**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 aA***k***and aA***k**†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 a***k***, a***k**†*, aA***k** *and aA***k**† *are all related to the thermal operators a***k***(b), a***k**†*(b),*

*aA***k***(b) and aA***k**†*(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 a***k***(t), a***k**†*(t), aA***k***(t) and aA***k**†*(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:*

### u

*a(b)*

*aA*††

*(b)*

### v

*4 ( 1 1 n0 b*)1 O2

### u

1*2f F*21

*2F*1

### v

### u

*a*

*aA*†

### v

, (2.1)*where a***k** *(a***k**†) 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

*a***k**N0 b 4 0 ,

*a***k**†N0 b 4 N1**k**b, etc . ,

*[a***k***, a***k8**†*] 4d***k , k8**.

*The tilde operators aA***k***, aA***k**†represent the quantum effect of the reservoir. We have

also

*aA***k**N0A b 40 ,

*aA*†

**k**N0A b 4N1A**k**b, etc . ,

*[aA***k***, aA***k8**†*] 4d***k , k8**.

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

*The operators a*

### (

*b(t)*

### )

*, a*†

### (

*b(t)*

### )

*, a*A

### (

*b(t)*

### )

*and a*A†

*(b(t) ) operate on the thermal vacuum:*

*a***k****(b) N0, bb 40 ,**

*a*††

**k** **(b) N0, bb 4N1****k***, bb , etc . ,*

and satisfy

*[a***k***(b), a***k8**††*(b) ] 4d***kk8**

*and similarly the operators aA***k***(b) and aA***k**††*(b). The symbol †† is used because (a*††)†c*a,*

*i.e. in general the transformation is not unitary (only when F 4f*1 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 b 4 a0Na*††

*(2.2)*

**(b) a(b) N0b .**As we can see using eq. (2.1)

*n0 b*4

*f*

*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:

.
/
´
*a***k***(t) 4e*
*iga (k , t)_{cosh u(k , t) a}*

**k**

*1 e*

*igb*†

**(k , t)**_{sinh u(k , t) a}**2k**,

*a***k**†*(t) 4e2igb (k , t)sinh u(k , t) a*

**k**

*1 e2iga*

**(k , t)****cosh u(k , t) a****2k**† , (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 2g*N*acos m 1u*
N
*tanh u sin m 40 ,*
*2g*N*btanh u sin g 1u*
N
*cos g 2g*N*asin m 1u*
N
*tanh u cos m 40 ,*
*with m : 42*

### s

*t*0

*t*

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

*unknown functions: g*N*b, g*

N

*a, g, u, u*

N

*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.*

*[a***k***(t), a***k8***(t) ] 40 , [a***k**†*(t), a***k8**†*(t) ] 40 ,*

and

as we can see from eqs. (3.1) and the initial condition (3.2),
*a***k***(t 40) 4a***k**,
*a*†
**k***(t 40) 4a***k**†,
*a***k**N0 b 4 0 , *a***k**†N0 b 4 N1**k**b ,
*a***k***(t) N0, tb 40 ,* *a***k**†*(t) N0, tb 4N1***k**b .

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

*0 t*4
*a 0 Na*†

**k***(t) a***k****(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 A***k***, A***k***(b) and A***k***(t), defined*

*by (using l as generic variable)*

**Ak***(l): 4*

## u

*a*

**k**

*(l)*

*a*†

**k**

*(l)*

*aA*

**k**

*(l)*

*aA*†

**k**

*(l)*

## v

. (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) A****k**,
*(4.2a)*
**Ak****(b) 4Y(k, b) A****k**,
*(4.2b)*
where
**V 4**

### u

**M**

**0**

**0**

**M**

### v

,**M 4**

### u

*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)}*

### v

,
**Y 4**

### u

**I**

**L**

**L**

**I**

### v

*( 1 1n0 b*) 1 O2

_{,}

**L 4**

### u

0*2f F*21

*2F*0

### v

.operator, such that

**Pa****k***4 a***2k**.

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 tb*as the mean value of the total number of created particles,
by

*n0 tb 4 a0Na*††

### (

*t , b(t)*

### )

*a*

### (

*t , b(t)*

### )

**N0b .**(4.3)

Then we need the transformation

**A**

### (

*t , b(t)*

### )

*therefore*

**4 Y(t) V(t) A»4 L(t) A ;***)1 O2*

**L 4 (11n**0 b*( 1 1n0 t*)1 O2

### u

**R**

**T**

**T**

**R**

### v

with**R 4**

### u

*e*

*iga*

*e2igb*

_{tanh uP}*eigb*

_{tanh uP}*e2iga*

### v

,**T 4**

### u

*2Fe*

*2igb*

_{tanh uP}*2f F*21

*eiga*

*2Fe2iga*

*2f F*21

*eigb*

_{tanh uP}### v

.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 aA***k** *and the operator a***2k** 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.

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(North-Holland, Amsterdam) 1982.

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[8] LACIANA

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