By N.N. Bogoliubov

On the finish of 1960 we made plans to write down a monograph in regards to the common rules of quantum box idea and their experimental implications. We meant essentially to offer an account of the development of the speculation of dispersion relatives because the visual appeal of the booklet of Bogolubov, Medvedev and Polivanov ([BMP]. As an advent we would have liked to incorporate a assessment of a number of the ways to axiomatic box idea. This advent needed to disguise not just the formula of Bogolubov, Medvedev and Polivanov, according to the equipment of sensible derivatives of the 5-matrix and the of microcausality, but in addition the sector formula linked to the names of Wightman, Haag, Lehmann, Symanzik, Zimmermann, and others. through the paintings the initiatives (and with them the scale) of the advent grew higher and bigger, until it built into this booklet.

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**Sample text**

The summation over J may also mean integration if some (or all) of the quantum numbers are continuous. The annihilation and creation operators satisfy the commutation relations a J , a+ = δJJ , J + [aJ , aJ ] = a+ = 0. 53) The vacuum state of the ﬁeld is deﬁned by aJ |0 = 0. 54) The states with particles are obtained by applying the creation operators to the vacuum state. For example, the state with one particle is Canonical quantization and the vacuum energy as a mode expansion |1 = a+ J |0 . e.

These are similar to boundary conditions for classical material boundaries. The simplest example is provided by the interval 0 ≤ x ≤ a whose initial and end points are identiﬁed by means of the following periodic conditions: ϕ(t, 0) = ϕ(t, a), ∂x ϕ(t, x)|x=0 = ∂x ϕ(t, x)|x=a . 42) is a circle of circumference a. Both manifolds are ﬂat, but their topologies are diﬀerent. Interval I [Fig. 42) possesses the topology of a circle S 1 [Fig. 1(b)]. 2). 3), new solutions are allowed, such that ϕ = 0 at the points x = 0, a.

5) in this chapter, which is devoted to simple models. A conﬁguration of two parallel planes of very large area S spaced a distance a apart is shown in Fig. 2. Mathematically, the area S of each plane is supposed to be inﬁnitely large. However, the results obtained are applicable for the condition √ a S. From classical electrodynamics, the electric ﬁeld and the magnetic induction, of both polarizations, satisfy the following boundary conditions on the surface of an ideal metal: E t (t, r)|S = B n (t, r)|S = 0.