Vectors on manifolds

The vector, the dual vector (one-form), components and inner products are defined and discussed. The difference between a vector and a one-form is carefully drawn out, with examples and diagrams. Contravariant and covariant components are described, and the way in which the metric can relate them is carefully explained. The transformation of vector components under a change of coordinate basis is derived.

Inner Product Groups and Riesz Representation Theorem

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.

Norms on Quotient Spaces of The 2-Inner Product Space

This paper discussed about construction of some quotients spaces of the 2-inner product spaces. On those quotient spaces, we defined an inner product with respect to a linear independent set. These inner products was derived from the -inner product. We then defined a norm which induced by the inner product in these quotient spaces.

Antilinear symmetry and the ghost problem in quantum field theory

The recognition that the eigenvalues of a non-Hermitian Hamiltonian could all be real if the Hamiltonian had an antilinear symmetry such as PT stimulated new insight into the underlying structure of quantum mechanics. Specifically, it led to the realization that Hilbert space could be richer than the established Dirac approach of constructing inner products out of ket vectors and their Hermitian conjugate bra vectors. With antilinear symmetry one must instead build inner products out of ket vectors and their antilinear conjugates, and it is these inner products that would be time independent in the non-Hermitian but antilinearly symmetric case even as the standard Dirac inner products would not be. Moreover, and in a sense quite remarkably, antilinear symmetry could address not only the temporal behavior of the inner product but also the issue of its overall sign, with antilinear symmetry being capable of yielding a positive inner product in situations such as fourth-order derivative quantum field theories where the standard Dirac inner product is found to have ghostlike negative signature. Antilinear symmetry thus solves the ghost problem in such theories by showing that they are being formulated in the wrong Hilbert space, with antilinear symmetry providing a Hilbert space that is ghost free. Antilinear symmetry does not actually get rid of the ghost states. Rather, it shows that the reasoning that led one to think that ghosts were present in the first place is faulty. Implications of our results for constructing unitary quantum theories of gravity are presented.

Sobolev Gradients for the Möbius Energy

Aiming to optimize the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to Sobolev inner products similar to the $$W^{3/2,2}$$ W 3 / 2 , 2 -inner product. This leads to optimization methods that are significantly more efficient and robust than standard techniques based on $$L^2$$ L 2 -gradients.

State-operator correspondence in celestial conformal field theory

The bulk-to-boundary dictionary for 4D celestial holography is given a new entry defining 2D boundary states living on oriented circles on the celestial sphere. The states are constructed using the 2D CFT state-operator correspondence from operator insertions corresponding to either incoming or outgoing particles which cross the celestial sphere inside the circle. The BPZ construction is applied to give an inner product on such states whose associated bulk adjoints are shown to involve a shadow transform. Scattering amplitudes are then given by BPZ inner products between states living on the same circle but with opposite orientations. 2D boundary states are found to encode the same information as their 4D bulk counterparts, but organized in a radically different manner.

When are maps preserving semi-inner products linear?

We observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.