scalar-field

↑ The line integral over a scalar field f can be thought of as the area under the curve C along a surface z = f(x,y), described by the field.

Line integral

In mathematics, a line integral (sometimes called a path integral, contour integral, or curve integral; not to be confused with calculating arc length using integration) is an integral where the function to be integrated is evaluated along a curve.

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals.

Energy-momentum tensor for a scalar field

Energy-momentum tensor for a scalar field

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It is claimed in [1] (3.2.1) that the momentum components of the energy-momentum tensor was found to be

\begin{equation}\label{eqn:noetherCurrentScalarField:20}
\Be_n \int d^3 x T^{0 n} = \int d^3 k \Bk a_k^\dagger a_k.
\end{equation}

I don’t see this result anywhere, so let’s calculate it.

First, from the Noether current for the scalar…

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arxiv.org
[1602.00708] Poisson algebras for non-linear field theories in the Cahiers topos

[ Authors ]
Marco Benini, Alexander Schenkel
[ Abstract ]
We develop an approach to construct Poisson algebras for non-linear scalar field theories that is based on the Cahiers topos model for synthetic differential geometry. In this framework the solution space of the field equation carries a natural smooth structure and, following Zuckerman’s ideas, we can endow it with a presymplectic current. We formulate the Hamiltonian vector field equation in this setting and show that it selects a family of observables which forms a Poisson algebra. Our approach provides a clean splitting between geometric and algebraic aspects of the construction of a Poisson algebra, which are sufficient to guarantee existence, and analytical aspects that are crucial to analyze its properties.

Hamiltonian for a scalar field

Hamiltonian for a scalar field

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In [1] it is left as an exersize to expand the scalar field Hamiltonian in terms of the raising and lowering operators. Let’s do that.

The field operator expanded in terms of the raising and lowering operators is

\begin{equation}\label{eqn:scalarFieldHamiltonian:20}
\phi(x) =
\int \frac{ d^3 k}{ (2 \pi)^{3/2} \sqrt{ 2 \omega_k } } \lr{
a_k…

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arxiv.org
[1602.01488] Entanglement Entropy Renormalization for the NC scalar field coupled to classical BTZ geometry

[ Authors ]
Tajron Jurić, Andjelo Samsarov
[ Abstract ]
In this work, we consider a noncommutative (NC) massless scalar field coupled to the classical nonrotational BTZ geometry. In a manner of the theories where the gravity emerges from the underlying scalar field theory, we study the effective action and the entropy derived from this noncommutative model. In particular, the entropy is calculated by making use of the two different approaches, the brick wall method and the heat kernel method designed for spaces with conical singularity. We show that the UV divergent structures of the entropy, obtained through these two different methods, agree with each other. It is also shown that the same renormalization condition that removes the infinities from the effective action can also be used to renormalize the entanglement entropy for the same system. Besides, the interesting feature of the NC model considered here is that it allows an interpretation in terms of an equivalent system comprising of a commutative massive scalar field, but in a modified geometry; that of the rotational BTZ black hole, the result that hints at a duality between the commutative and noncommutative systems in the background of a BTZ black hole.

arxiv.org
[1602.01289] A Scaling Relation in Inhomogeneous Cosmology with k-essence scalar fields

[ Authors ]
Debashis Gangopadhyay, Somnath Mukherjee
[ Abstract ]
We obtain a scaling relation for spherically symmetric k-essence scalar fields $\phi(r,t)$ for an inhomogeneous cosmology with the Lemaitre-Tolman- Bondi (LTB) metric. We show that this scaling relation reduces to the known relation for a homogeneous cosmology when the LTB metric reduces to the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric under certain identifications of the metric functions. A k-essence lagrangian is set up and the Euler-Lagrangian equations solved assuming $\phi(r,t)=\phi_{1}® + \phi_{2}(t)$. The solutions enable the LBT metric functions to be related to the fields. The LTB inhomogeneous universe exhibits late time accelerated expansion i.e.cosmic acceleration driven by negative pressure.

We develop an approach to construct Poisson algebras for non-linear scalar field theories that is based on the Cahiers topos model for synthetic differential geometry. In this framework the solution space of the field equation carries a natural smooth structure and, following Zuckerman's ideas, we can endow it with a presymplectic current. We formulate the Hamiltonian vector field equation in this setting and show that it selects a family of observables which forms a Poisson algebra. Our approach provides a clean splitting between geometric and algebraic aspects of the construction of a Poisson algebra, which are sufficient to guarantee existence, and analytical aspects that are crucial to analyze its properties.

http://arxiv.org/abs/1602.00708

arxiv.org
[1602.00809] Planck constraints on scalar-tensor cosmology and the variation of the gravitational constant

[ Authors ]
Junpei Ooba, Kiyotomo Ichiki, Takeshi Chiba, Naoshi Sugiyama
[ Abstract ]
Cosmological constraints on the scalar-tensor theory of gravity by analyzing the angular power spectrum data of the cosmic microwave background (CMB) obtained from the Planck 2015 results are presented. We consider the harmonic attractor model, in which the scalar field has a harmonic potential with curvature ($\beta$) in the Einstein frame and the theory relaxes toward the Einstein gravity with time. Analyzing the ${\it TT}$, ${\it EE}$, and ${\it TE}$ CMB data from Planck by the Markov Chain Monte Carlo method, we find that the present-day deviation from the Einstein gravity (${\alpha_0}^2$) is constrained as ${\alpha_0}^2<1.5\times10^{-4-20\beta^2}\ (2\sigma)$ and ${\alpha_0}^2<2.0\times10^{-3-20\beta^2}\ (4\sigma)$ for $0<\beta<0.45$. The time variation of the effective gravitational constant between the recombination and the present epochs is constrained as $G_{\rm rec}/G_0<1.0030\ (2\sigma)$ and $G_{\rm rec}/G_0<1.0067\ (4\sigma)$. We also find that the constraints are little affected by extending to nonflat cosmological models because the diffusion damping effect revealed by Planck breaks the degeneracy of the projection effect.

arxiv.org
[1602.00089] Casimir energy between two parallel plates and projective representation of Poincaré group

[ Authors ]
Takamaru Akita, Mamoru Matsunaga
[ Abstract ]
The Casimir effect is a physical manifestation of zero point energy of quantum vacuum. In a relativistic quantum field theory, Poincar'e symmetry of the theory seems, at first sight, to imply that non-zero vacuum energy is inconsistent with translational invariance of the vacuum. In the setting of two uniform boundary plate at rest, quantum fields outside the plates have (1+2)-dimensional Poincar'e symmtry. Taking a massless scalar field as an example, we have examined the consistency between the Poincar'e symmetry and the existence of the vacuum enegy. We note that, in quantum theory, symmetries are represented projectively in general and show that the Casimir energy is connected to central charges appearing in the algebra of generators in the projective representations.

arxiv.org
[1602.00699] The DBI Action, Higher-derivative Supergravity, and Flattening Inflaton Potentials

[ Authors ]
Sjoerd Bielleman, Luis E. Ibanez, Francisco G. Pedro, Irene Valenzuela, Clemens Wieck
[ Abstract ]
In string theory compactifications it is common to find an effective Lagrangian for the scalar fields with a non-canonical kinetic term. We study the effective action of the scalar position moduli of Type II D$p$-branes. In many instances the kinetic terms are in fact modified by a term proportional to the scalar potential itself. This can be linked to the appearance of higher-dimensional supersymmetric operators correcting the K"ahler potential. We identify the supersymmetric dimension-eight operators describing the $\alpha’$ corrections captured by the D-brane Dirac-Born-Infeld action. Our analysis then allows an embedding of the D-brane moduli effective action into an $\mathcal N = 1$ supergravity formulation. The effects of the potential-dependent kinetic terms may be very important if one of the scalars is the inflaton, since they lead to a flattening of the scalar potential. We analyze this flattening effect in detail and compute its impact on the CMB observables for single-field inflation with monomial potentials.

arxiv.org
[1602.00468] Hamiltonian constraint formulation of classical field theories

[ Authors ]
Vaclav Zatloukal
[ Abstract ]
Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined via the (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive the local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton-Jacobi equation applicable in the field theory then follows readily. In addition, we discuss the relation between symmetries and conservation laws, and derive a Hamiltonian version of the Noether theorem, where the Noether currents are identified as the classical momentum contracted with the symmetry-generating vector fields. The general formalism is illustrated by two examples: the scalar field theory, and the string theory.

arxiv.org
[1602.00262] Topological Excitations in Magnetic Materials

[ Authors ]
D. Bazeia, M.M. Doria, E.I.B. Rodrigues
[ Abstract ]
In this work we propose a new route to describe topological excitations in magnetic systems through a single real scalar field. We show here that spherically symmetric structures in two spatial dimensions, which map helical excitations in magnetic materials, admit this formulation and can be used to model skyrmion-like structures in magnetic materials.

arxiv.org
[1601.08182] Casimir entropy and internal energy of the objects in fluctuating scalar and electromagnetic fields

[ Authors ]
Marjan Jafari
[ Abstract ]
Casimir entropy is an important aspect of casimir effect.In this paper,we employ the path integral method to derive the total relation for casimir entropy and internal energy of arbitrary shaped objects in the presence of two,three and four dimensions scalar fields and electromagnetic field.We obtain the casimir entropy and internal energy of two nanoribbon immersed in scalar field and two nanospheres immersed in scalar field and electromagnetic field.The casmir entropy of two nanospheres immersed in the electromagnetic field in small interval of temperature variations,shown a different behavior.

arxiv.org
[1602.00916] Stress tensor for a scalar field in a spatially varying background potential: Divergences, &quot;renormalization,&quot; anomalies, and Casimir forces

[ Authors ]
Kimball A. Milton, Stephen A. Fulling, Prachi Parashar, Pushpa Kalauni, Taylor Murphy
[ Abstract ]
Motivated by a desire to understand quantum fluctuation energy densities and stress within a spatially varying dielectric medium, we examine the vacuum expectation value for the stress tensor of a scalar field with arbitrary conformal parameter, in the background of a given potential that depends on only one spatial coordinate. We regulate the expressions by incorporating a temporal-spatial cutoff in the (imaginary) time and transverse-spatial directions. The divergences are captured by the zeroth- and second-order WKB approximations. Then the stress tensor is “renormalized” by omitting the terms that depend on the cutoff. The ambiguities that inevitably arise in this procedure are both duly noted and restricted by imposing certain physical conditions; one result is that the renormalized stress tensor exhibits the expected trace anomaly. The renormalized stress tensor exhibits no pressure anomaly, in that the principle of virtual work is satisfied for motions in a transverse direction. We then consider a potential that defines a wall, a one-dimensional potential that vanishes for $z<0$ and rises like $z^\alpha$, $\alpha>0$, for $z>0$. The full finite stress tensor is computed numerically for the two cases where explicit solutions to the differential equation are available, $\alpha=1$ and 2. The energy density exhibits an inverse linear divergence as the boundary is approached from the inside for a linear potential, and a logarithmic divergence for a quadratic potential. Finally, the interaction between two such walls is computed, and it is shown that the attractive Casimir pressure between the two walls also satisfies the principle of virtual work (i.e., the pressure equals the negative derivative of the energy with respect to the distance between the walls).

arxiv.org
[1602.00682] A Perfect Fluid in Lagrangian Formulation due to Generalized Three-Form Field

[ Authors ]
Pitayuth Wongjun
[ Abstract ]
A Lagrangian formulation of perfect fluid due to a non-canonical three-form field is investigated. The thermodynamic quantities such as energy density, pressure and the four-velocity are obtained and then analyzed by comparing with the k-essence scalar field. The non-relativistic matter due to the generalized three-form field with the equation of state parameter being zero is realized while it might not be possible for the k-essence scalar field. We also found that non-adiabatic pressure perturbations can be possibly generated. The fluid dynamics of the perfect fluid due to the three-form field corresponds to the system in which the number of particles is not conserved. We argue that it is interesting to use this three-form field to represent the dark matter for the interaction theory between dark matter and dark energy.

arxiv.org
[1601.08140] Eternal Hilltop Inflation

[ Authors ]
Gabriela Barenboim, William H. Kinney, Wan-Il Park
[ Abstract ]
We consider eternal inflation in hilltop-type inflation models, favored by current data, in which the scalar field in inflation rolls off of a local maximum of the potential. Unlike chaotic or plateau-type inflation models, in hilltop inflation the region of field space which supports eternal inflation is finite, and the expansion rate $H_{EI}$ during eternal inflation is almost exactly the same as the expansion rate $H_*$ during slow roll inflation. Therefore, in any given Hubble volume, there is a finite and calculable expectation value for the lifetime of the “eternal” inflation phase, during which quantum flucutations dominate over classical field evolution. We show that despite this, inflation in hilltop models is nonetheless eternal in the sense that the volume of the spacetime at any finite time is exponentially dominated by regions which continue to inflate. This is true regardless of the energy scale of inflation, and eternal inflation is supported for inflation at arbitrarily low energy scale.