By N N Bogolubov, Nickolai N Bogolubov Jr

The linear polaron version is a wonderful instance of an precisely soluble, but nontrivial polaron process. It serves as an ordeal procedure or zero-level approximation in lots of refined equipment of polaron research. This publication analyzes, specifically, the opportunity of aid of the complete polaron Hamiltonian to the linear one, and introduces a different approach to calculating thermodynamical features in accordance with the calculation of the averages of T-products. This T-product formalism appears a more straightforward means of doing comparable calculations related to Feynman's course quintessential strategy.

This ebook follows a step by step strategy, from relatively basic actual rules to a transparent knowing of refined mathematical instruments of research in glossy polaron physics. The reader is ready to examine the actual standpoint with equipment proposed within the ebook, and while clutch the underlying arithmetic.

a few familiarity with quantum statistical mechanics is fascinating in examining this ebook.

**Contents: Linear Polaron version; Equilibrium Thermodynamic country of Polaron procedure; Kinetic Equations in Polaron idea.
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Sn . 6. Averaged Operator T-Product Calculus 47 It is interesting to note that, thanks to the definition of the T-product, the operators A(sj ) commute under the sign of the T-product. For example, T {A(s1 )A(s2 )} = T {A(s2 )A(s1 )}. 79)) and A(s) are linear forms composed of these Bose operators with coefficients dependent on the ordering parameter s. s1 T e ∞ ∫ ds A(s) s0 Γ = n s1 T ∫ ds A(s) . s0 n=0 Γ Keeping in mind that the Bloch–Dominicis theorem can be applied not only to the ordinary products but also to the T-products, we may repeat our previous reasoning and write down the final result at once: s1 T e ∫ ds A(s) s0 Γ = exp 2 s1 1 T 2 ∫ ds A(s) .

DRj . 128) for arbitrary real vectors R(s). These relations can be generalized for a broader set of functionals F (R) if one approximates these functionals by corresponding sequences of the above-mentioned “special functionals” with subsequent passage to the limit N → ∞. 129) 58 Ch. 1. Linear Polaron Model and approximate the functional F (R) in question by the form F (RN ), belonging obviously to the class of “special functionals”. We should like to stress here that the technique of functional integration was developed first by R.

On this contour, F (Ω) = O 1 L2 so we can see that ∫ F (Ω) dΩ = O C Therefore iε+∞ ∫ iε−∞ 1 L , → 0. F (Ω) dΩ = 0. The same considerations may be applied to prove that −iε+∞ ∫ −iε−∞ F (Ω) dΩ. In fact, F (Ω) — is a regular analytic function on the lower half-plane Im Ω −ε < 0. Therefore it is sufficient to choose the proper contour (see Fig. 3) and to repeat all the previous reasoning. 64) into account, we have Fint = 0. It should be stressed that this result follows entirely from the treatment of the dynamical system within the framework of classical mechanics.