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Particle creation amplification in curved space

due to thermal effects

C. E. LACIANA(*)(**)

Instituto de Astronomía y Física del Espacio

Casilla de Correo 67, Sucursal 28, 1428 Buenos Aires, Argentina

(ricevuto il 24 Settembre 1996; approvato il 3 Dicembre 1996)

Summary. — A physical system composed by a scalar field minimally coupled to gravity and a thermal reservoir, as in thermo field dynamics, all of them in curved space-time, is considered. When the formalism of thermo field dynamics is generalized to the above case, an amplification in the number of created particles is predicted.

PACS 04.60 – Quantum gravity.

PACS 04.62 – Quantum field theory in curved spacetime.

1. – Introduction

As is shown in ref. [1], thermo field dynamics (TFD) gives us, for thermal equilibrium at a given temperature, an elegant method to obtain mean values of physical magnitudes resulting at this level equivalent to the Schwinger-Keldysh closed-time path formalism (SKF). However, as is shown in ref. [2] TFD has advantages on the SKF when the generalization to nonequilibrium situations is performed, giving different results.

The main idea in TFD is the duplication of the degrees of freedom, introducing the tilde operators aAkand aAk†which operate on the quantum states of the reservoir [3].

In our generalization the system and the reservoir are in a curved background. The operators ak, ak, aAk and aAkare all related to the thermal operators ak(b), ak(b),

aAk(b) and aAk(b) by means of a Bogoliubov transformation. Moreover, due

to the curvature of the space-time we have a set of vacua related to the foliation. The operators related with those vacua are indicated by ak(t), ak(t), aAk(t) and aAk(t).

The parameter “t” labels the foliation. We have also a Bogoliubov transformation that relates those operators with the primitive ones (see ref. [4]). The particle creation will be due to two effects, one of them is the interaction with the geometry and the other

(*) The author of this paper has agreed to not receive the proofs for correction. (**) E-mail: lacianaHiafe.uba.ar


one related to the interaction with the thermal bath. The first is a dynamical effect and the second is a spontaneous phenomenon.

2. – Thermal creation of particles

Similarly as in ref. [5], we can relate operators at zero temperature with the ones at temperature T4T(b) by means of the following time-dependent transformation matrix:


a(b) aA††(b)


4 ( 1 1 n0 b )1 O2


1 2f F21 2F 1



a aA


, (2.1)

where ak (ak†) are quasiparticle annihilation (creation) operators for bosonic fields (in

eq. (2.1) we omitted the subindex k associated with the mode of the quasiparticle), defined by

akN0 b 4 0 ,

ak†N0 b 4 N1kb, etc . ,

[ak, ak8] 4dk , k8.

The tilde operators aAk, aAk†represent the quantum effect of the reservoir. We have


aAkN0A b 40 ,


kN0A b 4N1Akb, etc . ,

[aAk, aAk8] 4dk , k8.

We will use ]Nn, nAb( 4 ]Nnb(7 ]NnAb( as space of states. In the following we will call N0b: 4N0, 0A b.

The operators a




, a




, aA




and aA†(b(t) ) operate on the thermal vacuum:

ak(b) N0, bb 40 ,


k (b) N0, bb 4N1k, bb , etc . ,

and satisfy

[ak(b), ak8††(b) ] 4dkk8

and similarly the operators aAk(b) and aAk††(b). The symbol †† is used because (a††)†ca,

i.e. in general the transformation is not unitary (only when F 4f1 O2).

Now we will define the function n0 b(t) as the mean value of created particles due to the interaction with the reservoir, in the form

n0 b4 a0Na††(b) a(b) N0b . (2.2)


As we can see using eq. (2.1)

n0 b4


1 2f ,

f is in principle an arbitrary time-dependent function, in particular when f 4exp [2be]

we have Planckian spectrum.

The time-dependent function F is also arbitrary. One choice used in the literature [6] is known as thermal state condition, which corresponds to F 4fa with

0 EaE1. In particular, when a41O2, (a††)†4 a (then †† 4 †) and the transformation is unitary.

It is easier to prove also, using the inverse transformation: a0Na††(b) a(b) N0b 4 a0, bNa††aN0, bb .

3. – Particle creation in curved space

Following Parker (ref. [4]) we can relate the annihilation-creation operators at different times, by means of the following Bogoliubov transformation:

. / ´ ak(t) 4e iga(k , t)cosh u(k , t) a k1 e igb(k , t)sinh u(k , t) a2k,

ak(t) 4e2igb(k , t)sinh u(k , t) ak1 e2iga(k , t)cosh u(k , t) a2k† , (3.1)

where ga, gb and u are determined by the particle model used and by the field

equation. When an isotropic Robertson-Walker metric is considered, the functions introduced in eq. (3.1) satisfy the equations (see eqs. (27) and (39) of ref. [4]):

g N btanh u cos g 1u N sin g 2gNacos m 1u N tanh u sin m 40 , 2gNbtanh u sin g 1u N cos g 2gNasin m 1u N tanh u cos m 40 , with m : 42


t0 t

W(k , t 8) dt 8 and g:4ga1 gb. Therefore in eqs. (3.1) we have six

unknown functions: gNb, g


a, g, u, u


and m. Two initial conditions are added to the two equations (3.1). In ref. [4] those conditions are the following:

. / ´ u(k , 0 ) 40 , ga(k , 0 ) 40 , (3.2)

which give us as initial conditions a particular functional form (similar to WKB) for the field modes. To completely solve the problem two additional conditions must be introduced which determine definitively, at all instants, the particle model.

The annihilation-creation operators of eq. (3.1) satisfy the bosonic commutation relations, i.e.

[ak(t), ak8(t) ] 40 , [ak(t), ak8(t) ] 40 ,



as we can see from eqs. (3.1) and the initial condition (3.2), ak(t 40) 4ak, ak(t 40) 4ak†, akN0 b 4 0 , ak†N0 b 4 N1kb , ak(t) N0, tb 40 , ak(t) N0, tb 4N1kb .

We also define the mean value of created particles with k mode by nk

0 t4 a 0 Na

k(t) ak(t) N0b. In the following, the k index is supressed.

4. – Introduction of tilde operators in curved space-time

Next we will assume that the system and the reservoir are both in a curved background.

In order to use the transformations given by eqs. (2.1) and (3.1) on the same operators, we will introduce the quadrivectorial operators Ak, Ak(b) and Ak(t), defined

by (using l as generic variable)

Ak(l): 4


ak(l) ak(l) aAk(l) aAk(l)


. (4.1)

Let us also introduce the 4 34 matrices V and Y so that the two transformations can be written as Ak(t) 4V(k, t) Ak, (4.2a) Ak(b) 4Y(k, b) Ak, (4.2b) where V 4


M 0 0 M


, M 4


e iga(k , t)cosh u(k , t)

e2igb(k , t)sinh u(k , t) P

eigb(k , t)sinh u(k , t) P

e2iga(k , t)cosh u(k , t)


, Y 4




( 1 1n0 b) 1 O2, L 4


0 2f F21 2F 0




operator, such that

Pak4 a2k.

The total number of created particles comes from two sources, a geometric one due to the curvature of the space-time and the other one due to the thermal effects. Therefore we can define n0 tbas the mean value of the total number of created particles, by

n0 tb4 a0Na††


t , b(t)




t , b(t)


N0b . (4.3)

Then we need the transformation



t , b(t)


4 Y(t) V(t) A»4 L(t) A ; therefore L 4 (11n0 b)1 O2( 1 1n0 t)1 O2




with R 4


e iga e2igbtanh uP eigbtanh uP e2iga


, T 4


2Fe 2igbtanh uP 2f F21eiga 2Fe2iga 2f F21eigbtanh uP



When the calculation in eq. (4.3) is performed the following result is obtained:

n0 tb4 n0 t1 2 n0 bn0 t1 n0 b. (4.4)

This result is independent of the particle model. From eq. (4.4) we see that if at the beginning we had a curved geometry at zero temperature there would be a number n0 t of created particles at time “t”. When all the system is in interaction with a reservoir at temperature T the number of particles increases in agreement with eq. (4.4). 5. – Concluding remarks

As was shown in ref. [4] the transformation given by eq. (3.1) is a consistent way to obtain Heinsenberg annihilation-creation operators for a Robertson-Walker metric. For other metrics the Bogoliubov transformation may be different. However, the result obtained in eq. (4.4) is independent of the particle model used.

The increment in the number of created particles due to the temperature is equal to the one produced in curved space at zero temperature, when there is a particle distribution in the initial state, as is proved in ref. [4]. In other words, this fact is related with the analogy, which is shown in ref. [7], between the role of the tilde operator aAk and the operator a2k in curved space.

In our generalization of TFD to a curved space we said nothing about what the tilde modes are. We think that the role of tilde modes is played by the quantum


perturbations of the metric, this possibility is presently being studied by us. As a first approach we found that the conformal fluctuation of the metric can satisfy all the hypothesis in order to act as a natural tilde field (see ref. [8]).

* * *

This work was supported by the European Community DG XII and by the Departamento de Física de la Facultad de Ciencias Exactas y Naturales de Buenos Aires.


[1] TAKAHASHI Y. and UMEZAWA H., Collective Phenomena, 2 (1975) 55. [2] CHU H. and UMEZAWA H., Int. J. Mod. Phys. A, 9 (1994) 2363.

[3] UMEZAWAH., MATSUMOTOH. and TACHIKIM., Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam) 1982.

[4] PARKER L., Phys. Rev., 183 (1969) 1057.

[5] HARDMAN I., UMEZAWA H. and YAMANAKA Y., J. Math. Phys., 28 (1987) 2925. [6] ARIMITSU T., GUIDA M. and UMEZAWAH., Physica A, 148 (1988) 1.

[7] LACIANA C. E., Gen. Relativ. Gravit., 26 (1994) 363. [8] LACIANA C. E., Physica A, 216 (1995) 511.


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