The results of spin 1 symmetries of massless Dirac equation [21] ate proved ... The relationship between the massless Dirac equation and the Maxwell equa-.
BOSONIC SYMMETRIES OF THE
MASSLESS DIRAC EQUATION V. M. Simulik Institute of Electron Physics, Ukrainian National Academy o] Sciences, Universitetska 21, 29]t016 Uzhgorod, Ukraine and I. Yu. Krivsky Institute o] Electron Physics,. Ukrainian National Academy of Sciences, Universitetska P1, ~94016 Uzh9orod, Ukraine (Received:.December 14, 1997, Accepted:February 20, 1998) The results of spin 1 symmetries of massless Dirac equation [21] ate proved completely in the space of 4-component Dirac spinors on the basis of unitary operator in this space connecting this equation with the Maxwell equations containing gradient-like sources. Nonlocal representations of conformal group are found, which generate the transformations leaving the massless Dirac equation being invariant. The Maxwell equations with gradient-like sources are proved to be invariant with respect to fermionic representations of Poincar› and conformal groups and to be the kind of Maxwell equations with maximally symmetrical properties. Brief consideration of an application of these equations in physics is discussed. 1. I n t r o d u c t i o n The relationship between the massless Dirac equation and the Maxwell equations is a subject of interest of investigators [1-29] since the time of creation of quantum mechanics. In [10,13,15,16] one can find the beginning of investigation of the most interesting case where the mass is nonzero and the interaction
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Bosonic Symmetriesof the Massless .....
V.M. Simulikand I. Yu. Krivsky
in Dirac equation is also nonzero. A s a consequence, the hydrogen atom can be described [15,16,23,26-29] on the basis of the Ma• equations. Starting from [14] the Maxwell equations with gradient-like sources appeared in consideration. From our point of view it is the most interesting kind of the Maxwell equations especially in the problem of studying the relationship with the massless Dirac equation. That is the reason of studying these problems in details which we performed in our recent publications [19-29]. In [t9-21,24,25] we have found step by step the relationship between the symmetry properties of the Dirac and Maxwell equations, the connection between the conservation laws for the electromagnetic and spinor fields, the relationship between the Lagrangians for these fields and their quantization procedures. The relationship between the models of atom based on quantum mechanical Dirac and classical Maxwell equations was also investigated, see, e.g. [26-29]. Here we continue the investigation of these problems and the systematization of the results. Using the unitary operator [24,25] in the space of the 4-component complex spinors which connects the massless Dirac equation and the Maxwell equation with gradient-like sources, we are dealing with new possibilities in proving the symmetry properties of these equations. Here we apply the unitary relationship [24,25] to the investigation of additional symmetry properties first of all of the massless Dirac equation and then the Maxwell equations with gradient-like sources. A s a first step a new proof of the spin 1 (not 1/2) symmetries of the massless Dirac equation found in [20,21] is considered here. Moreover, we find new relations between the symmetry operators in the process of proving the corresponding theorems. After that the new symmetry properties of the massless Dirac equation are found. Here we underline that not only the Dirac equation has the bosonic symmetries but the Maxwell equations with gradien•-like sources has the fermionic symmetries too. Ir turns out that this form of the Maxwell equations is the most symmetrical one. 2. N o t a t i o n s Let us choose the 7 u matrices in the massless Dirac equation i~~o.r
= 0; ~ - ( ~ . )
0
, 9 R 4, ~, -- (~,~), o. -- - -oz,.
#----0,1,2,3, (1)
obeying the Clifford-Dirac algebra commutation relations 7~7 v + 7"7 ~ = 2g ~~, 7 ~t = g~~7~,
diag g = ( 1 , - i , - 1 , - 1 ) ,
(2)
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Advances in Applied Clifford Algebras 8, No. 1 (1998)
in the Pauli-Dirac representation (shortly: PD-representation):
~o 11 ol] ~~=t o ~kl =
0 -
'
- o -k 0
, k=1,2,3,
(3) =
10
'
= i O '
0-"
Conformal transformations (conformal group C(1,3) includes proper ortochronous Poincar› group P = ISL(2, C) ) in the space-time R 4 are infinitesimally represented in the form x~
>x
~#
(x) ------/(1
]- pa
@ aP91 --F -~w Upa -b nd + bPkp)x "~,
(4)
where f~ --= (aP,w pa, ~, bp) is the set of 15 independent real parameters of the C(1,3) D P group and the operators
0 , Mp~, - zpc% - z~,cgp, d -=- x~'O~,, kp =~ 2 x p d - Z20p cgp -- OxP
(5)
ate the generators of the representation of the C(1,3) group in R 4. Local C(1, 3) transformations
f(z) -> f' (x) ~ F ( ~ ) f ( G ~ I x ) ,
Ga E C ( 1 , 3 ) ,
f -___(ft,)
(6)
in a manifold {~ : R 4 > C 4} of 4-component complex-valued functions infinitesimally have the forro
f(~) --> f' (x) ~ ( i - aPap - -~wp 3pa - t~d- bPk#)f(x) A
A
(7)
A
In this case the generators (0, j , d, k) of the representation of real algebra of C(1, 3) D P group have the form (see, e.g. [20,21]) 0. = ~-~~., 3.,. = Mp~ + Spo, d'= d + r = kp = kp "b 2SpaX a -- 2"rXp :_ 2 x p d -
x~B~ +
r,
x20p "-I- 2Spax a ,
(8)
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Bosonic S y m m e t r i e s of the Massless . . . . .
V . M . Simulik and I. Yu. Krivsky
The generators (9) as well as the operators (5) obey the commutation relations
r^
A
1
A
A
r,, ^ 1
A
r,,
A 1
with arbitrary number r (the conformal number) and Spo matrices which are the generagors of the 4-dimensional represengation of the Lorentz group
SL(2, C). It is well known (see, e.g. [30,31]), that the equation (1) is invariant with respect to the transformations (6), (7) of the conformal group C(1, 3) in the case of fixing informulae (8) the conformal number r = 3/2, and choosing the 4• Spo matrices in the form S/~, --
41%,7o]ED
,0
@ 0,
,
(10)
where S/~ are the generators of spinor representation D ( 8 9 (0,89 of SL(2, C) group. Let us denote this representation of conformal group in the AI AI AI
manifold {f = ~} as C s D pS, while its generators - as (0, j ,d ,k ). It is well-known (see, e.g. [30]), that the equation (1) is invariant with respect to the real algebra As , whose generators have the form:
72C, {72C,
")'274C ,
i7274C, i, i74, 74, I,
(11)
where 74 -- 703,17273 and C is the operator of complex eonjugation: C ~ = ~*. Let us define in terms of the first six operators from (11) the following operators:
{ ,£237 : ~7 i 2C, S£II = - y 71 2C, So3 II s~~ = -s').
i 4, = - "g7
:
} (12)
SŸ = -~,i S• = _~~t 274C, S23H= ~i7~74C It is easy to verify that the operators (12) obey the same commutation relations as the generators (10) do, and, a s a eonsequence, they form another realization of the same spinor representation D (89 0) (9 (0, 89 of the SL(2, C) group. But, eontrary to operators (10), they themselves (without any differential angular momentum part) are the symmetry operators of the massless Dirac equation (1), i.e. they leave this equation being invariant.
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3. Local Bosonic S y m m e t r i e s of the Massless Dirac E q u a t i o n In order to consider the additional symmetries of the massless Dirac equation another representation (the so-called bosonic representation [24,25]) of the Dirac 3,~ matrices is preferable. This representation can be obtained from the 3'" matrices (3) in the PD-representation by the unitary relationship 7"
> ~~' =
(13)
VT" V ?,
where the unitary operator has the forro
V=
00 iC_ C+ C+ C_
0 0
00 iC+ C_ !| V ~ t C_ C+
C:t: =-
0 0 C+l. 2 ,
[ 0 0 C+ C_ C+ iC+ 0 0 I 0
0 C_ C+ C_ iC_ 0 0
/tv=
(14)
1.
The unitarity of the operator V can be proved by means of the relations Ca = (aC)* = a'C, (AC) t = C A t = A TC
(zs)
being valid for an arbitrary complex number a and the matrix A. The unitarity of the operator V i s the reason why the 3' matrices obey the same relations (2) of the Clifford-Dirac algebra as the 3,~ matrices (3) in the standard PDrepresentation do. It is shown in [21,24,25] that the massless Dirac equation ~ T . a . ~ ( x ) = o, ~ = vm,
(16)
can be considered not only as the equation for the spinor field but also as the equation for the bosonic field. Therefore the representation (13), (16) is called the bosonic representation (shortly: B-representation). Let us briefly recall with some details why the B-representation appears in consideration.
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Bosonic S y m m e t r i e s of the Massless . . . . .
V . M . Simulik and I. Yu. K r i v s k y
First of all note that after the substitution of r by any of the following eight columns
~[ --__
iE 3 - H o
iE 1 + E 2
iE 1 - E 2
-iE a _ H o -H 1 + iH 2
- H a +iE _H i _iH -E
o 2
_iH 1 _ H 2 iH 3 - E o
, Cv=
iH 3 + E ~
-H 3 + iE ~ _H 1 _ iH 2
-H
~ + iH t
-H , r
._~
iE 3 _ H o iE 1 _ E 2
1 + iH 2
H a q- i E o iE 1 + E ~ -lE a _ H o
(17)
_iH 1 - H 2
iH 3 + E ~ -H 2 + ~H 1
r
~ I I I _~
H 3 -k i E ~
1 + iE 2
E 3 _ iH o
,
E 3 + iH ~ E t + iE 2
,
iH 3 - E o
q VIII
-E
E 3 + iH ~ E l + iE 2
l --k i E 2
E 3 _ iH o
the Dirae equation (1) is transformed into equations for the system of electro-
m~~n~~i~(~,~')~nd ~~~l~r (~~ Oo-~ = curl]~ - gradE ~ d i v L ff = - O o E ~
~/~ ~old~.
007] = -curl-~ divT]
- gradH ~
(18)
= -OoH ~
(three other versions of these equations treatment are givert in [21,24,25], the proof that the complete set of linear connections between the Maxwell and the Dirac equations is given by formulae (17) see in [19,22,24,25]). In the notations (EU)=-
(E~
(H~
Eqs. (18) havethemanifestlycovariant
form O,E~ - O~E u + ~u~poOPH ~ =
0,
O , E u = O, O u H ~ = O.
(19)
In terms of the complex 4-component object E 1 _ iH 1 E 2 _ iH 2 g :
E3 _ iH 3 E o _ iH o
,
(20)
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75
Eqs. (18) have the form O#Eu -- cgvE# + ir
(21)
~ = O, O,E" = O.
The free Maxwell equations are obtained from Eqs. (18), (19), (21) in the case of E ~ = H ~ = 0. The unitary operator V (14) transform the CHz from (17) into the object
E (20):
i 11 r r
~o-
=~
Ca r
E a + E * a - Eo + E *o E 1 q- E *t q- iE 2 -.I-iE .2 E ~ + E *o _ E3 + E .3 _lE2 + is _ E1 + E,1
E1
E2
E3 = v e , (22) E0
T h a t is why the bosonic representation (13), (16) appeared in [21,24,25]. Now we are dealing with this representation because it appears to be the most convenient for the proof of the bosonic symmetry properties of the Dirac equation. Let us write down the explicit form of the Clifford-Dirac algebra generators in the bosonic representation
~0
i00 0 010 0 0010 000-i
0 0 0 0 0-i 0 i 0 -10 0
0 0 i0 0 0 01 -i 0 00 0 -100
0-i 0 0 i 0 0 0 C, 0 0 0 1 0 0 -1 0
1 0 0 C, 0
(23)
~2
~4 . = i , ~ 0 t . =
0
~oa=
iO0
-i 000 0 001 O 010
C, ~3
0 001 0 0 i0 0 -i00 1 0 00
O0-iO 0 , ~02 ~___ 0 •
i0 01
1 0 0 0 0
(24)
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Bosonic S y m m e t r i e s of t h e Massless . . . . .
V . M . Simulik a n d I. Yu. Krivsky
where ~u~... = ~u~,~.... The last relations in (24) are the transforms of the following PD-representation relations ,7#4 = --747 u, 7Jk = --~JklTOt4 (74 = 7~
(25)
In the B-representation, the complex number i is represented by the following matrix operator
= ViV t =iF,
0 i 0 -i0 0 000-1 0 0-1
s
0 0
=I't
(26)
r 2 = 1,
= F-l,
0
and the set (10) has the form _
S/~ -
_ _
41[5P'7~
S]k = -ieJk'S£
1
S£ = -57
-ol
9
(27)
Let us write down the explicit form of the S~• operators (11) in the Brepresentation:
0 Sor = {
0
0
O Oi! o
-1
-i
o
~--Sij=-VS~~V 1 : s j , =-i~~k's£ O0 -iO 0 0 0 -1 q- [ z ' s£
0 0
= 1
i 0 0-1
0 0
0 0
,o03=~
l_
0 i0 0 -i00 0 0 00-1 0 01 0
(28) Now we are able to introduce, in addition to the sets of generators (10), (12), the following two sets of matrices Spa:
•xu = S£ '-'Ok
rt Sok,
s ; [ = s~'o + s~'•
RHt --mn
[ + S..,, ti = S..,
(29) (a0)
T h e o r e m 1. The commutation relations (9b) of the Lorentz group are valid for each set ~.qt-zv of the Spa matrices. The sets (10), (12) (or (27) 1 (28)) pa ate the generators of the same (spinor) representation D (89 @ (0, 89 of the SL(2, C) group, the set (29) consists of the generators of the D (0, 1) (9 (0, 0)
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77
representation and the set (30) consists of the generators of the irreducible vector D (89 89 representation of the same group. P r o o f . The fact that the matrices (10) (or (27)) ate the generators of the spinor representation of the SL(2, C) group is well known. It is better to fulfil the proof of the nontrivial assertions of the theorem 1 in the B-representation where their validity can be seen directly from the explicit forro of the operators Sp~ even without the Casimir operators calculations. In fact, using the explicit forros of the matrices (27), (28) we find
Sp~, -. = c~'~oc ~
sT[
=
~'~o
0 00-I 0 O0 0 0 O0 0 =100 0
CS~~C, -.
=
, s~~
-,v Is~0 ~ 0 I: - - r o n ,
=
/~II[.'~III
S,~~ =
'~ok :
(31)
0 0 0 0 0 0 0 1 0 0 00 0-10 0
~IV
, 5o3
:
]80kO]
000 0 000 0 O00-1 001 0
, (32)
(33)
0 0 '
where :
i_ k m n
2 e Os n 0 --SkO, 0-1 ‰ ~ 1 0 0 , s~3= 0 0 - i ; 831 0 0 0 Ol 0
‰
0 01 -----
(34)
01 O0 O0
The unitarity of the C operator in relations (31), the direct calculation of correspondings commutators and the Casimir operators Sj: of the SL(2, C) group
S~ = -~(q + i T 2 ) ,
n
--
--~S ~' S . ~ . ,
T~_ =
-
r176
(35)
complete the proof of the theorem. QED. It is interesting to mark the following. Despite the fact that the matrices S-~~, and S'tp• are unitarily interconnected according to formulae (31), which in the PD-representation have the forro
s~,o-- 91 s ~ ~ c^, 91_
v'cv=
CO00 0100 OOCO 0001
(36)
78
B o s o n i c S y m m e t r i e s of t h e M a s s l e s s . . . . .
V . M . Simulik a n d I. Yu. K r i v s k y
the matrices S~a (10) (or (27)), as well as the matrices (29), (30) (or (32), (33)), contrary to the matrix operators (12) (or (28)) being taken themselves are not the invariance transformations of the massless Dirac equation (1) (or (16)). It is evident because the C operator does not commute (or anticommute) with the Diracian 7u0u. Nevertheless, due to the validity of the relations
[s.;;~,_.~~','",'vl,
=
0, ~, ~, p, ~ = (0,1, 2, 3),
not only the generktors (0, ]s) of the well known spinor representation the Poincar› group, but also the following generators
.?per'III'IV = Mpa "]- -pa'~III'IV
(37) pS of
(38)
are the transformations of invariance of the massless Dirac equation. It means the validity of the following assertion. T h e o r e m 2. The massless Dirac equation is invariant with respect to three different local representations pS pT pV of the Poincar› group P given by formula ~l/(x)
) xIl'(x) = FS-I*I(oa)~(A -1 (ac - a)),
(39)
where
FI(w) 9 D (0,89 @ ( 89 ( P = pS), FII(w) 9 D (0, 1) @ (0, 0), (P = pTs), FIH(w) 9 D(89189 ( P = P V ) .
(40)
P r o o f of the theorem follows from the theorem 1 and the above-mentioned consideration. QED. It is easy to construct the corresponding local C(1, 3) representations of conformal group, i. e. C s, CT and C v, but only one of them (the well-known local spinor representation C s) gives the transformations of invariance of the massless Dirac equation. 4. Lie-Backlund and N o n l o c a l S y m m e t r i e s
The simpliest of the Lie-Backlund symmetries are the transformations of invariance generated by the first-order differential operators with the matrix coefficients. The operators of the Maxwell and Dirac equations also belong to
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this class of operators. In order to complete the present eonsideration let us briefly reca].l our result [19,20]. T h e o r e m 3. The 128-d~mensional algebra A12s, whose generators are Qa = (0, ~t, ~I, ~r), Qb (11) and all their compositions Q a Q £ , a = (1,2, 3, ..., 15), b = (1, 2, 3, ..., 8), is the simplest algebra of invariance of the massless Dirac equation in the class of the first-order differential operators with matrix coefficients. P r o o f . For details see [19,20]. QED. In the class of nonlocal operators we ate able to represent here the new result. Despite the fact that above-mentioned local C Ÿ and C V are not the symmetries of the massless Dirac equation the corresponding symmetries can be constructed in the class of nonlocal operators. T h e o r e m 4. The massless Dirac equation (1) (or (16)) is invariant with respect to the representations ~T, ~ v of the conformal group C(1, 3). The corresponding generators are (0, " ~ r t X , l V ) being added by the following nonlocal operators dlll'[V
:
2
,3ok
, "~o
:
~
A ,~JOk
/
+
,
(41) m
~
,30m
J ,
where A = 0~. P r o o f . The validity of this theorem follows from the above-mentioned theorem 2 and the theorem 4 in [31].QED. 5. F e r m i o n i c S y m m e t r i e s of t h e M a x w e l l E q u a t i o n s Fermionic (i.e. spin 1/2) symmetry of the Maxwell equations is possible in the case of special kind of sources - the so-called gradient-like currents and charges. The corresponding forms of the Maxwell equations are presented in (18), (19) and (21).The reason of that fact is an assertion that all these forms are mathematically equivalent to the following equation 7~,a~g(~) = o, g = w " ~ ,
(42)
which is the Dirac-like form of the Maxwell equations with gradient-like sources. Here the notations g, ~u, i" ate the same as in (20), (23), (26). The validity of all four above-mentioned theorems for the Maxwell equations with gradient-Iike sources (42) is evident. It means that the kind of the Maxwell
80
Bosonic Symmetries of the Massless .....
V.M. Simulik and I. Yu. Krivsky
equations (18), (19), (21) and (42) is invariant not only to the ordinary-looking bosonic tensorial symmetry pT but also for the additional symmetries pS, pV. Of course, the most important is the fact of fermionic pS symmetry of such Maxwell equations, i.e. its symmetry with respect to the spin 1/2 representations of the Poincar› and conformal groups generated by the spinor D (89 @ (0, 89 representation of the Lorentz SL(2, C) group. Our experience (see [t9-21] and references therein) in investigating the symmetries of MaxwelI equations alIows us to suppose that the Maxwell equations with the gradient-like sources is the most symmetrical forro of these equations. 6. C o n c l u s i o n s What is the application of the most symmetrical Maxwell equations with the gradient-like sources to physics? W h a t is the meaning of these equations in physics? To answer these questions one can be referred to our pal~ers [24-29,32], where (i) using the solutions of these equations the speculation about the existence of longitudinal electromagnetic waves near the gradient-like currents and charges was suggested, (ii) on the basis of these equations the hydrogen spectrum was described completely within the framework of a classical electrodynamics without any appealing to probabilities and quantum mechanics (see also the review in [33]). Other, related, analysis of the massless Dirac to Maxwell equations can be found in the interesting approach of Vaz, Rodrigues et al. [34-38]. One of the conclusions of the investigation presented here is that a field equation itself does not answer the question, what kind of particle (Bose or Fermi) is described by this equation. To answ•r this question one needs to find all the representations of the Poincar› group under which the equation is invariant. If more than one such Poincar› representation is found, including the representations with integer and half-integer spins, then the given equation describes both Bose and Fermi particles, and both quantization types (Bose and Fermi) of the field function, obeying this equation, satisfy the microcausalit: condition (see, e.g. [24,25]).
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