By A. M. Polyakov

In line with his personal paintings, the writer synthesizes the main promising techniques and beliefs in box thought at the present time. He offers such matters as statistical mechanics, quantum box conception and their interrelation, non-stop worldwide symmetry, non-Abelian gauge fields, instantons and the quantam concept of loops, and quantum strings and random surfaces. This publication is aimed toward postgraduate scholars learning box concept and statistical mechanics, and for learn staff in non-stop worldwide conception.

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**Example text**

E. in the σ direction) while Pτµ is the momentum running transverse to it. 1): dP µ = Pτµ dσ + Pσµ dτ. 17) Actually, we can choose any slice of the world-sheet in order to compute this momentum. A most convenient one is a slice τ = constant, revealing the string in its original paramaterisation: P µ = Pτµ dσ, but any other slice will do. Similarly, one can deﬁne the angular momentum: M µν = (Pτµ X ν − Pτν X µ )dσ. 18) It is a simple exercise to work out the momenta for our particular Lagrangian: Pτµ = T Pσµ = T X˙ µ X 2 − X µ (X˙ · X ) (X˙ · X )2 − X˙ 2 X 2 X µ X˙ 2 − X˙ µ (X˙ · X ) (X˙ · X )2 − X˙ 2 X 2 .

The string spectrum has a massless sector separated by a gap (set by the tension) after which there is an inﬁnite tower of massive states. high energy (short distance, or ultra-violet (UV)) physics of the string. It is the massless part of the spectrum which is accessible at low energy and hence relevant to phenomenology. For example, closed string theories describe a massless spin two particle which is identiﬁed with the graviton. The questions of non-renormalisability which arose in quantum ﬁeld theory turn out to be circumvented by the remarkable ultra-violet properties of string theory, which give rise to an extremely well-behaved perturbative description of multi-loop processes involving gravitons† .

2 for a reminder of this in equations. The sourceless ﬁeld equations then follow from the recasting of the relative motion observed between frames on neighbouring geodesics in terms of an apparent ‘tidal’ force. The full statement of the ﬁeld equations to include sources is also guided by covariance, which means that it is a physical equation between tensors of the same type, and with the same divergenceless property (which is a physical statement of continuity). 1). e. ∇µ T µν = 0 uses the covariant derivative∗ , which is designed to yield a tensor after acting on one, say V : µ··· µ··· µ··· λ··· ∇κ Vν··· ≡ ∂κ Vν··· + Γµλκ Vν··· + · · · − Γλκν Vλ··· − ···.