Probabilities in the general boundary formulation

We give an introductory account of the general boundary formulation of quantum theory. We refine its probability interpretation and emphasize a conceptual and historical perspective. We give motivations from quantum gravity and illustrate them with a scenario for describing gravitons in quantum gravity.Comment: 7 pages, LaTeX + jpconf, contribution to proceedings of DICE2006, Piombino, Italy, September 2006; v2: typos corrected (including title) and references update

Discrete Dynamics: Gauge Invariance and Quantization

Gauge invariance in discrete dynamical systems and its connection with quantization are considered. For a complete description of gauge symmetries of a system we construct explicitly a class of groups unifying in a natural way the space and internal symmetries. We describe the main features of the gauge principle relevant to the discrete and finite background. Assuming that continuous phenomena are approximations of more fundamental discrete processes, we discuss -- with the help of a simple illustration -- relations between such processes and their continuous approximations. We propose an approach to introduce quantum structures in discrete systems, based on finite gauge groups. In this approach quantization can be interpreted as introduction of gauge connection of a special kind. We illustrate our approach to quantization by a simple model and suggest generalization of this model. One of the main tools for our study is a program written in C.Comment: 15 pages; CASC 2009, Kobe, Japan, September 13-17, 200

q-deformation of z→az+bcz+dz\to {az+b\over cz+d}

We construct the action of the quantum double of \uq on the standard Podle\'s sphere and interpret it as the quantum projective formula generalizing to the q-deformed setting the action of the Lorentz group of global conformal transformations on the ordinary Riemann sphere.Comment: LaTeX, 16 pages, we add a reference where an alternative construction of the q-Lorentz group action on the Podles sphere is give

Lorentz invariant and supersymmetric interpretation of noncommutative quantum field theory

In this paper, using a Hopf-algebraic method, we construct deformed Poincar\'e SUSY algebra in terms of twisted (Hopf) algebra. By adapting this twist deformed super-Poincar\'e algrebra as our fundamental symmetry, we can see the consistency between the algebra and non(anti)commutative relation among (super)coordinates and interpret that symmetry of non(anti)commutative QFT is in fact twisted one. The key point is validity of our new twist element that guarantees non(anti)commutativity of space. It is checked in this paper for N=1 case. We also comment on the possibility of noncommutative central charge coordinate. Finally, because our twist operation does not break the original algebra, we can claim that (twisted) SUSY is not broken in contrast to the string inspired $\mathcal{N}=1/2$ SUSY in N=1 non(anti)commutative superspace.Comment: 15 pages, LaTeX. v3:One section added, typos corrected, to appear in Int. J. Mod. Phys.

Codimension-2 surfaces and their Hilbert spaces: low-energy clues for holography from general covariance

We argue that the holographic principle may be hinted at already from low-energy considerations, assuming diffeomorphism invariance, quantum mechanics and Minkowski-like causality. We consider the states of finite spacelike hypersurfaces in a diffeomorphism-invariant QFT. A low-energy regularization is assumed. We note a natural dependence of the Hilbert space on a codimension-2 boundary surface. The Hilbert product is defined dynamically, in terms of transition amplitudes which are described by a path integral. We show that a canonical basis is incompatible with these assumptions, which opens the possibility for a smaller Hilbert-space dimension than canonically expected. We argue further that this dimension may decrease with surface area at constant volume, hinting at holographic area-proportionality. We draw comparisons with other approaches and setups, and propose an interpretation for the non-holographic space of graviton states at asymptotically-Minkowski null infinity.Comment: 13 pages, 9 eps figures. Added Section VI, improved presentation. Expanded and split the Introduction into two sections. Added Section VII. Added reference

Dual Computations of Non-abelian Yang-Mills on the Lattice

In the past several decades there have been a number of proposals for computing with dual forms of non-abelian Yang-Mills theories on the lattice. Motivated by the gauge-invariant, geometric picture offered by dual models and successful applications of duality in the U(1) case, we revisit the question of whether it is practical to perform numerical computation using non-abelian dual models. Specifically, we consider three-dimensional SU(2) pure Yang-Mills as an accessible yet non-trivial case in which the gauge group is non-abelian. Using methods developed recently in the context of spin foam quantum gravity, we derive an algorithm for efficiently computing the dual amplitude and describe Metropolis moves for sampling the dual ensemble. We relate our algorithms to prior work in non-abelian dual computations of Hari Dass and his collaborators, addressing several problems that have been left open. We report results of spin expectation value computations over a range of lattice sizes and couplings that are in agreement with our conventional lattice computations. We conclude with an outlook on further development of dual methods and their application to problems of current interest.Comment: v1: 18 pages, 7 figures, v2: Many changes to appendix, minor changes throughout, references and figures added, v3: minor corrections, 22 page

Causality and statistics on the Groenewold-Moyal plane

Quantum theories constructed on the noncommutative spacetime called the Groenewold-Moyal plane exhibit many interesting properties such as Lorentz and CPT noninvariance, causality violation and twisted statistics. We show that such violations lead to many striking features that may be tested experimentally. These theories predict Pauli forbidden transitions due to twisted statistics, anisotropies in the cosmic microwave background radiation due to correlations of observables in spacelike regions and Lorentz and CPT violations in scattering amplitudes.Comment: 12 pages, 1 figure. Based on the talk given by APB at the Workshop "Theoretical and Experimental Aspects of the Spin Statisics Connection and Related Symmetries", Stazione Marittima Conference Center, Trieste, Italy from the 21st to the 25th of October 200

Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes

In this article we study the quantization of a free real scalar field on a class of noncommutative manifolds, obtained via formal deformation quantization using triangular Drinfel'd twists. We construct deformed quadratic action functionals and compute the corresponding equation of motion operators. The Green's operators and the fundamental solution of the deformed equation of motion are obtained in terms of formal power series. It is shown that, using the deformed fundamental solution, we can define deformed *-algebras of field observables, which in general depend on the spacetime deformation parameter. This dependence is absent in the special case of Killing deformations, which include in particular the Moyal-Weyl deformation of the Minkowski spacetime.Comment: LaTeX 14 pages, no figures, svjour3.cls style; v2: clarifications and references added, compatible with published versio

Dual variables and a connection picture for the Euclidean Barrett-Crane model

The partition function of the SO(4)- or Spin(4)-symmetric Euclidean Barrett-Crane model can be understood as a sum over all quantized geometries of a given triangulation of a four-manifold. In the original formulation, the variables of the model are balanced representations of SO(4) which describe the quantized areas of the triangles. We present an exact duality transformation for the full quantum theory and reformulate the model in terms of new variables which can be understood as variables conjugate to the quantized areas. The new variables are pairs of S^3-values associated to the tetrahedra. These S^3-variables parameterize the hyperplanes spanned by the tetrahedra (locally embedded in R^4), and the fact that there is a pair of variables for each tetrahedron can be viewed as a consequence of an SO(4)-valued parallel transport along the edges dual to the tetrahedra. We reconstruct the parallel transport of which only the action of SO(4) on S^3 is physically relevant and rewrite the Barrett-Crane model as an SO(4) lattice BF-theory living on the 2-complex dual to the triangulation subject to suitable constraints whose form we derive at the quantum level. Our reformulation of the Barrett-Crane model in terms of continuous variables is suitable for the application of various analytical and numerical techniques familiar from Statistical Mechanics.Comment: 33 pages, LaTeX, combined PiCTeX/postscript figures, v2: note added, TeX error correcte

Jordanian Twist Quantization of D=4 Lorentz and Poincare Algebras and D=3 Contraction Limit

We describe in detail two-parameter nonstandard quantum deformation of D=4 Lorentz algebra $\mathfrak{o}(3,1)$, linked with Jordanian deformation of $\mathfrak{sl} (2;\mathbb{C})$. Using twist quantization technique we obtain the explicit formulae for the deformed coproducts and antipodes. Further extending the considered deformation to the D=4 Poincar\'{e} algebra we obtain a new Hopf-algebraic deformation of four-dimensional relativistic symmetries with dimensionless deformation parameter. Finally, we interpret $\mathfrak{o}(3,1)$ as the D=3 de-Sitter algebra and calculate the contraction limit $R\to\infty$ ($R$ -- de-Sitter radius) providing explicit Hopf algebra structure for the quantum deformation of the D=3 Poincar\'{e} algebra (with masslike deformation parameters), which is the two-parameter light-cone $\kappa$-deformation of the D=3 Poincar\'{e} symmetry.Comment: 13 pages, no figure