arXiv subjects
Publications and source records attributed to Gordon Chalmers.
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves generic nonlinear systems. Further properties characterized by the topology and geometry of the associated manifolds may define global properties of the solutions.
A formalism is given to count integer and rational solutions to polynomial equations with rational coefficients. These polynomials $P(x)$ are parameterized by three integers, labeling an elliptic curve. The counting of the rational solutions to $y^2=P(x)$ is facilitated by another elliptic curve with integral coefficients. The problem of counting is described by two elliptic curves and a map between them.
A method to the explict solutions of general systems of algebraic equations is presented via the metric form of affiliated Kähler manifolds. The solutions to these systems arise from sets of geodesic second order non-linear differential equations. Algebraic equations in various fields such as integers and rational numbers, as well as transcendental equations, are amenable. The case of Fermat's set of equations is a subset.
The Riemann hypothesis is identified with zeros of ${\cal N}=4$ supersymmetric gauge theory four-point amplitude. The zeros of the $ζ(s)$ function are identified with th complex dimension of the spacetime, or the dimension of the toroidal compactification. A sequence of dimensions are identified in order to map the zeros of the amplitude to the Riemann hypothesis.
A polynomial is presented that models a topological knot in a unique manner. It distinguishes all types of knots including the orientation and has a group theory interpretation. The topologies may be labeled via a number, which upon a base 2 expansion generate the polynomial; the equivalent numbers via Reidemeister moves are grouped into a superset polynomial with coefficients labeling the equivalent knots.
The perturbative dynamics of quantum field theories is described by a recursive expansion similar to the well known loop expansion. The equivalent formulation based on low-energy dynamics via an expansion in derivatives is well known in the literature; this is described by terms from low energy to high energy. The coefficients of these terms are presented in a simplified algebraic manner. In general, the dynamics of any bare Lagrangian, including additional higher dimension terms, is found by iteration in a discrete algorithm. Inversion of the quantized dynamics to the fundamental theory is possible.
A number theoretic algorithm is given for writing gauge theory amplitudes in a compact manner. It is possible to write down all details of the complete $L$ loop amplitude with two integers, or a complex integer. However, a more symmetric notation requires more integers, five or seven, depending on the type of theory. It is possible that in the symmetric form (or in the non-symmetric form) that a direct (or less direct) recursive algorithm or generating function can be developed to compute these numbers at arbitrary loop order. The existence of this function is implied by the recursive structure of loop amplitudes and their analyticity, i.e. multi-particle poles; a function requiring a finite number of computations such as a polynomial with derivable coefficients is desired.
A unified theory of four-dimensional gravity together with the standard model is presented, with supersymmetry breaking of M-theory at a TeV. Masses of the the known particles are derived. The cosmological constant is quantum generated to the observed value. Quantum corrections to the classical compactification are analyzed, and the scenario is stable.
Gauge theories with and without matter are formulated in the derivative expansion. Amplitudues are derived as a power series in the energy scales; there are simplifications as compared with the usual loop expansion. The incorporation and summation of Wilson loops is natural in this approach and allows for a derivation of the confining potential as well as the masses of the gauge invariant observables.
Maximally supersymmetric gauge theories have experienced renewed interest due to the AdS/CFT correspondence and its conjectured S-duality. These gauge theories possess a large amount of symmetry and have quasi-integrable properties. We derive the amplitudes in the derivative expansion of the spontaneously broken examples and perform all loop integrations. The S-matrix is found via an algebraic recursion and at each order is SL(2,Z) invariant.
The quantum correlations of scalar fields are examined as a power series in derivatives. Recursive algebraic equations are derived and determine the amplitudes; all loop integrations are performed. This recursion contains the same information as the usual loop expansion. The approach is pragmatic and generalizable to most quantum field theories.
We examine the form of the cosmological constant in the loop expansion of broken maximally supersymmetric supergravity theories, and after embedding, within superstring and M-theory. Supersymmetry breaking at the TeV scale generates values of the cosmological constant that are in agreement with current astrophysical data. The form of perturbative quantum effects in the loop expansion is consistent with this parameter regime.
The strongly coupled limit of string scattering and the automorphic construction of the graviton S-matrix is compared with the eleven dimensional formulation of M-theory. In a particular scaling limit at strong string coupling, M-theory is described by eleven-dimensional supergravity which does not possess a dilaton, but rather a perturbative expansion in the gravitational coupling and derivatives. The latter theory provides an off-shell description of the string, upon dimensional reduction.
The amplitudes in perturbative open string theory are examined as functions of the tachyon condensate parameter. The boundary state formalism demonstrates the decoupling of the open string modes at the non-perturbative minima of the tachyon potential via a degeneration of open world-sheets and identifies an independence of the coupling constants $g_s$ and $g_{YM}$ at general values of the tachyon condensate. The closed sector is generated at the quantum level; it is also generated at the classical level through the condensation of the propagating open string modes on the D-brane degrees of freedom.
Spontaneously broken gauge theories are described as a perturbation of selfdual gauge theory. Instead of the incorporation of scalar degrees of freedom, the massive component of the gauge field is obtained from an anti-selfdual field strength consisting of three components before gauge fixing. The interactions describe a massive gauge theory that is non-polynomial with an expansion containing an infinite number of terms. The Lagrangian generalizes the form of the axial anomaly in two dimensions. Unitary propagation of the tensor field occurs upon gauge fixing an additional symmetry.
A T-dualized selfdual inspired formulation of massive vector fields coupled to arbitrary matter is generated; subsequently its perturbative series modeling a spontaneously broken gauge theory is analyzed. The new Feynman rules and external line factors are chirally minimized in the sense that only one type of spin index occurs in the rules. Several processes are examined in detail and the cross-sections formulated in this approach. A double line formulation of the Lorentz algebra for Feynman diagrams is produced in this formalism, similar to color ordering, which follows from a spin ordering of the Feynman rules. The new double line formalism leads to further minimization of gauge invariant scattering in perturbation theory. The dualized electroweak model is also generated.
We show the existence of a global anomaly in the one-loop graphs of N=2 string theory, defined by sewing tree amplitudes, unless spacetime supersymmetry is imposed. The anomaly is responsible for the non-vanishing maximally helicity violating amplitudes. The supersymmetric completion of the N=2 string spectrum is formulated by extending the previous cohomological analysis with an external spin factor; the target space-time spin-statistics of these individual fields in a selfdual background are compatible with previous cohomological analysis as fields of arbitrary spin may be bosonized into one another. We further analyze duality relations between the open and closed string amplitudes and demonstrate this in the supersymmetric extension of the target space-time theory through the insertion of zero-momentum operators.
We calculate the genus-one three- and four-point amplitudes in the 2+2 dimensional closed N=(2,2) worldsheet supersymmetric string within the RNS formulation. Vertex operators are redefined with the incorporation of spinor helicity techniques, and the quantum scattering is shown to be manifestly gauge and Lorentz invariant after normalizing the string states. The continuous spin structure summation over the monodromies of the worldsheet fermions is carried out explicitly, and the field-theory limit is extracted. The amplitude in this limit is shown to be the maximally helicity violating amplitude in pure gravity evaluated in a two-dimensional setting, which vanishes, unlike the four-dimensional result. The vanishing of the genus-one N=2 closed string amplitude is related to the absence of one-loop divergences in dimensionally regulated IIB supergravity. Comparisons and contrasts between self-dual field theory and the N=2 string theory are made at the quantum level; they have different S-matrices. Finally, we point to further relations with self-dual field theory and two-dimensional models.