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Bosonic Dp-branes at finite temperature in TFD approach
ELSIZVIER
Nuclear Physics B (Proc. Suppl.) 127 (2004) 92-94
SUPPLEMENTS www.elsevierphysics.com
Bosonic Dpbranes at Finite Temperature in TFD Approach M. C. B. Abdalla a, A. L. Gadelha ‘, I. V. Vancea b “Instituto
de Fisica Tebrica, Unesp, Pamplona 145, Sk Paulo, SP, 01405-900, Brazil
bDepartamento de Fisica MatemBtica, Faculdade de Filosofia CMncias e Letras de Ribeirti Av. Bandeirantes, 3900, Ribeirti Preto, SP, 14040-901, Brazil A general formulation is presented and applied
of Therm0 Field Dynamics using transformation generators that form the SU(1, 1) group, to the closed bosonic string and for bosonic D,-brane with an external field.
The objective of this work is to summarize a systematic construction of bosonic strings and Bosonic D,-branes in the perturbative limit, at finite temperature, using the Therm0 Field Dynamics (TFD) [l-4]. In this context, the Dpbrane at finite temperature arises as a thermal boundary state constructed from the thermal vacuum of the Closed Bosonic String (CBS). In this way one must construct the thermal vacuum of the CBS in order to show how the thermal boundary states can be obtained out of it. We are dealing with a bosonic string in the light-cone gauge to avoid the existence OSspurious states [5]. To employ the TFD approach one needs first to double the degrees of freedom of the system by introducing an auxiliary system, identical to the original one, called tilde system and denoted by a tilde (-). In this way, one have a Hilbert space given by ?i = 3-18 %, where 3t denotes the original physical system and 5 is the Hilbert space of the tilde system. Operators acting at one space commute with the ones belonging to the other. There is a operation that maps elements of the original system to the tilde ones and viceversa, called tilde conjugation [7]. Once we double the system, a specific Bogoliubov transformation, called Thermal Bogoliubov Transformation (TBT), must be performed. The temperature dependence of the transformed space will came from the TBT parameters. Effectively, the temperature dependence arises from the mixing between elements of the original system and the tilde one. In fact, the thermal effects arise 0920-5632/s - see front matter 0 2004 Published by Elsevier doi:10.1016/j.nuc1physbps.2003.12.014
Preto, USP,
B.V
when one tries to take information about the original system using states that came from the Bogoliubov transformation of the doubled system
m
Either dealing with bosonic system, or as in our specific case, with CBS, the generator of TBT can both be constructed as a linear combination of generators that satisfy the ~(1, 1) algebra. Furthermore, with suitable choice of the parameters transformation one can go to spaces with different properties. In fact, as pointed out by Umezawa and Elmfors [8], considering 4 a linear map one have the following general possibilities
M2
: 4 (a+)# [4 (41+ A 4~(3 = [4bf,
M3 : 4 (a+) = [4 @41+ A 4 m # b$ (41.
(2) (3)
The M2 map is used in the usual general TFD formulation [7] and that choice tames from the fact that the tilde conjugation rules remains unchanged. This can be achieved, for CBS, using the following generator
for right-moving
modes, with
h(e) = iP2,, - fAk), x,,(e) = -i(el, + e2*), x,,(e) = ie3h, (5)
A4.C.B. Abdalla et al. /Nuclear Physics B (Pnx. Suppl.) 127 (2004) 92-94
and 81,) t&, 13s~E R, and a similar expression for the left-moving modes. The M3 is an alternative map and can be achieved by a generator with the same form of the above presented, but now denoted by Gg (y), together with the following X parameters h,
(7)
=
Ylr,
-
iY2,,
X2k (4
X3k(Y)
=
=
4,
(7)
>
(6)
Y3k>
with ~1~) 7sk, 7ak E R. This alternative approach was used in Ref. [3]. Originally the TFD was formulated using only one generator, consequently only one parameter, and just in this case one has
4 (a+)= Ma+
A
4m
= r4b)r.
29&
(a4)erlb
(B:.B:)
IO)),
(8)
k
where g E tr(@), IO)) = 10) IO), with IO) the vacuum of the CBS at zero temperature, r1l,
=
r3L
=
-Xi&senh (iAh) hk cash (iAk) + Xaksenh (in,) ’
(9)
AI,
and similarly for the left-moving operators. I is the identity operator. Consider now a D,-brane in the presence of an external field. Specifically we deal with a rigid brane located along the {X”} direction in the target-space, a = 1,2, . ..p at {Xi = yi}, where i = p + 1, . ..24. This object is analyzed in the presence of a constant field Fab [lO,ll]. In the non-perturbative limit of the string theory, we can obtain a thermal boundary state that satisfies the following operator equations [(I + F);A:N
+ (I - F);B:+(4]IBx(W
= 0,
[(I + F);A:+O)
+ (I-
= 0,
(7)
The temperature dependent states can be obtained by using a disentanglement theorem [9]. The formal expression for these states are identical in terms of X parameters and is given by
and A$ E (Xi, + Xi,Xzk). However the formal expression be the same, we must take care when we specify the case in use. If one choices the M2 case, the elements of the dual space must be redefined for a correct construction of the transformed Fock space. On the other hand if the choice is to work using the M3 map, one must redefine the tilde conjugation rules. The temperature dependent operators can be obtained using the following Bogoliubov matrix transformation,
F);B:(4]IBx(W)
[A;@) - B?(N]IBx(N)
= 0,
[A:(A) - B;(A)]IBx(X))
= 0,
V n > 0, and 6” IBx(X)) = [$ - Yi]lBX(4)
=o,
(13)
where F,b = Fab/T and T is the string tension, p” and ? are the momenta and coordinates of the center-of-mass of the CBS. The solution of the above equations is 2 (GZ+GB,) IBx (A))= en=1 IBx))
= Np”(F)6 Cdl)(4 _ $))&L) (j _ j-j>
(10)
ilk cash (iAk) + As* senh (iAk) ’
93
- 2 [A!,(X).M.Bf,(X)+~~(X).M.~~(X)] x e n=l x IO P>>T
(14
where the d-function transverse space. Np(F)
= /-det
is a normalization used
localizes the brane in the
(6+;), factor, and for short have we
(16) where BI, = cash (iAk)I + yfk;k)
(;;:I
-a;,);
(12)
as a suitable notation. The general solution (14) is a thermal boundary state interpreted as a D,brane at finite temperature.
94
M.C.B. Abdalla et al. /Nuclear Physics B (Pmt. Suppl.) 127 (2004) 92-94
The entropy of this state can be calculated in the framework of TFD using a general entropy operator as presented in [2,3]. For the CBS, the entropy operator is given by
(17)
K=Ka+KP=~(~g+Kf), k
Ka =with
Al,
Ai . AI, log
x2,
Tsinh2
(ihk)
k Xl,
x2,
1 + Tsinh’
(&)
k
>I ,
and similarly for Kfi by changing A by leftmovings B operators. The thermal vacuum expectation value of the entropy operator (17) times the Boltzman constant, leg, is interpreted as the entropy of the closed string and is given by S 0s =
2kB
c
[(9
+
nk)
1%
(9 +
nk>
REFERENCES
{ -nk
loi
(nk)
-
glog
(d])
>
(18)
where g E tr (@) as before, and nk
= =9
(“(~)~N~~~(x))=(~(x)~N~~~(x)) Al, X2k ;i-sinh2 k
e--POW2
(ihk)
=
1 _ e--,cowk,
(1%
I
is the thermal vacuum expectation value of the number operator of the CBS. Here, pc = (kBT)-‘. The factor two comes from the equal contribution of the right- and left-moving modes. Performing the same calculation for the expectation value of the entropy operator in the state given by (14)) one finds
(I+ 2nk) dk
SDb = -2kB c k
+scs, where S,, is entropy obtained dk is given by
where s?“’ = 0, 1, . ..oo. i = 1,2, . . .. n + 00, p, (T = 1,2, . ..) ;14, and B = Np (F, 19)t5(dl) (a - 9) . The entropy SDb is interpreted as the entropy of a D,-brane with an external constant abelian gauge field. In this work we summarize the construction of thermal vacuum for the closed bosonic string and D-brane in a constant external field using a general SU(1, 1) TFD approach. Boundary states at finite temperature were obtained by imposing boundary conditions in the bosonic thermal string solution. Those thermal boundary states were interpreted as a thermal D,-brane. The entropy for the closed bosonic string, as well as that of the Dp-brane, was calculated using a general entropy operator. We would like to thank Ademir E. Santana for fruitful discussions. I. V. V. was supported by the FAPESP Grant 02/05327-3. A. L. G. was supported by a FAPESP post-dot fellowship.
log (i%J}
(20) for the CBS and
1. I. V. Vancea, Phys. Lett. B 487 (2000) 175. 2. M. B. C. Abdalla, A. L. Gadelha and I. V. Vancea, Phys. Rev. D 64 (2001) 086005. 3. M. B. C. Abdalla, A. L. Gadelha and I. V. Vancea, Phys. Rev. D 66 (2002) 065005. 4. M. B. C. Abdalla, E. L. Graca and I. V. Vancea, Phys. Lett. B 536 (2002) 114. 5. M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Cambridge University Press, 1987. 6. Y. Takahasi and H. Umezawa, Collect. Phenom. 2 (1975) 55. 7. H. Umezawa, Advanced Field Theory: Micro, Macro and Thermal Field, American Institute of Physics, 1993. 8. P. Elmfors and H. Umezawa, Physica A 202 (1994) 577 9. K. Wodkiewicz and J. H. Eberly, J. Opt. Sot. Am. B 3 (1985) 485. 10. P. Di Vecchia, A. Liccardo, D-branes m String Theory I-II, hep-th/9912161, hepth/9912275. 11. I. V. Vancea, Introductory Lectures on Dbranes, hep-th/0109029.
Nuclear Physics B (Proc. Suppl.) 127 (2004) 92-94
SUPPLEMENTS www.elsevierphysics.com
Bosonic Dpbranes at Finite Temperature in TFD Approach M. C. B. Abdalla a, A. L. Gadelha ‘, I. V. Vancea b “Instituto
de Fisica Tebrica, Unesp, Pamplona 145, Sk Paulo, SP, 01405-900, Brazil
bDepartamento de Fisica MatemBtica, Faculdade de Filosofia CMncias e Letras de Ribeirti Av. Bandeirantes, 3900, Ribeirti Preto, SP, 14040-901, Brazil A general formulation is presented and applied
of Therm0 Field Dynamics using transformation generators that form the SU(1, 1) group, to the closed bosonic string and for bosonic D,-brane with an external field.
The objective of this work is to summarize a systematic construction of bosonic strings and Bosonic D,-branes in the perturbative limit, at finite temperature, using the Therm0 Field Dynamics (TFD) [l-4]. In this context, the Dpbrane at finite temperature arises as a thermal boundary state constructed from the thermal vacuum of the Closed Bosonic String (CBS). In this way one must construct the thermal vacuum of the CBS in order to show how the thermal boundary states can be obtained out of it. We are dealing with a bosonic string in the light-cone gauge to avoid the existence OSspurious states [5]. To employ the TFD approach one needs first to double the degrees of freedom of the system by introducing an auxiliary system, identical to the original one, called tilde system and denoted by a tilde (-). In this way, one have a Hilbert space given by ?i = 3-18 %, where 3t denotes the original physical system and 5 is the Hilbert space of the tilde system. Operators acting at one space commute with the ones belonging to the other. There is a operation that maps elements of the original system to the tilde ones and viceversa, called tilde conjugation [7]. Once we double the system, a specific Bogoliubov transformation, called Thermal Bogoliubov Transformation (TBT), must be performed. The temperature dependence of the transformed space will came from the TBT parameters. Effectively, the temperature dependence arises from the mixing between elements of the original system and the tilde one. In fact, the thermal effects arise 0920-5632/s - see front matter 0 2004 Published by Elsevier doi:10.1016/j.nuc1physbps.2003.12.014
Preto, USP,
B.V
when one tries to take information about the original system using states that came from the Bogoliubov transformation of the doubled system
m
Either dealing with bosonic system, or as in our specific case, with CBS, the generator of TBT can both be constructed as a linear combination of generators that satisfy the ~(1, 1) algebra. Furthermore, with suitable choice of the parameters transformation one can go to spaces with different properties. In fact, as pointed out by Umezawa and Elmfors [8], considering 4 a linear map one have the following general possibilities
M2
: 4 (a+)# [4 (41+ A 4~(3 = [4bf,
M3 : 4 (a+) = [4 @41+ A 4 m # b$ (41.
(2) (3)
The M2 map is used in the usual general TFD formulation [7] and that choice tames from the fact that the tilde conjugation rules remains unchanged. This can be achieved, for CBS, using the following generator
for right-moving
modes, with
h(e) = iP2,, - fAk), x,,(e) = -i(el, + e2*), x,,(e) = ie3h, (5)
A4.C.B. Abdalla et al. /Nuclear Physics B (Pnx. Suppl.) 127 (2004) 92-94
and 81,) t&, 13s~E R, and a similar expression for the left-moving modes. The M3 is an alternative map and can be achieved by a generator with the same form of the above presented, but now denoted by Gg (y), together with the following X parameters h,
(7)
=
Ylr,
-
iY2,,
X2k (4
X3k(Y)
=
=
4,
(7)
>
(6)
Y3k>
with ~1~) 7sk, 7ak E R. This alternative approach was used in Ref. [3]. Originally the TFD was formulated using only one generator, consequently only one parameter, and just in this case one has
4 (a+)= Ma+
A
4m
= r4b)r.
29&
(a4)erlb
(B:.B:)
IO)),
(8)
k
where g E tr(@), IO)) = 10) IO), with IO) the vacuum of the CBS at zero temperature, r1l,
=
r3L
=
-Xi&senh (iAh) hk cash (iAk) + Xaksenh (in,) ’
(9)
AI,
and similarly for the left-moving operators. I is the identity operator. Consider now a D,-brane in the presence of an external field. Specifically we deal with a rigid brane located along the {X”} direction in the target-space, a = 1,2, . ..p at {Xi = yi}, where i = p + 1, . ..24. This object is analyzed in the presence of a constant field Fab [lO,ll]. In the non-perturbative limit of the string theory, we can obtain a thermal boundary state that satisfies the following operator equations [(I + F);A:N
+ (I - F);B:+(4]IBx(W
= 0,
[(I + F);A:+O)
+ (I-
= 0,
(7)
The temperature dependent states can be obtained by using a disentanglement theorem [9]. The formal expression for these states are identical in terms of X parameters and is given by
and A$ E (Xi, + Xi,Xzk). However the formal expression be the same, we must take care when we specify the case in use. If one choices the M2 case, the elements of the dual space must be redefined for a correct construction of the transformed Fock space. On the other hand if the choice is to work using the M3 map, one must redefine the tilde conjugation rules. The temperature dependent operators can be obtained using the following Bogoliubov matrix transformation,
F);B:(4]IBx(W)
[A;@) - B?(N]IBx(N)
= 0,
[A:(A) - B;(A)]IBx(X))
= 0,
V n > 0, and 6” IBx(X)) = [$ - Yi]lBX(4)
=o,
(13)
where F,b = Fab/T and T is the string tension, p” and ? are the momenta and coordinates of the center-of-mass of the CBS. The solution of the above equations is 2 (GZ+GB,) IBx (A))= en=1 IBx))
= Np”(F)6 Cdl)(4 _ $))&L) (j _ j-j>
(10)
ilk cash (iAk) + As* senh (iAk) ’
93
- 2 [A!,(X).M.Bf,(X)+~~(X).M.~~(X)] x e n=l x IO P>>T
(14
where the d-function transverse space. Np(F)
= /-det
is a normalization used
localizes the brane in the
(6+;), factor, and for short have we
(16) where BI, = cash (iAk)I + yfk;k)
(;;:I
-a;,);
(12)
as a suitable notation. The general solution (14) is a thermal boundary state interpreted as a D,brane at finite temperature.
94
M.C.B. Abdalla et al. /Nuclear Physics B (Pmt. Suppl.) 127 (2004) 92-94
The entropy of this state can be calculated in the framework of TFD using a general entropy operator as presented in [2,3]. For the CBS, the entropy operator is given by
(17)
K=Ka+KP=~(~g+Kf), k
Ka =with
Al,
Ai . AI, log
x2,
Tsinh2
(ihk)
k Xl,
x2,
1 + Tsinh’
(&)
k
>I ,
and similarly for Kfi by changing A by leftmovings B operators. The thermal vacuum expectation value of the entropy operator (17) times the Boltzman constant, leg, is interpreted as the entropy of the closed string and is given by S 0s =
2kB
c
[(9
+
nk)
1%
(9 +
nk>
REFERENCES
{ -nk
loi
(nk)
-
glog
(d])
>
(18)
where g E tr (@) as before, and nk
= =9
(“(~)~N~~~(x))=(~(x)~N~~~(x)) Al, X2k ;i-sinh2 k
e--POW2
(ihk)
=
1 _ e--,cowk,
(1%
I
is the thermal vacuum expectation value of the number operator of the CBS. Here, pc = (kBT)-‘. The factor two comes from the equal contribution of the right- and left-moving modes. Performing the same calculation for the expectation value of the entropy operator in the state given by (14)) one finds
(I+ 2nk) dk
SDb = -2kB c k
+scs, where S,, is entropy obtained dk is given by
where s?“’ = 0, 1, . ..oo. i = 1,2, . . .. n + 00, p, (T = 1,2, . ..) ;14, and B = Np (F, 19)t5(dl) (a - 9) . The entropy SDb is interpreted as the entropy of a D,-brane with an external constant abelian gauge field. In this work we summarize the construction of thermal vacuum for the closed bosonic string and D-brane in a constant external field using a general SU(1, 1) TFD approach. Boundary states at finite temperature were obtained by imposing boundary conditions in the bosonic thermal string solution. Those thermal boundary states were interpreted as a thermal D,-brane. The entropy for the closed bosonic string, as well as that of the Dp-brane, was calculated using a general entropy operator. We would like to thank Ademir E. Santana for fruitful discussions. I. V. V. was supported by the FAPESP Grant 02/05327-3. A. L. G. was supported by a FAPESP post-dot fellowship.
log (i%J}
(20) for the CBS and
1. I. V. Vancea, Phys. Lett. B 487 (2000) 175. 2. M. B. C. Abdalla, A. L. Gadelha and I. V. Vancea, Phys. Rev. D 64 (2001) 086005. 3. M. B. C. Abdalla, A. L. Gadelha and I. V. Vancea, Phys. Rev. D 66 (2002) 065005. 4. M. B. C. Abdalla, E. L. Graca and I. V. Vancea, Phys. Lett. B 536 (2002) 114. 5. M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Cambridge University Press, 1987. 6. Y. Takahasi and H. Umezawa, Collect. Phenom. 2 (1975) 55. 7. H. Umezawa, Advanced Field Theory: Micro, Macro and Thermal Field, American Institute of Physics, 1993. 8. P. Elmfors and H. Umezawa, Physica A 202 (1994) 577 9. K. Wodkiewicz and J. H. Eberly, J. Opt. Sot. Am. B 3 (1985) 485. 10. P. Di Vecchia, A. Liccardo, D-branes m String Theory I-II, hep-th/9912161, hepth/9912275. 11. I. V. Vancea, Introductory Lectures on Dbranes, hep-th/0109029.