Abstract
We generalize the DeWitt-Virasoro (DWV) construction of arXiv:0912.3987 [hep-th] to tensor representations of higher ranks. A rank-ntensor state, which is by itself coordinate invariant, is expanded in terms of position eigenstates that transform as tensors of the same rank. The representation of the momentum operator in these basis states is then obtained by generalizing DeWitt's argument (1952). Such a representation is written in terms of certain bivector of parallel displacement and its covariant derivatives. With this machinery at hand we find tensor representations of the DWV generators defined in the previous work. The results differ from those in spin-zero representation by additional terms involving the spin connection. However, we show that the DWV algebra found earlier as a scalar expectation value remains the same, as required by consistency, as all the additional contributions conspire to cancel in various ways. In particular, vanishing of the anomaly term requires the same condition of Ricci-flatness for the background.
Highlights
Us to write down the classical Virasoro generators in terms of the phase-space variables of the infinite-dimensional/particle description
In that work the momentum operator in position space was obtained by demanding that the canonical commutation relations and the hermiticity conditions hold true as expectation values between two position eigenstates of spin zero
A natural generalization of the orthonormality condition for higher rank basis states is given in terms of bi-vector of parallel displacement
Summary
The analysis of [2] was done using an infinite-dimensional language. We will establish that the scope of the existing framework in [3, 2] is as follows, Non-trivial invariant matrix elements can be constructed only in scalar representation. We define a tensor state to be one whose wavefunction is a tensor in the infinite-dimensional sense The latter can be obtained either by multiplying a tensor field with a scalar wavefunction or by applying covariant derivatives on it. Proceeding in the similar way one can argue that the only non-trivial (i.e. involving momentum operator) invariant matrix elements that can be constructed are of the type (2.2).
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