- Email: [email protected]
Remarks about UV regularization of basic commutators in string theories
UCLEAR PHYSIC'.
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 104 (2002) 165-168
www.elsevier.com/Iocate/npe
Remarks about UV regularization of basic commutators in string theories Alexander Yu. Kamenshchik a., Isaak M. Khalatnikov bt and Maurizio Martellini c aL.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences, Kosygin str. 2, 117334 Moscow, Russia Landau Network - Centro Volta, Villa Olmo, via Cantoni 1, 22100 Como, Italy b L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences, Kosygin str. 2, 117334 Moscow, Russia Landau Network - Centro Volta, Villa Olmo, via Cantoni 1, 22100 Como, Italy Tel Aviv University, Raymond and Sackler Faculty of Exact Sciences; School of Physics and Astronomy, Ramat Aviv, 69978, Israel cUniversity of Insubria, via Valleggio 11, 22100 Como, Italy Landau Network - Centro Volta, Villa Olmo, via Cantoni 1, 22100 Como, Italy The zeta regularization of basic commutators in string theories, recently proposed by Hwang, Marnelius and Saltsidis, is generalized to the string models with non-trivial vacuums. It is shown that implementation of this regularization implies the cancellation of dangerous terms in the commutators between Virasoro generators, which break the Jacobi identity.
It is well known what role is played by the problem of regularization and renormalization of ultraviolet divergences in quantum field theory. However, the importance of the problem of a proper treatment of ultraviolet divergences arising from quantization of string theories is less understood. Perhaps, it is connected with the fact that calculating the central extensions of Virasoro algebra in string theories it is possible to avoid the need to consider infinite sums, by virtue of the tacitly assumed and very natural regularization providing their cancellation [1]. However, in the cases when conventional vacuum state is absent (for example in the case of tensionless conformal string [2]) explicit ultraviolet regularization of basic commutators of the theory becomes unavoidable. In recent papers by Hwang, Marnelius and Saltsidis [3] a convenient version of ultraviolet zeta regularization of basic commutators in string theories was elaborated. This scheme was applied *Supported by the CARIPLO Scientific Foundation and by RFBR via grant 99-02-18409 ?Supported by RFBR via grant 99-02-18409
to Hamiltonian B R S T quantization [4] of string theories. The well-known results for critical dimensions obtained by the same method earlier [5] were reproduced in the new context. Here, we consider traditional closed bosonic string model but with non-trivial vacuums. These vacuums are obtained by means of redefinition of creation and annihilation operators due to Bogoliubov transformation. As a result of this redefinition, in the expression for central extension infinite sums arise while the traditional finite part of contribution to the central charge acquires the structures responsible for violation of Jacobi identities. The direct generalization of the regularization suggested in Ref. [3] allows to calculate the finite contributions of infinite sums, which have the structure providing the total cancellation of pathological terms and restoration of Jacobi identities. One can hope that this scheme can be applied to more complicated theories such as quantum cosmology where the necessity of ultraviolet regularization is getting obvious [6]. The detailed presentation of the results obtained is given in our paper [7].
- see front matter © 2002 ElsevierScience B.V. All rights reserved. PII S0920-5632(01)01605-X
0920-5632/02/$
A. Yu. Kamenshchik et al. /Nuclear Physics B (Proc. Suppl.) 104 (2002) 165-168
166
To begin with let us consider a closed bosonic string. The constraints in the Hamiltonian formalism have the following form: HI
=
;p2+ ;qt2,
H = w’,
(2)
where HI is the so called super-Hamiltonian constraint, H is supermomentum, q is the coordinate of string, while p is conjugate momentum. Now let us expand q and its conjugate momentum p via creation and annihilation operators and zeromode harmonics
+a-+eikz + Si;e-ikz),
_ akeiJ= +a-keikx
+ c-ke-ikz),
(4
where ok, &, at and sik+are the annihilation and creation operators for left- and right-hand oriented excitations respectively, while go and pe are zero modes. Substituting Eqs. (3)-(4) into Eqs. (l)- (2) one can write down the expressions for the harmonics of super-Hamiltonian and supermomentum
-ac
(akane_k
+
1
(k,,/san+kak
+-+ + ak an+k)
k(n + k)
an+kat
-+ + aka,+k)
k>O
$zzz
+
k(n + k) @izz
1
(6)
(
-
(n + k)k
$zzz
k) +
k(n - k)2 wk
>
- JWX)
(n+k)/a.
(7)
If we make the traditional choice of creation and annihilation operators, then the second infinite sum in the expression (7) disappears, while the finite sum gives the well-known result:
k>O
+;cc
Ldk(n
4
+:Fo
+fcc
- -+
ak%+k
As usual one can define the Virasoro constraints as L, = ~(H_L~ + H,),L, = i(Hln - H,,). Traditional choice of dispersion relation between frequency wk and wave number k : wk = k provides diagonality of Hamiltonian Hlo. Besides the - choice wk = k, L does not depend on a, a+ and L does not depend on a, a+ respectively. Other choices of dispersion relations or, in other terms, other choices of creation and annihilation operators obtained by means of Bogoliubov transformations make Hamiltonian non-diagonal and the structure of Virasoro constraints becomes more involved. Nevertheless it is useful to write down the general expression for the central extension for the commutators of Virasoro generators in this case as well:
c$%~_~)
O
-
-~~+kfik - tiL+kat).
=-
7
Hln = ipo(i%i,+-a,)
-ak%+k
(5)
[L,, L-&.,. = 4 xc<&,, k(n- k> = +(n2 - 1). However, the other choice of vacuum immediately implies that in the result of finite summation in the formula for central extension (7) one has terms incompatible with Jacobi identities. On the
d. Yu. Kamenshchik et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 165-168
It is easy to calculate the finite sum in the expression (13) using the well known summation formulae for finite sums. Then one has
other hand the infinite sum in Eq. (7) becomes divergent and needs an explicit regularization. It is interesting to check how regularization of quantum commutators suggested in Ref. [3] works for these cases. The main idea of the algorithm [3] consists in replacing the commutator
[a., a+m] = 6rim
(
_
(9)
where f is a real function which satisfies the condition f(n,O) = 1. This regularization should give final results when the regulator is removed ( s --* 0). The regulator which we choose slightly differs from that of Ref. [3] and has the following form:
(151
k=l
This function is finite at s > 1 and can be analytically continued to the values s < 1. In particular, we shall need the values ~R(0) = --1,~R(--1) = 1 ~R(--2) = 0. At s = 1 the Riemann ¢12' function has the pole: ~ R ( l + s ) = ~ + c o n s t + 0 ( s ) . Now, expanding the factor depending on s in the infinite sum in expression (13) in inverse degrees of k one can get the following expression:
where a is a positive number. Application of this regularization to the formula (7) implies t h a t the infinite sum should be multiplied by the factor (11)
1
1
- k -2s - 2sk -2s-1
(czn+ )
(n + k + a ) ~ (k + a ) ~
because this infinite sum arises as a sum of prodor the ucts of commutators [ak,a+][ak+n, %+~] + corresponding commutators with g, g+. Now, let us choose for simplicity the vacuum and creation and annihilation operators defined by the dispersion relation
Wk = wo,
(14)
48w0
CR(s) = ~ 1 k--;"
(10)
f('~)(k, s)f('~)(n + k, s),
+
This expression contains terms proportional to n 2 and n 4, which as is well known lead to violation of Jacobi identities for commutators. Now we should evaluate an infinite sum in the expression (13) using ~-function technique [8]. Let us recall the definition of Riemann ~function:
with the regularized commutator
f('~)(n, s) = (n + a) -~,
1 wo(n - k) + 4o n WO Won(n -- 1) n2(n + 1) 2 8
(s)
[~, ~+ml = 6nmY(n, ~),
167
-s(2s + 1)(2s + 2)a - ~ - ~ (~ + ~)33
_sk_2S_2
n 2 + s(s +
T
(12)
1)k-2 -3
n n2 + 3) 7"
(16)
Contracting expression (16) with the multiplier ( ?2030 n2k nk2 wo wo ]~ in the expression (13) one can
where w0 is a constant. Substituting (12) into (7), and multiplying the terms in an infinite sum by the factor (11) one has
get the limit s --* 0 using the corresponding formulae. As a result one can get the renormalized contribution of the infinite sum in the expression 1 kinwo- k)2 ) [L'~,L-'~]~.e. = ~o<~k
1(
k>O 1 (n+k+a)s
{k~>ol ( WO 1 ( k + a ) s"
WO /
n°30
(13)
=n
n2k
nk2~}
WO
°d0 ]
q-
renormMized
168
A.Yu. Kamenshchik et aL /Nuclear Physics B (Proc. Suppl.) 104 (2002) 165-168
+n2
(
1 48-w0
80 )
n4 48w0"
(17)
It is easy to see that dangerous terms proportional to n 2 and n 4 in the regularized expression for the infinite sum (17) exactly cancel those in the expression for the finite sum (14). Thus, in such a way, Ultraviolet renormalization given by the regulator (9) and (10) provides the restoration of Jacobi identities. One can consider also another choice of creation and annihilation operators defined by the dispersion relation wk = A k 2.
(18)
One can easily check that all results of calculations for the case given by the dispersion relation (18) coincide with those obtained in the case given by the dispersion relation (12). (All the formulae for the latter case can be obtained from the formulae for the former case by substitution A ~ 1/a~0). Thus, we have seen that the regularization of commutators suggested in Ref. [3], when applied to the string model with non-standard definitions of creation and annihilation operators, allows to preserve fundamental symmetry properties encoded into the Jacobi identities. This could be interpreted as an additional argument in favour of this regularization. However, we should recognize that it is not true for any choice of vacuum. For example, for the vacuum defined by the dispersion relation B wk = -k
(19)
(for this case all calculations also could be carried out in explicit form) we have got the term proportional to n s, which cannot be cancelled with the help of regularization of creation and annihilation operators. We can now present a clear explanation of why for different types of vacuum the efficiency of the regularization of commutation relations is not the same. The qualitative ilustration of the effect could be given by use of Bogoliubov transformations [7]. One can easily check that the value Jj3k]2 (where fl is the standard Bogoliubov coef-
ficient) responsible for the description of the difference between the chosen vacuum and standard one and describing such processes as the particle generation (see e.g. Ref. [9]) behaves at k ~ oc I kl 2 ~ k,
(20)
while for the dispersion relation (19) the ultraviolet behavior of this value is stronger IZkl 2 ~
k ~.
(21)
Thus, one can guess that the regularization suggested in Ref. [3] is not strong enough to cope with ultraviolet divergences in this case. Notice in conclusion that non-traditional dispersion relations which we have used for calculational purposes, have recently attracted attention in the context of trans-Planckian physics [10]. REFERENCES
1. L. Brink and M. Henneaux, Principles of string theory, Plenum Press, New York, 1988. 2. H. Gustafsson, U. LindstrSm, P. Saltsidis, B. Sundborg and R.v. Unge, Nucl. Phys. B440 (1995) 495. 3. S. Hwang, R. Marnelius and P. Saltsidis, hepth/9804003; J. Math. Phys. 40 (1999) 4639. 4. E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B55 (1975) 224; I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B69 (1977) 309; I.A. Batalin and E.S. Fradkin, Ann. Inst. Henri Poincard 49 (1988) 145. 5. S. Hwang, Phys. Rev. D28 (1983) 2614. 6. A.Yu. Kamenshchik and S.L. Lyakhovich, Nucl. Phys. B495 (1997) 309. 7. A.Yu. Kamenshchik, I.M. Khalatnikov and M. Martellini, Phys. Rev. D59 (1999) 046005. 8. S.W. Hawking, Commun. Math. Phys. 55 (1977) 133. 9. B.S. DeWitt, Phys. Rep. 19 (1975) 295. 10. A.A. Starobinsky, Pisma Zh. Eksp. Teor. Fiz. 73 (2001) 415.
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 104 (2002) 165-168
www.elsevier.com/Iocate/npe
Remarks about UV regularization of basic commutators in string theories Alexander Yu. Kamenshchik a., Isaak M. Khalatnikov bt and Maurizio Martellini c aL.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences, Kosygin str. 2, 117334 Moscow, Russia Landau Network - Centro Volta, Villa Olmo, via Cantoni 1, 22100 Como, Italy b L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences, Kosygin str. 2, 117334 Moscow, Russia Landau Network - Centro Volta, Villa Olmo, via Cantoni 1, 22100 Como, Italy Tel Aviv University, Raymond and Sackler Faculty of Exact Sciences; School of Physics and Astronomy, Ramat Aviv, 69978, Israel cUniversity of Insubria, via Valleggio 11, 22100 Como, Italy Landau Network - Centro Volta, Villa Olmo, via Cantoni 1, 22100 Como, Italy The zeta regularization of basic commutators in string theories, recently proposed by Hwang, Marnelius and Saltsidis, is generalized to the string models with non-trivial vacuums. It is shown that implementation of this regularization implies the cancellation of dangerous terms in the commutators between Virasoro generators, which break the Jacobi identity.
It is well known what role is played by the problem of regularization and renormalization of ultraviolet divergences in quantum field theory. However, the importance of the problem of a proper treatment of ultraviolet divergences arising from quantization of string theories is less understood. Perhaps, it is connected with the fact that calculating the central extensions of Virasoro algebra in string theories it is possible to avoid the need to consider infinite sums, by virtue of the tacitly assumed and very natural regularization providing their cancellation [1]. However, in the cases when conventional vacuum state is absent (for example in the case of tensionless conformal string [2]) explicit ultraviolet regularization of basic commutators of the theory becomes unavoidable. In recent papers by Hwang, Marnelius and Saltsidis [3] a convenient version of ultraviolet zeta regularization of basic commutators in string theories was elaborated. This scheme was applied *Supported by the CARIPLO Scientific Foundation and by RFBR via grant 99-02-18409 ?Supported by RFBR via grant 99-02-18409
to Hamiltonian B R S T quantization [4] of string theories. The well-known results for critical dimensions obtained by the same method earlier [5] were reproduced in the new context. Here, we consider traditional closed bosonic string model but with non-trivial vacuums. These vacuums are obtained by means of redefinition of creation and annihilation operators due to Bogoliubov transformation. As a result of this redefinition, in the expression for central extension infinite sums arise while the traditional finite part of contribution to the central charge acquires the structures responsible for violation of Jacobi identities. The direct generalization of the regularization suggested in Ref. [3] allows to calculate the finite contributions of infinite sums, which have the structure providing the total cancellation of pathological terms and restoration of Jacobi identities. One can hope that this scheme can be applied to more complicated theories such as quantum cosmology where the necessity of ultraviolet regularization is getting obvious [6]. The detailed presentation of the results obtained is given in our paper [7].
- see front matter © 2002 ElsevierScience B.V. All rights reserved. PII S0920-5632(01)01605-X
0920-5632/02/$
A. Yu. Kamenshchik et al. /Nuclear Physics B (Proc. Suppl.) 104 (2002) 165-168
166
To begin with let us consider a closed bosonic string. The constraints in the Hamiltonian formalism have the following form: HI
=
;p2+ ;qt2,
H = w’,
(2)
where HI is the so called super-Hamiltonian constraint, H is supermomentum, q is the coordinate of string, while p is conjugate momentum. Now let us expand q and its conjugate momentum p via creation and annihilation operators and zeromode harmonics
+a-+eikz + Si;e-ikz),
_ akeiJ= +a-keikx
+ c-ke-ikz),
(4
where ok, &, at and sik+are the annihilation and creation operators for left- and right-hand oriented excitations respectively, while go and pe are zero modes. Substituting Eqs. (3)-(4) into Eqs. (l)- (2) one can write down the expressions for the harmonics of super-Hamiltonian and supermomentum
-ac
(akane_k
+
1
(k,,/san+kak
+-+ + ak an+k)
k(n + k)
an+kat
-+ + aka,+k)
k>O
$zzz
+
k(n + k) @izz
1
(6)
(
-
(n + k)k
$zzz
k) +
k(n - k)2 wk
>
- JWX)
(n+k)/a.
(7)
If we make the traditional choice of creation and annihilation operators, then the second infinite sum in the expression (7) disappears, while the finite sum gives the well-known result:
k>O
+;cc
Ldk(n
4
+:Fo
+fcc
- -+
ak%+k
As usual one can define the Virasoro constraints as L, = ~(H_L~ + H,),L, = i(Hln - H,,). Traditional choice of dispersion relation between frequency wk and wave number k : wk = k provides diagonality of Hamiltonian Hlo. Besides the - choice wk = k, L does not depend on a, a+ and L does not depend on a, a+ respectively. Other choices of dispersion relations or, in other terms, other choices of creation and annihilation operators obtained by means of Bogoliubov transformations make Hamiltonian non-diagonal and the structure of Virasoro constraints becomes more involved. Nevertheless it is useful to write down the general expression for the central extension for the commutators of Virasoro generators in this case as well:
c$%~_~)
O
-
-~~+kfik - tiL+kat).
=-
7
Hln = ipo(i%i,+-a,)
-ak%+k
(5)
[L,, L-&.,. = 4 xc<&,, k(n- k> = +(n2 - 1). However, the other choice of vacuum immediately implies that in the result of finite summation in the formula for central extension (7) one has terms incompatible with Jacobi identities. On the
d. Yu. Kamenshchik et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 165-168
It is easy to calculate the finite sum in the expression (13) using the well known summation formulae for finite sums. Then one has
other hand the infinite sum in Eq. (7) becomes divergent and needs an explicit regularization. It is interesting to check how regularization of quantum commutators suggested in Ref. [3] works for these cases. The main idea of the algorithm [3] consists in replacing the commutator
[a., a+m] = 6rim
(
_
(9)
where f is a real function which satisfies the condition f(n,O) = 1. This regularization should give final results when the regulator is removed ( s --* 0). The regulator which we choose slightly differs from that of Ref. [3] and has the following form:
(151
k=l
This function is finite at s > 1 and can be analytically continued to the values s < 1. In particular, we shall need the values ~R(0) = --1,~R(--1) = 1 ~R(--2) = 0. At s = 1 the Riemann ¢12' function has the pole: ~ R ( l + s ) = ~ + c o n s t + 0 ( s ) . Now, expanding the factor depending on s in the infinite sum in expression (13) in inverse degrees of k one can get the following expression:
where a is a positive number. Application of this regularization to the formula (7) implies t h a t the infinite sum should be multiplied by the factor (11)
1
1
- k -2s - 2sk -2s-1
(czn+ )
(n + k + a ) ~ (k + a ) ~
because this infinite sum arises as a sum of prodor the ucts of commutators [ak,a+][ak+n, %+~] + corresponding commutators with g, g+. Now, let us choose for simplicity the vacuum and creation and annihilation operators defined by the dispersion relation
Wk = wo,
(14)
48w0
CR(s) = ~ 1 k--;"
(10)
f('~)(k, s)f('~)(n + k, s),
+
This expression contains terms proportional to n 2 and n 4, which as is well known lead to violation of Jacobi identities for commutators. Now we should evaluate an infinite sum in the expression (13) using ~-function technique [8]. Let us recall the definition of Riemann ~function:
with the regularized commutator
f('~)(n, s) = (n + a) -~,
1 wo(n - k) + 4o n WO Won(n -- 1) n2(n + 1) 2 8
(s)
[~, ~+ml = 6nmY(n, ~),
167
-s(2s + 1)(2s + 2)a - ~ - ~ (~ + ~)33
_sk_2S_2
n 2 + s(s +
T
(12)
1)k-2 -3
n n2 + 3) 7"
(16)
Contracting expression (16) with the multiplier ( ?2030 n2k nk2 wo wo ]~ in the expression (13) one can
where w0 is a constant. Substituting (12) into (7), and multiplying the terms in an infinite sum by the factor (11) one has
get the limit s --* 0 using the corresponding formulae. As a result one can get the renormalized contribution of the infinite sum in the expression 1 kinwo- k)2 ) [L'~,L-'~]~.e. = ~o<~k
1(
k>O 1 (n+k+a)s
{k~>ol ( WO 1 ( k + a ) s"
WO /
n°30
(13)
=n
n2k
nk2~}
WO
°d0 ]
q-
renormMized
168
A.Yu. Kamenshchik et aL /Nuclear Physics B (Proc. Suppl.) 104 (2002) 165-168
+n2
(
1 48-w0
80 )
n4 48w0"
(17)
It is easy to see that dangerous terms proportional to n 2 and n 4 in the regularized expression for the infinite sum (17) exactly cancel those in the expression for the finite sum (14). Thus, in such a way, Ultraviolet renormalization given by the regulator (9) and (10) provides the restoration of Jacobi identities. One can consider also another choice of creation and annihilation operators defined by the dispersion relation wk = A k 2.
(18)
One can easily check that all results of calculations for the case given by the dispersion relation (18) coincide with those obtained in the case given by the dispersion relation (12). (All the formulae for the latter case can be obtained from the formulae for the former case by substitution A ~ 1/a~0). Thus, we have seen that the regularization of commutators suggested in Ref. [3], when applied to the string model with non-standard definitions of creation and annihilation operators, allows to preserve fundamental symmetry properties encoded into the Jacobi identities. This could be interpreted as an additional argument in favour of this regularization. However, we should recognize that it is not true for any choice of vacuum. For example, for the vacuum defined by the dispersion relation B wk = -k
(19)
(for this case all calculations also could be carried out in explicit form) we have got the term proportional to n s, which cannot be cancelled with the help of regularization of creation and annihilation operators. We can now present a clear explanation of why for different types of vacuum the efficiency of the regularization of commutation relations is not the same. The qualitative ilustration of the effect could be given by use of Bogoliubov transformations [7]. One can easily check that the value Jj3k]2 (where fl is the standard Bogoliubov coef-
ficient) responsible for the description of the difference between the chosen vacuum and standard one and describing such processes as the particle generation (see e.g. Ref. [9]) behaves at k ~ oc I kl 2 ~ k,
(20)
while for the dispersion relation (19) the ultraviolet behavior of this value is stronger IZkl 2 ~
k ~.
(21)
Thus, one can guess that the regularization suggested in Ref. [3] is not strong enough to cope with ultraviolet divergences in this case. Notice in conclusion that non-traditional dispersion relations which we have used for calculational purposes, have recently attracted attention in the context of trans-Planckian physics [10]. REFERENCES
1. L. Brink and M. Henneaux, Principles of string theory, Plenum Press, New York, 1988. 2. H. Gustafsson, U. LindstrSm, P. Saltsidis, B. Sundborg and R.v. Unge, Nucl. Phys. B440 (1995) 495. 3. S. Hwang, R. Marnelius and P. Saltsidis, hepth/9804003; J. Math. Phys. 40 (1999) 4639. 4. E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B55 (1975) 224; I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B69 (1977) 309; I.A. Batalin and E.S. Fradkin, Ann. Inst. Henri Poincard 49 (1988) 145. 5. S. Hwang, Phys. Rev. D28 (1983) 2614. 6. A.Yu. Kamenshchik and S.L. Lyakhovich, Nucl. Phys. B495 (1997) 309. 7. A.Yu. Kamenshchik, I.M. Khalatnikov and M. Martellini, Phys. Rev. D59 (1999) 046005. 8. S.W. Hawking, Commun. Math. Phys. 55 (1977) 133. 9. B.S. DeWitt, Phys. Rep. 19 (1975) 295. 10. A.A. Starobinsky, Pisma Zh. Eksp. Teor. Fiz. 73 (2001) 415.