Quantum Mechanics of Fundamental Systems 1 by Stephen L. Adler (auth.), Claudio Teitelboim (eds.)

By Stephen L. Adler (auth.), Claudio Teitelboim (eds.)

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3. THE HAMILTONIAN Weshall work in an n-dimensional Riemannian manifold and weshall use the Misner-Thorne- Wheeler convention (-, -,-) [6]. We shall study a neutral scalar field with mass m and the field equation (2) where R is the curvature scalar and g a coupling constant. L, v = 0, 1, ... , n - 1) be the ordinary energy-momentum tensor of the field; we define our observer-dependent Hamiltonian, at an orthogonal surface to the fluid ~. for the observer's system with vectors u and v: H'2:. vv~-'-d~v = J'2:.

SUMMARY AND OUTLOOK Wehave been able to identify a unique covariant tensor 6 in Cß which is invariant under local reparametrizations. However, we also found that 6 is necessarily degenerate, and this spoils the possibility of formulating astring field theory as a metric theory on Cß in analogy with the conventional point-field case in equation (9). The degeneracy of 6 means that, owing to the required invariance under Jocal reparametrizations, one Iacks in Cß the usual isomorphism between tangent and cotangent bundles (obtained by "raising" and "lowering"indices with an invertible tensor).

Teracting part R~+I is an exterior (p; + 1)-forrn function of the potentials BK and of their exterior derivatives, that is to say a function of B(K) and G(KJ with K 'i' i. Rp+I must be suchthat the Gp+ 1's satisfy Bianchi identities [1, 2]. , Yang-Mills) gauge fields contained in the set of fields B, and not more than linearly [ 1]. The field strengths Gp+I contains the (} and 8 variations of all classical and ghost components of BP, through the term (s + s)BP. To determine these variations one imposes that all terms in Gp+I with nonzero ghost number vanish.

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