Quantum theory of a strongly-dissipative scalar field

0.11 MB·

by  Marjan Jafari

Keywords: Scalar field; Diagonalization; Response functions; Correlation functions; Dissipa-
tion
Quantum theory of a strongly-dissipative scalar field
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Marjan Jafari and Fardin Kheirandish
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1Department of Physics, Faculty of Science,
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Imam Khomeini International University, 34148 - 96818, Ghazvin, Iran

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a 2Department of Physics, University of Kurdistan,
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6 P.O.Box 66177-15175, Sanandaj, Iran

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h Abstract
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t The properties of a quantum dissipative scalar field is analyzed by Caldeira-Leggett model in
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strong-coupling regime. The Lagrangian of the total system is canonically quantized and the
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[ full Hamiltonian is diagonalized using Fano technique. A mode-dependent probability density is

1 introduced. The steady state energy and correlation functions at finite temperature are calculated
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in terms of the probability density.
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PACS numbers: 05.40.Jc;05.40.-a;05.30.-d
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Keywords:
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I. INTRODUCTION
Dissipative scalar field theories have an important place in quantum field theory and appear
in various problems in physics. Even in quantum mechanics, introducing dissipation in the
Hamiltonianofaquantummechanical system, withoutinvoking toareservoir oraheatbath,
is not a straightforward task and usually leads to inconsistencies in the formulation such as
violation of Heisenberg uncertainty principle [1–5]. There are two important approaches to
take into account dissipation in quantum theory, namely phenomenological and canonical
approach. In these approaches, the interaction between the system and its environment
is defined such that an irreversible energy flow from system to its environment results [6–
8]. In the canonical approach, which we follow here, the whole system is described by a
total Lagrangian. That is, the environment or reservoir is modeled by a collection of non-
interacting harmonic oscillators with a continuum of frequencies. In this model, known
as Caldeira-Leggett model, each oscillator in the reservoir is coupled to the main system
through appropriate coupling functions [9–12]. This model has been applied to a variety
of dissipative quantum systems [13]. In weak-coupling regime, one can apply Markovian
approximation [14] but in strong-coupling regime, non-Markovianity of the system should
be taken into account [15]. The quantum theory of a damped harmonic oscillator in strong
coupling regime has been investigated in [16, 17].
In the present paper, following Cladeira-Leggett model, the environment or background
of a scalar field is modeled by a continuum of harmonic oscillators and the quantum theory
of the scalar field in the presence of such a medium is investigated in strong-coupling regime.
WestartfromatotalLagrangianandquantizethetotalsystemintheframeworkofcanonical
quantization. TheHamiltonianofthetotalsystemisdiagonalizedusingFanodiagonalization
technique [18]. The main ingredient in this investigation is the appearance of a mode
dependent probability density function describing the statistical properties of the system in
its ground state. The quantum dynamics, energy and the correlation functions of dynamical
variables related to the scalar field, are obtained in terms of the probability density function.
The layout of the paper is as follows: In Sec. 2, a classical Lagrangian is proposed for
the total system and using Euler-Lagrange equations, the equations of motion for the scalar
field and the reservoir are obtained. In Sec. 3, the system is canonically quantized and
Hamiltonian is diagonalized using Fano diagonalization technique. In Sec. 4, the energy of
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the dissipative scalar field in ground state is obtained. In Sec. 5, the quantum dynamics of
a dissipative scalar field is investigated and the correlation functions of dynamical variables
are obtained in a thermal state.
II. CLASSICAL DYNAMICS
Consider a real massive scalar field that is interacted with an environment defined by real
scalar fields X (x,t). We assume that the scalar field is defined in 1+1-dimensional space-
ω
time and use natural units. The Lagrangian density for the field-environment system is
(t) = + + . (1)
s R int
L L L L
The first term is the Lagrangian of a massive scalar field or main system
s
L
1 1
= ∂ φ(x,t)∂µφ(x,t) m2φ2(x,t), (2)
s µ
L 2 − 2
the second term, is the Lagrangian density of a reservoir or heat bath consisting of a con-
tinuum of harmonic oscillators with displacements X and frequencies ω [0, )
ω
∈ ∞
∞
1
= dω[X˙2(x,t) ω2X2(x,t)], (3)
LR 2 Z ω − ω
0
and the last term is the interaction between the scalar field and its reservoir that is assumed
to be linear and given by
∞
= dωf(ω)φ(x,t)X (x,t), (4)
int ω
L Z
0
where f(ω) is the coupling function between the scalar field and reservoir. For simplicity we
work in the reciprocal space and write the fields in terms of their spatial Fourier transforms
∞
dk
φ(x,t) = eikxφ(k,t)+e−ikxφ∗(k,t) , (5)
Z 2π
(cid:2) (cid:3)
0
∞
dk
X (x,t) = eikxX (k,t)+e−ikxX∗(k,t) . (6)
ω Z 2π ω ω
(cid:2) (cid:3)
0
where we have used the reality of the fields and restricted the range of the variable k to the
half space [0, ). In the reciprocal half space the Lagrangian components can be expressed
∞
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as
∞
dk
˙ 2 2 2
L = φ(k,t) ω φ(k,t) , (7)
s Z π (cid:16)| | − k| | (cid:17)
0
2 2 2
where ω = (m +k ).
k
∞
dk
L = X˙ (k,t) 2 ω2 X (k,t) 2 , (8)
R Z π | ω | − | ω |
(cid:16) (cid:17)
0
∞ ∞
dk
L = dωf(ω)(φ(k,t)X∗(k,t)+φ∗(k,t)X (k,t)), (9)
int Z π Z ω ω
0 0
where φ∗(k,t) = φ( k,t) and X∗(k,t) = X ( k,t). From Euler-Lagrange equations, the
− ω ω −
classical equations of motion for φ(k,t) and X (k,t) are
ω
∞
2 2
(∂ +ω )φ(k,t) = dωf(ω)X (k,t), (10)
t k ω
Z
0
2 2
(∂ +ω )X (k,t) = f(ω)φ(k,t). (11)
t ω
III. CANONICAL QUANTIZATION
From the Lagrangian density with measure dk/π
L
∞
= φ˙(k,t)φ˙∗(k,t)+ dωX˙ (k,t)X˙∗(k,t),
L Z ω ω
0
∞
ω2φ˙(k,t)φ˙∗(k,t) dωω2X˙ (k,t)X˙∗(k,t)
− k −Z ω ω
0
∞
+ dω[φ(k,t)X∗(k,t)+φ∗(k,t)X (k,t)], (12)
ω ω
Z
0
the conjugate momenta corresponding to the fields φ and X within the half k-space are
ω
defined by
∂
˙
π(k,t) = L = φ(k,t), (13)
∂φ˙∗
∂
Π (k,t) = L = X˙ (k,t). (14)
ω ∂X˙∗ ω
ω
To quantize the theory canonically, the following equal-time commutation relations are be
imposed on the fields and their conjugate momenta
[φˆ†(k,t),πˆ(k′,t)] = iδ(k k′), (15)
−
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[Xˆω†(k,t),Πˆω′(k′,t)] = iδ(k −k′)δ(ω −ω′), (16)
with all other equal-time commutators being zero. Using the canonical momenta and the
Lagrangian density, we find the Hamiltonian density of the field-environment system as
follows
∞
= πˆ(k,t) 2 +ω2 φˆ(k,t) 2 + dω Πˆ (k,t) 2 +ω2 Xˆ (k,t) 2
H h| | k| | i Z (cid:16)| ω | | ω | (cid:17)
0
∞
dωf(ω)[φˆ†(k,t)Xˆ (k,t)+φˆ(k,t)Xˆ†(k,t)]. (17)
− Z ω ω
0
We can rewrite the Hamiltonian density in a minimal-coupling form
2 2 ˆ 2
= πˆ(k,t) +Ω φ(k,t)
H | | k| |
h i
∞
f(ω)
+ dω Πˆ (k,t) 2 +ω2 Xˆ (k,t) φˆ(k,t) 2 , (18)
Z (cid:20)| ω | | ω − ω2 | (cid:21)
0
where
∞
2
f (ω)
2 2
Ω = ω dω . (19)
k k −Z ω2
0
The Hamiltonian is positive only if this quantity is positive. The positivity of the Hamilto-
nian induce a physical restriction on the strength of the coupling
∞
2
f (ω)
2
ω > dω . (20)
k Z ω2
0
To facilitate the calculations, let us introduce the following annihilation operators
ω i
k ˆ
aˆ (t) = φ(k,t)+ πˆ(k,t) , (21)
k
r 2 (cid:20) ω (cid:21)
k
ω i
aˆ†(t) = k φˆ†(k,t) πˆ†(k,t) , (22)
k r 2 (cid:20) − ω (cid:21)
k
ω i
ˆ ˆ ˆ
b (ω,t) = X (k,t)+ Π (k,t) , (23)
k ω ω
r2 (cid:20) ω (cid:21)
ω i
ˆb†(ω,t) = Xˆ†(k,t) Πˆ†(k,t) , (24)
k r2 (cid:20) ω − ω ω (cid:21)
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satisfying bosonic equal-time commutation relations
[aˆk(t),aˆk′(t)] = [aˆ†k(t),aˆ†k′(t)] = 0,
[aˆ (t),aˆ† (t)] = δ(k k′), (25)
k k′ −
[ˆbk(ω,t),ˆbk′(ω′,t)] = [ˆb†k(ω,t),ˆb†k′(ω′,t)] = 0,
[ˆb (ω,t),ˆb† (ω′,t)] = δ(k k′)δ(ω ω′). (26)
k k′ − −
The commutation relations (25,26) in contrast to the previous relations (15,16), which were
correct only in the half k-space, are now valid in the whole reciprocal space. Using the
equations (21-24), we can write the canonical variables φ and X in terms of the creation
ω
and annihilation operators as
1
φˆ(k,t) = aˆ (t)+aˆ† (t) , (27)
√2ω k −k
k h i
ω
πˆ(k,t) = i k aˆ (t) aˆ† (t) , (28)
− r 2 k − −k
h i
1
Xˆ (k,t) = ˆb (ω,t)+ˆb† (ω,t) , (29)
ω √2ω h k −k i
ω
Πˆ (k,t) = i ˆb (ω,t) ˆb† (ω,t) . (30)
ω − r2 k − −k
h i
Hamiltonian in terms of creation and annihilation operators is
∞ ∞ ∞
dk dk
† ˆ† ˆ
H = ω aˆ (t)aˆ (t)+ dωωb (ω,t)b (ω,t)
Z π k k k Z π Z k k
−∞ −∞ 0
∞
dk ∞
dωv(ω,k)[a†(t)+aˆ (t)][ˆb† (ω,t)+ˆb (ω,t)], (31)
− Z π Z0 k −k −k k
−∞
where the coupling is given by v(ω,k) = f(ω)/2√ωω .
k
A. Hamiltonian diagonalization
Hamiltonian can be diagonalized by introducing a complete set of eigenoperators Cˆ (ω,t)
k
and Cˆ†(ω,t), obeying bosonic commutation relation
k
Cˆ (ω,t),Cˆ†(ω′,t) = δ(ω ω′)δ(k k′). (32)
k k′ − −
h i
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These operators satisfy the equations
Cˆ (ω,t),Hˆ = ωCˆ (ω,t), (33)
k k
h i
Cˆ†(ω,t),Hˆ = ωCˆ†(ω,t), (34)
k − k
h i
∞ ∞
dk
Hˆ = dωωCˆ†(ω,t)Cˆ (ω,t). (35)
Z π Z k k
−∞ 0
From now on, for notational convenience, we do not write the explicit time dependence of
operators. The eigen-operators can be written as a superposition of a complete set of scalar
field and reservoir operators
ˆ †
C (ω) = α (ω)aˆ +β (ω)aˆ
k k k k −k
∞
+ dω′ γ (ω,ω′)ˆb (ω′)+δ (ω,ω′)ˆb† (ω′) , (36)
Z h k k k −k i
0
by substituting (36) in (33) we find the coefficients as
ω ω
k
β (ω) = − α (ω), (37)
k k
ω +ω
k
1 2ω
δ (ω,ω′) = v(ω′,k) − k α (ω) (38)
k (cid:18)ω +ω′(cid:19) ω +ω k
k
and
P 2ω
γ (ω,ω′) = +Y (ω)δ(ω ω′) v(ω′,k) − k α (ω), (39)
k (cid:18)ω ω′ k − (cid:19) ω +ω k
k
−
where
∞
2 2
1 (ω ω ) P 1
Y (ω) =  k − dω′ v2(ω′,k) (40)
k v2(ω,k) 2ω −Z (cid:18)ω ω′ − ω +ω′(cid:19)
k
0 −
 
If we insert Cˆ (ω), expressed in terms of α (ω), into the commutation relation (32), we find
k k
2
(ω +ω ) 1
2 k
α (ω) = . (41)
| k | 4ω2v2(ω,k) (cid:18)Y2(ω)+π2(cid:19)
k k
The field operators in terms of eigen-operators are as follows
∞
aˆ = dω α∗(ω)Cˆ (ω) β (ω)Cˆ† (ω) (42)
k Z (cid:16) k k − k −k (cid:17)
0
∞
aˆ† = dω α (ω)Cˆ†(ω) β∗(ω)Cˆ (ω) (43)
k Z (cid:16) k k − k −k (cid:17)
0
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The requirement that these operators satisfy the familiar boson commutation relation,
[aˆ ,aˆ† ] = δ(k k′), leads to
k k′ −
∞ ∞
4ω ω
2 2 2 k
dω α (ω) β (ω) = dω α (ω) = 1. (44)
Z | k | −| k | Z | k | (ω +ω )2
(cid:2) (cid:3) k
0 0
The integrand in (44) is positive for all frequencies and hence has the mathematical form of
a frequency probability distribution
4ω ω
2 k
P (ω) = α (ω) . (45)
k | k | (ω +ω )2
k
So having this distribution, we can find the average value of an arbitrary function of fre-
quency
∞
f(ω) = dωf(ω)P (ω). (46)
hh iik Z k
0
IV. GROUND-STATE
The ground-state 0 of the eigenoperators Cˆ (ω) is defined by
k
| i
Cˆ (ω) 0 = 0, (47)
k
| i
the ground state of the scalar field can be obtained by tracing out the environmental degree
of freedom
ρˆ = tr ( 0 0 ). (48)
s R
| ih |
Following [19], we use the characteristic function to determine the form of the mixed state
χ(k,η) = tr[ρ exp(ηaˆ† η∗aˆ )] = 0 exp(ηaˆ† η∗aˆ ) 0 . (49)
−k − k h | −k − k | i
†
By inserting aˆ and aˆ in terms of the eigenoperators into (49) we find
k k
∞
1
χ(k,η) = exp dω ηα (ω)+η∗β (ω) 2, (50)
k k
−2 Z | |
0
 
which can be rewritten the charactristic function in terms of probability density
∞
1 ω ω
2 k 2
χ(k,η) = exp P (ω) η + η , (51)
−2 Z k (cid:18)ω r ω i(cid:19)
k
0
 
1 ω
= exp hh iikη2 + ω−1 ω η2 , (52)
(cid:20)−2 (cid:18) ω r hh iik k i(cid:19)(cid:21)
k
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where η and η are imaginary and real part of η. The following relations can be easily
i r
obtained
ˆ
0 φ(k,0) 0 = 0, (53)
h | | i
0 πˆ(k,0) 0 = 0, (54)
h | | i
0 φˆ2(k,0) 0 = ω−1 /2, (55)
h | | i hh iik
2
0 πˆ (k,0) 0 = ω /2. (56)
h | | i hh iik
The mean energy of the scalar field for k-mode is
1 ω ω ω
π2(k,0)+(k2 +m2)φ2(k,0) = k hh iik +ω ω−1 > k, (57)
2h i 4 (cid:18) ω khh iik(cid:19) 2
k
which is more than the ground state energy of the free mode due to the amount of energy
to be paid to decouple the scalar field from bath [20]. The mean-square value of the scalar
field in space-time is given by
1 ω 1
0 φˆ2(x,t) 0 = dk hh iik cos2(kx) ω−1 . (58)
h | | i Z 2π2ω (cid:20) 2ω − (cid:18)ω −hh iik(cid:19)(cid:21)
k k k
V. SCALAR FIELD DYNAMICS
From the diagonalised Hamiltonian it is straightforward to evaluate the time-evolution
of the scalar field. The time-evolution of eigenoperators are
Cˆ (ω,t) = Cˆ (ω,0)e−iωt, (59)
k k
Cˆ†(ω,t) = Cˆ†(ω,0)eiωt. (60)
k k
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Using these expressions, we find the time-evolution of the k-mode annihilation operator of
the scalar field
∞
aˆ (t) = dω[α∗(ω)Cˆ (ω,0)e−iωt β (ω)Cˆ† (ω,0)eiωt],
k Z k k − k −k
0
∞
= dωα∗(ω)e−iωt α (ω)aˆ (0)+β (ω)aˆ† (0)
Z k { k k k −k
0
∞
+ dω′(γ (ω,ω′)ˆb (ω′,0)+δ (ω,ω′)ˆb† (ω′,0))
Z k k k −k }
0
∞
dωβ (ω)eiωt α∗(ω)aˆ†(0)+β∗(ω)aˆ (0)
− Z k { k k k −k
0
∞
+ dω′(γ∗(ω,ω′)ˆb†(ω′,0)+δ (ω,ω′)ˆb (ω′,0)) . (61)
Z k k k −k }
0
When the environment is a stationary state, we have ˆb (ω,0) = ˆb†(ω,0) = 0, and the
h k i h k i
expectation values of the position and momentum operators are
1
φˆ(x,t) = cos(ωt) φˆ(x,0) + ω−1sin(ωt) πˆ(x,0) , (62)
k k
h i hh ii h i 2ω hh ii h i
k
ω
πˆ(x,t) = cos(ωt) πˆ(x,0) k ω−1sin(ωt) φˆ(x,0) . (63)
k k
h i hh ii h i− 2 hh ii h i
In thermal equilibrium, we have
hCˆk†(ω,0)Cˆk′(ω′,0)i = N(ω)δ(k −k′)δ(ω −ω′), (64)
where
1
N(ω) = . (65)
exp(~ω/k T) 1
B
−
The expectation value of k-mode number operator of the scalar field in a thermal state is
1 ω N(ω) ω ω 1 1
† k k k
aˆ (t)aˆ (t) = ωN(ω) + + hh ii + , (66)
h k k ith 2ω hh iik 2 hh ω iik 4ω 4 ω − 2
k k k
hh ii
which consists of zero and finite temperature contributions.
VI. CONCLUSION
The quantum theory of a dissipative scalar field in strong-coupling regime and in the
framework of canonical quantization investigated. The Hamiltonian of the system deter-
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