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E. 45) b=1 (iv) covariant derivatives of metric (inverse metric) tensors are equal to zero: Da g bc = 0 One can deﬁne a contravariant derivative acting by the inverse metric tensor: D a = g ab Db Naturally, covariant (as well as contravariant) derivatives do not commute. Let us check this with the operators Da Db and Db Da acting on a scalar function f . 46) which is already a covariant vector. 43). 3. Two main radii of a two-dimensisonal surface at a given point A. is called the Riemann tensor.
The reader has to remember, however, that these descriptions relate different objects (the oscillator and the string in the given case). 1. In order to strengthen the case for the equivalence, let us calculate one more physical quantity, the partition function. 39) which converges. 41) where Z 0 depends linearly on L. 42) which is the well known expression for the harmonic oscillator. So the equivalence holds. Exercise Consider a system of one-dimensional acoustic phonons. This system can be represented as a system of harmonic oscillators with frequencies ω(q) = c|q|, where q is the wave vector of a phonon.
These relations are convenient if our calculational procedure naturally provides us with Green’s functions in frequency momentum representation. This is not always the case, however. Sometimes we can work only in real space (see the chapters on one-dimensional systems). Then it is better not to calculate D(iωn ) ﬁrst and continue it analytically, but to skip this intermediate step and to express the retarded functions directly in terms of D(τ ). 32) (the upper sign is for bosons, the lower one for fermions).