SUPERSYMMETRY AND DIMENSIONALITY IN THE SUPERSTRING THEORY

2009 ◽  
Vol 24 (23) ◽  
pp. 4373-4388 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Momentum Tensor ◽  
Einstein Equations ◽  
Quadratic Term ◽  
World Sheet ◽  
Superstring Theory ◽  
Original Source ◽  
Heterotic String Theory ◽  
Quartic Term ◽  
Dimensional Minkowski Space ◽  
Global Supersymmetry

The realization of non-linear global supersymmetry in the superstring theory requires the quadratic fermionic Lagrangian [Formula: see text], defined from the D-dimensional, Minkowski-space energy–momentum tensor Tmn, to have the same form as the quadratic gravitational contribution [Formula: see text] to the superstring Lagrangian. Here, we prove that this condition is only satisfied for the heterotic string theory after reduction to D = 4, irrespective of whether the original source of [Formula: see text] in ten or twenty-six dimensions is the quadratic term [Formula: see text] or the quartic term [Formula: see text]. If [Formula: see text] derives from [Formula: see text], the solution is D = 4 (or the unphysical value D = 1), while if we suppose that D≠4 and [Formula: see text] dominates, we obtain the (singular) solution (D-2)3 = 0. The world sheet is also discussed. The bosonic string and type-II superstring, on the other hand, yield solutions for D which are complex, non-integral, or at the singular point D = 2, where the Einstein equations hold identically.

ON THE QUARTIC HIGHER-DERIVATIVE GRAVITATIONAL TERMS IN THE HETEROTIC SUPERSTRING THEORY

2006 ◽  
Vol 21 (02) ◽  
pp. 373-404 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Dimensional Space ◽  
Flat Space ◽  
Momentum Tensor ◽  
Space Time ◽  
Einstein Equations ◽  
Quadratic Term ◽  
Additional Contribution ◽  
Superstring Theory ◽  
Weyl Spinor ◽  
Higher Derivative

The quartic higher-derivative gravitational terms [Formula: see text] in the heterotic-superstring effective Lagrangian [Formula: see text], defined from the Riemann ten-tensor [Formula: see text], are expanded, after reduction to the conformally-flat physical D-space gij, in terms of the Ricci tensor Rij and scalar R. The resulting quadratic term [Formula: see text] is tachyon-free and agrees exactly with the prediction from global supersymmetry in the nonlinear realization of Volkov and Akulov of the flat-space, quadratic fermionic Lagrangian [Formula: see text] for a massless Dirac or Weyl spinor, only when D = 4, assuming the Einstein equation [Formula: see text] for the energy–momentum tensor. This proves that the heterotic superstring has to be reduced from ten to four dimensions if supersymmetry is to be correctly incorporated into the theory, and it rules out the bosonic string and type-II superstring, for which [Formula: see text] has the different a priori forms ±(R2-4RijRij) derived from [Formula: see text], which also contain tachyons (that seem to remain after the inclusion of a further contribution to [Formula: see text] from [Formula: see text]). The curvature of space–time introduces a mass into the Dirac equation, [Formula: see text], while quadratic, higher-derivative terms [Formula: see text] make an additional contribution to the Einstein equations, these two effects causing a difference between [Formula: see text] and [Formula: see text] on the one hand, and the predictions from [Formula: see text] and [Formula: see text] on the other. The quartic terms [Formula: see text] still possess some residual symmetry, however, enabling us to estimate the radius-squared of the internal six-dimensional space [Formula: see text] in units of the Regge slope-parameter α′ as B r ≈ 1.75, indicating that compactification occurs essentially at the Planck era, due to quantum mechanical processes, when the action evaluated within the causal horizon is S h ~ 1. This symmetry is also discussed with regard to the zero-action hypothesis. The dimensionality D = 4 of space–time is rederived from the Wheeler–DeWitt equation (Schrödinger equation) of quantum cosmology in the mini-superspace approximation, by demanding invariance and positive-semi-definiteness of the potential [Formula: see text] under Wick rotation of the time coordinate, which also determines the three-space to be flat, so that K = 0, and again involves the nonlinearity of gravitation.

scholarly journals Inverse problem: What can we tell about the matter and energy in our Universe from its dimensionality and evolution

2018 ◽  
Vol 27 (07) ◽  
pp. 1841005
Author(s):  
Hanna Makaruk ◽  
James Langenbrunner
Keyword(s):  
Energy Momentum Tensor ◽  
Physical Space ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Time Dimension ◽  
Superstring Theory ◽  
Anisotropic Energy ◽  
The One ◽  
Multidimensional Models ◽  
Additional Space

The most popular theories of everything are various versions of the superstring theory. The theories require existence of additional space dimensions, vibrations of which create the material particles in [Formula: see text] space. The additional space dimensions are understood as being currently smaller than the Planck Length and due to this not directly observable. We search for multidimensional models of the Universe (one time dimension; three isotropic, flat external dimensions, and [Formula: see text]-internal dimensions), which satisfy the multidimensional Einstein equations and which started from the same radius of all of the internal and external dimensions, with an anisotropic energy–momentum tensor. Analytical solution of [Formula: see text]-dimensional Einstein equation in a reparameterized time is reminded and discussed. The energy–momentum tensor is solely responsible for expansion of the external dimensions and shrinking of the internal ones; and to obtain this behavior of the space the tensor needs to fulfill some conditions i.e. the energy–momentum tensor cannot include only radiation, vacuum and baryonic matter. For the behavior of the physical space consistent with the one observed in our Universe, the dark energy and/or dark matter have to exist.

scholarly journals A more natural composite Higgs model

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Hsin-Chia Cheng ◽  
Yi Chung
Keyword(s):  
Standard Model ◽  
Higgs Model ◽  
Quadratic Term ◽  
Beyond The Standard Model ◽  
Higgs Potential ◽  
Little Higgs ◽  
Composite Higgs ◽  
Partial Compositeness ◽  
Quartic Term ◽  
Natural Composite

Abstract Composite Higgs models provide an attractive solution to the hierarchy problem. However, many realistic models suffer from tuning problems in the Higgs potential. There are often large contributions from the UV dynamics of the composite resonances to the Higgs potential, and tuning between the quadratic term and the quartic term is required to separate the electroweak breaking scale and the compositeness scale. We consider a composite Higgs model based on the SU(6)/Sp(6) coset, where an enhanced symmetry on the fermion resonances can minimize the Higgs quadratic term. Moreover, a Higgs quartic term from the collective symmetry breaking of the little Higgs mechanism can be realized by the partial compositeness couplings between elementary Standard Model fermions and the composite operators, without introducing new elementary fields beyond the Standard Model and the composite sector. The model contains two Higgs doublets, as well as several additional pseudo-Nambu-Goldstone bosons. To avoid tuning, the extra Higgs bosons are expected to be relatively light and may be probed in the future LHC runs. The deviations of the Higgs couplings and the weak gauge boson couplings also provide important tests as they are expected to be close to the current limits in this model.

The Einstein pseudo-tensor and the flux integral for perturbed static space-times

1991 ◽  
Vol 435 (1895) ◽  
pp. 645-657 ◽  
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Second Variation ◽  
Common Procedure ◽  
Static Vacuum ◽  
Linear Perturbations ◽  
Maxwell Space ◽  
The Common ◽  
Static Space

The flux integral for axisymmetric polar perturbations of static vacuum space-times, derived in an earlier paper directly from the relevant linearized Einstein equations, is rederived with the aid of the Einstein pseudo-tensor by a simple algorism. A similar earlier effort with the aid of the Landau–Lifshitz pseudo-tensor failed. The success with the Einstein pseudo-tensor is due to its special distinguishing feature that its second variation retains its divergence-free property provided only the equations governing the static space-time and its linear perturbations are satisfied. When one seeks the corresponding flux integral for Einstein‒Maxwell space-times, the common procedure of including, together with the pseudo-tensor, the energy‒momentum tensor of the prevailing electromagnetic field fails. But, a prescription due to R. Sorkin, of including instead a suitably defined ‘Noether operator’, succeeds.

scholarly journals Kodama Time, Entropy Bounds, the Raychaudhuri  Equation, and the Quantum Interest Conjecture

2021 ◽  
Author(s):  
◽  
Gabriel Abreu
Keyword(s):  
General Relativity ◽  
Coordinate System ◽  
Negative Energy ◽  
Specific Form ◽  
Einstein Equations ◽  
Surface Gravity ◽  
Quantum Field ◽  
Raychaudhuri Equation ◽  
Dimensional Minkowski Space ◽  
Entropy Bounds

<p>General Relativity, while ultimately based on the Einstein equations, also allows one to quantitatively study some aspects of the theory without explicitly solving the Einstein equations. These geometrical notions of the theory provide an insight to the nature of more general spacetimes. In this thesis, the Raychaudhuri equation, the choice of the coordinate system, the notions of surface gravity and of entropy, and restrictions on negative energy densities on the form of the Quantum Interest Conjecture, will be discussed. First, using the Kodama vector, a geometrically preferred coordinate system is built. With this coordinate system the usual quantities, such as the Riemann and Einstein tensors, are calculated. Then, the notion of surface gravity is generalized in two different ways. The first generalization is developed considering radial ingoing and outgoing null geodesics, in situations of spherical symmetry. The other generalized surface gravity is a three-vector obtained from the spatial components of the redshifted four acceleration of a suitable set of fiducial observers. This vectorial surface gravity is then used to place a bound on the entropy of both static and rotating horizonless objects. This bound is obtain mostly by classical calculations, with a minimum use of quantum field theory in curved spacetime. Additionally, several improved versions of the Raychaudhuri equation are developed and used in different scenarios, including a two congruence generalization of the equation. Ultimately semiclassical quantum general relativity is studied in the specific form of the Quantum Inequalities, and the Quantum Interest Conjecture. A variational proof of a version of the Quantum Interest Conjecture in (3 + 1)–dimensional Minkowski space is provided.</p>

scholarly journals Kodama Time, Entropy Bounds, the Raychaudhuri  Equation, and the Quantum Interest Conjecture

2021 ◽  
Author(s):  
◽  
Gabriel Abreu
Keyword(s):  
General Relativity ◽  
Coordinate System ◽  
Negative Energy ◽  
Specific Form ◽  
Einstein Equations ◽  
Surface Gravity ◽  
Quantum Field ◽  
Raychaudhuri Equation ◽  
Dimensional Minkowski Space ◽  
Entropy Bounds

<p>General Relativity, while ultimately based on the Einstein equations, also allows one to quantitatively study some aspects of the theory without explicitly solving the Einstein equations. These geometrical notions of the theory provide an insight to the nature of more general spacetimes. In this thesis, the Raychaudhuri equation, the choice of the coordinate system, the notions of surface gravity and of entropy, and restrictions on negative energy densities on the form of the Quantum Interest Conjecture, will be discussed. First, using the Kodama vector, a geometrically preferred coordinate system is built. With this coordinate system the usual quantities, such as the Riemann and Einstein tensors, are calculated. Then, the notion of surface gravity is generalized in two different ways. The first generalization is developed considering radial ingoing and outgoing null geodesics, in situations of spherical symmetry. The other generalized surface gravity is a three-vector obtained from the spatial components of the redshifted four acceleration of a suitable set of fiducial observers. This vectorial surface gravity is then used to place a bound on the entropy of both static and rotating horizonless objects. This bound is obtain mostly by classical calculations, with a minimum use of quantum field theory in curved spacetime. Additionally, several improved versions of the Raychaudhuri equation are developed and used in different scenarios, including a two congruence generalization of the equation. Ultimately semiclassical quantum general relativity is studied in the specific form of the Quantum Inequalities, and the Quantum Interest Conjecture. A variational proof of a version of the Quantum Interest Conjecture in (3 + 1)–dimensional Minkowski space is provided.</p>

scholarly journals Semi-Classical Einstein Equations: Descend to the Ground State

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 74
Author(s):  
Zbigniew Haba
Keyword(s):  
Ground State ◽  
Energy Momentum Tensor ◽  
Local Maximum ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Thermal Dissipation ◽  
Stationary Probability Distribution ◽  
Expectation Value ◽  
Warm Inflation ◽  
The Right

The time-dependent cosmological term arises from the energy-momentum tensor calculated in a state different from the ground state. We discuss the expectation value of the energy-momentum tensor on the right hand side of Einstein equations in various (approximate) quantum pure as well as mixed states. We apply the classical slow-roll field evolution as well as the Starobinsky and warm inflation stochastic equations in order to calculate the expectation value. We show that, in the state concentrated at the local maximum of the double-well potential, the expectation value is decreasing exponentially. We confirm the descent of the expectation value in the stochastic inflation model. We calculate the cosmological constant Λ at large time as the expectation value of the energy density with respect to the stationary probability distribution. We show that Λ ≃ γ 4 3 where γ is the thermal dissipation rate.

scholarly journals Gravitational energy in the framework of embedding and splitting theories

2018 ◽  
Vol 27 (02) ◽  
pp. 1750188 ◽  
Author(s):  
D. A. Grad ◽  
R. V. Ilin ◽  
S. A. Paston ◽  
A. A. Sheykin
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Gravitational Energy ◽  
Einstein Equations ◽  
Field Energy ◽  
Energy Localization ◽  
Embedding Theory ◽  
Isometric Embeddings ◽  
Definition Of ◽  
Noether Current

We study various definitions of the gravitational field energy based on the usage of isometric embeddings in the Regge–Teitelboim approach. For the embedding theory, we consider the coordinate translations on the surface as well as the coordinate translations in the flat bulk. In the latter case, the independent definition of gravitational energy–momentum tensor appears as a Noether current corresponding to global inner symmetry. In the field-theoretic form of this approach (splitting theory), we consider Noether procedure and the alternative method of energy–momentum tensor defining by varying the action of the theory with respect to flat bulk metric. As a result, we obtain energy definition in field-theoretic form of embedding theory which, among the other features, gives a nontrivial result for the solutions of embedding theory which are also solutions of Einstein equations. The question of energy localization is also discussed.

TYPE I SUPERSTRING THEORY IN THE FORM OF OPEN TYPE IIB ONE WITH APPROPRIATE CHOICE OF BOUNDARY CONDITIONS

2009 ◽  
Vol 24 (15) ◽  
pp. 2857-2865
Author(s):  
BOJAN NIKOLIĆ ◽  
BRANISLAV SAZDOVIĆ
Keyword(s):  
Boundary Conditions ◽  
Type I ◽  
World Sheet ◽  
Superstring Theory ◽  
Open Type

We improve relation between type IIB and type I superstring theory. In fact we obtain type I theory as type IIB on the solution of appropriately chosen boundary conditions. We find that type I theory, which is symmetric under world-sheet parity transformation Ω : σ → - σ, beside Ω even fields of the type IIB theory, contains squares of Ω odd fields.

scholarly journals POINT CHARGE SELF-ENERGY IN THE GENERAL RELATIVITY

2005 ◽  
Vol 20 (11) ◽  
pp. 2288-2293
Author(s):  
M. B. GOLUBEV ◽  
S. R. KELNER
Keyword(s):  
Point Charge ◽  
Momentum Tensor ◽  
Generalized Functions ◽  
Einstein Equations ◽  
Classical Electrodynamics ◽  
Classical Solutions ◽  
Complex Nature ◽  
Einstein Tensor ◽  
Self Energy ◽  
Divergence Problem

Singularities in the metric of the classical solutions to the Einstein equations (Schwarzschild, Kerr, Reissner – Nordström and Kerr – Newman solutions) lead to appearance of generalized functions in the Einstein tensor that are not usually taken into consideration. The generalized functions can be of a more complex nature than the Dirac δ-function. To study them, a technique has been used based on a limiting solution sequence. The solutions are shown to satisfy the Einstein equations everywhere, if the energy-momentum tensor has a relevant singular addition of non-electromagnetic origin. When the addition is included, the total energy proves finite and equal to mc2, while for the Kerr and Kerr–Newman solutions the angular momentum is mca. As the Reissner–Nordström and Kerr–Newman solutions correspond to the point charge in the classical electrodynamics, the result obtained allows us to view the point charge self-energy divergence problem in a new fashion.

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