March 20, 2003 9:48 WSPC/146-MPLA 00796

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

fhym@mail.sjtu.edu.cn

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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March 20, 2003 9:48 WSPC/146-MPLA 00796

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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March 20, 2003 9:48 WSPC/146-MPLA 00796

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

1. Memorial issue for H. Umezawa, Int. J. Mod. Phys. B10 (1996) and references therein.2. H. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dynamics and Condensed

States (North-Holland, 1982).3. Y. Takahashi and H. Umezawa, Collective Phenomena 2, 55 (1975).4. H.-Y. Fan and Yue Fan, Phys. Lett. A246, 242 (1998); A282, 269 (2001).5. A. Mann, M. Revzen, H. Umezawa and Y. Yamanaka, Phys. Lett. A140, 475 (1989);

A. Mann, M. Revzen, K. Nakamura, H. Umezawa and Y. Yamanaka, J. Math. Phys.

30, 2883 (1989).6. A. Mann and M. Revzen, Phys. Lett. A134, 273 (1989); A134, 273; 531 (1989).7. H. Umezawa, Micro, Macro, and Thermal Physics (AIP, 1993).8. Bateman Manuscript Project, Higher Transcendental Functions, ed. A. Erdelyi

(McGraw-Hill, 1953).

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