Quantum dissipation

ANNALS OF PHYSICS 215, 156-170 (1992)

Quantum Dissipation

E. CELECHINI

Dipartimento di Fisica and Serione I.N.F.N., Universitci di Fir&,-e, Florence, Italy

M. RASETTI

Dipartimento di Fisica and Unitci I.N.F.M., Politecnico di Torino, Turin, Italy

AND

G. VITIELLO

Dipartimento di Fisica, Universitci di Salerno, and I.N.F.N., Sezione di Napoli, Naples. Ita!,>

Received September 30, 1991

We discuss some aspects of dissipation in quantum field theory starting from the example of the quantum mechanical damped harmonic oscillator. We show that the set of states of the system splits into unitarily inequivalent representations of the canonical commutation relations. At quantum level the irreversibility of time evolution is expressed as tunneling among the unitarily inequivalent representations. Statistical and thermodynamical properties of the formalism are analysed and canonical quantization is shown to lead to time dependent SU( 1, 1) coherent states, well known in high energy physics as well as in quantum optics and thermal field theory. $0 1992 Academic Press. Inc.

1. INTRODUCTION

The study of dissipative systems in quantum theory is of strong theoretical interest and of great relevance in practical applications. As a matter of fact, any microscopic system is always embedded in some macroscopic environment and therefore it is never really isolated. Dissipation effects play a relevant role in high energy physics (e.g., in quark-gluon physics), in the early universe physics, in the vacuum structure in the presence of gravitational background, as well as in condensed matter physics, in phase transition phenomena, and in general in quantum field theory (QFT) at non-zero temperature [l].

Also, it has been recently shown [2] that the squeezed states of light entering quantum optics [3] can be identified, up to elements of the group 9 of automorphisms of su(1, l), with the states of the damped quantum harmonic oscillator.

156 CKJO3-4916/92 $7.50 Copyright 0 1992 by Academic Press. Inc. All rights of reproduction in any form reserved.

QUANTUM DISSIPATION 157

A major di~culty that appears in the study of dissipative systems in quantum mechanics is that the canonical commutation relations (CCR) are not preserved by time evolution due just to damping terms. Then one introduces fluctuating forces in order to preserve the quantum mechanical consistency, namely the canonical structure. Another way to handle the problem is to start from the beginning with a Hamiltonian that describes the system, the bath, and the system-bath interaction. Subsequently, one eliminates the bath variables which originate both damping and fluctuations, thus obtaining the reduced density matrix.

The purpose of the present paper is to discuss some aspects of dissipation in QFT, resorting just to the relatively simple example of the damped harmonic oscillator, whose classical equation of motion is

rn2-b y.? + KX = 0. (1)

In order to implement a canonical quantization scheme for the system (1 ), one must first double the phase-space dimensionso as to deal with an effective isolated system [4]. The new degrees of freedom thus introduced may be assumed to represent by a single equivalent (collective) degree of freedom for the bath, which absorbs the energy dissipated by the oscillator. In Section 2, closely following the methods of Refs. [2,4], we carry out this program in the framework of quantum mechanics and we show that the dynamical group structure associated with our problem is that of SU( 1, 1). We also show that time evolution would lead out of the Hilbert space of states and that quantum mechanical treatment does not provide a unitary irreducible representation of SU( 1, 1). These pathologies may find their counterpart in the more traditional approach where the CCR are not preserved in time due to dissipation, as observed above. The preliminary pedagogical discussion in Section 2 is instructive, since it also points out a possible cure for these pathologies; it suggests indeed that the canonical quantization can possibly be achieved in a consistent way in a second quantization scheme, i.e., by moving to QFT, where infinitely many unitarily inequivalent representations of the CCR are allowed (in the infinite volume or thermodynamic limit) and where the bath may find its most appropriate description in terms of a field. In Section 3 we show that, indeed, in this case the set of the states of the damped oscillator splits into unitarily inequivalent representations (i.e. into disjoint folia, in the C*-algebra formalism) each one representing the states of the system at time t : in a more conventional language, the time evolution may be described as tunnelZing between unitarily inequivalent representations. A remarkable feature of our description thus emerges: at a microscopic level the irreversibility of time evolution (the arrow of time) of a damped oscillator is expressed by the non unitary evolution across the unitarily inequivalent representations of the CCR. In Section 4 we analyze the connection of our formalism with thermal QFT and we realize that the states constructed in Section 3 are indeed time-dependent thermo-field dynamics (TFD) states [S]. It is interesting to observe that, although statistical concepts were not introduced a priori, the statistical nature of dissipative phenomena naturally emerges in our treatment. Finally,

158 CELEGHINI, RASETTI, AND VITIELLO

Section 5 is devoted to conclusions and further remarks concerning, e.g., the non- perturbative character of quantum dissipation and the breakdown of symmetries related with SU(1, 1) and time-reversal transfo~ations.

2. CANONICAL QUANTIZATION

In this section we thoroughly review the approach of Feshbach and Tikochinsky [4] recently recast in the Holstein-P~makoff representation in Ref. [2], Following Ref. [4], in order to deal with an isolated system, as the canonical quantization scheme requires, a procedure of doubling of the phase-space dimension is necessary. The lagrangian for system (1) is written as

L = mc@ + &(xj - iY) - tcxy, (2)

where y denotes the position variable for the doubled system. Intuitively one expects the y variable to grow as rapidly as the x solution decays; in this sense y may be thought of as describing an effective degree of freedom for the heat bath to which the system (1) is coupled. (1) is obtained by varying (2) with respect to y, whereas variation with respect to x gives indeed

a aL aL ----= ataz ax

o 0 mj;-yj+Icy=O, (3)

which appears in fact to be the time reversed (y -+ - y) of (1). The canonical momenta p+; and p.,, (the collection of dynamical variables (x, pIl, y, p,,} spans the new phase-space) are then given by p, E aLi% = m-G - ;yY; p? = aL,@ = rn.2 + iyx.

The hamiltonian hence reads

H=p,f+pyj- L

1 1 =,P.~P,,+Zmi’tYP.~-xp.,)+ K-f xy.

( 1 (4)

Canonical quantization may then be performed by introducing the commutators [x, P,~] = ih = [y, py J, [x, y] = 0 = [p,, pp], and the corresponding sets of annihilation and creation operators

(5)

with [a, a’] = 1 = [b, !I+], [a, b] = 0 = [a, b+], (6)


where we have introduced

the common frequency of the two oscillators (1) and (3), assuming Q real, hence ti > y2/4m, corresponding classically to the the case of no overdamping. Resorting to (5) and (6) and performing the linear canonical transformation A = (l/$)(a + b), B = (l/fi)(a - h), the quantum hamiltonian is obtained

3?=&+3q,

X0 = hQ(A+A - B+B), -lu; = ihT(A+B+ - AB), (7)

where r= y/2m is the decay constant for the classical variable x(t). Equation (7) is a remarkable result in many ways. As pointed out in Refs. [2,4],

only if one restricts the Hilbert space of states to the subspace even with respect to time-reversal (x-y), i.e., if one retains only those states, I$), which are annihilated by B: B I$) = 0, then the eigenstates of J? merge into those of the simple undamped harmonic oscillator when y -+O ( Q +o= a). Therefore the states generated by Bt represent the sink where the energy dissipated by the quantum damped oscillator flows: the B-oscillator thus represents the (single-mode) reservoir or heat bath coupled to the A-oscillator.

It is easy to realize that the dynamical group structure associated with our system of coupled quantum oscillators is that of SU( 1, 1). The two mode realization of the algebra SU( 1, 1) is indeed generated by

J =A+B+, + J-=J:=AB, J,=;(A+A+B+B+l), (8)

corresponding to Casimir operator %’ defined as: V’ = $ + J$ - i(J+ Jp + Jp J,) = b(AtA - BtB)2. The hamiltonian (7) is then rewritten as

x0 = 2?U2 %, i$= iW(J+ - Jp) = -2hTJ,. (9)

Thus [%I, #I = 0, (10)

as y% is in the center of the dynamical algebra. Let us denote by {In,, n,)} the set of simultaneous eigenvectors of AtA and

BtB, with nA, nB non-negative integers. One can see that the eigenvalue of X0 in this frame is the constant (conserved) quantity 2fiQ(n, - nB). The eigenstates of X, can be written in the standard basis, in terms of the eigenstates of (J3 - i) in the representation labelled by the value j E Z,,, of W, { lj, m ); m > xl jl }:

~l.i,m)=jlj,m), j=t(n,-n,);

V-i) lj,m>=m km>, m = i(n, + ne). (11)


Use of the su( 1, 1) algebra

[J+,J-]= -253, CJ3,J+l= fJ,> (12)

shows that the kets defined as j$j,M) =exp(+(z/2)J,) lj, m) (recall that J, = $(J+ +J-), J2 = -(i/2)(J+ -J-)), satisfy the equation J2 Ill/,,,) =p l$j,j.,) with pure imaginary p = i(m + f).

It is worth pointing out here that: (i) the states i$j,i,,> are not normalizable [2,4,6], i.e., their norm

(ii/j,, / ej+) is divergent, as explicitly shown in Ref. [4]; (ii) the transformation which generates lll/j,,,) from the standard vector

/j, m} is not a proper rotation in SU(1, I), but is rather a pseudorotation in SL(2, C), which in fact corresponds to a non-unitary transformation in SU( 1, 1). Thus Itij,,> does not provide a unitary irreducible representation (UIR). This is consistent with the feature, proved by Lindblad and Nagel [7], that in any UIR of SU( 1, 1) J2 should have a purely continuous and real spectrum (which is not the case for i+j,cli,m)). On the other hand, a UIR with a continuous and real spectrum has nothing to do with the description of dissipation and therefore it is ruled out.

These features are retrieved here in the apparent contradiction inherent to the fact that, although J2 appears to be hermitian, it has a pure imaginary discrete spectrum in I$j,m).

In order to emend the origin of this pathology, the hermitian conjugation of J, therefore has to be defined in the Hilbert space of states by endowing the latter with a suitable inner product, in such a way that Ilcl,.,) has a finite norm. To this aim we introduce in the Hilbert space itself a new metric, in analogy with what is done for the inde~nite metric theory [S], following the procedure proposed in Refs. [2, 41.

We notice first that under the antiunitary operation of time reversal F, (A, B)* (--A*, -Bt). Therefore, we introduce the conjugation operation induced by (Il/j,ml E [S ltij,,)]‘, with F itij,,,) E I~j._,,+l)) (which can be interpreted as a prescription for continuation to negative m’s). In the metric corresponding to this conjugation operation for kets, the above difficulty is bypassed, since the hermitian of

($j,-im+,jl J~=~&j~m+d pF= -iC-(m+l)+$]= --p*=p.

The solution to the Schrodinger equation corresponding to the initial pure state lj, mo) is given by [Z, 4)

IY(t))=exp[-(i252j+T)t] l@(t)>.

( >

(13) I@,(t)) = z exp( -2Fmt){j, ml exp -;J1 lj, mo)etK’2tJi lj, m>,

~3 IA


which asymptotically (i.e., for large t) behaves as

I~(t))N(-)mo+i2.i+11* [(m;;j)]‘~2 ev( -W) Ix>,

( > Ix)=exp 55, Iid-$exp(J+) lj,j),

(14)

where Ix) is the generalized coherent state [9] for SU( 1, l), related to squeezed states as discussed in [2].

If, in particular, the initial state is the vacuum In, = 0 , n, = 0) = IO), such that A IO) = 0 = B IO), i.e., j= 0, wzO = 0, its time-evolution is given by

lO(r))=exp( -if:) /O)=exp( -it?) IO)

1 = cosh( rt ) exp(tanh(rl)J+) IO>, (15)

namely a two-mode Glauber coherent state [lo] (i.e., once more a generalized coherent state for su(1, 1)). Notice that Eq. (15) is formally correct as far as the time interval [0, t] is finite.

We also observe that, due to Eq. (lo), the interaction picture is exact in the sense that

where .&, is the hamiltonian in the interaction picture, whereas Y& is the hamiltonian in the Schrodinger representation. Equation (15) shows that at every time t the state IO(t)) has unit norm; however, as t -+ 00 it gives rise to an asymptotic state which is orthogonal to the initial state (0):

(O(t) I O(t)) = 1, lim (O(t) IO) = lim exp( -In cash(R)) + 0. ,+‘X r+m

(17)

(18)

Equation (18) expresses the instability (decay) of the vacuum under the time evolution operator @ = exp( - i~(%~/h)) in the infinite time limit. We thus reach the conclusion that, as an effect of damping (recall that X, + 0 as y + 0), 2, acts as the time evolution generator which is well defined as long as a finite time interval is considered. However, as t + co, time evolution leads out of the original Hilbert space of states. This translates in physical terms the mathematical difficulties pointed out in our previous discussion, focusing the obstruction which is to be bypassed in producing a canonical quantization scheme for the damped harmonic oscillator. One should notice that so far the problem was tackled within the framework of quantum mechanics, namely within a scheme where the von


Neumann theorem only allows unitarily equivalent representations of CCR. In what follows, we intend to show how the pathology exhibited by (18) can be controlled if one operates in a different scheme, giving up, in particular, the condition of finiteness of the number of degrees of freedom. This, in turn, is obviously equivalent to moving to a second quantization scheme, i.e., to a quantum field theory, where the infinite number of degrees of freedom allows the coexistence of infinitely many unitarily inequivalent representations of the CCR. This is connected with the possibility of continuously mapping representations one onto the other by intertwining operators [ 111; it is also physically more realistic, because an infinite number of degrees of freedom is more adequate to describe the heat bath.

3. THE QUANTUM FIELD THEORY FRAMEWORK

A straightforward extension to QFT of the hamiltonian given -by Eq. (7), describing an (infinite) collection of damped harmonic oscillators, is

X=&~-+X*,

(19)

where K labels the field degrees of freedom, e.g. spatial momentum. As usual, the computational strategy is now to work at linite volume of the system V, and to perform at the end the limit V -+ cx). Notice that as far as time and volume are finite there arises no problems with unitarity. The commutation relations are:

[A,, Af] =&h.,l = LB,, Bf]; [A,, &?I =O= EA,, B,l (20)

The group structure survives in the QFT, but with SU(1, 1) replaced by OK SU( 1, 1 ),, each SU( 1, 1 ), denoting a copy of SU( I,1 ), as one can check using Eqs. (20). Here and in the sequel we assume that difficulties arising with (continuous) tensor products of groups may be overcome as in conventional QFT [ 121. In particular, we still have [&$, X1] = 0 (cf. Eq. (lo)), and corresponding to Eq. (15) we have (formally, at finite volume V),

lo(t)) = r*I coshll. t) exp(tanh(~~f)~~))lO}, K

(21)

with JtK) z A ‘t Bt + h Y’ Moreover (O(t,,b(t,) = 1 vt, (22)

(O(t)~O)=exp - c c (23)


which shows how, provided 2, rK > 0,

Thus, for finite volume, the situation is the same as in the previous section. Now, however, we may also consider the limit V + co. Using the customary continuous limit relation CK H ( V/(~Z)~) 1 d 3~, in the infinite-volume limit we have (for i d3~ r, finite and positive)

We notice that the time-evolution transformations

A.~A~(t)=e-“‘~‘~‘A~e’(“~‘.~~=A~cosh(f~t)- BLsinh(r,t);

B, i--+ B,(t) = P - ictih)~*BKe’(“h)xf= -A: sinh(r,t) + B, cosh(r,t), (26)

and their hermitian conjugates can each be implemented (formally at finite volume), for every K, as an inner automorphism for the algebra su( 1, 1 ), [ 131. Such an automorphism is nothing but the well-known Bogolubov transformations. The transformations (26) are canonical, as they preserve the CCR (20). In other words, at every time t we have a copy {AK(t), AL(t), B,(t), B:(t); IO(t)) 1 VK} of the original algebra and of its highest weight vector {AK, Ai, B,, BL; IO) 1 VK), induced by the time evolution operator (i.e, we have a bona fide quantum realization of the operator algebra at each time t, which can be implemented by Gel’fand-Naimark-Segal construction in the C*-algebra formalism [ 143). The time evolution operator can therefore be thought of as a generator of the group of automorphisms of 0, su( 1, 1), parametrized by time t.

It is very important to point out that the various copies need not be unitarily equivalent representations of the CCRs; as a matter of fact, they do become unitarily inequivalent in the infinite-volume limit, as shown in (25). This implies that the automorphisms (26) are defined up to arbitrary intertwining operators; in fact, one should more accurately say that the dynamical algebra is given globally by the doubly-continuous direct sum @ ( OK d” (” 0 su( 1, 1 ), , where JzP(‘) denotes the intertwining operator connecting the representations realizable at time t. Since d” can be explicitly constructed in terms of vertex operators acting (over the suitable Verma module) in the Fock representation, it is obviously labelled by K, and it is K which is varying at each step with t. The intertwining operators for SU(1, 1) (and X(2, C)) which allow transforming one representation into the


other Cl1 ] can be realized as integral operators whose kernel may be singular, even though in general globally convergent.’

As a direct check, one can easily verify how at each time t one has

A,(t)lO(t))=O=B,(t)IO(t)), vt. On the other hand, the commutativity of X0 with Ji; ensures that under time

evolution the number (nAK -n,) is a constant of motion for any K. Thus the su(l, 1) structure can also be clearly recognized by recalling [15] that the group SU( 1, 1) is realized on Q: x C as the set of all unimodular 2 x 2 matrices leaving invariant the hermitian form (2;~~ -z:z2ft Z,E c, M = 1,2, and that the elements of SU(1, 1) can be parametrized by a pair of (complex) numbers C, S such that ICI*- ISI*= 1 (cf. Eq. (26), w h ere C = cosh(r,t) and S = sinh(r, t)).

The number of modes of type A, is given, at each instant t, by

.NiK= (O(t)~A~A./O(f))=sinh*(~,t),

and similarly for the modes of type B,. Moreover,

(27)

A:WlW)) =coshjr K t) A,tlOW =sinhfr.t) Bti lo(t)>, h

(28)

Bt(~)lO(t))=coshjTt)~!lO(t))=sinhfr~l)A,lO(~)). L h

Equation (21) shows that IO(t)) is a two-mode Glauber coherent state with equal numbers of modes A, and B, condensed in it for each h: and each t. Equations (28) show that the creation of a mode A, is equivalent to the destruction of a mode B, and vice versa. This leads us to interpreting the B, modes as the holes for the modes A, and confirms that, also in the QFT scheme, the B-system can be considered as the sink where the energy dissipated by the A-system flows.

The state IO(t)>, on the other hand may be written as

lO(f)>=exp -IY j9)=exp -!Y IX>, ( 2 4 ! 2 4

where

/9)=exp 1 A:B: IO> ( > K

’ The .C’s are explicitly given by integrals, over an appropriate element (cycle) of the homology group Ii, of the a-dimensional manifold AZ z {zr, ,.., z, / zA E (C)“; z1 # z,,, l< 1 <p < a], for some uoZ+, of a vertex operator parametrized by the set (zn) over the product fl;=, z;+ ‘. Here u is the integer- or half-integer-valued solution (if any) of the consistency equation, expressing the commutation of the action of the algebra with the intertwining operator, which connects a and Y with the labels K of the corresponding intertwined representations.


is the invariant (not normalizable) vector introduced by Umezawa and Takahashi in Ref. [S]. Here

YA= -C ~A~AtiInsinh2(~,t)-A.A,tlncosh2(T,f)); (30)

and $ given by the same expression with B, and 3: replacing A, and AZ, respectively. As AK’s and B,‘s commute (see (20)), due to (29) we shall simply write Y for either S$ or YB. The representation (29) realizes the infinite tensor product in (21) by explicit use of the Baker-Camp~l~-Hausdorff formula [163. Equations (29) and (30) show that time dependence of IO(l)) is controlled solely by the exponential of 4L$ (or, respectively, $S’$) whose operator part depends uniquely on the A (B) variables: thus Eq. (29) may be regarded as the projection on the (sub)system A (B) with the elimination of the B (A ) variables from the A (B) system time evolution. This can be thought of as a realization of the procedure whereby one obtains the reduced density matrix by integrating out bath variables. Moreover, from (29) one derives the expansion

where n denotes the multi-index {n,), with

Note that the expansion necessarily contains only terms for which nA, equals nBK for all K’S, This is interesting in that

Equation (31) leads us therefore to interpreting 9 as the entr5py for the dissipative system. On the other hand, we have, for the time variation of IO(r)) at finite volume V,

; IO(t)> = -; q IO(t)>

Use of Eqs. (19), (30), and (28) shows then that

f IO(f)> = -if g) IO(f)>, (331

166 CELE~HINI, RASETTI, AND VITIELLO

which may also be derived directly from (29) [17, 181. Equation (33) shows that i($(aY/&)) is the generator of time-translations, namely time-evolution is controlled by the entropy variations. It appears suggestive to us that for a dissipative quantum system the same operator Y that controls time evolution could be interpreted as defining a dynamical variable whose expectation value is formally an entropy: we conjecture that these features of Y correctly reflect the irreversibility of time evolution characteristic of dissipative motion. Damping (or, more generally, dissipation) implies indeed the choice of a privileged direction in time evolution (time arrow) with a consequent breaking of time-reversal invariance.

We also observe that (O(t)/ Y IO(r)) grows monotonically with t from value 0 at t = 0 to infinity at t = 00; i.e., the entropy for both A and I3 increases as the system evolves in time towards the stability condition at t = co. Moreover, the difference YA - YB of the A- and B-entropies is constant in time:

[ya--Y~,&?j=.O. (34)

Since the B-particles are the holes for the A-particles, YA - YB turns out to be, in fact, the (conserved) entropy for the complete system,

In conclusion, the system in its evolution runs over a variety of representations of the CCR which are unitarily inequivalent to each other for t # t’ in the inlinite- volume limit. It is in fact the non-unitary character of time-evolution implied by damping which is recovered, in a consistent scheme, in the unitary inequivalence among representations at different times in the infinite-volume limit.

The above analysis has striking similarities with results on decaying particles in QFT [17], on the vacuum structure for QFT in curved space-time and on the Hawking radiation for black-hole solutions [18]. As somewhat already implied in those papers and as it emerges from the above arguments, the statistical nature of dissipative phenomena naturally emerges from our formalism, even though no statistical concepts were introduced a priori; for example, the entropy operator enters the picture as the time evolution generator (see Eq. (33)). It is therefore an interesting question to ask ourselves whether and how such statistical features may actually be related to thermal concepts.

4. THERMAL FIELD THEORY

Let us focus, for the sake of definiteness, on the A-modes and introduce the functional

EF(OU~l (e+#w). /I is a strictly positive function of time to be determined; XA is the part of &$

relative to the A-modes only, namely, &” =E:, f&trA~A,. Heretofore we have followed the formulation of TFD as presented, e.g., in Ref. [S], even though the

QUAN~M DISSIPATION 167

present framework is quite different from that of Ref. [S], where thermal and statistical concepts are introduced from the very beginning. We write 9, = f,t, and look for the values of 9, rendering 9, stationary:

(36)

Condition (36) is clearly a stability condition to be satisfied for each representation. Setting E, s h&2,, it gives

B(t),?, = -In tanh’(8,). (37)

From (37) we then have

(38)

which is the Bose distribution for A, at time t, provided we assume /I(t) represents the inverse temperature p(t) = l/k, r(t) at time t (k, denotes the Boltzmann constant). This allows us to recognize (IO(t))) as a representation of the CCR at finite temperature, equivalent-up to an arbitrary choice of the temperature scale, which is a differentiable function of time-with the TFD representation {lO(fl))} of Umezawa and Takahashi [S].

We can now therefore interpret .!& as the free energy and yiva as the average number of activated A-modes at the temperature defined by time t through the function b(t). As time evolves, the change in the energy E, = C, E, Jl/rA, is given by

dE, =I E,u&,” dt; Wa)

as one should expect, since the time evolution induces transitions over different representations, which in turn imply changes in the number of activated modes. Time derivative of MA, is of course obtained from Eq. (27). As shown in Ref. [S], Eq. (39a) can also be obtained through (32),

= if (O(t)1 C& %I l@tt> dt W’bt

168 C~LE~HINI, RASETTI, AND VITIELLO

which also illustrates the feature of running over the representations {lo(t))} as time evolves. We can also compute, resorting to Eqs. (30), (27), and (37), the change in entropy

= fidE,(r).

Equation (40) shows that

dE,-;dY,=O.

(40)

(41)

When changes in inverse temperature are slow, namely

aB 1 dT - ---.---...~(),

at- kBT2 at

(which is the case for adiabatic variations of temperature, at T high enough), Eq. (41) can be obtained directly by minimizing the free energy (351,

which-by reference to conventionai thermodynamics-allows us to recognize E, as the internal energy of the system. Equation (42) also expresses the first principle of thermodynamics for a system coupled with the environment at constant temperature and in absence of mechanical work. We may also as usual define heat as dQ = (l/p) dS. We thus see that dNA, the change in time of the number of particles condensed in the vacuum, turns into heat dissipation dQ.

5. COMMENTS AND CONCLUSION

The total Hamiltonian (19) is invariant under the transfo~ations generated by .I2 = $, .Ip’. The vacuum, however, is not invariant under J, (see Eq. (21)) in the infinite volume limit. Moreover, at each time t, the representation [O(t) > may be characterized by the expectation value in the state IO(t)) of, e.g., Jy’ - f; thus the total number of particles nA + n, = 2n can be taken as an order parameter. There- fore, at each time t the symmetry under JZ transformations is spontaneously broken. On the other hand, A$ is proportional to J2. Thus, in addition to the breakdown of time-reversal (discrete) symmetry, already mentioned in the previous

QUANTUM DI~IPATION 169

sections, we also have the spontaneous breakdown of time translation (continuous) symmetry.

In other words, we led dissipation, as described in this paper, to a consequence breakdown of time translation and time-reversal symmetry. It is an interesting question to ask which is the zero-frequency mode playing the role of the Goldstone mode, related with the breakdown of continuous time translation symmetry; we observe that, since nA - nB is constant in time, the condensation (annihilation and/or creation) of AB-pairs does not contribute to the vacuum energy, so that the M-pair may play the role of a zero-frequency mode.

From the point of view of boson condensation, time evolution in the presence of damping then may be thought of as a sort of continuous transition among different phases, each phase corresponding, at time t, to the representation IO(t)) and characterized by the value of the order parameter at the same time r. The damped oscillator thus provides an archetype of a system undergoing continuous phase transition.

We also observe that transformations implemented by Jz are unbounded in time (see Eq.(26)). This fact may be related with the singularity that Bose distribution exhibits at fl=O and E, different from zero. It has been noted in Section 2 that the interaction picture is exact in the sense specified by Eq. (16). In the standard formalism of perturbation theory [ 12] a basic role is played by the adiabatic ~~p~~~e~~~ by which the interaction may be switched off at t = & co. It is such a possibility which allows the definition and the introduction of free, i.e., non- interacting fields. In the case of the damped oscillator the switching off of the interaction in the infinite time limit is not possible, since time evolution is intrinsically non-unitary and the adiabatic hypothesis thus fails. As a matter of fact, we have seen that the set of annihilation and creation operators changes at each time t and the same concept of a non-interacting field thus loses meaning. On the other hand, normal ordering of operator fields, which is a crucial tool in perturbation calculations, depends on the representation of CCR and thus needs to be redefined at each t (and for each value of temperature 7’) [19].

We thus conclude that damping and dissipation require a non-perturbative approach and perturbation methods may be used only for Iocal (in time and temperature variables) fluctuations. An example of such a situation is provided by the description of unstable particles in QFT [ 171 and by the problem of the quantization of the matter field in curved space-time [18].

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Quantum dissipation

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