arXiv subjects

Philip D. Mannheim

Publications and source records attributed to Philip D. Mannheim.

At least 37 records · Page 2Linked to original sources

arXiv2021

Antilinear Symmetry and the Ghost Problem in Quantum Field Theory

The recognition that the eigenvalues of a non-Hermitian Hamiltonian could all be real if the Hamiltonian had an antilinear symmetry such as $PT$ stimulated new insight into the underlying structure of quantum mechanics. Specifically, it lead to the realization that Hilbert space could be richer than the established Dirac approach of constructing inner products out of ket vectors and their Hermitian conjugate bra vectors. With antilinear symmetry one must instead build inner products out of ket vectors and their antilinear conjugates, and it is these inner products that would be time independent in the non-Hermitian but antilinearly symmetric case even as the standard Dirac inner products would not be. Moreover, and in a sense quite remarkably, antilinear symmetry could address not only the temporal behavior of the inner product but also the issue of its overall sign, with antilinear symmetry being capable of yielding a positive inner product in situations where the standard Dirac inner product is found to have ghostlike negative signature. Antilinear symmetry thus solves the ghost problem in quantum field theory by showing that when a theory has ghost states it is being formulated in the wrong Hilbert space, with antilinear symmetry providing a Hilbert space that is ghost free. Antilinear symmetry does not actually get rid of the ghost states. Rather, it shows that the reasoning that led one to think that ghosts were present in the first place is faulty.

hep-th↗

arXiv2021

Cosmological Fluctuations on the Light Cone

In studying temperature fluctuations in the cosmic microwave background Weinberg has noted that some ease of calculation and insight can be achieved by looking at the structure of the perturbed light cone on which the perturbed photons propagate. In his approach Weinberg worked in a specific gauge and specialized to fluctuations around the standard Robertson-Walker cosmological model with vanishing spatial three-curvature. In this paper we generalize this analysis by providing a gauge invariant treatment in which no choice of gauge is made, and by considering geometries with non-vanishing spatial three-curvature. By using the scalar, vector, tensor fluctuation basis we find that the relevant gauge invariant combinations that appear in the light cone temperature fluctuations have no explicit dependence on the spatial curvature even if the spatial curvature of the background geometry is nonvanishing. We find that a not previously considered, albeit not too consequential, temperature fluctuation at the observer has to be included in order to enforce gauge invariance. As well as working with comoving time we also work with conformal time in which a background metric of any given spatial three-curvature can be written as a time-dependent conformal factor (the comoving time expansion radius as written in conformal time) times a static Robertson-Walker geometry of the same spatial three-curvature. For temperature fluctuations on the light cone this conformal factor drops out identically. Thus the gauge invariant combinations that appear in the photon temperature fluctuations have no explicit dependence on either the conformal factor or the spatial three-curvature at all.

gr-qc↗

arXiv2021

External Field Effect in Gravity

In both Newtonian gravity and Einstein gravity there is no force on a test particle located inside a spherical cavity cut out of a static, spherically symmetric mass distribution. Inside the cavity exterior matter is decoupled and there is no external field effect that could act on the test particle. However, for potentials other than the Newtonian potential or for geometries other than Ricci flat ones this is no longer the case, and there then is an external field effect. We explore this possibility in various alternate gravity scenarios, and suggest that such (Machian) external field effects can serve as a diagnostic for gravitational theory.

gr-qc↗

arXiv2021

Exact solution to perturbative conformal cosmology from recombination until the current era

In a previous paper (P. D. Mannheim, Phys. Rev. D 102, 123535 (2020)) we studied cosmological perturbation theory in the cosmology associated with the fourth-order derivative conformal gravity theory, and provided an exact solution to the theory in the recombination era. In this paper we present an exact solution that holds all the way from recombination until the current era.

gr-qc↗

arXiv2020

Exact solution to perturbative conformal cosmology in the recombination era

We study cosmological perturbation theory in the cosmology associated with conformal gravity, establish the validity of the decomposition theorem for it, and then use the theorem to provide an exact solution to the theory in the recombination era. Central to our approach is the use of a fully gauge invariant formulation of the cosmological fluctuations equations. In the recombination era not only is perturbation theory applicable, because of its specific structure in the conformal case the fluctuation equations are found to greatly simplify. Using a master equation for scalar, vector and tensor fluctuation modes we show that the radial equations for the three-dimensional vector and tensor modes are respectively the same as those of scalar modes in five and seven spatial dimensions. This enables us to construct normalization conditions for the three-dimensional modes.

gr-qc↗

arXiv2020

Comparing light-front quantization with instant-time quantization

We compare light-front quantization and instant-time quantization both at the level of operators and at the level of their Feynman diagram matrix elements. At the level of operators light-front quantization and instant-time quantization lead to equal light-front time commutation (or anticommutation) relations that appear to be quite different from equal instant-time commutation (or anticommutation) relations. Despite this we show that at unequal times instant-time and light-front commutation (or anticommutation) relations actually can be transformed into each other, with it only being the restriction to equal times that makes the commutation (or anticommutation) relations appear to be so different. While our results are valid for both bosons and fermions, for fermions there are subtleties associated with tip of the light cone contributions that need to be taken care of. At the level of Feynman diagrams we show for non-vacuum Feynman diagrams that the pole terms in four-dimensional light-front Feynman diagrams reproduce the three-dimensional light-front on-shell Hamiltonian Fock space formulation in which the light-front energy and light-front momentum are on shell. However, because of circle at infinity contributions we show that this equivalence fails for four-dimensional light-front vacuum tadpole diagrams. Then, precisely because of these circle at infinity contributions, light-front vacuum tadpole diagrams are not only nonzero, they are actually equal to instant-time vacuum tadpole diagrams. Light-front vacuum diagrams are not correctly describable by the on-shell Hamiltonian formalism, and thus not by the closely related infinite momentum frame prescription either. With the transformation from instant-time fields to light-front fields being a spacetime translation, not only are instant-time quantization and light-front quantization equivalent, they are unitarily equivalent.

hep-ph↗

arXiv2020

Equivalence of light-front quantization and instant-time quantization

Commutation or anticommutation relations quantized at equal instant time and commutation or anticommutation relations quantized at equal light-front time not only cannot be transformed into each other, they take completely different forms. While they would thus appear to describe different theories, we show that this is not in fact the case. By looking not at equal times but at unequal times, we show that unequal instant-time commutation or anticommutation relations are completely equivalent to unequal light-front time commutation or anticommutation relations. Light-front quantization and instant-time quantization are thus the same, with it being only the restriction to equal times that makes them look different. However for fermions there is a caveat, as the light-front anticommutation relations involve projection operators acting on the fermion fields. Nonetheless, not only can one still derive fermion unequal light-front time anticommutators starting from unequal instant-time ones, one can even derive unequal instant-time fermion anticommutators starting from unequal light-front time anticommutators even though the fermion projection operators that are relevant in the light-front case are not invertible. To establish the equivalence for gauge fields we present a quantization procedure that does not involve zero-mode singularities that are commonly encountered in light-front gauge field studies. We also study time-ordered products of fields, and again show the equivalence despite the fact that there are additional terms in the fermion light-front case. We establish our results first for free theories, and then to all orders in interacting theories though comparison of the instant-time and light-front Lehmann representations. Finally, we compare instant-time Hamiltonians and light-front Hamiltonians, and show that in the instant-time rest frame they give identical results.

hep-ph↗

arXiv2020

Ghost Problems from Pauli-Villars to Fourth-Order Quantum Gravity and their Resolution

We review the history of the ghost problem in quantum field theory from the Pauli-Villars regulator theory to currently popular fourth-order derivative quantum gravity theories. While these theories all appear to have unitarity-violating ghost states with negative norm, we show that in fact these ghost states only appear because the theories are being formulated in the wrong Hilbert space. In these theories the Hamiltonians are not Hermitian but instead possess an antilinear symmetry. Consequently, one cannot use inner products that are built out of states and their Hermitian conjugates. Rather, one must use inner products built out of states and their conjugates with respect to the antilinear symmetry, and these latter inner products are positive. In this way one can build quantum theories of gravity in four spacetime dimensions that are unitary.

hep-th↗

arXiv2020

Light-front quantization is the same as instant-time quantization

Commutation or anticommutation relations quantized at equal instant time and commutation or anticommutation relations quantized at equal light-front time cannot be transformed into each other. While they would thus appear to describe different theories, we show that this is not in fact the case. In instant-time quantization unequal instant-time commutation or anticommutation relations for free scalar, fermion, or gauge boson fields are c-numbers. We show that when these unequal instant-time commutation or anticommutation relations are evaluated at equal light-front time they are identical to the equal light-front time commutation or anticommutation relations. Light-front quantization and instant-time quantization are thus the same and thus describe the same physics.

hep-th↗

arXiv2019

Three-dimensional and four-dimensional scalar, vector, tensor cosmological fluctuations and the cosmological decomposition theorem

In cosmological perturbation theory it is convenient to use the scalar, vector, tensor (SVT) basis as defined according to how these components transform under 3-dimensional rotations. In attempting to solve the fluctuation equations that are automatically written in terms of gauge-invariant combinations of these components, the equations are taken to break up into separate SVT sectors, the decomposition theorem. Here, without needing to specify a gauge, we solve the fluctuation equations exactly for some standard cosmologies, to show that in general the various gauge-invariant combinations only separate at a higher-derivative level. To achieve separation at the level of the fluctuation equations themselves one has to assume boundary conditions for the higher-derivative equations. While asymptotic conditions suffice for fluctuations around a dS background or a $k=0$ RW background, for fluctuations around a $k\neq 0$ RW background one additionally has to require that the fluctuations be well-behaved at the origin. We show that in certain cases the gauge-invariant combinations themselves involve both scalars and vectors. For such cases there is no decomposition theorem for the individual SVT components themselves, but for the gauge-invariant combinations there still can be. Given the lack of manifest covariance in defining a basis with respect to 3-dimensional rotations, we introduce an alternate SVT basis whose components are defined according to how they transform under 4-dimensional general coordinate transformations. With this basis the fluctuation equations greatly simplify, and while one can again break them up into separate gauge-invariant sectors at the higher-derivative level, in general we find that even with boundary conditions we do not obtain a decomposition theorem in which the fluctuations separate at the level of the fluctuation equations themselves.

gr-qc↗

arXiv2019

Is dark matter fact or fantasy? -- clues from the data

We discuss arguments both in favor of and against dark matter. With the repeated failure of experiment to date to detect dark matter we discuss what could be done instead, and to this end look for clues in the data themselves. We identify various regularities in galactic rotation curve data that correlate the total gravitational potential with luminous matter rather than dark matter. We identify a contribution to galactic rotation curves coming from the rest of the visible Universe, and suggest that dark matter is just an attempt to describe this global effect in terms of standard local Newtonian gravity within galaxies. Thus the missing mass is not missing at all -- it has been hiding in plain sight all along as the rest of the visible mass in the Universe.

astro-ph.GA↗

arXiv2019

Structure of light front vacuum sector diagrams

We study the structure of scalar field light front quantization vacuum graphs. In instant time quantization both non-vacuum and vacuum graphs can equivalently be described by either the off-shell four-dimensional Feynman diagram approach or the on-shell three-dimensional Fock space approach, with this being the case since the relevant Feynman diagrams are given entirely by pole terms. This is also the case for light front quantization non-vacuum graphs. However this is the not the case for light front vacuum sector diagrams, since then there are also circle at infinity contributions to Feynman diagrams. These non-pole contributions cause vacuum diagrams to be nonzero and to not be given by a light-front Hamiltonian Fock space analysis. The three-dimensional approach thus fails in the light front vacuum sector. In consequence, the closely related infinite momentum frame approach also fails in the light front vacuum sector.

hep-ph↗

arXiv2019

Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories

In recent work Alexandre, Ellis, Millington and Seynaeve have extended the Goldstone theorem to non-Hermitian Hamiltonians that possess a discrete antilinear symmetry such as $PT$. They restricted their discussion to those realizations of antilinear symmetry in which all the energy eigenvalues of the Hamiltonian are real. Here we extend the discussion to the two other realizations possible with antilinear symmetry, namely energies in complex conjugate pairs or Jordan-block Hamiltonians that are not diagonalizable at all. In particular, we show that under certain circumstances it is possible for the Goldstone boson mode itself to be one of the zero-norm states that are characteristic of Jordan-block Hamiltonians. While we discuss the same model as Alexandre et al. our treatment is quite different, though their main conclusion that one can have Goldstone bosons in the non-Hermitian case remains intact. In their paper Alexandre et al. presented a variational procedure for the action in which the surface term played an explicit role, to thus suggest that one has to use such a procedure in order to establish the Goldstone theorem in the non-Hermitian case. However, by taking certain fields that they took to be Hermitian to actually either be anti-Hermitian or be made so by a similarity transformation, we show that we are then able to obtain a Goldstone boson using a completely standard variational procedure. Since we use a standard variational procedure we can readily extend our analysis to a continuous local symmetry by introducing a gauge boson. We show that when we do this the gauge boson acquires a non-zero mass by the 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↗

arXiv2019

Cosmological perturbations in conformal gravity II

In this paper we continue a study of cosmological perturbations in the conformal gravity theory. In previous work we had obtained a restricted set of solutions to the cosmological fluctuation equations, solutions that were required to be both transverse and synchronous. Here we present the general solution. We show that in a conformal invariant gravitational theory fluctuations around any background that is conformal to flat (backgrounds that include the cosmologically interesting Robertson-Walker and de Sitter geometries) can be constructed from the (known) solutions to fluctuations around a flat background. For this construction to hold it is not necessary that the perturbative geometry associated with the fluctuations itself be conformal to flat. Using this construction we show that in a conformal Robertson-Walker cosmology early universe fluctuations grow as $t^4$. We present the scalar, vector, tensor decomposition of the fluctuations in the conformal theory, and compare and contrast our work with the analogous treatment of fluctuations in the standard Einstein gravity theory.

gr-qc↗

arXiv2018

Radial Acceleration and Tully-Fisher Relations in Conformal Gravity

In 2016 McGaugh, Lelli and Schombert established a universal Radial Acceleration Relation for centripetal accelerations in spiral galaxies. Their work showed a strong correlation between observed centripetal accelerations and those predicted by luminous Newtonian matter alone. Through the use of the fitting function that they introduced, mass discrepancies in spiral galaxies can be constrained in a uniform manner that is completely determined by the baryons in the galaxies. Here we present a new empirical plot of the observed centripetal accelerations and the luminous Newtonian expectations, which more than doubles the number of observed data points considered by McGaugh et al. while retaining the Radial Acceleration Relation. If this relation is not to be due to dark matter, it would then have to be due to an alternate gravitational theory that departs from Newtonian gravity in some way. In this paper we show how the candidate alternate conformal gravity theory can provide a natural description of the Radial Acceleration Relation, without any need for dark matter or its free halo parameters. We discuss how the empirical Tully-Fisher relation follows as a consequence of conformal gravity.

astro-ph.GA↗

arXiv2018

Unitarity of loop diagrams for the ghost-like $1/(k^2-M_1^2)-1/(k^2-M_2^2)$ propagator

With fourth-order derivative theories leading to propagators of the generic ghost-like $1/(k^2-M_1^2)-1/(k^2-M_2^2)$ form, it would appear that such theories have negative norm ghost states and are not unitary. However on constructing the associated quantum Hilbert space for the free theory that would produce such a propagator, Bender and Mannheim found that the Hamiltonian of the free theory is not Hermitian but is instead $PT$ symmetric, and that there are in fact no negative norm ghost states, with all Hilbert space norms being both positive and preserved in time. Even though perturbative radiative corrections cannot change the signature of a Hilbert space inner product, nonetheless it is not immediately apparent how such a ghost-like propagator would not then lead to negative probability contributions in loop diagrams. Here we obtain the relevant Feynman rules and show that all states obtained in cutting intermediate lines in loop diagrams have positive norm. Also we show that due to the specific way that unitarity (conservation of probability) is implemented in the theory, negative signatured discontinuities across cuts in loop diagrams are cancelled by a novel and unanticipated contribution of the states in which tree approximation (no loop) graphs are calculated, an effect that is foreign to standard Hermitian theories. Perturbatively, the fourth-order derivative theory is thus viable. The theory associated with the pure massless $1/k^4$ propagator is equally shown to be perturbatively viable.

hep-th↗

arXiv2017

Appropriate Inner Product for PT-Symmetric Hamiltonians

A Hamiltonian $H$ that is not Hermitian can still have a real and complete energy eigenspectrum if it instead is $PT$ symmetric. For such Hamiltonians three possible inner products have been considered in the literature, the $V$ norm, the $PT$ norm, and the $C$ norm. Here $V$ is the operator that implements $VHV^{-1}=H^{\dagger}$, the $PT$ norm is the overlap of a state with its $PT$ conjugate, and $C$ is a discrete linear operator that always exists for any Hamiltonian that can be diagonalized. Here we show that it is the $V$ norm that is the most fundamental as it is always chosen by the theory itself. In addition we show that the $V$ norm is always equal to the $PT$ norm if one defines the $PT$ conjugate of a state to contain its intrinsic $PT$ phase. We discuss the conditions under which the $V$ norm coincides with the $C$ operator norm, and show that in general one should not use the linear $C$ operator but for the purposes that it is used one can instead use the antilinear $PT$ operator itself.

quant-ph↗

arXiv2017

Universal Properties of Galactic Rotation Curves and a First Principles Derivation of the Tully-Fisher Relation

In a recent paper McGaugh, Lelli, and Schombert showed that in an empirical plot of the observed centripetal accelerations in spiral galaxies against those predicted by the Newtonian gravity of the luminous matter in those galaxies the data points occupied a remarkably narrow band. While one could summarize the mean properties of the band by drawing a single mean curve through it, by fitting the band with the illustrative conformal gravity theory with fits that fill out the width of the band we show here that the width of the band is just as physically significant. We show that at very low luminous Newtonian accelerations the plot can become independent of the luminous Newtonian contribution altogether, but still be non-trivial due to the contribution of matter outside of the galaxies (viz. the rest of the visible universe). We present a new empirical plot of the difference between the observed centripetal accelerations and the luminous Newtonian expectations as a function of distance from the centers of galaxies, and show that at distances greater than 10 kpc the plot also occupies a remarkably narrow band, one even close to constant. Using the conformal gravity theory we provide a first principles derivation of the empirical Tully-Fisher relation.

astro-ph.GA↗