CC BY
43
UDC 530.1; 539.1
Quantum Equivalence of Massive and Massless p-form Models in Curved
D-Dimensional Space
E. N. Kirillova
Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk, 634061, Russia.
E-mail: kirillovaenQtspu.edu.ru
We consider massive and massless p-forms in arbitrary D-dimensional curved space-time. Quantization of these models and evaluation of effective actions have been performed. Results are presented in terms of d'Alembertians acting on p-forms. The massive theories of p-forms do not possess gauge invariance, in contrary of massless theories. The gauge invariance is restored with help of multi-step Stuckelberg procedure. Comparising effective actions of classically equivalent theories (massless quantum theory of p-forms and theory of (D-p-2)-forms, massive quantum theory of p-forms and theory of (D-p-l)-forms), we demonstrate a quantum equivalence of corresponding models. A zeta-function excluding zero modes has been used.
Keywords: quantum Gelds in curved space-time, p-forms, gauge field theories, effective, action, quantum equivalence.
1 Introduction
Totally antisymmetric tensor fields (ATFs, p-forms) have been investigated since the sixties ( fl] etc.). ATFs have applications in string, supersymmetry and supergravity models, in quantum chromodynamics. ATGFs are considered in context of quantum equivalence of different field representations by [2], [3] and so on.
At present p-forms arouse interest for the most part in connexion with supergravity, M or string theory ( [4] etc.). Some behavior of massive ATFs has been investigated by [5], [6] and others.
In this paper, we consider massive and massless p-form theories in arbitrary D-dimensional curved spacetime. Sections 2 and 3 are devoted to describing and quantization of massive and massless p-form models. In section 4, we study the questions of quantum equivalence of classically equivalent massive and massless p-form theories with investigation of EAs structure in terms of a generalized zeta-functions associated with corresponding d’Alembertians. In conclusion, presented results are dicussed briefly.
2 Models
(p)
We consider p-form B, p < D, in arbitrary D-dimensional curved space:
(p)
1
B = T p!
E
B
Ml—Mp
(x) dxMl A ... A dxMp,
Ml—Mp
with an exterior derivative d,
(p)
and a co-derivative S, (SB)Ml...Mp-1 = —B^...^ .
Combination of d with S yields de Rham Laplasian A = dS + Sd = — □, □ being the usual d’Alembertian.
Generalizing Maxwell action to p-forms, one has a classical action for ATGFs, massless p-forms, in arbitrary D-dimensional curved space:
(p) 1 (p) (p)
S [ B ] = - 2(dB,dB ).
(1)
This action is invariant under the gauge transformations (GTs), because of d2 = 0,
(k) (k) (k) (k-i)
B ^ B' = B —d B , 1 < k < p, p < D. (2)
Adding to the action (1) a massive term, we get a
generalized Proca action, which is not invariant under
GTs (2) now:
(p)
1 (p) (p)
S m[ B ] = — 2(dB,dB) + —(B, B).
To quantize this theory it is convenient to restore the gauge invariance with help of Stuckelberg procedure
(p-i)
[7]. Introducing an extra field C , we get new action
(p) (p-i) (p) 1 (p-i)
Sm[B, C ] = Sm[B+-d C ], (3)
m
which is invariant under the interrelated GTs of the
(p) (p-i)
BC
d B = (p | 1)! y~] (dB)Mi...Mp+i (x) dxMl A...AdxMp+1.
(k) (k) (k) (fc-1)
B ^ B' = B —d B ,
(k-1) (k-1) (k-1) (k-1)
C ^ C' = c +m B , 1 < k < p.
(4)
Ml ...Mp+i
Now we proceed to quantization of models (1) and (3).
3 Quantization of massive and massless p-form models in curved D-dimensional space
To quantize theories (1) and (3) we use Faddeev-Popov multistep procedure in a form adapted for Abelian theories with linearly dependent generators [3,8].
(p) (p-i)
BC
corresponding to the classical action (1),
(p) f (p) , (p) , ,
I[B] = Ip = DB expi(S[B]), (5)
and to the classical action (3),
(p) (p-1) f (p) (p-1) (p) (p-1)
Im[B, C ] = / D B D C exp(iS[B, C ]), (6)
contain infinite gauge volumes, related to the GTs (2) and (4) correspondingly.
A gauge fixing function in massless case has usual (p-1) (p)
form: K = SB. Separating a gauge group volume
from path integral (5), one should input to this integral
(p-1)
K
(p) (p)
and, in view of a relation Z[B] = exp(*r[B]), an effective action
. p+1 *
2
rp = -ôE(-1)p+1-k(P +1 - k)Tr ln □ fc.
(8)
k=0
In massive theory (6), we choose the gauge fixing
(p-1) (p) (p-1) (p-2) function as Km = SB +m( C +d C ), last term
is absent if (p — 2) < 0, at lower quantization level.
Constructing a “senior” Faddeev-Popov determinant
i (p-1) (p-2) ~ (p-1)
A-- = J D B D C ~[ K' ],
A -1
related to the lower GTs levels: p — 2, p — 3, p — 4,..., up
(0 ) (0 ) (0 ) to the A0 = Det(^ 0+m2^d K' = K +(□ 0+m2) B.
After extracting from (6) infinite gauge volumes, related to all levels GTs (4), we obtain p-order generating functional in massive theory (with k0(p) from (7)):
ym
Zp
+ 1/2 (p-1)/2-n-ko(p)
n n
n=-1/2 s = 0
So, the effective action be
Detn(^p-2S-„-1/2 + m2).
(p) (p-1)
(p)
Ap-W D B ¿[ K ] exp*(S[B]),
here ~[... ] being delta- function (tilde distinguishes it
~ (p-1)
from co-derivative S). However, it appears that ~[ K ] is ill defined, because of co-derivative obeys S2 = 0.
(p-1)
K
generalizing method presented in [3,8]. Denoting new generalized delta- function as S [...], we obtain:
(k) k-2 (fc-3+fc0)/2 f (2s + 1-ko )
D V x
i=0
_ (2s+3-ko ) (2s + 1-ko) - (1)
x5[5 V +d V ] • (ko + (1 — ko) • ¿[J V]),
where Ak = ]k (Det □ s)(k s+1) ( 1)
ko(k) = (1 + ( — 1)k)/2.
(fc-s)
and
+1/2
rm = —* E
n=-1/2
(p-1)/2-n-ko(p)
E nTr ln (□p-2s-n-1/2 + m2). (9)
4 Quantum equivalence
It is known that massless p-form in D-dimensional D p— 2
massive p-form has 0 D— 1 ) degrees. It occurs that
quantum theories of p-forms reserve the properties of equivalence of classically equivalent theories.
Deriving a difference between EAs of massless p-forms and of (D-p-2)-forms, A rp = rp — rD-p-2, we obtain with the help of (8):
(7)
A rp = 2 E( —1)p+k(—k + p + 1)Tr ln □ k.
k=0
Further we use in (10) a relation between opera-(—□ k)
zeta-functions, Trln( — □) = —(Zk(0) + ln(^2)Zk(0)), similar relation is hold for massive case. Zeta-function is defined by a formula, which excludes zero modes [9]:
D
(10)
Using this new generalized delta-function, we eliminate from path integral (5) infinite gauge volumes, tied with GTs (2) of all levels Ar, 1 < k < p. Then we obtain an expression for p-order generating functional
Zp = 1](Det □k)^ ( —1)p+1-k, p < D, k=0
Ck(s)
1
r(s)
dtts-1 Tr(e k — Pk )
p
CXJ
0
Pk is a projector onto the space of the zero modes of □k
Using some properties of zeta-functions [9], we can demonstrate that A rp (10) is equal to zero, that implies quantum equivalence of massless theories of p-forms and of (D-p-2)-forms.
Under the usual definition of the zeta-function that includes zero modes this difference gives the Gauss-Bonnet topological invariant. Corresponding energy-momentum tensors coincide.
In massive case, a difference between EAs (9) of massive p-forms and of (D-p-l)-forms, A rm = rm —
pm
r D-p-1>
be
A rm
2 <-i)p t.
k=0
(-1)kTr ln (□ k + m2).
(H)
Using an expansion of zeta-function at non-zero mass in power series in massless zeta-function, one gets a relation between operators ( —□ p + m2) and generalized zeta-functions. Then we obtain for (11):
D
A rm = -^(-1)p£(-1)k[Zk(0,m)+ln(M2)Cfc(0,m)j.
k=0
With relations for zeta-functions [91 we arrive to a con-
clusion that A rm
0
sive p-form models and of (D-p-l)-form models are coincides, and these theories are quantum equivalent.
5 Conclusion
Quantization scheme of massive and massless p-form models in arbitrary D-dimensional curved space has been presented. Gauge invariance in massive theories is restored with the help of the Stiickelberg multistep procedure. Effective actions are evaluated in terms of d‘Alembertians acting on a p-forms.
We demonstrate a quantum equivalence of classically equivalent theories by means of comparison of EAs of corresponding models. A zeta-function excluding zero modes has been used. It was proven in D-dimensional curved space that massless quantum theory of p-forms is equivalent to theory of (D-p-2)-forms and massive quantum theory of p-forms is equivalent to theory of (D-p-l)-forms.
Acknowledgement
Author is grateful to professor IX. Buchbinder for a statement of the problem.
References
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Received 01.10.2012
E.H. Кириллова
КВАНТОВАЯ ЭКВИВАЛЕНТНОСТЬ МАССИВНЫХ И БЕЗМАССОВЫХ МОДЕЛЕЙ Р-ФОРМ В ИСКРИВЛЕННОМ D-MEPHOM ПРОСТРАНСТВЕ
Рассматриваются массивные и безмассовые р-формы в произвольном D-мерном искривленном пространстве-времени. Выполнено квантование данных моделей, произведена оценка эффективного действия. Результаты представлены в терминах Даламбертианов, действующих на р-формы. Массивные теории р-форм не обладают калибровочной инвариантностью, в отличие от безмассовых теорий. Калибровочная инвариантность восстанавливается с помощью многоступенчатой процедуры Штюкельберга. Сравнение эффективного действия классически эквивалентных теорий (безмассовые квантовые теории р-форм и (Б-р-2)-форм, массивные квантовые теории р-форм и (Б-р-1)-форм) доказывает квантовую эквивалентность соответствующих моделей. Для доказательства используется дзета-функция с исключенными нулевыми модами.
Ключевые слова: к вантовые поля в искривленном пространстве-времени, р-формы, калибровочные полевые теории, эффективное действие, квантовая эквивалентность.
Кириллова E. H., кандидат физико-математических наук, доцент.
Томский государственный педагогический университет.
Ул. Киевская, 60, Томск, Россия, 634061.
E-mail: [email protected]
