F
Frankie Doddato

  • May 15, 2023
  • Joined Dec 14, 2022
  • In Product manifolds as realizations of general linear symmetries

    https://www.worldscientific.com/doi/10.1142/S0219887822400060

    This paper considers the relationship between geometry, symmetry and fundamental interactions — gravity and those mediated by gauge fields. We explore product spacetimes which (a) have the necessary symmetries for gauge interactions and four-dimensional gravity and (b) reduce to an N-dimensional isotropic universe in their flat space limit. The key technique is looking at orbits of the operator form of symmetric rank-two tensors under changes of coordinate system. Orbits containing diagonal matrices are seen to correspond to product manifolds. The GL(N,R) symmetry of the decompactified universe acts nonlinearly on such a product spacetime. We explore the resulting Kaluza–Klein theories, in which the internal symmetries act indirectly on space of the extra dimensions, and give two examples: a six-dimensional model in which the gauge symmetry is U(1) and a seven-dimensional model in which it is SU(2). We identify constraints that can be placed on any rank-two symmetric tensor to obtain such spacetimes: relationships between polynomial invariants. The multiplicities of its eigenvalues determine the dimensionalities of the factor spaces and hence the gauge symmetries. If the tensor in question is the Ricci tensor, other than two-dimensional factor spaces all the factor spaces are Einstein manifolds. This situation represents the classical vacuum of the Kaluza–Klein theory.

  • In Is time a cyclic dimension?

    https://www.sciencedirect.com/science/article/abs/pii/S0003491622002925

    If “quantization is an art” then it can be greatly refined by adopting cyclic time formalism. In past papers we have proven the effectiveness of a formulation of physics based on cyclic relativistic time. Now we are able to demonstrate in a general way, by using theorems of Geometric Quantization, that the Poisson brackets of intrinsically cyclic time dynamics directly imply the ordinary canonical commutation relations and the other Dirac’s rules of canonical Quantum Mechanics. In other words, according to our result, the canonical quantization is an implicit way of imposing intrinsically cyclic time dynamics without explicitly saying that time is a cyclic dimension.

  • In The Physical Foundation of Quantum Theory

    I would like to thank Mehran Shaghaghi for initiating a discussion on their recent paper, “The Physical Foundations of Quantum Theory”, based on their recent paper published in February 2023 in Springer Link: https://link.springer.com/article/10.1007/s10701-023-00673-2.

    Abstract:

    The number of independent messages a physical system can carry is limited by the number of its adjustable properties. In particular, systems with only one adjustable property cannot carry more than a single message at a time. We demonstrate that this is true for the photons in the double-slit experiment, and that this is what leads to the fundamental limit on measuring the complementary aspect of the photons. Next, we illustrate that systems with a single adjustable property exhibit other quantum behaviors, such as noncommutativity and no-cloning. Finally, we formulate a mathematical theory to describe the dynamics of such systems and derive the standard Hilbert space formalism of quantum mechanics as well as the Born probability rule. Our derivation demonstrates the physical foundation of quantum theory.