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Deformed systems at finite temperature
PmlCA B
Physica A 206 (1994) 253-266 North-Holland SSDZ 0378-4371(94)00018-O
Deformed systems at finite temperature M. R-Monteiro
and I. Roditi
Centro Brasileiro de Pesquisas Fisicas - CPBFKNPq, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil Received 11 October 1993 Revised manuscript received
10 December
1993
We apply the formalism of thermo-field dynamics to two kinds of deformed Hamiltonians. These Hamiltonians are made up of operators which satisfy deformed Heisenberg algebras. The entropy and the specific heat are computed for small q - 1, where q is the deformation parameter.
0. Introduction There has been an intense activity in the area of deformed algebras and groups by physicists and mathematicians. These mathematical structures, quasi-triangular Hopf algebras [ l-41 #l, are sometimes called quantum groups and have found applications in several areas of physics [ 4-81 such as: inverse scattering method, vertex models, anisotropic spin chain Hamiltonians, knot theory, conformal field theory, heuristic phenomenology of deformed molecules and nuclei, non-commutative approach to quantum gravity and anyon physics. Because of the relevance played by Heisenberg algebra in physics, deformed Heisenberg algebras have attracted a lot of attention in the last few years. Their study is not a new subject [ 91 and recently their connection with quantum algebras and superalgebras have been established [ lo- 121, together with their derivation from the contraction of quantum algebra [ 111. A two-parameter Heisenberg algebra has also recently been obtained [ 13 ] by Schwinger’s contraction procedure for the two-parameter quantum semigroup SC,,~(2) [ 141. The algebra of Se,, (2) is called quantum semigroup because it is not possible to find an antipode function on the bialgebra SC,,$ (2) to endow it with a Hopf algebra structure [ 141.
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254
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
The thermodynamic properties of deformed systems have also started to be investigated and many attempts to find a direct physical application have been made [ 15-2 11. On the other hand thermo-field dynamics (TFD) [22] is a formalism whereby the usual field theory defined in real space-time can be generalized to the case of finite temperature. In this formalism the Feynman diagram method can be easily formulated by means of the real-time causal Green’s function [ 23 1, which are expressed in terms of “temperature-dependent vacuum” expectation values, and all the operator relations of T = 0 field theory are preserved. Therm0 field dynamics has been extensively developed and has been applied to problems in condensed matter physics as well as in high-energy physics [23,24]. In this paper we apply the formalism of TFD to two kinds of deformed Hamiltonians. These Hamiltonians are made up of operators which satisfy deformed Heisenberg algebras (q-oscillators). In section 1, we review the subject of deformed oscillators and extend the results of ref. [20] which come out of the application of TFD to a simple deformed Hamiltonian. In section 2 we consider a more complicated deformed Hamiltonian and compute, its entropy and the specitic heat. Section 3 is devoted to some final comments.
1. Thermo-field dynamics of deformed systems In the formalism of TFD [ 22-241 one constructs a temperature dependent vacuum 10(/? )), in which the statistical average of 6 coincides with the vacuum expectation value, using this new vacuum 10(/3 ) ). For instance if one uses the canonical ensemble one has (b) E 2-l
(/?) Tr[ e-aH6]
= (O(p)]d]O(p))
(1.11
with p = (kaT) -l and ks the Boltzmann constant. Let { 1n)} be the orthonormal basis of the state vector space ti consisting of eigenstates of the Hamiltonian H,
Hln) = EnIn) ,
@In) = &?l,n.
(1.2)
In order to construct such a state 10(p)) one introduces a fictitious system (tilde system) characterized by the Hamiltonian R and the state vector space fi spanned by {In)} obeying
fib) =&In) ,
(Fzp?z) =
a,,, .
(1.3)
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
255
The thermal vacuum ]0(/3)) belongs to the tensor product space ‘FI8 fi and is given by: IO(p))=
G
Z-‘12(j3) Z-1/2 (j?)
e+En12 In) 8 In)
&
e-bEn12 In, ii) .
(1.4)
n
Ifoneuses
(1.4) in (1.1) onehas
C
(0(/3)l6lO(jl)) = Z-‘(p)
edpEn” emgEmi2(n”, nldlm, 62)
n,m =
C
Z-‘(p)
e-BE~(n181n) =
(8)
(1.5)
n
which is the result claimed in ( 1.1) . This doubling of degrees of freedom has a sensible physical interpretation [ 22,201 and is related to the algebraic formulation of Statistical Mechanics developed by Haag, Hugenholtz and Winnink
[251. This formalism can be extended to the case of statistical averages of systems made up of q-oscillators [20]. One calls bosonic q-oscillators the associative algebra generated by the elements d, A+ and N satisfying the relations [10,11,26] [N,d+]
=A+,
14 A+ 1A
=fd(N)
[N,d]
= -A,
.
(1.6)
We are going to consider here the following forms of the above algebra ( 1.6):
[a,a+],saa+ - qa+a [A,A+]AsAA+
[%a+Ia
- q2A+A
sa(Y+ -s-‘q
= q-N,
(1.7a)
= 1)
(1.7b)
cx+(Y = (sq)_N
.
(1.7c)
The first two algebras are related to each other via [26] A = qN12a
,
A+ = a+qN12,
(1.8)
and the last one is the two-parameter deformed Heisenberg [ 131 algebra obtained from the quantum semi-group ,S.& (2) [ 141 by the Schwinger contraction.
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
256
It is possible to construct the representation of the relations ( 1.7 ) in the Fock space 3 spanned by the normalized eigenstates In) of the number operator N as
AIO)= 0 , In)
= d&
Nln) = nln)
n = 0,1,2...
9
(1.9)
(A+ )“I@ 3
where [n]d! = [n]d’.. [l]d, [n], = (q”-q-“)/(4-q-‘), [n]~ = (q2”1)/(q2 - 1) and [n], = ((s-‘q)” - (sq)-“)/(.s-‘q - (sq)-‘). In the Fock space F it is possible to express the deformed oscillators in terms of the standard bosonic ones b, b+ as [26] d
=
( [y+1;d)1’2b, d+
=
b+
( [NN’+1:“)‘/2 ,
(1.10)
and it can easily be shown in F that
dd+ = [N+
11.4,
d+d
(1.11)
= [N]d.
Let us now consider an ensemble of q-bosons with Hamiltonian H=wN
(1.12)
with eigenvalue on (n = 0, 1,2,. . -) on F. We introduce the tilde system as A=COiV, where the tilde q-operators
[R,d+]=d+,
given by
the Hamiltonian
of
(1.13) we are considering
satisfy the following relations:
[RT,d] = -A,
[A, d+ ]A = f~(fi) .
(1.14)
In the cases we are going to consider here, we have
(1.15)
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
and [&A] = [A,d+] thus given by
= 0. The temperature
2
IO(p))= Z-‘/*(j9)
eeBnwi2
dependent
vacuum
251
IO@)) is
. &(d+,“ca+,.jo)
n=O = ( 1 - emDo) ‘I2 exp,
( e-pm/* A+ A+ ) 10)
(1.16)
withexp,,x = C~,o(l/[n]~!)~“theq-exponential [27],and]O) = ]O)@]O). In the formula ( 1.16) we used the information that the partition function of q-bosons [ 191 corresponding to the Hamiltonian ( 1.12) coincides with the usual one for harmonic oscillators. We can easily see that the non-deformed case is recovered in the q + 1 limit. We denote up
=
(1
_
e-b
)-1/2 3
2.‘8
=
(
eBO-1)-l/*,
GB = -i6,
[ ( ~~:l:r)1’2
- ($--)I”
( &YJ2~
(&-)“*d+/t+]
where cash 8, c UP. With these definitions dependent vacuum IO(B)), (1.16), as IO(p)) = exp(-iGB)]O)
(1.17) we can rewrite the temperature
E BIO) .
Let us now define the temperature
dependent
(1.18) operators;
they are given by:
&=exp(-iGB)dexp(iGB) &=exp(iG~)dexp(iG~)
.
(1.19)
It is interesting to observe that this transformation algebra (1.6),( 1.7), i.e.
preserves the q-Heisenberg
aSaS+- qaS+ag = qmNfl,
ApA; - q2Ag+Ag = 1, a&
- s-‘qa;cy~
= (sq)-4
.
(1.20)
258
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
This can be seen by the use of relation resembles a q-Bogoliubov transformation. d#(P))
= &O(P))
(1.21)
= 0
and the Fock space can be constructed (1.18) on (1.9), i.e.
for n = 0, 1, . . .. us now compute depends the deformation given by: (d+d)
( 1. lo), thus the B transformation Obviously
by applying
the B-transformation
average of d+d which, as we are to see, considered. In the TFD approach this average is
= (WP)Id+.W(P))
= (O(P)\
[NIAIO(P)) .
In order to perform this calculation we go to the basis of non-deformed operators. In the non-deformed case one has [22] bb =exp(-iG,g)bexp(iG,g)
= ubb -va6’,
68 =exp(-iGB)?)exp(iGB)
= up6 - vab+,
(1.23) bosonic
(1.24)
with the inverse given by
b=upbfi
+ wpbs’,
(1.25)
where
Gg = -8,(&J
- 6+b+)
(1.26)
.
Now using ( 1.25 ) one finds the following expression for the number operator N N =
w$($ + 1) + u;Ng + u@J&$~;
+6&A.
(1.27)
Here we recall that the generators of SU ( 1, 1) algebra, J+, JO, can be realized A la Schwinger [ 28 ] as 5 Jo = ;(Ng + & + 1). (1.28) J+ = b;bj+ , J_ = babs ,
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
259
Thus using (1.28) we can rewrite (1.27) as
N =
04; +
+ upJp(J+
L)
;c
+
(1.29)
with (1.30)
C&v/y-&?-l.
Notice that C commutes with the first two terms of the right-hand side of expression ( 1.29 ) . Using now (1.29) and taking 4”’ = expel, the relevant terms in the calculations of ( 1.23 ) have the form
(O(P)1eAN IO(P)) = ((-J(P)1 e~[(U:+“:)JO+Usvs(J++J-)+(1/2)Cl
[(-j(p)).
(1.31)
This last expression can be computed by means of the Baker-CampbellHausdorff (BCH) formula, which can be derived for the SU ( 1,l) algebra [ 29 1. The BCH formula for the case we are considering is given by: (1.32) where
‘=
cosh(l/2)
2~s sinh(i2/2) - (z$ + $) sinh(A/2)
y = -2ln[cosh(A/2)
The above procedure obtain
= p/2
- (~5 + $) amounts
’
sinh(i2/2)]
.
(1.33)
to normal ordering eq. ( 1.3 1). Using this we
2q-m/2 + q-m/2 - ($ + $) (qd
- q-m/2) ’
(1.34)
Finally using (1.23) and (1.34) we can see that
(w/vl~+4w-w
= e2bw -(q
ebw -1 + q-l)
eaW
+l’
( 1.35a)
260
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
eBw1_q2’
(wPN~+4wP)) = (O(P)l~+~]O(P))
(1.35b)
=4( eBo-l)[(s-2fS-1(q+q-1)
+ l)(@W-1)2
-2(sm2 - l)( e2p0 -1) + (s-2 -s-l
+ l)(eS”
(q + q-l)
+1)2]-1,(l.35C)
where ( 1.35a,b) agree with the results given in ref. [ 191.
2. Entropy of deformed systems In this section we are going to consider, using the formalism of TFD, the entropy of a deformed system. As we have noticed in the previous section the partition function corresponding to the Hamiltonian ( 1.12) coincides with the usual one for standard harmonic oscillators and as we are not changing the statistical averages the thermodynamic functions are not modified, therefore we take another Hamiltonian given by H = e[NIA-,uN.
(2.1)
The form given by (2.1) is a general deformed Hamiltonian that describes in the q + 1 limit the usual harmonic oscillator. Note that both [N] and N are deformed but, as we shall see later, their statistical properties are different because only [N] affects the extensivity. This choice permits to follow where the deformation modifies the thermodynamic functions (e.g. when E = 0 one recovers the non-deformed quantities). This is indeed the case as one can easily check later. For a collection of oscillators, such as q-boson gas, the second term in (2.1) corresponds to the chemical potential. For free systems, in the TFD approach, one can define an entropy operator, R, as follows. Let us consider the thermal vacuum ]0(/3)) which, as we have seen, is IO(p))= Z-1/2(p)
C
= ,Inz-“~(p)-~H
’
emBEJ2In, ii)
CIrz,ii) n
G
e-k12CIn,i)
(2.2)
n
with g = lnZ(j3)
+pH,
(2.3)
M. R-Monteiro, I. Roditi / Deformed system at finite temperature
where the entropy, K, is obtained K =
261
from I? as
b(OtP)l@XP)) .
(2.4)
One can see that if one writes [22]
(2.5)
with w,, = e- BE, /Z (/I), we have
(OtB)I~lO(P))= Cwn[lnz(P) + PE.1 = - CwJnw.
(2.6)
n
n
where C, wn = 1 by construction. Thus K given by (2.4) recovers the usual Shannon entropy. For the deformed system we are considering, (2.1)) one can be convinced that the entropy is given by the Shannon entropy with Wn
=
(2.7)
(p)
@b'ld-Pn)/Z
where Z(p)
=
2
e-~(e[nld-Pn)
.
(2.8)
n=O
As the partition function cannot be computed exactly we take q = eal, s = ea2 and consider the case where (Yis small. Defining j = j? (E - p ) we may rewrite Z (8) in terms of the non-deformed partition function as z(p)
= ~~e([-~ld+a^)Zo(jj)
where 8 = a/a/?, Zo(j)
=
2
(2.9)
and e-j”
(2.10)
n=O
the non-deformed partition function. Performing an expansion in Q!and taking into account only the lowest order in CL: we have for Z(p)
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
262
1
Z(p)-‘d,‘(B)
2PE 1 - CY+Q(~)-
Q”‘)
+ O(CE;) ,
(2.1 la)
Z(P)-‘=Z,‘(li)[l
+ 44Qc2)+
Q”‘,
+ Ok&,
(2.1 lb)
2PE 1 - a2@Q (1)_,1_-(Q(3)_
Z(p)-‘=Z,‘(j3)
[P&Q
+ &P& +
forthealgebras
2Q’2)
+
Q(3)
the exponential
and (1.6)(1.7c)
PEQ(2))]
respectively,where
.
(2.12) with the Q(j)‘s is
aA’) =
(j +
,
a!‘) r+l
(2.7) we a: 1 - +W?
=
- 1) + O(c&
[ e-~(e[nlA-~n) = e-/j” [l - culPsn(~ - 1) + Q(aT)], e-8(e[nla-w)
+
aj’) are found by the recursion law:
ja,!f_i’) +
e-8(e[nl,-Pn)
+
(1.6)(1.7b)
for the calculation
where the coefficients
=
;(Q”’
(2.1 lc)
(1.6)(1.7a),
A useful expression
I
(I)* _
m$&
Q”’ = z,(j)-‘ajZ,(j)
,!i)
Q”‘)
9
a: a212 + +n2-
= e-s”
+ #n((n
- U2 -pm)]
Substituting (2.7) and (2.10)-(2.12) forward calculation S, =SO - a:kgjklj
0.
1,
(2.13a) (2.13b)
1)
+ O(&;)}
.
(2.13~)
in the entropy we get after a straight-
em2j(1 - eCs)-4(
emB +2) + Q(at),
(2.14a)
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature S, =A’, - 4alkB/3cg
e-2$
S, = S,, - a2kB/3cj
d(
-cu:kBj3cB c2j( -a;k&?c +fj[2
1 - e-a)-3
+ 0(a;),
263
(2.14b)
1 - e-j)_2 e-8 +2)(1
- e-8)-4
e-j{ @E( 1 - e-b)-2 e-S(2
+ 7 e-8)(1
-_B&(l + e-j)(l
- e-$-4
- e-$-s]}
(2.14~)
with S’s the entropy of the non-deformed system and S,, SA and S, the entropies corresponding to the algebras ( 1.6), ( 1.7a,b,c) respectively. It has been suggested [ 301 that there is an underlying lattice structure with lattice spacing cx = q - 1 for deformed theories with deformation parameter q. We see from the above formulae that the entropies of deformed systems are lower than the standard ones for qi > 1, i.e. the deformations introduce a correlation which could be interpreted as an underlying lattice structure. When the deformation parameter is lower than one we do not have the lattice interpretation anymore and for the (2.14b,c) cases, due to a linear contribution in (~1and (~2respectively, the entropy increases with respect to the non-deformed case. This kind of oscillators ( 1.7) with q < 1 were shown to be related to the Brownian motion [ 3 1 ] which could explain the increase of entropy. After a long but straightforward calculation the specific heat can be found as: c(a) = cc + a: CL;’ + . . .
(2.15a)
CA) = c, + a!1 C@) (I, + ...
(2.15b)
I
0)
= c, + a@) a* + a:cy
with Cc, the standard
+ cy:c$
(2.1%)
+ **. ,
specific heat and
C(U) = C’“’ = 2kBp2c(c -p) a: a: -kBP3c(c -,u)~ e-‘j(2
ee2b(2 + ees)(l + e-8)2(1
- eB )-4
- e-b)-5,
(2.16a)
C$‘) = 4kB/?‘c(c - ,u) em2fi( 1 - ea)-3 x[2 + e-s-2/3(c-~)(1
- e-j)-l(l
+ ePS)],
(2.16b)
264
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
c;y
e-)(1
=2k#&(&-p)
-k#&(&-j# cl;)
- e-b)-*
e-Q1
= -2kj3jP&(&
+ e_fi)(l-
- p) e-*$(2
e-$-3,
(2.16~)
+ 7 e-8) (1 - e-s )-4
2 _k&*c*
e-a (1 - e-b)-*
+4kBp3c2(c--) -kBp3c(&
e-j(l
-p)*
e-$1
+ e-$(1
- e-a)-3
- e-a)_5
x (20 - 64 e-a +35 e-*‘j -27 e-3a) -;kB/34c2(c
- /.L)*
e&(1
- e-B)-4(1
+ 4 e-b + e-2b). (2.16d)
If we consider the high-temperature approximation, p --) 0, one can see that when the deformation parameter is bigger than one the specific heat is lower than the standard one, Co. As before, in this limit, when the contribution of the deformation for the specific heat is linear and the deformation parameter is lower than one the specific heat increases with respect to Cc. It is interesting to notice that the system has the memory of the deformation in the /I + 0 limit. The difficulty in computing the exact value of the thermodynamic potentials in the case of the Hamiltonian (2.1) can be understood in the formalism of TFD in the following way: Consider the thermal vacuum for the case of the Hamiltonian (2.1) and call it [0(/3)),, where IO(B)),
Z-‘/*(p)
=
C
e-(8/2)(E[nlA-Pn)
112, fi)
,
(2.17)
n then it is possible to express (2.15 ) in terms of the thermal vacuum for the Hamiltonian (1.12) IO(p))
IO@>) = Z,“‘(B) C
e-an IO)
(2.18)
n as
lOCP>),=
2-1'2t8)~(~~)Z~'2~B)10(P^))
(2.19)
with F(x)
=
e-Be((l/*)[-*Xla+x)
(2.20)
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
265
Showing that the transformation from the non-deformed thermal vacuum to the deformed one (2.17) is highly non-trivial, being simple only when the deformed and non-deformed Hamiltonians have the same spectrum.
3. Final remarks We have considered in this paper the approach of TFD for two different Hamiltonians, each one made up of a different kinds of operators satisfying deformed Heisenberg algebras. The first Hamiltonian is linear in the number operator N and does not present any difftculty, however the second Hamiltonian generates a partition function and thermodynamic potentials which cannot be exactly computed for arbitrary values of the deformation parameter. Even if it is more complicated the second Hamiltonian is more interesting; it is non-interacting with non-additive energy. We have also seen that the exact computation of mean values for general deformed Hamiltonians is a rather complicated issue, the reason being the non-trivial relation (2.17). We think that it would be very useful to develop a variational approach to deal with such systems. The non-additivity of energy seems to be a deep property of deformed systems and its is already present in the deformed Poincare group [ 321. There is still no strong evidence of the presence of the quantum group structure in physical systems, we believe that the non-additivity of energy present in deformed systems could give evidence of the presence of such structures as discussed in [ 321 for the case of deformed Poincare group. The same issue is analysed in the context of the generalized statistical mechanics [ 33,341 in ref. [ 2 11. We also consider that it would be interesting to develop TFD in the context of the generalized statistical mechanics [ 33,341, which would permit the analysis of the models we have discussed within this approach.
Acknowledgements Some of the ideas contained cussions with C. Tsallis.
in the present paper have benefited
References [ 1] V.G. Drinfeld, Sov. Math. Dold. 32 (1985) 254. [2] M. Jimbo, Lett. Math. Phys. 10 (1985) 63; 11 (1986) 247.
from dis-
266
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
[3] L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtadzhyan,
Algebra and Analysis 1 (1987) 178. [4] S. Majid, Int. J. Mod. Phys. A 5 (1990) 1; P. Aschieri and L. Castellani, Int. J. Mod. Phys. A 8 (1993) 1667; M. R-Monteiro, Introduction to quantum groups, preprint CBPF-NF-061/93, to appear in the Proceedings of XIV ENFPC, Caxambi, Brazil. [5] C. Zachos, Contemp. Math. 134 (1992) 351 (and references therein). [6] A. Lerda and S. Sciuto, Nucl. Phys. B 401 (1993) 613. [7] R. Caracciolo and M. R-Monteiro, Phys. Lett. B 308 ( 1993) 58; M. Frau, M. R- Monteiro and S. Sciuto, J. Phys. A 27 (1994) 801; J.L. Matheus-Valle and M.R-Monteiro, Anyonic construction of the &(2) algebra, Preprint CBPF-NF-062193. [8] M. Chaichian, R. Gonzalez Felipe and C. Montonen, On the class of possible non-local anyon-like operators and quantum groups. [9] V.V. Kuryshkin, Ann. Found. L. de Broglie 5 (1980) 111. [lo] A.J. Macfarlane, J. Phys. A 22 (1989) 4581; L.C. Biedenham, J. Phys. A 22 (1989) L873. [ 111 M. Chaichian and P. Kulish, Phys. Lett. B 234 (1990) 72. [12] T. Hayashi, Comm. Math. Phys. 127 (1990) 129. C.P. Sun and H.C. Fu, J. Phys. A 22 (1989) L983. [13] S. Jing, Mod. Phys. Lett. A 8 (1993) 453. [ 141 C. Burdik and L. Hlavaty, J. Phys. A 24 (1991) L165. [ 151 M.A. Martin-Delgado, J. Phys. A 24 (1991) 1285. [ 161 P.V. NeSkovic and B.V. UrosSevic, Int. J. Mod. Phys. A 7 (1992) 3379. [ 171 V.I. Man’ko, G. Marmo, S. Solimeno and F. Zaccaria, Int. J. Mod. Phys. A 8 (1993) 3577. [ 181 V.I. Man’ko, G. Marmo, S. Solimeno and F. Zaccaria, Phys. Lett. A 176 (1993) 173. [ 191 M. Chaichian, R. Gonzalez Felipe and C. Montonen, J. Phys. A 26 (1993) 4025. [20] M.R-Monteiro and I. Roditi, Mod. Phys. Lett. B 7 (1993) 1843. [ 2 1] C. Tsallis, Non-extensive physics. A possible connection between generalized statistical mechanics and quantum groups, Preprint CBPF-NF-049/93; Extensive versus nonextensive physics, Preprint CBPF-NF-056/93. [22] Y. Takahashi and H. Umezawa, Collective Phenomena 2 (1975) 55. [23] H. Umezawa, H. Matsumoto and M. Tachiki, Therm0 Field Dynamics and Condensed States (North-Holland, Amsterdam, 1982). [24] N.P. Landsman and Ch. G. van Weert, Phys. Rep. 145 (1987) 141. [25] R. Haag, N.M. Hugenholtz and M. Winnink, Commun. Math. Phys. 5 (1967) 215. [26] P. Kulish and E.V. Damaskinsky, J. Phys. A 23 (1990) L415; A. Polychronakos, Mod. Phys. Lett. A 5 ( 1990) 2325. [27] H. Exton, q-Hypergeometric Functions and Applications (Horwood, Chichester, 1983). [28] J. Schwinger, in: Quantum Theory of Angular Momentum, L.C. Biedenharn and H. Van Dam, eds. ( Academic Press, New York, NY, USA, 1965). [29] R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974). [ 301 A. Dimakis and F. Miiller-Hoissen, Phys. Lett. B 295 (1992) 242. [ 3 1 ] M. Boiejko and R. Speicher, Commun. Math. Phys. 137 ( 199 1) 5 19. 1321 H. Bacry, Do we have to believe in dark matter?, Preprint CPT-93/P.2911. [33] C. Tsallis, J. Stat. Phys. 52 (1988) 479. [34] E.M.F. Curado and C. Tsallis, J. Phys. A 24 ( 1991) L69.
Physica A 206 (1994) 253-266 North-Holland SSDZ 0378-4371(94)00018-O
Deformed systems at finite temperature M. R-Monteiro
and I. Roditi
Centro Brasileiro de Pesquisas Fisicas - CPBFKNPq, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil Received 11 October 1993 Revised manuscript received
10 December
1993
We apply the formalism of thermo-field dynamics to two kinds of deformed Hamiltonians. These Hamiltonians are made up of operators which satisfy deformed Heisenberg algebras. The entropy and the specific heat are computed for small q - 1, where q is the deformation parameter.
0. Introduction There has been an intense activity in the area of deformed algebras and groups by physicists and mathematicians. These mathematical structures, quasi-triangular Hopf algebras [ l-41 #l, are sometimes called quantum groups and have found applications in several areas of physics [ 4-81 such as: inverse scattering method, vertex models, anisotropic spin chain Hamiltonians, knot theory, conformal field theory, heuristic phenomenology of deformed molecules and nuclei, non-commutative approach to quantum gravity and anyon physics. Because of the relevance played by Heisenberg algebra in physics, deformed Heisenberg algebras have attracted a lot of attention in the last few years. Their study is not a new subject [ 91 and recently their connection with quantum algebras and superalgebras have been established [ lo- 121, together with their derivation from the contraction of quantum algebra [ 111. A two-parameter Heisenberg algebra has also recently been obtained [ 13 ] by Schwinger’s contraction procedure for the two-parameter quantum semigroup SC,,~(2) [ 141. The algebra of Se,, (2) is called quantum semigroup because it is not possible to find an antipode function on the bialgebra SC,,$ (2) to endow it with a Hopf algebra structure [ 141.
*I For reviews, see for instance 0378-437 l/94/$07.00
[ 41.
@ 1994-Elsevier
Science B.V. All rights reserved
254
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
The thermodynamic properties of deformed systems have also started to be investigated and many attempts to find a direct physical application have been made [ 15-2 11. On the other hand thermo-field dynamics (TFD) [22] is a formalism whereby the usual field theory defined in real space-time can be generalized to the case of finite temperature. In this formalism the Feynman diagram method can be easily formulated by means of the real-time causal Green’s function [ 23 1, which are expressed in terms of “temperature-dependent vacuum” expectation values, and all the operator relations of T = 0 field theory are preserved. Therm0 field dynamics has been extensively developed and has been applied to problems in condensed matter physics as well as in high-energy physics [23,24]. In this paper we apply the formalism of TFD to two kinds of deformed Hamiltonians. These Hamiltonians are made up of operators which satisfy deformed Heisenberg algebras (q-oscillators). In section 1, we review the subject of deformed oscillators and extend the results of ref. [20] which come out of the application of TFD to a simple deformed Hamiltonian. In section 2 we consider a more complicated deformed Hamiltonian and compute, its entropy and the specitic heat. Section 3 is devoted to some final comments.
1. Thermo-field dynamics of deformed systems In the formalism of TFD [ 22-241 one constructs a temperature dependent vacuum 10(/? )), in which the statistical average of 6 coincides with the vacuum expectation value, using this new vacuum 10(/3 ) ). For instance if one uses the canonical ensemble one has (b) E 2-l
(/?) Tr[ e-aH6]
= (O(p)]d]O(p))
(1.11
with p = (kaT) -l and ks the Boltzmann constant. Let { 1n)} be the orthonormal basis of the state vector space ti consisting of eigenstates of the Hamiltonian H,
Hln) = EnIn) ,
@In) = &?l,n.
(1.2)
In order to construct such a state 10(p)) one introduces a fictitious system (tilde system) characterized by the Hamiltonian R and the state vector space fi spanned by {In)} obeying
fib) =&In) ,
(Fzp?z) =
a,,, .
(1.3)
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
255
The thermal vacuum ]0(/3)) belongs to the tensor product space ‘FI8 fi and is given by: IO(p))=
G
Z-‘12(j3) Z-1/2 (j?)
e+En12 In) 8 In)
&
e-bEn12 In, ii) .
(1.4)
n
Ifoneuses
(1.4) in (1.1) onehas
C
(0(/3)l6lO(jl)) = Z-‘(p)
edpEn” emgEmi2(n”, nldlm, 62)
n,m =
C
Z-‘(p)
e-BE~(n181n) =
(8)
(1.5)
n
which is the result claimed in ( 1.1) . This doubling of degrees of freedom has a sensible physical interpretation [ 22,201 and is related to the algebraic formulation of Statistical Mechanics developed by Haag, Hugenholtz and Winnink
[251. This formalism can be extended to the case of statistical averages of systems made up of q-oscillators [20]. One calls bosonic q-oscillators the associative algebra generated by the elements d, A+ and N satisfying the relations [10,11,26] [N,d+]
=A+,
14 A+ 1A
=fd(N)
[N,d]
= -A,
.
(1.6)
We are going to consider here the following forms of the above algebra ( 1.6):
[a,a+],saa+ - qa+a [A,A+]AsAA+
[%a+Ia
- q2A+A
sa(Y+ -s-‘q
= q-N,
(1.7a)
= 1)
(1.7b)
cx+(Y = (sq)_N
.
(1.7c)
The first two algebras are related to each other via [26] A = qN12a
,
A+ = a+qN12,
(1.8)
and the last one is the two-parameter deformed Heisenberg [ 131 algebra obtained from the quantum semi-group ,S.& (2) [ 141 by the Schwinger contraction.
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
256
It is possible to construct the representation of the relations ( 1.7 ) in the Fock space 3 spanned by the normalized eigenstates In) of the number operator N as
AIO)= 0 , In)
= d&
Nln) = nln)
n = 0,1,2...
9
(1.9)
(A+ )“I@ 3
where [n]d! = [n]d’.. [l]d, [n], = (q”-q-“)/(4-q-‘), [n]~ = (q2”1)/(q2 - 1) and [n], = ((s-‘q)” - (sq)-“)/(.s-‘q - (sq)-‘). In the Fock space F it is possible to express the deformed oscillators in terms of the standard bosonic ones b, b+ as [26] d
=
( [y+1;d)1’2b, d+
=
b+
( [NN’+1:“)‘/2 ,
(1.10)
and it can easily be shown in F that
dd+ = [N+
11.4,
d+d
(1.11)
= [N]d.
Let us now consider an ensemble of q-bosons with Hamiltonian H=wN
(1.12)
with eigenvalue on (n = 0, 1,2,. . -) on F. We introduce the tilde system as A=COiV, where the tilde q-operators
[R,d+]=d+,
given by
the Hamiltonian
of
(1.13) we are considering
satisfy the following relations:
[RT,d] = -A,
[A, d+ ]A = f~(fi) .
(1.14)
In the cases we are going to consider here, we have
(1.15)
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
and [&A] = [A,d+] thus given by
= 0. The temperature
2
IO(p))= Z-‘/*(j9)
eeBnwi2
dependent
vacuum
251
IO@)) is
. &(d+,“ca+,.jo)
n=O = ( 1 - emDo) ‘I2 exp,
( e-pm/* A+ A+ ) 10)
(1.16)
withexp,,x = C~,o(l/[n]~!)~“theq-exponential [27],and]O) = ]O)@]O). In the formula ( 1.16) we used the information that the partition function of q-bosons [ 191 corresponding to the Hamiltonian ( 1.12) coincides with the usual one for harmonic oscillators. We can easily see that the non-deformed case is recovered in the q + 1 limit. We denote up
=
(1
_
e-b
)-1/2 3
2.‘8
=
(
eBO-1)-l/*,
GB = -i6,
[ ( ~~:l:r)1’2
- ($--)I”
( &YJ2~
(&-)“*d+/t+]
where cash 8, c UP. With these definitions dependent vacuum IO(B)), (1.16), as IO(p)) = exp(-iGB)]O)
(1.17) we can rewrite the temperature
E BIO) .
Let us now define the temperature
dependent
(1.18) operators;
they are given by:
&=exp(-iGB)dexp(iGB) &=exp(iG~)dexp(iG~)
.
(1.19)
It is interesting to observe that this transformation algebra (1.6),( 1.7), i.e.
preserves the q-Heisenberg
aSaS+- qaS+ag = qmNfl,
ApA; - q2Ag+Ag = 1, a&
- s-‘qa;cy~
= (sq)-4
.
(1.20)
258
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
This can be seen by the use of relation resembles a q-Bogoliubov transformation. d#(P))
= &O(P))
(1.21)
= 0
and the Fock space can be constructed (1.18) on (1.9), i.e.
for n = 0, 1, . . .. us now compute depends the deformation given by: (d+d)
( 1. lo), thus the B transformation Obviously
by applying
the B-transformation
average of d+d which, as we are to see, considered. In the TFD approach this average is
= (WP)Id+.W(P))
= (O(P)\
[NIAIO(P)) .
In order to perform this calculation we go to the basis of non-deformed operators. In the non-deformed case one has [22] bb =exp(-iG,g)bexp(iG,g)
= ubb -va6’,
68 =exp(-iGB)?)exp(iGB)
= up6 - vab+,
(1.23) bosonic
(1.24)
with the inverse given by
b=upbfi
+ wpbs’,
(1.25)
where
Gg = -8,(&J
- 6+b+)
(1.26)
.
Now using ( 1.25 ) one finds the following expression for the number operator N N =
w$($ + 1) + u;Ng + u@J&$~;
+6&A.
(1.27)
Here we recall that the generators of SU ( 1, 1) algebra, J+, JO, can be realized A la Schwinger [ 28 ] as 5 Jo = ;(Ng + & + 1). (1.28) J+ = b;bj+ , J_ = babs ,
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
259
Thus using (1.28) we can rewrite (1.27) as
N =
04; +
+ upJp(J+
L)
;c
+
(1.29)
with (1.30)
C&v/y-&?-l.
Notice that C commutes with the first two terms of the right-hand side of expression ( 1.29 ) . Using now (1.29) and taking 4”’ = expel, the relevant terms in the calculations of ( 1.23 ) have the form
(O(P)1eAN IO(P)) = ((-J(P)1 e~[(U:+“:)JO+Usvs(J++J-)+(1/2)Cl
[(-j(p)).
(1.31)
This last expression can be computed by means of the Baker-CampbellHausdorff (BCH) formula, which can be derived for the SU ( 1,l) algebra [ 29 1. The BCH formula for the case we are considering is given by: (1.32) where
‘=
cosh(l/2)
2~s sinh(i2/2) - (z$ + $) sinh(A/2)
y = -2ln[cosh(A/2)
The above procedure obtain
= p/2
- (~5 + $) amounts
’
sinh(i2/2)]
.
(1.33)
to normal ordering eq. ( 1.3 1). Using this we
2q-m/2 + q-m/2 - ($ + $) (qd
- q-m/2) ’
(1.34)
Finally using (1.23) and (1.34) we can see that
(w/vl~+4w-w
= e2bw -(q
ebw -1 + q-l)
eaW
+l’
( 1.35a)
260
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
eBw1_q2’
(wPN~+4wP)) = (O(P)l~+~]O(P))
(1.35b)
=4( eBo-l)[(s-2fS-1(q+q-1)
+ l)(@W-1)2
-2(sm2 - l)( e2p0 -1) + (s-2 -s-l
+ l)(eS”
(q + q-l)
+1)2]-1,(l.35C)
where ( 1.35a,b) agree with the results given in ref. [ 191.
2. Entropy of deformed systems In this section we are going to consider, using the formalism of TFD, the entropy of a deformed system. As we have noticed in the previous section the partition function corresponding to the Hamiltonian ( 1.12) coincides with the usual one for standard harmonic oscillators and as we are not changing the statistical averages the thermodynamic functions are not modified, therefore we take another Hamiltonian given by H = e[NIA-,uN.
(2.1)
The form given by (2.1) is a general deformed Hamiltonian that describes in the q + 1 limit the usual harmonic oscillator. Note that both [N] and N are deformed but, as we shall see later, their statistical properties are different because only [N] affects the extensivity. This choice permits to follow where the deformation modifies the thermodynamic functions (e.g. when E = 0 one recovers the non-deformed quantities). This is indeed the case as one can easily check later. For a collection of oscillators, such as q-boson gas, the second term in (2.1) corresponds to the chemical potential. For free systems, in the TFD approach, one can define an entropy operator, R, as follows. Let us consider the thermal vacuum ]0(/3)) which, as we have seen, is IO(p))= Z-1/2(p)
C
= ,Inz-“~(p)-~H
’
emBEJ2In, ii)
CIrz,ii) n
G
e-k12CIn,i)
(2.2)
n
with g = lnZ(j3)
+pH,
(2.3)
M. R-Monteiro, I. Roditi / Deformed system at finite temperature
where the entropy, K, is obtained K =
261
from I? as
b(OtP)l@XP)) .
(2.4)
One can see that if one writes [22]
(2.5)
with w,, = e- BE, /Z (/I), we have
(OtB)I~lO(P))= Cwn[lnz(P) + PE.1 = - CwJnw.
(2.6)
n
n
where C, wn = 1 by construction. Thus K given by (2.4) recovers the usual Shannon entropy. For the deformed system we are considering, (2.1)) one can be convinced that the entropy is given by the Shannon entropy with Wn
=
(2.7)
(p)
@b'ld-Pn)/Z
where Z(p)
=
2
e-~(e[nld-Pn)
.
(2.8)
n=O
As the partition function cannot be computed exactly we take q = eal, s = ea2 and consider the case where (Yis small. Defining j = j? (E - p ) we may rewrite Z (8) in terms of the non-deformed partition function as z(p)
= ~~e([-~ld+a^)Zo(jj)
where 8 = a/a/?, Zo(j)
=
2
(2.9)
and e-j”
(2.10)
n=O
the non-deformed partition function. Performing an expansion in Q!and taking into account only the lowest order in CL: we have for Z(p)
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
262
1
Z(p)-‘d,‘(B)
2PE 1 - CY+Q(~)-
Q”‘)
+ O(CE;) ,
(2.1 la)
Z(P)-‘=Z,‘(li)[l
+ 44Qc2)+
Q”‘,
+ Ok&,
(2.1 lb)
2PE 1 - a2@Q (1)_,1_-(Q(3)_
Z(p)-‘=Z,‘(j3)
[P&Q
+ &P& +
forthealgebras
2Q’2)
+
Q(3)
the exponential
and (1.6)(1.7c)
PEQ(2))]
respectively,where
.
(2.12) with the Q(j)‘s is
aA’) =
(j +
,
a!‘) r+l
(2.7) we a: 1 - +W?
=
- 1) + O(c&
[ e-~(e[nlA-~n) = e-/j” [l - culPsn(~ - 1) + Q(aT)], e-8(e[nla-w)
+
aj’) are found by the recursion law:
ja,!f_i’) +
e-8(e[nl,-Pn)
+
(1.6)(1.7b)
for the calculation
where the coefficients
=
;(Q”’
(2.1 lc)
(1.6)(1.7a),
A useful expression
I
(I)* _
m$&
Q”’ = z,(j)-‘ajZ,(j)
,!i)
Q”‘)
9
a: a212 + +n2-
= e-s”
+ #n((n
- U2 -pm)]
Substituting (2.7) and (2.10)-(2.12) forward calculation S, =SO - a:kgjklj
0.
1,
(2.13a) (2.13b)
1)
+ O(&;)}
.
(2.13~)
in the entropy we get after a straight-
em2j(1 - eCs)-4(
emB +2) + Q(at),
(2.14a)
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature S, =A’, - 4alkB/3cg
e-2$
S, = S,, - a2kB/3cj
d(
-cu:kBj3cB c2j( -a;k&?c +fj[2
1 - e-a)-3
+ 0(a;),
263
(2.14b)
1 - e-j)_2 e-8 +2)(1
- e-8)-4
e-j{ @E( 1 - e-b)-2 e-S(2
+ 7 e-8)(1
-_B&(l + e-j)(l
- e-$-4
- e-$-s]}
(2.14~)
with S’s the entropy of the non-deformed system and S,, SA and S, the entropies corresponding to the algebras ( 1.6), ( 1.7a,b,c) respectively. It has been suggested [ 301 that there is an underlying lattice structure with lattice spacing cx = q - 1 for deformed theories with deformation parameter q. We see from the above formulae that the entropies of deformed systems are lower than the standard ones for qi > 1, i.e. the deformations introduce a correlation which could be interpreted as an underlying lattice structure. When the deformation parameter is lower than one we do not have the lattice interpretation anymore and for the (2.14b,c) cases, due to a linear contribution in (~1and (~2respectively, the entropy increases with respect to the non-deformed case. This kind of oscillators ( 1.7) with q < 1 were shown to be related to the Brownian motion [ 3 1 ] which could explain the increase of entropy. After a long but straightforward calculation the specific heat can be found as: c(a) = cc + a: CL;’ + . . .
(2.15a)
CA) = c, + a!1 C@) (I, + ...
(2.15b)
I
0)
= c, + a@) a* + a:cy
with Cc, the standard
+ cy:c$
(2.1%)
+ **. ,
specific heat and
C(U) = C’“’ = 2kBp2c(c -p) a: a: -kBP3c(c -,u)~ e-‘j(2
ee2b(2 + ees)(l + e-8)2(1
- eB )-4
- e-b)-5,
(2.16a)
C$‘) = 4kB/?‘c(c - ,u) em2fi( 1 - ea)-3 x[2 + e-s-2/3(c-~)(1
- e-j)-l(l
+ ePS)],
(2.16b)
264
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
c;y
e-)(1
=2k#&(&-p)
-k#&(&-j# cl;)
- e-b)-*
e-Q1
= -2kj3jP&(&
+ e_fi)(l-
- p) e-*$(2
e-$-3,
(2.16~)
+ 7 e-8) (1 - e-s )-4
2 _k&*c*
e-a (1 - e-b)-*
+4kBp3c2(c--) -kBp3c(&
e-j(l
-p)*
e-$1
+ e-$(1
- e-a)-3
- e-a)_5
x (20 - 64 e-a +35 e-*‘j -27 e-3a) -;kB/34c2(c
- /.L)*
e&(1
- e-B)-4(1
+ 4 e-b + e-2b). (2.16d)
If we consider the high-temperature approximation, p --) 0, one can see that when the deformation parameter is bigger than one the specific heat is lower than the standard one, Co. As before, in this limit, when the contribution of the deformation for the specific heat is linear and the deformation parameter is lower than one the specific heat increases with respect to Cc. It is interesting to notice that the system has the memory of the deformation in the /I + 0 limit. The difficulty in computing the exact value of the thermodynamic potentials in the case of the Hamiltonian (2.1) can be understood in the formalism of TFD in the following way: Consider the thermal vacuum for the case of the Hamiltonian (2.1) and call it [0(/3)),, where IO(B)),
Z-‘/*(p)
=
C
e-(8/2)(E[nlA-Pn)
112, fi)
,
(2.17)
n then it is possible to express (2.15 ) in terms of the thermal vacuum for the Hamiltonian (1.12) IO(p))
IO@>) = Z,“‘(B) C
e-an IO)
(2.18)
n as
lOCP>),=
2-1'2t8)~(~~)Z~'2~B)10(P^))
(2.19)
with F(x)
=
e-Be((l/*)[-*Xla+x)
(2.20)
M. R-Monteiro, I. Roditi / Deformed systems at finite temperature
265
Showing that the transformation from the non-deformed thermal vacuum to the deformed one (2.17) is highly non-trivial, being simple only when the deformed and non-deformed Hamiltonians have the same spectrum.
3. Final remarks We have considered in this paper the approach of TFD for two different Hamiltonians, each one made up of a different kinds of operators satisfying deformed Heisenberg algebras. The first Hamiltonian is linear in the number operator N and does not present any difftculty, however the second Hamiltonian generates a partition function and thermodynamic potentials which cannot be exactly computed for arbitrary values of the deformation parameter. Even if it is more complicated the second Hamiltonian is more interesting; it is non-interacting with non-additive energy. We have also seen that the exact computation of mean values for general deformed Hamiltonians is a rather complicated issue, the reason being the non-trivial relation (2.17). We think that it would be very useful to develop a variational approach to deal with such systems. The non-additivity of energy seems to be a deep property of deformed systems and its is already present in the deformed Poincare group [ 321. There is still no strong evidence of the presence of the quantum group structure in physical systems, we believe that the non-additivity of energy present in deformed systems could give evidence of the presence of such structures as discussed in [ 321 for the case of deformed Poincare group. The same issue is analysed in the context of the generalized statistical mechanics [ 33,341 in ref. [ 2 11. We also consider that it would be interesting to develop TFD in the context of the generalized statistical mechanics [ 33,341, which would permit the analysis of the models we have discussed within this approach.
Acknowledgements Some of the ideas contained cussions with C. Tsallis.
in the present paper have benefited
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