Quantum Brownian Representation for the Quantum Field Modes

Table 1

Detail of the equivalent linear QBM system for a 𝝀𝝓^𝟒 quantum field theory. The 𝝀𝝓^𝟒 model has been chosen for concreteness, but the correspondence would be analogous for any other field theory model. The symbol (=) indicates that the original and equivalent quantities are indeed identical despite the name change.

	Original system	Equivalent linear QBM
System	Two field modes	Two identical oscillators
System d.o.f.	𝜙𝐩, 𝜙−𝐩	2 copies of 𝑞
Environment	Other modes 3-d field	2 1-d field
Env. d.o.f.	𝜙𝐪, 𝐪≠±𝐩	𝜑𝑝
Frequency	𝐸𝐩	Ω (=)
Coupling	(𝜆/3!)∑𝐪𝐪′𝜙±𝐩𝜙𝐪𝜙𝐪′𝜙∓𝐩−𝐪−𝐪′	𝑔^2∑𝑝ℐ(𝑝)𝑞𝜑̇𝑝
2-point function	𝐺𝑎𝑏(𝑡,𝑡^′;𝐩)	𝐺𝑎𝑏(𝑡,𝑡^′)(=)