arXiv subjects

Philip D. Mannheim

Publications and source records attributed to Philip D. Mannheim.

At least 19 recordsLinked to original sources

arXiv2026

Critique of Breit-Wigner resonance scattering

In the standard Breit-Wigner approach to scattering the phase shift is to have a form $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ at a real energy resonance. This leads to complex energy poles in the scattering amplitude at $E_{\rm BW}=E_1-iΓ_1$, poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ is not always a good description of the real energy scattering amplitude, that $Γ_1$ can be negative, that $E_{\rm BW}$ is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear $PT$ symmetry solutions to the square well Schrödinger equation appear in complex conjugate energy pairs $E_{\mp}=E_2\mp i Γ_2$ with $E_- \neq E_{\rm BW}$, doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two.

hep-ph↗

arXiv2026

PT symmetry and the square well potential: Antilinear symmetry rather than Hermiticity in scattering processes

A real potential Hamiltonian has real energy bound states below the scattering threshold and complex energy resonances above it. Scattering states are not square integrable, being instead delta function normalized. This lack of square integrability breaks the connection between Hermiticity and real eigenvalues, to thus allow for real bound state sector eigenvalues and complex scattering sector eigenvalues. When written as contour integrals delta functions take support in the complex plane, with the scattering amplitude being able to take support in the complex plane too. However, the scattering amplitude is CPT symmetric. For resonance scattering this antilinear symmetry requires the presence of a complex conjugate pair of energies, one to describe the excitation of the resonance and the other to describe its decay, with it being their interplay that enforces probability conservation. Each complex pair of energy eigenvalues corresponds to only one observable resonance not two, to thus modify the standard pure decaying complex energy pole discussion of resonances. We show that the non-relativistic real potential square-well Schrödinger equation possesses C and PT symmetry in both the bound and scattering sectors, with there being complex conjugate pairs of energy eigenvalue solutions in the scattering sector. The Hamiltonian thus acts as a Hermitian operator below the scattering threshold and as a non-Hermitian one above it. For those values of the potential for which bound states lie right at the top of the well the scattering amplitude threshold branch point is an exceptional point, a characteristic of systems with antilinear symmetry at which there are more independent solutions to the Schrödinger equation than there are eigenstates of a then non-Hermitian Hamiltonian. The square well provides an explicit realization of how antilinearity is more general than Hermiticity.

quant-ph↗

arXiv2026

States that grow linearly in time, exceptional points, and zero norm states in the simple harmonic oscillator

The simple harmonic oscillator has a well-known normalizable, positive energy, bound state spectrum. We show that degenerate with each such positive energy eigenvalue there is a non-normalizable positive energy eigenstate whose eigenfunction is orthogonal to that of the standard energy eigenfunction. This class of states is not built on the vacuum that $a$ annihilates, but is instead built on the vacuum that $a^{\dagger} a$ annihilates. These non-normalizable but nonetheless stationary energy eigenstates are accompanied by yet another set of non-normalizable states, states whose wave functions however are not stationary but instead grow linearly in time. With these states not being energy eigenstates, the eigenbasis of the Hamiltonian is incomplete; with the full Hilbert space containing states that are not energy eigenstates. Thus each energy eigenvalue of the harmonic oscillator is an exceptional point at which the Hamiltonian becomes of non-diagonalizable, and thus manifestly non-Hermitian, Jordan-block form. Such non-Hermitian structures occur for Hamiltonians that have an antilinear $PT$ symmetry. As is characteristic of such systems, one can construct a probability conserving inner product that despite the linear in time growth is nonetheless time independent, and not only that, it leads to states with zero norm. In addition, as is again characteristic of $PT$ symmetry, these non-normalizable states can be made normalizable by a continuation into a so-called Stokes wedge domain in the complex plane. In this domain one has a completely consistent quantum theory, one that lives alongside the standard normalizable energy eigenspectrum sector. This thus not quite so simple harmonic oscillator provides an explicit realization of our general contention that antilinearity is more basic to quantum theory than Hermiticity.

quant-ph↗

arXiv2025

Creating a Universe from Nothing as an Alternative to the Cosmological Principle

In the cosmological Robertson-Walker geometry required of the cosmological principle both the Weyl tensor $C^{μλνκ}$ and the Bach tensor $W^{μν}=[2\nabla_κ\nabla_λ-R_{λκ}]C^{μλνκ}$ vanish. In general, in perturbations around the cosmological background neither of the fluctuating $δC^{μλνκ}$ or $δW^{μν}$ would vanish. However, it is possible for $δW^{μν}$ to vanish even as $δC^{μλνκ}$ does not. In this paper we construct an explicit model in which this is the case. The model consists of a 3-tensor gravitational wave fluctuating around a background with a constant negative 3-curvature. The model is exactly solvable and consists purely of geometric quantities with no matter fields at all (i.e., $G^{μν}=0$, $δG^{μν}=0$, $W^{μν}=0$, $δW^{μν}=0$, where $G^{μν}$ is the Einstein tensor). The model can thus be created out of nothing, with creating a universe from nothing thus being an alternative principle to the cosmological principle. The fluctuating gravitational wave contributes to the temperature anisotropy in the cosmic microwave background and its $B$ mode polarization in a calculable manner, one for which we provide a simple analytic way of treating spatial modes that is based on the use of a spatial mode addition theorem. In addition, we provide a treatment of the anisotropy that is based on properties of bandwidth limited functions. Classically by ``nothing" we mean that there are no $T^{μν}$ or $δT^{μν}$ matter field terms. Quantum-mechanically by ``nothing" we mean that all fields other than the gravitational field are in a negative energy mode vacuum state, with the only occupied positive energy modes being graviton modes. As well as use the Bach tensor as a diagnostic, we consider dynamics based on it.

gr-qc↗

arXiv2025

Properties of associated Legendre conical functions

We present some new properties of associated Legendre conical functions of the first and second kind, $P^{-1/2-K}_{-1/2+i τ}(χ)$ and $Q^{-1/2-K}_{-1/2+i τ}(χ)$. In particular we show that with the $τ$-independent $\mathcal{R}^{K}_{n}(χ)=(2π)^{-3/2}\tanh^{-K}χ\sinh^{-1/2}χ[Γ(1+K)]^{-1}\int_{0}^{2π} dω\left (1-\cos ω/\coshχ\right)^{K}e^{inω}$ for any general $K$, we can set $P^{-1/2-K}_{-1/2+i τ}(χ)=2\sum_{n}\mathcal{R}^{K}_{n}(χ)\sin[(τ-in)χ)]/(τ-in)$, where $n$ ranges from $-\ell$ to $\ell$ in unit steps when $K$ is a non-negative integer $\ell$, and from $-\infty$ to $\infty$ in unit steps otherwise. Also we can set $Q^{-1/2-K}_{-1/2+i τ}(χ)=-π\sum_n\mathcal{R}^K_{n-K}(χ)e^{-i χ(τ-i(n-K))}/(τ-i(n-K))$, where $n$ ranges from $0$ to $2\ell$ in unit steps when $K$ is a non-negative integer $\ell$, and from $0$ to $\infty$ in unit steps otherwise. With these forms isolating the entire $τ$ dependence, and especially its associated pole structure, we can use these forms to determine closed form expressions for integrals over $τ$ of associated Legendre conical functions and their products. The $Q^{-1/2-K}_{-1/2+i τ}(χ)$ have an integral representation containing the integral $\int_χ^{\infty}dωe^{iωτ}(\coshω-\coshχ)^{K}$, an integral that only converges at $ω=\infty$ if ${\rm Re}[K]<{\rm Im}[τ]$. We show how to use the divergence of this integral outside of this range in order to characterize the complex $τ$ plane pole structure of $Q^{-1/2-K}_{-1/2+i τ}(χ)$. We present a new treatment of the Borwein integral and the Nyquist-Shannon sampling theorem.

math-ph↗

arXiv2025

Time Advance and Probability Conservation in PT-Symmetric Quantum Mechanics

When excited states decay the time evolution operator $U(t)=e^{-iHt}$ does not obey $U^{\dagger}(t)U(t)=I$. Nonetheless, probability conservation is not lost if one includes both excitation and decay, though it takes a different form. Specifically, if the eigenspectrum of a Hamiltonian is complete, then due to $CPT$ symmetry, a symmetry that holds for all physical systems, there must exist an operator $V$ that effects $VHV^{-1}=H^{\dagger}$, so that $V^{-1}U^{\dagger}(t)VU(t)=I$. In consequence, the time delay associated with decay must be accompanied by an equal and opposite time advance for excitation. Thus when a photon excites an atom the spontaneous emission of a photon from the excited state must occur without any decay time delay at all. An effect of this form together with an associated negative time delay appear to have recently been reported by Sinclair et. al., PRX Quantum \textbf{3}, 010314 (2022) and Angulo et. al., arXiv:2409.03680 [quant-ph].

quant-ph↗

arXiv2025

Exact Solution to Standard Model Hydrodynamic Cosmological Perturbation Theory and its Implications for Acoustic Oscillations

We present an exact solution to standard model cosmological perturbation theory in a matter-dominated, adiabatic, hydrodynamic era. The solution is in the form of hypergeometric functions. While such functions can oscillate with the sound velocity, they can only do so at high frequency. There is thus a maximum wavelength to these oscillations, with this maximum wavelength serving to provide a horizon for acoustic oscillations.

gr-qc↗

arXiv2025

Physics on and off the light cone

We study light-front physics and conformal symmetry, and their interplay both on and off the light cone. The full symmetry of the light cone is conformal symmetry not just Lorentz symmetry. Spontaneously breaking conformal symmetry gives masses to particles and takes them off the light cone. Canonical quantization specifies equal-time commutators on the light cone. Equal instant-time and equal light-front-time commutators look very different, but can be shown to be equivalent by looking at unequal-time commutators. We discuss the connection of the light-front approach to the infinite momentum frame approach, and show that vacuum graphs are outside this framework. We show that there is a light-front structure to both AdS/CFT and the eikonal approximation. While mass generation involves scale breaking mass scales, we show that such mass scales can arise via dynamical symmetry breaking in the presence of scale invariant interactions at a renormalization group fixed point.

hep-th↗

arXiv2024

Pauli-Villars and the ultraviolet completion of Einstein gravity

Through use of the Pauli-Villars regulator procedure we construct a second- plus fourth-order-derivative theory of gravity that serves as an ultraviolet completion of standard second-order-derivative quantum Einstein gravity that is ghost-free, unitary and power counting renormalizable.

hep-th↗

arXiv2023

Determining the normalization of the quantum field theory vacuum, with implications for quantum gravity

In a standard quantum field theory the norm $\langle Ω\vert Ω\rangle$ of the vacuum state is taken to be finite. In this paper we provide a procedure, based on constructing an equivalent wave mechanics, for determining whether or not $\langle Ω\vert Ω\rangle$ actually is finite. We provide an example based on a second-order plus fourth-order scalar field theory, a prototype for quantum gravity, in which it is not. In this example the Minkowski path integral with a real measure diverges though the Euclidean path integral does not. Thus in this example contributions from the Wick rotation contour cannot be ignored. Since $\langle Ω\vert Ω\rangle$ is not finite, use of the standard Feynman rules is not valid. And while these rules not only lead to states with negative norm, they in fact lead to states with infinite negative norm. However, if the fields in that theory are continued into the complex plane, we show that then there is a domain in the complex plane known as a Stokes wedge in which one can define an appropriate time-independent, positive and finite inner product, viz. the $\langle L\vert R\rangle$ overlap of left-eigenstates and right-eigenstates of the Hamiltonian; with the vacuum state then being normalizable, and with there being no states with negative or infinite $\langle L\vert R\rangle$ norm. In this Stokes wedge it is the Euclidean path integral that diverges while the Minkowski path integral does not. The concerns that we raise in this paper only apply to bosons since the matrices associated with their creation and annihilation operators are infinite dimensional. Since the ones associated with fermions are finite dimensional, the fermion theory vacuum is automatically normalizable. Our results are relevant to the construction of a consistent, unitary and renormalizable quantum theory of gravity.

hep-th↗

arXiv2023

Normalization of the vacuum and the ultraviolet completion of Einstein gravity

Second-order-derivative plus fourth-order-derivative gravity is the ultraviolet completion of second-order-derivative quantum Einstein gravity. While it achieves renormalizability through states of negative Dirac norm, the unitarity violation that this would entail can be postponed to Planck energies. As we show in this paper the theory has a different problem, one that occurs at all energy scales, namely that the Dirac norm of the vacuum of the theory is not finite. To establish this we present a procedure for determining the norm of the vacuum in any quantum field theory. With the Dirac norm of the vacuum of the second-order-derivative plus fourth-order-derivative theory not being finite, the Feynman rules that are used to establish renormalizability are not valid, as is the assumption that the theory can be used as an effective theory at energies well below the Planck scale. This lack of finiteness is also manifested in the fact that the Minkowski path integral for the theory is divergent. Because the vacuum Dirac norm is not finite, the Hamiltonian of the theory is not Hermitian. However, it turns out to be $PT$ symmetric. And when one continues the theory into the complex plane and uses the $PT$ symmetry inner product, viz. the overlap of the left-eigenstate of the Hamiltonian with its right-eigenstate, one then finds that for the vacuum this norm is both finite and positive, the Feynman rules now are valid, the Minkowski path integral now is well behaved, and the theory now can serve as a low energy effective theory. Consequently, the theory can now be offered as a fully consistent, unitary and renormalizable theory of quantum gravity.

hep-th↗

arXiv2023

How to quantize gravity and how not to quantize gravity

Taking the quantization of electromagnetism as the paradigm, we show how this procedure cannot work for Einstein gravity. However, it does work for conformal gravity, a fourth-order derivative, renormalizable theory of gravity that Bender and Mannheim have shown to be ghost free. We show that in any gravity theory gravity cannot be quantized canonically. Rather, because of an interplay between the zero-point energy of gravity and that of its matter source, gravity is quantized purely by its coupling to a quantized matter source, with gravity being intrinsically quantum mechanical. Treating the zero-point energy this way provides a solution to the cosmological constant problem. With the gravitational zero-point energy issue having been ignored in standard Einstein gravity, it is not possible to solve the cosmological constant problem in standard gravity, since without the zero-point contribution gravity does not know where the zero of energy is.

gr-qc↗

arXiv2023

Exceptional points for associated Legendre functions of the second kind

We consider the complex $ν$ plane structure of the associated Legendre function of the second kind $Q^{-1/2-K}_ν(\coshρ)$. We find that for any noninteger value for $K$ $Q^{-1/2-K}_ν(\coshρ)$ has an infinite number of poles in the complex $ν$ plane, but for any negative integer $K$ there are no poles at all. For $K=0$ or any positive integer $K$ there is only a finite number of poles, with there only being one single pole (at $ν=0$) when $K=0$. This pattern is characteristic of the exceptional points that appear in a wide variety of physical contexts. However, unusually for theories with exceptional points, $Q^{-1/2-K}_ν(\coshρ)$ has an infinite number of them. Other than in the $PT$-symmetry Jordan-block case, exceptional points usually occur at complex values of parameters. While not being Jordan-block exceptional points themselves, the exceptional points associated with the $Q^{-1/2-K}_ν(\coshρ)$ nonetheless occur at real values of $K$.

math-ph↗

arXiv2022

Imprint of galactic rotation curves and metric fluctuations on the recombination era anisotropy

In applications of the conformal gravity theory it has been shown that a scale of order 105 Mpc due to large scale inhomogeneities such as clusters of galaxies is imprinted on the rotation curves of galaxies. Here we show that this same scale is imprinted on recombination era anisotropies in the cosmic microwave background. We revisit an analysis due to Mannheim and Horne, to show that in the conformal gravity theory the length scale of metric signals that originate in the primordial nucleosynthesis era at $10^{9\circ}$K can fill out the entire recombination era sky. Similarly, the length scale of acoustic signals that originate at $10^{13\circ}$K can also fill out the entire recombination era sky. We show that the amplitudes of metric fluctuations that originate in the nucleosynthesis era can grow by a factor of $10^{12}$ at recombination, and by a factor of $10^{18}$ at the current time. In addition we find that without any period of exponential expansion a length scale as small as $10^{-33}$ cm can grow to the size of the recombination sky if it begins to grow at a temperature of order $10^{33}$ degrees.

astro-ph.CO↗

arXiv2022

Critique of the use of geodesics in astrophysics and cosmology

Since particles obey wave equations, in general one is not free to postulate that particles move on the geodesics associated with test particles. Rather, for this to be the case one has to be able to derive such behavior starting from the equations of motion that the particles obey, and to do so one can employ the eikonal approximation. To see what kind of trajectories might occur we explore the domain of support of the propagators associated with the wave equations. For a minimally coupled massless scalar field the domain of support in curved space is shown to not be restricted to the light cone, while for a conformally coupled massless scalar field the curved space domain is only restricted to the light cone if it propagates in a conformal to flat background. Consequently, eikonalization does not in general lead to null geodesics for curved space massless rays even though it does lead to straight line trajectories in flat spacetime. Equal remarks apply to the massless conformal invariant Maxwell equations. However, for massive particles one does obtain standard geodesic behavior this way, since they do not propagate on the light cone to begin with. Thus depending on how big the curvature actually is, in principle, even if not necessarily in practice, the standard null-geodesic-based gravitational bending formula and the general behavior of propagating light rays are in need of modification in regions with high enough curvature. We show that relativistic eikonalization has an intrinsic light-front structure, and show that eikonalization in a theory with local conformal symmetry leads to trajectories that are only globally conformally symmetric. Normals to wavefronts follow the eikonal trajectories, with these trajectories being the trajectories along which energy and momentum are transported.

gr-qc↗

arXiv2022

Structure of conformal gravity in the presence of a scale breaking scalar field

We revisit the structure of conformal gravity in the presence of a c-number, conformally coupled, long range, macroscopic scalar field. And in the static, spherically symmetric case discuss two classes of exact exterior solutions, in one of which the scalar field has a constant value and in the other, which is due to Brihaye and Verbin, it has a radial dependence. In light of these two solutions Horne and then Hobson and Lasenby raised the concern that the fitting of conformal gravity to galactic rotation curves had been misapplied and thus called the successful fitting of the conformal theory into question. We show that the analysis of Brihaye and Verbin is not actually general, but is nonetheless valid in the particular case that they studied. For the analyses of Horne and of Hobson and Lasenby we show that this macroscopic scalar field is not related to the mass generation that is required in a conformal theory. Rather, not just in conformal gravity, but also in standard Einstein gravity, the presence of such a long range scalar field would lead to test particles whose masses would be of the same order as the masses of the galaxies around which they orbit. Since particle masses are not at all of this form, such macroscopic fields cannot be responsible for mass generation; and the existence of any such mass-generating scalar fields can be excluded, consistent with there actually being no known massless scalar particles in nature. Instead, mass generation has to be due to c-number vacuum expectation values of q-number fields. Such expectation values are microscopic not macroscopic and only vary within particle interiors, giving particles an extended, baglike structure, as needed for localization in a conformal theory. And being purely internal they have no effect on galactic orbits, to thus leave the good conformal gravity fitting to galactic rotation curves intact.

gr-qc↗

arXiv2021

Solution to the ghost problem in higher-derivative gravity

With standard Einstein gravity not being renormalizable at the quantum level there is much interest in studying higher-derivative quantum gravity theories. Thus just as a Ricci-scalar-based action produces a propagator that behaves as a non-renormalizable $1/k^2$ at large $k^2$, an action based on the square of the Ricci scalar behaves as a renormalizable $1/k^4$ at large $k^2$. An action based on both the Ricci scalar and its square leads to a renormalizable propagator of the generic Pauli-Villars form. However, given the form of the Hamiltonian and the propagator such theories are thought to be plagued by either energies that are unbounded from below or states of negative Dirac norm (the overlap of a ket with its Hermitian conjugate bra). But when one constructs the quantum Hilbert space one finds (Bender and Mannheim) that in fact neither of these problems is actually present. The Hamiltonian turns out to not be Hermitian but to instead have an antilinear $PT$ symmetry, and for this symmetry the needed inner product is the overlap of a ket with its $PT$ conjugate bra. And this inner product is positive definite. Moreover, for the pure $1/k^4$ propagator the Hamiltonian turns out to not be diagonalizable, and again there are no states of negative energy or of negative norm. Instead there are states of zero norm, non-standard but perfectly acceptable states that serve to maintain probability conservation. With the locally conformal invariant fourth-order derivative conformal gravity theory being in this category, it can be offered as a candidate theory of quantum gravity that is renormalizable and unitary in four spacetime dimensions.

hep-th↗

arXiv2021

Extension of the Goldstone and the Englert-Brout-Higgs mechanisms to non-Hermitian theories

We discuss the extension of the Goldstone and Englert-Brout-Higgs mechanisms to non-Hermitian Hamiltonians that possess an antilinear PT symmetry. We study a model due to Alexandre, Ellis, Millington and Seynaeve and show that for the spontaneous breakdown of a continuous global symmetry we obtain a massless Goldstone boson in all three of the antilinear symmetry realizations: eigenvalues real, eigenvalues in complex conjugate pairs, and eigenvalues real but eigenvectors incomplete. In this last case we show that it is possible for the Goldstone boson mode to be a zero-norm state. For the breakdown of a continuous local symmetry the gauge boson acquires a non-zero mass by the Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the fact that the continuous local symmetry has been spontaneously broken, the gauge boson remains massless.

hep-th↗