 x — m] ¥i(x)= 0,	(13.21)
[iY x3 x + m] ^2(x)= 0.	(13.22)
In fact, he studied all properties of this relativistic quantum model (while he did not know yet the quantum field theory in 1937). Next, he added and subtracted these equations. What did he obtain?
iY x3 x^(x)— mx(x) = 0,	(13.23)
iY x3 xX(x) — m9(x)= 0,	(13.24)
thus, cp and x solutions can be presented as some superpositions of the Dirac 4-spinors u— and v—. These equations, of course, can be identified with the equations
for the Majorana-like À— and p— we presented in ref. [11].'
iy^AS(x) - mpA(x) = 0, — mAS(x) = 0, iy^9^AA(x) + mpS(x) = 0, iy^9^pS(x) + mÀA(x) = 0.
(13.25)
(13.26)
(13.27)
(13.28)
Neither of them can be regarded as the Dirac equation. However, they can be written in the 8-component form as follows:
[ir^ — m] ¥(+) (x) = 0
[ir^ + m] ¥(_) (x) = 0
(13.29)
(13.30)
with
(x)
PA (x) AS(x)

)(x)=	, and r^
0 y^ Y^ 0
(13.31)
You may say that all this is just related to the basis rotation (unitary transformations). However, in the previous papers I explained: The connection with the Dirac spinors has been found [11,13].7 For instance,
/ATS(P)\ AS(P) AA(P)
VaA(p)/
/
i
-i 1 1 —i 1
—1 i —i —1 —1 —i i —V
/U+1/2(P)\
U—1/2 (p) V+1/2 (p) \V— 1/2(P)Z
(13.32)
Thus, we can see that the two 4-spinor systems are connected by the unitary transformations, and this represents itself the rotation of the spin-parity basis. However, the A— and p— spinors describe the neutral particles, meanwhile u— and v— spinors describe the charged particles. Kirchbach [13] found the amplitudes for neutrinoless double beta decay 00v|3 in this scheme. It is obvious from (13.32) that there are some additional terms comparing with the standard formulation.
One can also re-write the above equations into the two-component form. Thus, one obtains the Feynman-Gell-Mann [12] equations. As Markov wrote himself, he was expecting "new physics" from these equations.
Barut and Ziino [9] proposed yet another model. They considered y5 operator as the operator of the charge conjugation. Thus, the charge-conjugated Dirac equation has the different sign comparing with the ordinary formulation:
[iY^ + m]¥Bz = 0,
(13.33)
and the so-defined charge conjugation applies to the whole system, fermion+electro-magnetic field, e —» —e in the covariant derivative. The superpositions of the ¥BZ
Of course, the signs at the mass terms depend on, how do we associate the positive- or
negative- frequency solutions with À and p.
7 I also acknowledge personal communications from D. V. Ahluwalia on these matters.
6
and also give us the "doubled Dirac equation", as the equations for A— and p— spinors. The concept of the doubling of the Fock space has been developed in Ziino works (cf. [10,14]) in the framework of the quantum field theory. In their case the charge conjugate states are simultaneously the eigenstates of the chirality. Next, it is interesting to note that for the Majorana-like field operators we have
ML , n „ ML t , 11 .
V	Cv t (x^) /2 =
d3p
(2n)3 2E
L
0

0

(13.34)
ML	ML t , „.'
v (x^) - Cv t (x^)
/2 =
d3p
(2n)3 2Ep
L
0
)


+
-ie^LVh at(p^)eip• x
(13.35)
which, thus, naturally lead to the Ziino-Barut scheme of massive chiral fields, ref. [9].
Finally, I would like to mention that, in general, in the Weyl basis the y-matrices are not Hermitian, = y°y^Y°. The energy-momentum operator is obviously Hermitian. So, the question, if the eigenvalues of the Dirac operator (the mass, in fact) would be always real, and the question of the complete system of the eigenvectors of the non-Hermitian operator deserve careful consideration [15]. Bogoliubov and Shirkov [5, p.55-56] used the scheme to construct the complete set of solutions of the relativistic equations, fixing the sign of p° = +Ep.
The conclusion is: the doubling of the Fock space and the corresponding solutions of the Dirac equation got additional mathematical bases in this talk presentation. Similar conclusion can be deduced for the higher-spin equations. I appreciate the discussions with participants of several recent Conferences.
References
1.	P. A. M. Dirac, Proc. Roy. Soc. Lond. A 117 (1928) 610.
2.	J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, 1967.
3.	L. H. Ryder, Quantum Field Theory, Cambridge University Press, Cambridge, 1985.
4.	C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill Book Co., 1980, p. 156.
5.	N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, 2nd Edition, Nauka, Moscow, 1973.
6.	V. V. Dvoeglazov, J. Phys. Conf. Ser. 284 (2011) 012024, arXiv:1008.2242.
7.	V. V. Dvoeglazov, Hadronic J. Suppl. 18 (2003) 239, physics/0402094; Int. J. Mod. Phys. B 20 (2006) 1317.
8.	M. Markov, ZhETF 7 (1937) 579; ibid. 603; Nucl. Phys. 55 (1964) 130.
9.	A. Barut and G. Ziino, Mod. Phys. Lett. A 8 (1993) 1099; G. Ziino, Int. J. Mod. Phys. A 11 (1996) 2081.
10.	I. M. Gelfand and M. L. Tsetlin, ZhETF 31 (1956) 1107; G. A. Sokolik, ZhETF 33 (1957) 1515.
11.	V. V. Dvoeglazov, Int. J. Theor. Phys. 34 (1995) 2467; Nuovo Cim. A 108 (1995) 1467; Hadronic J. 20 (1997) 435; Acta Phys. Polon. B 29 (1998) 619.
12.	R. P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958) 193.
13.	M. Kirchbach, C. Compean and L. Noriega, Eur. Phys. J. A 22 (2004) 149.
14.	V. V. Dvoeglazov, Int. J. Theor. Phys. 37 (1998) 1915.
15.	V. A. Ilyin, Spektralnaya Teoriya Differencialnyh Operatorov. (Nauka, Moscow, 1991); V. D. Budaev, Osnovy Teorii Nesamosopryazhennyh Differencialnyh Operatorov. (SGMA, Smolensk, 1997).