CC BY
31
UDC 530.1; 539.1
INTERACTION OF MASSIVE FERMIONIC HIGHER SPIN FIELDS WITH CONSTANT
ELECTROMAGNETIC FIELD
V. A. Krykhtin
Department of Theoretical Physics, Tomsk State Pedagogical University, Kievskava str., 60, 634061 Tomsk, Russia.
E-mail: krvkhtinQtspu.edu.ru
We develop a universal general gauge-invariant method of Lagrangian construction based on the BRST approach for halfinteger higher spin fields interacting with constant electromagnetic field in Minkowski space of any dimension. No off-shell constraints for the fields and gauge parameters are imposed from the very beginning.
Keywords: higher spin Gelds, BRST approach, fermionic fields.
1 Introduction
Constructing of Lagrangians for interacting higher spin fields is a problem which has been attracting much attention for a long time due to the hope of finding new possibilities and approaches to the unification of the fundamental interactions and is one of the unsolved general problems of classical field theory (see, e.g., the reviews [1]).
In the present paper we investigate the interaction problem of fermionic fields. This problem has been studied much less than the problem of interacting bosonic fields. In particular it was studied construction of the cubic vertices in the light cone framework [2], and different problems of interaction with gravitational and elctromagnetic field (see e.g. [3-10]).
In this paper we develop a gauge-invariant approach based on the BRST construction to a Lagrangian construction for totally symmetric fermionic higher spin fields interacting with constant electromagnetic field in Minkowski space of any dimension and solve the problem in the linear in approximation.
The paper is organized as follows. In section 2, we give brief reminder about BRST approach to Lagrangian construction for free massive fermionic field in Minkowski space. Then in section 3 we modify this procedure so as to construct interaction of the fermionic fields with constant electromagnetic field and realize it in the linear in approximation.
It is known that totally symmetric tensor-spinor field (the Dirac index is suppressed) describing
the irreducible massive spin s = n +1/2 representation of the Poincare group must satisfy the following conditions
(iYvdv - m)^Ml...Mn = 0
dMl Wi...M„ =0.
YMl VVi ...Mn =0,
(1)
with |ym, yv} = -2nMV.
In order to avoid explicit manipulation with the indices it is convenient to introduce an auxiliary Fock space generated by creation and annihilation operators a+, av satisfying the commutation relations
[a+,«v ] = nMv.
Eqs (1) are realized in this Fock space as follows
(2)
(iYvdv - Ym)|^„) =0,
=0,
№n) = ^Mi-Mna+Ml ...a+Mn|0).
Y^mK ) =0,
(3)
Here we introduce Grassmann odd "gamma-marix like objects" ym and 7 which are connected with the usual Grassmann even gamma-matrices ym by relation [11]
2 Free massive fermionic fields in Minkowski space
Lagrangian construction for massive fermionic field in Minkowski space based on BRST approach was carried out in [11] (see also [12,13]). In this section we briefly remind the result1.
YM = YMY, (4)
{7M, Yv} = -2nMV, {YM, 7} = 0, {Y, 7} = -2. (5)
Next following the procedure described e.g. in [11-13] on the base of the operators (3) we construct all the operators involved in the BRST charge. These
1 Unlike [11] the signature of the metrics in this paper is "mostly plus". There are also other differences in the notation.
operators are
to = - 7m, (6)
lo = d2 - m2, (7)
Zi = iaMdM + mbi, (8)
1+ = + m6+, (9)
ti = - Ybi + f+b2 - 2(6+62 + h)f, (10)
t+ = «+7M - Y6+ + f + - 26+ f, (11)
12 = 1 + 2 62 + (6+62 + f+f + h)62, (12)
1+ = 2 a+M«+ + 16+2 + 6+, (13)
go = a+aM + 6+61 + 26+62 + f +f + + h. (14)
To construct the operators above we enlarge the Fock
space with two pairs of bosonic (6+, 61 and 6+, 62) and f+ f
operators with the standard commutation relations
[6i,6+] = 1,
[b2,b+] = 1, {/,/ +} = 1
(*№>„ew = (*i|K |*2>,
CO ..
Kh = E n (|n>(n|C(n,h)
n=0
and possess the standard ghost number distribution, gh(Ci) = -gh(Pj) = 1, providing the property gh(Q) = 1.
Extracting from the BRST operator first dependence on ghosts nG PG and then no, Po and qo, po one obtains
Q = Q + nG (N + ^ + h) + (2q+qi - n+n2)PG,
N = a+ aM + 6+6i + 26+62 + f +f + q+ipi
- ip+qi + n+Pi + P+ni + 2n+P2 + 2P+n2,
Q = qoto + nolo + AQ
+ (q+ni - n+qi)ipo + (n+ni - q2)po,
AQ = q+ti + qit+ + n+li + nil+n+12 + n2l+
h
below, is associated with spin of the field h = 2 - s -1. Note that the set of the operators is invariant under Hermitian conjugation if we change the scalar product in the (62, f)-sector of the Fock space as follows
+t1 + + + + + + 2q+2P2 + 2q2P+ + q+n2ip+ - n+qiipi
- n+niPi - n+n2P +, to = to + 2q+P i +2qiP +,
where Q is independent of nG PG and AQ to are also independent of no, Po and qo, po.
Next following the procedure of [11] we choose the following representation of the Hilbert space
(p0,qi,pi, Po, Pg, ni, Pi,n2, P2) |0> = 0,
(18)
(15)
(16)
and suppose that the vectors and gauge parameters do not depend on nG,
|X>
(
ki
(qo)kl (q+)k2 (p+)k3 (no)k4(/+)k5 (n+ )k6 x
- 2/ + |n>(n|/C(n + 1,h)) ,
C(n, h) = h(h +1) • • • (h + n - 1), C(0, h) = 1, |n> = (6+)n|0>.
The BRST operator constructed on the base of operators (6)-(14) is
Q = qo to + q+ti + qit+ + nolo + n+1i + ni1+
+ n+12 + n2l+ + nGgo + 2qo(q+ P i + qiP +) + (q+ni - n+qi)iPo + (n+ni - q2)Po + 2q+2P2 + 2q2P+ + q+n2ip+ - n2+ qi»Pi - n2+niPi - n+n2P+ + (2q+ qi - n+n2)Pg + nG(q+ipi - qiip+ + n+Pi - niP+ + 2n++P2 - 2n2P +).
Here, qo, qi, q+ and no, n+, ni, n+ nG are, respectively, the bosonic and fermionic ghost "coordinates" corresponding to their canonically
po p+ p1 Po P1 P+ P2 P +, PG. They obey the (anti)commutation relations
{ni, P + } ={Pi,n+} = {n2, P +} = {P2,n++} ={no, P o} = {nG, P g} = 1, [qo,po] =[qi,p+] = [q+,pi] = i (17)
x (P+)k7(n+ )k8(P2+)k9(6+)kl0(6+)kl1 x
x a
1
+Ml
,+Mfco
Xr X (x)|0>.
(19)
The sum in (19) is taken over k0,ki,k^ k3, ki0, kii, running from 0 to infinity, and over k4, k5, k6, k7, k8, k9, running from 0 to 1. Then, we derive from the equations that determine the physical vector, Q|x> = 0, as well as from the reducible gauge transformations, J|x> = Q|A>, a sequence of relations
Q|x> =0, <%> = Q|A>, <S|A> = Q|A(i) >, ¿|A(i-i) > = Q|A(i
(a + h) |x> =0, (20)
(a + h)|A> =0, (21)
(a + h)|A(i) > =0, (22)
(a + h)|A(i)> =0, (23)
The middle equation in (20) presents the equations for h
h = 2 - s - d. (24)
By fixing the value of spin, we also fix the parameter hh must substitute it into each of the expressions (20)-(23). (See [11-13] for more details.)
As was shown in [11] the equation of motion (20) Q|x) = 0 indeed reproduce equations (1) up
to the reducible gauge transformations (21)-(23), but this equation of motion can't be obtained from a Lagrangian.
To extract from Q|x) = 0 lagrangian set of equations of motion we decompose the state vector and gauge parameters in qo, no ghosts
lx) = £ ?o (IXo) + nolxk)),
k=0 to
|A) = £ q° (|A°) + no|Ak)).
o=o
Then following the procedure described in [12] we remove some of the fields with the help of a part of the gauge transformation and a part of the equations of motion and the remaining fields will be |xo) and |Xo)- Their equations of motion and reducible gauge transformations are
AQ|xo) + 2 {to ,n+ni}|Xo) =0, to|xo) + AQ|Xo) =0,
5|xo) = AQA) + 1{to, n+ni}|Ao), 5|x1) = to|Ao) +AQ|AJ),
¿|A(i)o) = AQ|A(i+1)o) + ^ Tfo,n1hn^ |A(i+1)o),
¿|A(i)o) = Tfo|A(i+1)o) + AQ|A(i+1)J),
that the trace of a field and its traceless part are independent each other and therefore we can shift the trace of the field so that the traceless condition remains unchanged. Thus we suppose that the operators related with the traceless condition t1; t+, Z2, Z+ and go also remains unchanged. Moreover the (62, /)-sector of the Fock space was introduced to modify only these operators [see (10)—(14)] and therefore we also suppose that the creation and annihilation operators of this part of the Fock space will not take part in construction of enlarged expressions for the rest operators Lo, L1; L+.
Since we are going to consider only linear in FMV approximation we take the following ansatz for the operators
where |A(o)o)„ = |Ao), and |A(o)J)„ = |Aj).
Equations (26) can be obtained from Lagrangian
L = <xS|Kh{to |xo) + AQ|x1)} + <xo|Kfc{ AQ|xo) + 2 {io,nin1}|xo^. (31)
Here {t o, n1n1} = to^^ + n+n^ o and is operator (16).
In the next section we generalize the above construction to the case of massive fermionic fields interacting with constant electromagnetic field.
3 Fermionic fields interacting with constant EM field in Minkowski space
Let us try to construct interaction of the fermionic fields with constant EM field FMV = const by the following way. First we replace all the partial derivatives by the covariant ones DM = — ieAM and enlarge the expressions of the operators (6)-(14) by terms vanishing at FMV ^ 0 limit (we will denote these enlarged expressions of the operators by the corresponding upper case letters) and demand that the new expressions for the operators form an algebra.
Before writing ansatz for the new expressions for the operators we give some comments. It is known
L1 = iaaDa + mb1 + aaFaff D" £ /oo 6+° 6°
o=o
to
+ 77tD" £ /2° 6+°6°+1
o=o
to
+ a+^F^ D" £ /40 6+°6°+2
o=o
to
+ y £ doo 6+°6°+1
o=o
to
+ YY"Fffaaa £ d2° 6+°6°
o=o
to
+ o+^FMaaa £ dso 6+°6°+1
o=o
to
+ 77"£ d4° 6+°6°+2, (32)
°=o
to
To = iYMDM — Ym + ytFt"D" £ co° 6+°6°
°=o
to
+ YaaFa"D" £ C4° (6+)°+16°
°=o
to
+ 7 D" £ C5° 6+°6°+1
°=o
to
+ 7 7MVFMv £ ao° 6+° 6°
°=o
to
+ Yo+MFMaaa £ a4° 6+°6°
°=o
to
+ Y"F"aaa £ a2° (6+)°+16° °=o
TO
+ 7"F"Ma+^ £ a,3° 6+°6°+1, (33)
°=o
L+ = ia+MDM + m6+ + a+MFM"D" £ /1° 6+°6°
°=o
+ 77tD7 E f3k (b+)fc+1 b
1=0 TO
+ aaFa(TD7 E /sfc (b+)fc+2&i
1=0
to
+ 7MVE dii (b+)1+1bl
1=0
to
+ 777E d3i
i=0
TO
+ E d9i (b+)1+1bl
i=0
TO
+ 77"Fffaaa E d5i (b+)fc+2b!
(34)
1=0
where ajk, cjk, cik are arbitrary complex constants and by definition we suppose that L0 = T2. The rest operators (10)-(14) are unchanged. One should also note that the ansatz for the operators Li, TO, L+ (32)-(34) are not general.
L0
(Li)+ = L+ with respect to the "new" scalar product2 (15) the arbitrary coefficients in (32)-(34) are found from the condition that the new operators form an algebra in the linear approximation
a0(0) ie 8m - —Co, 2m0 a0(fc) = 0,
a2(0) ie 2m m a2(1) = (-2)1 k! e£, m
a3(0) ie 2m - ^1, m a3(fc) = (-2)1 k! ea, m
a4(0) ie 2m 2ie + — Z0, m a4(1) = 0,
d0(0) ie 8m ie + 2m (Z0 + ^ d0(fc) = (-2)1-1 e ^ k! m 1
d1(0) ie 8m ie + 2- (Z0 - i£0, d1(fc) = (-2)1-k! -eC1, m
d2(0) ie 4m 2— d2(1) (-2)1-k! 1 "(k + 1) m
d3(0) ie 4m + 2m d3(fc) = - (T (k + 1)e«1- k!m
d4(1) = ("f 'iL k>0, k!m
d5(fc
d8(0
d9(0
(-2)1 e
k!
m
k > 0,
ie ie . „
---(2Z0 + i&),
mm
ie ie . „
---(2Z0 - i&),
mm
d8(l)
J9(fc)
(-2)i e
k! m (—2)k e
£1,
k!
£1,
c0(0
c4(0
c5(0 f0(0
f0(1 f1(0
f1(1 f2(0 f3(0 f4(0 f5(0
ie
2m2
e m
e
(2Z1 + £1),
^ (-2Z0 + i&),
— (2Z0 + i&), m2
(Z0 + iZ1),
m2
e
— (2Z0 - Í&), m2
(-Z0 + iZ1),
m2
e
-^ (2Z0 + ), ^ (Z0 -
c0(1) = ( -2)1 ie
k! m2
c4(1) = ( -2)1 ie
k! m2
c5(1) = ( -2)1 ie
k! m2
¿1,
a,
m
2
ei
-2 (Z0 + 2 £1),
m2 2
--2 ^
m2
zrn C0,
/0(1) — -
f1(fc) = -f2(1) = f3(1) = -
(-2)
i1
k!
^1,
(-2)
1-1
k! m (-2)1-1 ie
ra
k!
(-2)
i-1
k!
^1,
/4(1) — 0 k > 1, /5(1) = 0 k > 1
Here Z0, C^ are arbitrary real dimensionless constants. Note that similar problem was considered in [3], but we found two more arbitrary constant due to rejection of some condition on the coefficients in (32)-(34). The new operators form algebra which coincides with the algebra in the free case.
Further construction of Lagrangians go in the usual way [11,12].
4 Conclusion
Thus in the present paper we have developed a universal general gauge-invariant method of Lagrangian construction based on the BRST approach for half-integer higher spin fields interacting with constant electromagnetic field in Minkowski space of any dimension. In this procedure no off-shell constraints for the fields and gauge parameters are imposed from the very beginning.
Acknowledgement
V. A. K. is grateful to the grant for LRSS, projects 88.2014.2 and RFBR grants, project 12-02-00121-a and 13-02-90430 for partial support. His research was also supported by grant of Russian Ministry of Education and Science, project TSPU-122.
2In the (aM, bi)-sector of the Fock space in which the operators Lo, Li, L+ are constructed scalar product (15) coincide with the usual scalar product in the Fock space.
m
2
m
m
m
References
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Received 03.11.2014
В. А. Крыхтин
ВЗАИМОДЕЙСТВИЕ МАССИВНЫХ ФЕРМИОННЫХ ПОЛЕЙ ВЫСШИХ СПИНОВ С ПОСТОЯННЫМ ЭЛЕКТРОМАГНИТНЫМ ПОЛЕМ
Развивается БРСТ подход к построению лагранжианов для фермионных полей высших спинов, взаимодействующих с постоянным электромагнитным полем в пространстве Минковского произвольной размерности. В предлагаемой процедуре построения лагранжианов не предполагается никаких ограничений на поля и калибровочные параметры.
Ключевые слова: поля высших спинов, БРСТ подход, ферм,ионные поля.
Крыхтин В. А., доктор физико-математических наук, доцент. Томский государственный педагогический университет.
Ул. Киевская, 60, 634061 Томск, Россия. E-mail: krykhtin@tspu.edu.ru
