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Register a new userA Universal Formula for Deformation Quantization on Kahler Manifolds
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Added by
Niels Leth Gammelgaard
Publication date
2010
fields
Physics
and research's language is
English
Authors
Niels Leth Gammelgaard
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We give an explicit local formula for any formal deformation quantization, with separation of variables, on a Kahler manifold. The formula is given in terms of differential operators, parametrized by acyclic combinatorial graphs.
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Deformation quantization conventionally is described in terms of multidifferential operators. Jet manifold technique is well-known provide the adequate formulation of theory of differential operators. We extended this formulation to the multidifferential ones, and consider their infinite order jet prolongation. The infinite order jet manifold is endowed with the canonical flat connection that provides the covariant formula of a deformation star-product.
We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined subsymbol of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus.
We consider formal deformations of the Poisson algebra of functions (with singularities) on $T^*M$ which are Laurent polynomials of fibers. Tn the case: $dim M=1$ ($M=S^1, {bf R}$), there exists a non-trivial $star$-product on this algebra non-equivalent to the standard Moyal product.
We use a deformed differential structure to obtain a curved metric by a deformation quantization of the flat space-time. In particular, by setting the deformation parameters to be equal to physical constants we obtain the Friedmann-Robertson-Walker (FRW) model for inflation and a deformed version of the FRW space-time. By calculating classical Einstein-equations for the extended space-time we obtain non-trivial solutions. Moreover, in this framework we obtain the Moyal-Weyl, i.e. a constant non-commutative space-time, by a consistency condition.
In this semi-expository paper we study two examples of coherent states based on the Weyl- Heisenberg group and the group of $2 times 2$ upper triangular matrices. It is known that sometimes the coherent states provide us with a Kahler embedding of a coadjoint orbit into the projective Hilbert space ${mathbb C}P^n$ or ${mathbb C}P^{infty}$. We show an explicit computation of this in the above two examples. We also note the presence of other coadjoint orbits which only embed symplectically into the projective Hilbert space. These correspond to squeezed states, which have several applications in physics. Our exposition includes a detailed study of the geometric quantisation of the coadjoint orbits of the Lie Algebra of upper triangular matrices. This reveals the presence of distinguished orbits which correspond to coherent states, as well as others corresponding to squeezed states. The coadjoint orbit of $SUT^{+}$ we consider is intimately connected to the $2$-dimensional Toda system.
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