Chapter 24
Quantum Fields at Non-zero
Temperature
The general structure discussed above provides a neat and unique prescription for
the quantization of relativistic fields at non-zero temperature (thermofield theory).
For simplicity, we consider the case of a relativistic scalar field (see Example 12.1)
φ(x) = (2π)
−3/2
d
3 k (2ω k )
−1
[a k e
−ikx
+ a
∗
k e
ikx
],
(24.1)
where kx = k 0 x 0 − k · x, k 0 = ω k = (k
2
+ m
2
)
1/2 and the annihilation and creation
operators have been so normalized that they obey the canonical commutation relations (CCR) in relativistically covariant form
[a k , a
∗
k ] = 2ω k δ(k − k
), [a k , a k ] = 0.
(24.2)
This fixes the algebraic structure.
The equilibrium (gauge invariant) state at inverse temperature β is characterized
by the KMS condition, (22.4),
< a
∗
k a q > β ≡ ω β (a
∗
k a q ) = ω β (a q α −iβ (a
∗
k )) = e
−β ω k < a q a
∗
k > β ,
where for simplicity we have considered the case of zero chemical potential.
On the other hand, the CCR give
< a
∗
k a q > β =< a q a
∗
k > β −2ω k δ(k − q).
In conclusion, one has
< a
∗
k a q >= 2ω k N (ω k ) δ(k − q),
(24.3)
N (ω k ) ≡ e
−β ω k (1 − e
−β ω k )
−1
= (e
β ω k − 1)
−1
.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_24
167