arXiv subjects
Publications and source records attributed to David Ritz Finkelstein.
Starting from the vacuum, iterated Grassmann-algebra formation consecutively introduces vector spaces and groups with the structure first of charge, then of isospin, then of color.
Quantum field theory can be physically regularized by modularizing it on several levels of aggregation. Since computation is already thoroughly modularized, physical experiments are treated here as quantum relativistic cellular computations with spins for cells, address, memory, and control registers. For regularity the modules are taken to be iterated Fermi-Dirac assemblies. These are shown to be spins in various dimensions. Bose statistics are expressed as approximations to the Palev statistics of pairs of Fermi-Dirac quanta.
A modular quantum architecture is given for the space-time, particles, and fields of the Standard Model and General Relativity. It assumes a right-handed neutrino, so that based on their multiplet structure all fundamental fermions have isospin 1/2. This opens the possibility that the Higgs field can be identified with the Yang $i$-field of 1947. The quantum gravitational metric form proposed is a quantification of the Killing form of the quantum space-time cell. There is no trace of the black hole phenomenon at the one-cell quantum level.
A finite relativistic quantum space-time is constructed. Its unit cell has Palev statistics defined by a spin representation of an orthogonal group. When the Standard Model and general relativity are physically regularized by such space-time quantization, their gauges are fixed by nature; the cell groups remain.
Set theory reduces all processes to assembly and disassembly. A similar architecture is proposed for nature as quantum computer. It resolves the classical space-time underlying Feynman diagrams into a quantum network of creation and annihilation processes, reducing kinematics to quantum statistics, and regularizing the Lie algebra of the Einstein diffeomorphism group. The usually separate and singular Lie algebras of kinematics, statistics, and conserved currents merge into one regular statistics Lie algebra.
Present-day quantum field theory can be regularized by a decomposition into quantum simplices. This replaces the infinite-dimensional Hilbert space by a high-dimensional spinor space and singular canonical Lie groups by regular spin groups. It radically changes the uncertainty principle for small distances. Gaugeons, including the gravitational, are represented as bound fermion-pairs, and space-time curvature as a singular organized limit of quantum non-commutativity. Keywords: Quantum logic, quantum set theory, quantum gravity, quantum topology, simplicial quantization.
Quantum gauge theories with finite-dimensional representation spaces are constructed that can have canonical gauge field theories as singular limits. They describe nature as a recursive quantum assembly by iterating Fermi-Dirac quantification. Six iterations are necessary and sufficient for present physics. The gauge structure, the spin-statistics correlation, the space-time metric, and the Higgs field are modeled.
Duerer's engraving ``MELENCOLIA I'' was circulated in two versions not previously distinguished. Besides their conspicuous early Renaissance scientific instruments and tools, they contain numerous apparently unreported concealments whose detection reveals heresies expressed in the work. The main one is encoded in the motto {\em MELENCOLIA I} itself: Natural Philosophy, not Mathematical Philosophy or Theological Philosophy, is the way to knowledge. Unusual optical illusions and subliminal images, differing between the two versions, declare the relativity and ambiguity of perception, and indicate that the work was a Humanist document intended for a Humanist viewership.
I construct a finite-dimensional quantum theory from general relativity by a homotopy method. Its quantum history is made up of at least two levels of fermionic elements. Its unitary group has the diffeomorphism group as singular limit. Its gravitational metrical form is the algebraic square. Its spinors are multivectors.
Arbitrarily small changes in the commutation relations suffice to transform the usual singular quantum theories into regular quantum theories. This process is an extension of canonical quantization that we call general quantization. Here we apply general quantization to the time-independent linear harmonic oscillator. The unstable Heisenberg group becomes the stable group SO(3). This freezes out the zero-point energy of very soft or very hard oscillators, like those responsible for the infrared or ultraviolet divergencies of usual field theories, without much changing the medium oscillators. It produces pronounced violations of equipartition and of the usual uncertainty relations for soft or hard oscillators,and interactions between the previously uncoupled excitation quanta of the oscillator, weakly attractive for medium quanta, strongly repulsive for soft or hard quanta.
We general-quantize the dynamics of the quantum harmonic oscillator to obtain a covariant finite quantum dynamics in a finite quantum time. The usual central (``superselected'') time results from a self-organization. Unitarity necessarily fails, imperceptibly for middle times and grossly near the beginning and end of time. Time and energy interconvert during space-time decondensation or melt-down, at a rate governed by a constant like the Planck power.
Segal's hypothesis that physical theories drift toward simple groups follows from a general quantum principle and suggests a general quantization process. I general-quantize the scalar meson field in Minkowski space-time to illustrate the process. The result is a finite quantum field theory over a finite quantum space-time with higher symmetry than the singular theory. Multiple quantification connects the levels of the theory.
A Clifford algebra over the binary field 2 = {0,1} is a second-order classical logic that is substantially richer than Boolean algebra. We use it as a bridge to a Clifford algebraic quantum logic that is richer than the usual Hilbert space quantum logic and admits iteration. This leads to a higher-order Clifford-algebraic logic. We formulate a toy Dirac equation with this logic. It isexactly Lorentz-invariant, yet it approximates the usual Dirac equation as closely as desired and all its variables have finite spectra. It is worth considering as a Lorentz-invariant improvement on lattice space-times.
Segal proposed ultraquantum commutation relations with two ultraquantum constants hbar' and hbar'' besides Planck's quantum constant hbar with a variable i. The Heisenberg quantum algebra is a contraction - in a more general sense than that of Inonu and Wigner - of the Segal ultraquantum algebra. The usual constant i arises as a vacuum order-parameter in the quantum limit where hbar' and hbar'' approach zero. One physical consequence is a discrete spectrum for canonical variable and space-time coordinates. Another is an interconvention of time and energy accompnying space-time meltdown (disorder), with a fundamental conversion factor of some kilograms of energy per second.
We develop a computation model for solving Boolean networks by implementing wires through quantum ground-mode computation and gates through identities following from angular momentum algebra and statistics. Gates are represented by three-dimensional (triplet) symmetries due to particle indistinguishability and are identically satisfied throughout computation being constants of the motion. The relaxation of the wires yields the network solutions. Such gates cost no computation time, which is comparable with that of an easier Boolean network where all the gate constraints implemented as constants of the motion are removed. This model computation is robust with respect to decoherence and yields a generalized quantum speed-up for all NP problems.
We develop a computation model for solving Boolean networks that implements wires through quantum ground-state computation and implements gates through identities following from angular momentum algebra and statistics. The gates are static in the sense that they contribute Hamiltonian 0 and hold as constants of the motion; only the wires are dynamic. Just as a spin 1/2 makes an ideal 1-bit memory element, a spin 1 makes an ideal 3-bit gate. Such gates cost no computation time: relaxing the wires alone solves the network. We compare computation time with that of an easier Boolean network where all the gate constraints are simply removed. This computation model is robust with respect to decoherence and yields a generalized quantum speed-up for all NP problems.
Systems of spin 1, such as triplet pairs of spin-1/2 fermions (like orthohydrogen nuclei) make useful three-terminal elements for quantum computation, and when interconnected by qubit equality relations are universal for quantum computation. This is an instance of quantum-statistical computation: some of the logical relations of the problem are satisfied identically in virtue of quantum statistics, which takes no time. We show heuristically that quantum-statistical ground-mode computation is substantially faster than pure ground-mode computation when the ground mode is reached by annealing.