168
24 Quantum Fields at Non-zero Temperature
Then, the CCR and the above equation give for the two-point function of φ, putting
z ≡ x − y,
< φ(x) φ(y) > β =
1
(2π) 3
d
3 k
(2ω k )
e
−ik·z
[e
iω k z 0 N (ω k ) + e
−iω k z 0 (1 + N (ω k ))].
By using the identity
N (ω) = −(1 + N (−ω)),
one can cast the above two-point function in the form
143
< φ(x) φ(y) > β = (2π)
−4
d
4 k δ(k
2
− m
2
)e
−ikz
ε(k 0 ) (1 + N (k 0 )).
(24.4)
The relativistic spectral condition yields a larger analyticity domain than in the nonrelativistic case; in fact, the two-point function has an analytic continuation to the
domain {z ∈ C
4
; Imz ∈ V + ∩ (β, 0) + V − } where V ± denote the forward and backward cones (relativistic KMS condition).
144
Historically, the quantization of fields at non-zero temperature has been done with
different strategies, based on the functional integral approach. The same results can
be obtained more directly by exploiting the KMS condition. For example, the socalled imaginary time (Matsubara) formulation can be obtained if one i) analytically
continues the correlation functions to purely imaginary time and ii) introduces a
complex time ordering of products of operators with respect to a fixed complex time
contour.
As an example we consider the two-point function < A A z >≡ Ω(A A z ), where A
denotes a field variable at time zero and A z , z = iτ , −β < τ < β, the corresponding
variable after an imaginary time translation (as in (22.5)). A (complex) time ordered
expectation is defined by
Δ
T
(τ ) = θ(τ ) Tr (e
−β H A A z ) + θ(−τ ) Tr(e
−β H A z A),
(24.5)
where θ denotes the Heaviside step function and −β < Imz = τ < β. Thus, the
KMS condition (22.4) gives
Δ
T
(τ + β) = Δ
T
(τ ).
The periodicity implies that only discrete frequencies occur in the Fourier transform
of Δ
T .
143 J. Bros and D. Buchholz, Z. Phys. C-Particles and Fields 55, 509 (1992); Nucl. Phys. B429, 291
(1994).
144 J. Bros and D. Buchholz, loc. cit., previous footnote.
24 Quantum Fields at Non-zero Temperature
Then, the CCR and the above equation give for the two-point function of φ, putting
z ≡ x − y,
< φ(x) φ(y) > β =
1
(2π) 3
d
3 k
(2ω k )
e
−ik·z
[e
iω k z 0 N (ω k ) + e
−iω k z 0 (1 + N (ω k ))].
By using the identity
N (ω) = −(1 + N (−ω)),
one can cast the above two-point function in the form
143
< φ(x) φ(y) > β = (2π)
−4
d
4 k δ(k
2
− m
2
)e
−ikz
ε(k 0 ) (1 + N (k 0 )).
(24.4)
The relativistic spectral condition yields a larger analyticity domain than in the nonrelativistic case; in fact, the two-point function has an analytic continuation to the
domain {z ∈ C
4
; Imz ∈ V + ∩ (β, 0) + V − } where V ± denote the forward and backward cones (relativistic KMS condition).
144
Historically, the quantization of fields at non-zero temperature has been done with
different strategies, based on the functional integral approach. The same results can
be obtained more directly by exploiting the KMS condition. For example, the socalled imaginary time (Matsubara) formulation can be obtained if one i) analytically
continues the correlation functions to purely imaginary time and ii) introduces a
complex time ordering of products of operators with respect to a fixed complex time
contour.
As an example we consider the two-point function < A A z >≡ Ω(A A z ), where A
denotes a field variable at time zero and A z , z = iτ , −β < τ < β, the corresponding
variable after an imaginary time translation (as in (22.5)). A (complex) time ordered
expectation is defined by
Δ
T
(τ ) = θ(τ ) Tr (e
−β H A A z ) + θ(−τ ) Tr(e
−β H A z A),
(24.5)
where θ denotes the Heaviside step function and −β < Imz = τ < β. Thus, the
KMS condition (22.4) gives
Δ
T
(τ + β) = Δ
T
(τ ).
The periodicity implies that only discrete frequencies occur in the Fourier transform
of Δ
T .
143 J. Bros and D. Buchholz, Z. Phys. C-Particles and Fields 55, 509 (1992); Nucl. Phys. B429, 291
(1994).
144 J. Bros and D. Buchholz, loc. cit., previous footnote.