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Modern Physics Letters AVol. 18, No. 10 (2003) 677–682c© World Scientific Publishing Company
NEW MINIMUM UNCERTAINTY STATES FOR
THERMO FIELD DYNAMICS
HONGYI FAN
Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China
Department of Material Science and Engineering, University of Science and
Technology of China, Hefei, Anhui 230026, China
Received 29 July 2002Revised 8 March 2003
In Takahashi–Umezawa Thermo Field Dynamics theory every operator a acting on realfield space is accompanied by its image a acting on fictitious field space, we propose anew uncertainty relation which embodies the finite temperature effect and derive thecorresponding minimum uncertainty state — a new thermolized two-variable Hermitepolynomial state.
Keywords: Thermo field dynamics; minimum uncertainty state.
PACS Nos.: 05.70.-a, 03.65.-w
Thermo field dynamics (TFD) was invented by Takahashi and Umezawa1–3 to
convert the calculations of ensemble averages at finite temperature into equivalent
pure state expectation calculations. In this formalism a density operator can be
represented by a pure state vector in the extended Hilbert space at the expense
of introducing a fictitious field (or a so-called tilde-conjugate field). Thus every
state |n〉 in the original Hilbert space H is accompanied by a tilde state |n〉 in H.
A similar rule holds for operators: every operator a acting on H has an image a
acting on H. At finite temperature T the thermal vacuum |0(β)〉, (β = 1kT , k is
the Boltzmann constant), is defined such that the vacuum expectation value agrees
with the statistical average, i.e.
〈0(β)|A|0(β)〉 =Tr(Ae−βH)
Tr(e−βH), (1)
where H is the system’s Hamiltonian. For the ensemble of free bosons with Hamil-
tonian H0 = ωa†a, the Takahashi–Umezawa thermal vacuum state |0(β)〉 is
677
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678 H. Fan
|0(β)〉 = S(θ)|0, 0〉 = sech θ exp[a†a† tanh θ]|0, 0〉 ,
tanh θ = exp
(
− ~ω
2kT
)
,
(2)
where the vacuum state |0, 0〉 is annihilated by either a or a, S(θ) is named the
thermal operator, since it changes the zero temperature vacuum to the thermal
vacuum
S(θ) = exp[θ(a†a† − aa)] , (3)
the parameter θ is determined by the number of thermalized photons, the latter is
given by the Bose–Einstein distribution, i.e.
sinh2 θ = 〈0(β)|a†a|0(β)〉 = n =
[
exp
(
~ω
kT
)
−1
]−1
or tanh θ = exp
(
− ~ω
2kT
)
.
(4)
Some quantum mechanical representations for thermal excitation and de-excitation
in TFD are introduced in Ref. 4. In TFD formalism a series of papers by Umezawa
group dealt with minimum uncertainty state in thermal situations5,6: Their outline
of the papers are summarized in the book7 which contains the essential features of
recent TFD.
In this paper we show that for the finite temperature case of TFD there exists
another new minimum uncertainty state which has been overlooked for a long time.
That is, since every operator a acting on H is accompanied by a acting on H, for
examining the Heisenberg uncertainty relation in TFD theory we should consider
not only Q = a+a†, P = a−a† as the two quadratures, but also the following pair
of Hermitian observables
Y1 ≡ 1
2(a†a† + aa) , Y2 ≡ i
2(a†a† − aa) (5)
as a pair of new quadratures, where each Bose operator in real field space is accom-
panied by its image in fictitious field space. From the commutative relation
Y3 ≡ [Y1, Y2] =i
2(a†a+ a†a+ 1) (6)
we know that the corresponding uncertainty relation is
∆Y1∆Y2 ≥ 1
4〈a†a+ a†a+ 1〉 , (7)
where (∆Y1)2 = 〈Y 2
1 〉 − 〈Y1〉2. We now search for a state, denoted as |ψ〉λ, which
makes this uncertainty relation minimum. According to the theory of determining
the minimum uncertain state in quantum mechanics, we set up the equation
(Y1 + iλY2)|ψ〉λ = β|ψ〉λ , (8)
where β is a complex number, λ is a positive number. By introducing the generalized
thermal operator
S(ξ) = exp(ξ∗aa− ξa†a†) , ξ = reiθ , (9)
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New Minimum Uncertainty States for Thermo Field Dynamics 679
and the state
|ψ′〉 = S(ξ)|ψ〉λ , (10)
we can derive the equation obeyed by |ψ′〉{
1
2sinh 2r
[
1
2(1 − λ)e−iθ +
1
2(1 + λ)eiθ
]
(a†a+ a†a+ 1)
+
[
1
2(1 − λ) cosh2 r +
1
2(1 + λ)e2iθ sinh2 r
]
a†a†
+
[
1
2(1 + λ) cosh2 r +
1
2(1 − λ)e−2iθ sinh2 r
]
aa
}
|ψ′〉 = β|ψ′〉 . (11)
Taking
tanh2 r = e−2iθ(λ − 1)/(λ+ 1) , (12)
which enforces the coefficient of a†a† in Eq. (11) to vanish, and letting
θ =π
2, cosh r =
(
1 + λ
2λ
)1/2
, sinh r =
(
1− λ
2λ
)1/2
, for 0 < λ < 1 , (13)
θ = 0 , cosh r =
(
1 + λ
2
)1/2
, sinh r =
(
λ− 1
2
)1/2
, for λ ≥ 1 , (14)
respectively, then Eq. (11) reduces to[
i
2(1 − λ2)1/2(a†a+ a†a+ 1) + aa
]
|ψ′〉 = β|ψ′〉 , for 0 < λ < 1 , (15)
and
λaa+1
2(λ2 − 1)1/2(a†a+ a†a+ 1)|ψ′〉 = β|ψ′〉 , for λ ≥ 1 , (16)
respectively. Expanding |ψ′〉 in terms of two-mode Fock states
|ψ′〉 =
∞∑
n,m=0
Cn,m|n, m〉 , (17)
and then substituting (17) into (15) and (16) respectively yields the recursive
relation
Cn+1,m+1 =2β − i
√1 − λ2(n+m+ 1)
2[(n+ 1)(m+ 1)]1/2Cn,m , for 0 < λ < 1 (18)
and
Cn+1,m+1 =2β −
√λ2 − 1(n+m+ 1)
2λ[(n+ 1)(m+ 1)]1/2Cn,m , for λ ≥ 1 . (19)
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680 H. Fan
Now we are restricted to taking some special values of β so that the series of Cn,m
can be cutoff. For 0 < λ < 1 case we set β = i2
√1− λ2(M + 1), where M is non-
negative, from (18) we see that |ψ′〉 is a sum of terms either starting from C0,k|0, k〉,or a sum of terms starting from Ck,0|k, 0〉. The former is denoted as |ψ′
1(k, λ,M)〉,
|ψ′1(k, λ,M)〉 = C0,k|0, k〉 + C0,k|1, 1 + k〉 t(M − k)
2[k + 1]1/2
+C0,k|2, 2 + k〉 t2(M − k)(M − k − 2)
22[2(k + 1)(k + 2)]1/2+ · · ·
+C0,k|l, l+ k〉 tl(M − k)(M − k − 2) · · · (M − k − 2l)
2l[1 · 2 · 3 · · · l(k + 1)(k + 2) · · · (k + l)]1/2+ · · ·
= C0,k
√k!
h∑
l=0
tlh!
l!(l + k)!(h− l)!a†la†l+k|0, 0〉
≡h
∑
l=0
Cl,l+k |l, l+ k〉 , (20)
where
h =1
2(M − k) , t = i
√
1 − λ2 . (21)
The latter is denoted as |ψ′2(k, λ,M)〉
|ψ′2(k, λ,M)〉 =
h∑
l=0
Cl+k,l|l + k, l〉 . (22)
By introducing a new pair of parameters p and q such that
p+ q = M , max(p, q) − min(p, q) = k , (23)
we can compact (20)–(22) as a unified expression
|ψ′(p, q, λ)〉 = N√
|p− q|!{max(p, q)}!
min(p,q)∑
v=0
q!p!tmin(p,q)−v
v!(p− v)!(q − v)!a†p−v a†q−v |0, 0〉 , (24)
where N = C0,k for p ≤ q; N = Ck,0 for p > q. Examine the Heisenberg uncertainty
relation for TFD theory in a way which automatically includes the temperature
effect. Recalling the definition of the two-variable Hermite polynomials8
Hm,n(α, β) =
min(m,n)∑
l=0
(−1)l n!m!
l!(n− l)!(m− l)!αm−lβn−l , (25)
we obtain
|ψ′(p, q, 0 < λ < 1)〉 = Cp,q(λ)Hp,q(µa†, µa†)|0, 0〉 , (26)
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New Minimum Uncertainty States for Thermo Field Dynamics 681
where
µ2 = −t = −i√
1 − λ2 , (27)
Cp,q(λ) = N√
|p− q|!{max(p, q)}! (−1)min(p,q)|µ||p−q| . (28)
For λ ≥ 1 case, from (19) we select 2β =√λ2 − 1(M + 1), taking the similar
procedures as for deriving (26) we can deduce
|ψ′(p, q, λ ≥ 1)〉 = Cp,q(λ)Hp,q(µa†, µa†)|0, 0〉 , (29)
where
µ2 = −√λ2 − 1
λ. (30)
Combining (26) and (29), we obtain a unified expression for (10),
|ψ〉λ = S−1(ξ)|ψ′〉 = Cp,q(λ)S−1(ξ)Hp,q(µa
†, µa†)|0, 0〉 , (31)
where Cp,q(λ) can be determined by λ〈ψ|ψ〉λ = 1 and the generating function
formula of Hp,q
∑
m,n=0
sms′n
m!n!Hm,n(α, α∗) = exp(−ss′ + sα+ s′α∗) , (32)
the result is
Cp,q(λ) =
min(p,q)∑
l=0
(
p
l
) (
q
l
)
p!q!|µ|2(p+q−2l)
−1/2
. (33)
Equation (32) is a thermal two-variable Hermite polynomial state, since S−1(ξ) is
a generalized thermalizing operator. It is remarkable that both µa† and µa† appear
in the argument of Hp,q , which again exhibit that every operator a acting on real
field space is accompanied by its image a acting on fictitious field space.
When one thinks of that in TFD the expectation values we need is ones,
sandwiched by two states which are invariant under the tilde transformation,
then in (26), (29) and (31) we only take those states with p = q, i.e. a subset
Hp,p(µa†, µa†)|0, 0〉, meaning invariant under the exchange between a† and a†.
In summary, by taking nonzero temperature effect into account, we have pro-
posed a pair of new quadratures, Y1 = 12 (a†a†+aa), Y2 = i
2 (a†a†−aa) for examining
the corresponding Heisenberg uncertainty relation, and we have derived the mini-
mum uncertainty state, which turns out to be a thermalized two-variable Hermite
polynomial state.
Acknowledgment
The author hopes to take this opportunity to thank Prof. Umezawa for the hospi-
tality extended to him during his stay at University of Alberta in Canada in 1987.
This work was supported by the National Natural Science Foundation of China
under Grant 10175057 and the President of Chinese Academy of Science.
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682 H. Fan
References
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