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NOTE BREVI

Possible generalized statistics in curved space-time

R. SCIPIONI(*)

School of Physics and Chemistry, Lancaster University - Lancaster LA1 4YB, UK Dipartimento di Fisica, Università di Roma I “La Sapienza”

P.le Aldo Moro 2, 00185 Roma, Italy

(ricevuto il 3 Giugno 1996; approvato il 30 Ottobre 1996)

Summary. — We consider the generalized guon algebra in a dynamically evolving curved space-time. By considering the Bogoliubov transformations associated with the particle creation in the expanding space-time, we obtain that when the evolution operator of the ground state is discontinuous, a transition of statistics may be produced. PACS 05.30 – Quantum statistical mechanics.

PACS 04.60 – Quantum gravity.

The problem of allowed statistics, such as Bose, Fermi, Para-Bose, Para-Fermi, Anyonic or infinite statistics, and their role in physics is of great importance [1-4]. Recently, in a remarkable paper by Goodison and Toms, a proof has been given about the non-existence of q-statistics in curved space-time evolving dynamically [5]. It has been claimed that this should rule out a quantum field theory based on q-deformed algebras.

More recently, different generalizations of statistics have been proposed with new interesting properties [6, 7]; in particular, one of these has been applied in order to show that the results of ref. [5] are not so general [8].

In this paper, using a different approach to generalized statistics proposed by the author and others [9-13], we want to discuss the behaviour of a system of identical particles in a space-time which is dynamical. As we shall see, if the system has a discontinuous ground state, then we may observe a transition of statistics.

Let us briefly review the formalism of g-statistics; the operators of creation and annihilation are supposed to satisfy the algebra

aiak†2 gak†ai4 dik, aiak2 gakai4 0 ,

(1)

where g is an operator both Hermitian and unitary.

(*) E-mail: rsHluna.ph.lancs.ac.uk

Suppose now, as in ref. [5], to have a time-dependent space-time, which for t Et1is

flat and which for tDt2 is also flat. The space-time can be dynamic for t1EtEt2.

As a specific example we would consider a spatially flat Robertson-Walker space-time with metric

ds2_{4 dt}2_{2 R}2(t)

(

where R(t) 4R1for t Gt1and R(t) 4R2for t Ft2. The portion with t Et1will be called

the in region, while that with t Dt2 the out region.

In the in region we may expand the field operator f(x) (which, for simplicity, we suppose to be real scalar) in terms of creation and annihilation operators as

f(x) 4

!

i

(

)

(3)

Fi(x) is a complete set of positive-frequency solutions of the Klein-Gordon equation.

We will assume that the operators in (3) satisfy

aiak†2 ginak†ai4 dik, aiaj2 ginajai4 0 .

(4)

An expansion similar to (3) may be imposed in the out region, f(x) 4

!

i

(

)

(5)

with Gi(x) a complete set of positive-frequency solutions. The operators biand bi †

will in general differ from ai and ai

†

if there is particle creation due to the expansion of the universe. We assume that

bibj†2 goutbj†bi4 dij, bibj2 goutbjbi4 0 ;

(6)

being complete the set Fi(x) and Gi(x), we may expand one in terms of the other,

Gi(x) 4

!

Fj(x) 1bijF *j (x) ] .

(7)

The expansion coefficients in (7) are called Bogoliubov coefficients.

They were first used to study particle creation in the expanding universe by Parker [14]. Using (3) and (5) with (7), we may get

ai4

!

bj1 b *ijbj†) ,

(8)

for the metric (2) the Bogoliubov coefficients are diagonal. Therefore, aij4 aidij, bij4 bidij.

(9)

Substitution of (8) and its complex conjugate in (4) with use of (6) gives dik4

(

)

(10)

1b *i a *j (bi†bj†2 ginbj†bi†) 1 (12goutgin) b *i bjbi†bj†1 (gout2 gin) aia* bj j†bi.

We define the vacuum state in the out region by biN0 , outb 4 0 ,

taking the expectation value of (10) with N0, outb,

1 4NaiN22 a 0 , outNginN0 , outb NbiN2.

(12)

If we define

Nkl , outb 4 bk†bi†N0 , outb ,

(13)

it is easily shown by using (11) that

akl , outNmn, outb 4dmkdln1 dlmdkma 0 , outNgoutN0 , outb .

(14)

Suppose now that

Na 0 , outNgoutN0 , outb N E 1 ;

(15)

then as in ref. [5] it is easy to show that the states (13) are linearly independent. Calculating (10) on N0, outb and using (11) we have

dijN0 , outb 4

(

)

(16)

From the cited linear independence of states (13) the second term in the right-hand side has to be zero, so we have to satisfy one of the following conditions:

b* aj * 40j

(17) or

(bi†bj†2 ginbj†bi†) N0, outb 40 .

(18)

In the first case we obtain from (16)

NaiN22 ginNbiN24 1

(19)

and the relation (10) becomes

0 4 (12goutgin) bi* bj* bi†bj1 (gout2 gin) aia *j bj†bi

(20)

from which it is immediate to get

gout4 gin.

(21)

The use of (19) yields moreover

gin4 6 1

(22)

which contradicts (15).

In the case in which (18) is satisfied, we get directly by a comparison with (6) that the relation

gin4 gout

(23)

result (22). In such a way we have proved that

Na 0 , outNgoutN0 , outb N 4 NqoutN 4 1

(24)

and that gin4 gout4 g .

That is the statistics is Bose or Fermi; in analogous way, repeating the discussion for the operators a in the in region gives

Na 0 , inNginN0 , inb N 4 1 .

(25)

So we have shown that only the Bose and Fermi statistics are allowed in the in and out regions. It is now necessary to find the relation between these two cases. In other words, is it possible that a symmetric statistics in the in region will become an anti-symmetric statistics in the out one?

From (23) and (24) we have

Na 0 , outNginN0 , outb 4 a 0 , outNgoutN0 , outb 6 1 .

(26)

Suppose now to introduce the unitary operator U which connects N0, inb to N0, outb: UN0, inb 4N0, outb ;

(27)

from this relation we get that

a 0 , inNginN0 , inb 4 a 0 , outNUginU†N0 , outb 4 a 0 , outNUgoutU†N0 , outb 4 61 ;

(28)

then if the operator U is continuous the relation

a 0 , inNginN0 , inb 4 a 0 , outNgoutN0 , outb 4 61

(29)

has to be satisfied, meaning that the statistics cannot change.

There are, however, situations in which this could be not true. To explain how this can happen, suppose that our system of particles is thermalized and due to its thermodynamic nature that it undergoes a first-order transition [15]. The ground state in the precise time when T becomes the critical temperature changes abruptly and U is no longer continuous. If this is the case, the constraint (29) may be not valid and we can have a transition of the statistics. If the transition actually occurs depends on the nature of the system; what we have shown is that, when the system of identical particles which satisfies the generalized g-statistics undergoes a first-order trans-ition, then there is not a constraint which forces the statistics to be invariant.

This seems to be a quite important result showing that in some systems there could be a transition fermionD boson due to the dynamics of curved space-time, which might appear in the form of a first-order transition.

A possible application of these results could be Cosmology. In fact, at the early times the Universe can be schematized as a system of bosons and fermions; so it could be that after the big bang some statistics transitions have occurred producing a new amount of new interesting phenomena to be studied.

Another point is worth remarking, as recently noted by Altherr and Grandou [16]: deformed algebras do not have problems of local commutativity, if we consider the context of thermal field theory. This can be immediately generalised to the g-statistics, meaning that in thermal field theory we have no problems connected with local commutativity. This can suggest that our approach can bring us to consider a new type of particles which could have existed at the beginning of time, with non-defined

statistics; the possible consequences of these ideas are very important. We hope to investigate such aspects in forthcoming publications.

* * *

I wish to thank the International Centre of Cultural Co-operation NOOPOLIS (Italy) for funding.

R E F E R E N C E S

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