Searcharxiv⌕ Search

arXiv subjects

Philip D. Mannheim

Publications and source records attributed to Philip D. Mannheim.

At least 55 records · Page 3Linked to original sources
arXiv2017

Living Without Supersymmetry -- the Conformal Alternative and a Dynamical Higgs Boson

We show that key results of supersymmetry can be achieved via conformal symmetry. We propose that the Higgs boson be a dynamical bound state rather than an elementary scalar, so that there is no quadratic divergence self-energy problem for it and no need to invoke supersymmetry to resolve it. We study a conformal invariant theory of interacting fermions and gauge bosons, in which there is scaling with anomalous dimensions and dynamical symmetry breaking, with the dynamical dimension of $\barψψ$ being reduced from 3 to 2. With this reduction we augment the theory with a then renormalizable 4-fermion interaction with dynamical dimension equal to 4. We reinterpret the theory as a renormalizable version of the Nambu-Jona-Lasinio (NJL) model, with the gauge theory sector with its now massive fermion being the mean field and the 4-fermion interaction being the residual interaction. It is this residual interaction that generates dynamical Goldstone and Higgs states, states that, as noted by Baker and Johnson, the gauge theory sector itself does not possess. The Higgs boson is found to be a narrow resonance just above threshold. We couple the theory to conformal gravity, with the interplay between conformal gravity and the 4-fermion interaction taking care of the vacuum energy problem. With conformal gravity being a consistent quantum gravity theory there is no need for string theory with its supersymmetric underpinnings. With conformal gravity fits to galactic rotation curves and the accelerating universe not needing dark matter, there is no need to introduce supersymmetry for either the vacuum energy problem or to provide a potential dark matter candidate. We propose that it is conformal symmetry rather than supersymmetry that is fundamental, with the theory of nature being a locally conformal, locally gauge invariant, non-Abelian NJL theory.

hep-ph↗
arXiv2017

Antilinearity Rather than Hermiticity as a Guiding Principle for Quantum Theory

Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear $PT$ symmetry, since such Hamiltonians can still lead to the time-independent evolution of scalar products, and can still have an entirely real energy spectrum. However, such theories can also admit of energy spectra in which energies come in complex conjugate pairs, and can even admit of Hamiltonians that cannot be diagonalized at all. Hermiticity is just a particular realization of $PT$ symmetry, with $PT$ symmetry being the more general. These $PT$ theories are themselves part of an even broader class of theories, theories that can be characterized by possessing some general antilinear symmetry, as that requirement alone is a both necessary and sufficient condition for the time-independent evolution of scalar products, with all the different realizations of the $PT$ symmetry program then being obtained. Use of complex Lorentz invariance allows us to show that the antilinear symmetry is uniquely specified to be $CPT$, with the $CPT$ theorem thus being extended to the non-Hermitian case. For theories that are separately charge conjugation invariant, the results of the $PT$-symmetry program then follow. We show that in order to construct the correct classical action needed for a path integral quantization one must impose $CPT$ symmetry on each classical path, a requirement that has no counterpart in any Hermiticity condition since Hermiticity of a Hamiltonian is only definable after the quantization has been performed and the quantum Hilbert space has been constructed. We show that whether or not a $CPT$-invariant Hamiltonian is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Thus Hermiticity never needs to be postulated at all.

hep-th↗
arXiv2017

Anomalous Dimensions and the Renormalizability of the Four-Fermion Interaction

We show that when the dynamical dimension of the $\barψψ$ operator is reduced from three to two in a fermion electrodynamics with scaling, a $g(\barψψ)^2+g(\barψiγ^5ψ)^2$ four-fermion interaction which is dressed by this electrodynamics becomes renormalizable. In the fermion-antifermion scattering amplitude every term in an expansion to arbitrary order in $g$ is found to diverge as just a single ultraviolet logarithm (i.e. no log squared or higher), and is thus made finite by a single subtraction. While not necessary for renormalizability per se, the reduction in the dimension of $\barψψ$ to two leads to dynamical chiral symmetry breaking in the infrared, with the needed subtraction then automatically being provided by the theory itself through the symmetry breaking mechanism, with there then being no need to introduce the subtraction by hand. Since the vector and axial vector currents are conserved, they do not acquire any anomalous dimension, with the four-fermion $(\barψγ^μψ)^2$ and $(\barψγ^μγ^5ψ)^2$ interactions instead having to be controlled by the standard Higgs mechanism.

hep-th↗
arXiv2017

Is the Cosmological Constant Problem Properly Posed?

In applications of Einstein gravity one replaces the quantum-mechanical energy-momentum tensor of sources such as the degenerate electrons in a white dwarf or the black-body photons in the microwave background by c-number matrix elements. And not only that, one ignores the zero-point fluctuations in these sources by only retaining the normal-ordered parts of those matrix elements. There is no apparent justification for this procedure, and we show that it is precisely this procedure that leads to the cosmological constant problem. We suggest that solving the problem requires that gravity be treated just as quantum-mechanically as the sources to which it couples, and show that one can then solve the cosmological constant problem if one replaces Einstein gravity by the fully quantum-mechanically consistent conformal gravity theory.

hep-th↗
arXiv2017

Critical Scaling and a Dynamical Higgs Boson

In a quantum electrodynamics theory that is realized by critical scaling and anomalous dimensions, the action is not chiral invariant and there are no dynamical Goldstone or Higgs boson bound states. In the mean-field approximation to a chiral invariant four-fermion theory the associated mean-field sector action is not chiral invariant either and it also possesses no dynamical bound states, with Goldstone and Higgs bosons instead being generated by an accompanying four-fermion residual interaction. In this paper we show that if a critical scaling electrodynamics in which the dimension $d_θ$ of $\barψψ$ is reduced from three to two is augmented with a four-fermion interaction, precisely because it possesses no dynamical bound states the electrodynamic sector can be reinterpreted as a mean-field approximation to a larger theory that is chiral symmetric. And with $d_θ=2$ we show in this larger theory there is a residual interaction that then does generate dynamical Goldstone and Higgs bosons in scattering amplitudes that are completely finite. While the dynamically generated Goldstone boson is massless, the dynamically induced Higgs boson is found to be a narrow resonance just above threshold, with its width being a diagnostic that could potentially enable one to distinguish between a dynamical Higgs boson and an elementary one.

hep-ph↗
arXiv2017

Mass Generation, the Cosmological Constant Problem, Conformal Symmetry, and the Higgs Boson

In 2013 the Nobel Prize in Physics was awarded to Francois Englert and Peter Higgs for their work in 1964 along with the late Robert Brout on the mass generation mechanism (the Higgs mechanism) in local gauge theories. This mechanism requires the existence of a massive scalar particle, the Higgs boson, and in 2012 the Higgs boson was finally discovered at the Large Hadron Collider after being sought for almost half a century. In this article we review the work that led to the discovery of the Higgs boson and discuss its implications. We approach the topic from the perspective of a dynamically generated Higgs boson that is a fermion-antifermion bound state rather than an elementary field that appears in an input Lagrangian. In particular, we emphasize the connection with the Barden-Cooper-Schrieffer theory of superconductivity. We identify the double-well Higgs potential not as a fundamental potential but as a mean-field effective Lagrangian with a dynamical Higgs boson being generated through a residual interaction that accompanies the mean-field Lagrangian. We discuss what we believe to be the key challenge raised by the discovery of the Higgs boson, namely determining whether it is elementary or composite, and through study of a conformal invariant field theory model as realized with critical scaling and anomalous dimensions, suggest that the width of the Higgs boson might serve as a suitable diagnostic for discriminating between an elementary Higgs boson and a composite one. We show that the contribution to the cosmological constant due to a composite Higgs boson is more tractable and under control than the contribution due to an elementary Higgs boson, and is potentially completely under control if there is an underlying conformal symmetry not just in a critical scaling matter sector, but equally in the gravity sector to which the matter sector couples.

hep-ph↗
arXiv2016

CPT Symmetry Without Hermiticity

In the literature the $CPT$ theorem has only been established for Hamiltonians that are Hermitian. Here we extend the $CPT$ theorem to quantum field theories with non-Hermitian Hamiltonians. Our derivation is a quite minimal one as it requires only the time independent evolution of scalar products and invariance under complex Lorentz transformations. The first of these requirements does not force the Hamiltonian to be Hermitian. Rather, it forces its eigenvalues to either be real or to appear in complex conjugate pairs, forces the eigenvectors of such conjugate pairs to be conjugates of each other, and forces the Hamiltonian to admit of an antilinear symmetry. The latter requirement then forces this antilinear symmetry to be $CPT$, with Hermiticity of a Hamiltonian thus only being a sufficient condition for $CPT$ symmetry and not a necessary one. $CPT$ symmetry thus has primacy over Hermiticity, and it rather than Hermiticity should be taken as a guiding principle for constructing quantum theories. With conformal gravity being a non-Hermitian theory, our approach allows us to construct a positive, ghost-free norm for the theory, to thereby establish the unitarity of conformal gravity. Since our approach allows for complex energies and decays, our work justifies the use of the $CPT$ theorem in establishing the equality of the lifetimes of unstable particles that are charge conjugates of each other. In the quantum-mechanical limit where charge conjugation is separately conserved, the key results of the $PT$ symmetry program of Bender and collaborators are recovered.

hep-th↗
arXiv2016

Conformal Invariance and the Metrication of the Fundamental Forces

We revisit Weyl's metrication (geometrization) of electromagnetism. We show that by making Weyl's proposed geometric connection be pure imaginary, not only are we able to metricate electromagnetism, an underlying local conformal invariance makes the geometry be strictly Riemannian and prevents observational gravity from being complex. Via torsion we achieve an analogous metrication for axial-vector fields. We generalize our procedure to Yang-Mills theories, and achieve a metrication of all the fundamental forces. Only in the gravity sector does our approach differ from the standard picture of fundamental forces, with our approach requiring that standard Einstein gravity be replaced by conformal gravity. We show that quantum conformal gravity is a consistent and unitary quantum gravitational theory, one that, unlike string theory, only requires four spacetime dimensions.

gr-qc↗
arXiv2016

Comment on "Problems with Mannheim's conformal gravity program"

Recently in Phys. Rev. D 88, 027504 (2013) Yoon has suggested that there may be problems for the non-relativistic limit of the conformal gravity theory. Here we show that Yoon's results only hold because of the assumption that gravitational sources can be treated the same way that they are treated in standard Newton-Einstein gravity. Since such an assumption violates the theory's underlying conformal invariance,Yoon's conclusions are invalidated.

gr-qc↗
arXiv2016

Colloquium on the Higgs Boson

In 2013 the Nobel Prize in Physics was awarded to Francois Englert and Peter Higgs for their development in 1964 of the mass generation mechanism (the Higgs mechanism) in local gauge theories. This mechanism requires the existence of a massive scalar particle, the Higgs boson, and in 2012 the Higgs boson was finally observed at the Large Hadron Collider after being sought for almost half a century. In this pedagogical article we review the work of these Nobel recipients and discuss its implications. We approach the topic from the perspective of a dynamically generated Higgs boson that is a fermion-antifermion bound state rather than an elementary field that appears in an input Lagrangian. In particular, we emphasize the connection with the BCS theory of superconductivity. We identify the double-well Higgs potential not as a fundamental potential but as a mean-field effective Lagrangian with a dynamical Higgs boson being generated through a residual interaction that accompanies the mean-field Lagrangian. We discuss what we believe to be the key challenge raised by the discovery of the Higgs boson, namely determining whether it is elementary or composite, and suggest that the width of the Higgs boson might serve as a suitable diagnostic for discriminating between an elementary Higgs boson and a composite one.

physics.pop-ph↗
arXiv2015

Extension of the $CPT$ Theorem to non-Hermitian Hamiltonians and Unstable States

We extend the $CPT$ theorem to quantum field theories with non-Hermitian Hamiltonians and unstable states. Our derivation is a quite minimal one as it requires only the time independent evolution of scalar products, invariance under complex Lorentz transformations, and a non-standard but nonetheless perfectly legitimate interpretation of charge conjugation as an anti-linear operator. The first of these requirements does not force the Hamiltonian to be Hermitian. Rather, it forces its eigenvalues to either be real or to appear in complex conjugate pairs, forces the eigenvectors of such conjugate pairs to be conjugates of each other, and forces the Hamiltonian to admit of an anti-linear symmetry. The latter two requirements then force this anti-linear symmetry to be $CPT$, while forcing the Hamiltonian to be real rather than Hermitian. Our work justifies the use of the $CPT$ theorem in establishing the equality of the lifetimes of unstable particles that are charge conjugates of each other. We show that the Euclidean time path integrals of a $CPT$ symmetric theory must always be real. In the quantum-mechanical limit the key results of the $PT$ symmetry program of Bender and collaborators are recovered, with the $C$-operator of the $PT$ symmetry program being identified with the linear component of the charge conjugation operator.

quant-ph↗
arXiv2015

Advancing the case for $PT$ Symmetry -- the Hamiltonian is always $PT$ Symmetric

While a Hamiltonian can be both Hermitian and $PT$ symmetric, it is $PT$ symmetry that is the more general, as it can lead to real energy eigenvalues even if the Hamiltonian is not Hermitian. We discuss some specific ways in which $PT$ symmetry goes beyond Hermiticity and is more far reaching than it. We show that simply by virtue of being the generator of time translations, the Hamiltonian must always be $PT$ symmetric, regardless of whether or not it might be Hermitian. We show that the reality of the Euclidean time path integral is a necessary and sufficient condition for $PT$ symmetry of a quantum field theory, with Hermiticity only being a sufficient condition. We show that in order to construct the correct classical action needed for a path integral quantization one must impose $PT$ symmetry on each classical path, a requirement that has no counterpart in any Hermiticity condition since Hermiticity of a Hamiltonian is only definable after the quantization has been performed and the quantum Hilbert space has been constructed. With the spacetime metric being $PT$ even we show that a covariant action must always be $PT$ symmetric. Unlike Hermiticity, $PT$ symmetry does not need to be postulated as it is derivable from Poincare invariance. Hermiticity is just a particular realization of $PT$ symmetry, one in which the eigenspectrum is real and complete.

quant-ph↗
arXiv2015

$PT$ Symmetry, Conformal Symmetry, and the Metrication of Electromagnetism

We present some interesting connections between $PT$ symmetry and conformal symmetry. We use them to develop a metricated theory of electromagnetism in which the electromagnetic field is present in the geometric connection. However, unlike Weyl who first advanced this possibility, we do not take the connection to be real but to instead be $PT$ symmetric, with it being $iA_μ$ rather than $A_μ$ itself that then appears in the connection. With this modification the standard minimal coupling of electromagnetism to fermions is obtained. Through the use of torsion we obtain a metricated theory of electromagnetism that treats its electric and magnetic sectors symmetrically, with a conformal invariant theory of gravity being found to emerge. An extension to the non-Abelian case is provided.

hep-th↗
arXiv2014

Torsion, Magnetic Monopoles and Faraday's Law via a Variational Principle

Even though Faraday's Law is a dynamical law that describes how changing $\bf{E}$ and $\bf {B}$ fields influence each other, by introducing a vector potential $A_μ$ according to $F_{μν}=\partial_μA_ν-\partial_νA_μ$ Faraday's Law is satisfied kinematically, with the relation $(-g)^{-1/2}ε^{μνστ}\nabla_νF_{στ}=0$ holding on every path in a variational procedure or path integral. In a space with torsion $Q_{αβγ}$ the axial vector $S^μ=(-g)^{1/2}ε^{μαβγ}Q_{αβγ}$ serves as a chiral analog of $A_μ$, and via variation with respect to $S_μ$ one can derive Faraday's Law dynamically as a stationarity condition. With $S_μ$ serving as an axial potential one is able to introduce magnetic monopoles without $S_μ$ needing to be singular or have a non-trivial topology. Our analysis permits torsion and magnetic monopoles to be intrinsically Grassmann, which could explain why they have never been detected. Our procedure permits us to both construct a Weyl geometry in which $A_μ$ is metricated and then convert it into a standard Riemannian geometry.

hep-th↗
arXiv2014

Continuity of the torsionless limit as a selection rule for gravity theories with torsion

While one can in principle augment gravity theory with torsion, it is generally thought that any such torsion affects would be too small to be of consequence. Here we show that this cannot in general be the case. We show that the limit of vanishing torsion is not necessarily a continuous one, with the theory obtained in the limit not necessarily coinciding with the theory in which torsion had never been present at all. However, for a standard torsion tensor that is antisymmetric in two of its indices we have found two cases in which the vanishing torsion limit is in fact continuous, namely Einstein gravity and conformal gravity. For other gravity theories of common interest to possess a continuous limit the torsion tensor would need to be antisymmetric in all three of its indices.

gr-qc↗
arXiv2012

PT symmetry as a necessary and sufficient condition for unitary time evolution

While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution. In this paper we provide conditions that are both necessary and sufficient. We show that $PT$ symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution. For any $PT$-symmetric Hamiltonian $H$ there always exists an operator $V$ that relates $H$ to its Hermitian adjoint according to $VHV^{-1}=H^{\dagger}$. When the energy spectrum of $H$ is complete, Hilbert space norms $<ψ_1|V|ψ_2>$ constructed with this $V$ are always preserved in time. With the energy eigenvalues of a real secular equation being either real or appearing in complex conjugate pairs, we thus establish the unitarity of time evolution in both cases. We also establish the unitarity of time evolution for Hamiltonians whose energy spectra are not complete. We show that when the energy eigenvalues of a Hamiltonian are real and complete the operator $V$ is a positive Hermitian operator, which has an associated square root operator that can be used to bring the Hamiltonian to a Hermitian form. We show that systems with $PT$-symmetric Hamiltonians obey causality. We note that Hermitian theories are ordinarily associated with a path integral quantization prescription in which the path integral measure is real, while in contrast non-Hermitian but $PT$-symmetric theories are ordinarily associated with path integrals in which the measure needs to be complex, but in which the Euclidean time continuation of the path integral is nonetheless real. We show that through $PT$ symmetry the fourth-order derivative Pais-Uhlenbeck theory can be stabilized against transitions to states negative frequency.

hep-th↗
arXiv2012

Galactic rotation curves in conformal gravity

We review some recent work by Mannheim and O'Brien on the systematics of galactic rotation curves in the conformal gravity theory. In this work the conformal theory was applied to a comprehensive, high quality sample of spiral galaxies whose rotation curves extend well beyond the galactic optical disks. On galactic scales the conformal gravitational theory departs from the standard Newtonian theory in two distinct ways. One is a local way in which local matter sources within galaxies generate not just Newtonian potentials but linear potentials as well. The other is a global way in which two universal global potentials, one linear the other quadratic, are generated by the rest of the matter in the universe. The study involves a broad set of 138 spiral galaxies of differing luminosities and sizes, and is augmented here through the inclusion of an additional three tidal dwarf galaxies. With its linear and quadratic potentials the conformal theory can account for the systematics of an entire 141 galaxy sample without any need for galactic dark matter, doing so with only one free parameter per galaxy, namely the visible galactic mass to light ratio.

astro-ph.CO↗