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 A VACUUM-LIKE CONFIGURATION IN GENERAL RELATIVITY AS A MANIFESTATION OF A LORENTZ-INVARIANT MODE OF FIVE-DIMENSIONAL GRAVITY

2007 ◽  
Vol 16 (04) ◽  
pp. 711-736
Author(s):  
VALENTIN D. GLADUSH
Keyword(s):  
General Relativity ◽  
Dimensional Space ◽  
Flat Space ◽  
Mass Function ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Integral Of Motion ◽  
Birkhoff Theorem ◽  
Conformally Invariant ◽  
Dimensional Gravity

A Lorentz-invariant cosmological model is constructed within the framework of five-dimensional gravity. The five-dimensional theorem which is analogical to the generalized Birkhoff theorem is proved, that corresponds to the Kaluza's "cylinder condition." The five-dimensional vacuum Einstein equations have an integral of motion corresponding to this symmetry, the integral of motion is similar to the mass function in general relativity (GR). Space closure with respect to the extra dimensionality follows from the requirement of the absence of a conical singularity. Thus, the Kaluza–Klein (KK) model is realized dynamically as a Lorentz-invariant mode of five-dimensional general relativity. After the dimensional reduction and conformal mapping the model is reduced to the GR configuration. It contains a scalar field with a vanishing conformally invariant energy–momentum tensor on the flat space–time background. This zero mode can be interpreted as a vacuum configuration in GR. As a result the vacuum-like configuration in GR can be considered as a manifestation of the Lorentz-invariant empty five-dimensional space.

THE SUPERSTRING WORLD SHEET IN CURVED SPACE–TIME

2008 ◽  
Vol 17 (01) ◽  
pp. 95-110 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Energy Density ◽  
Dimensional Space ◽  
Flat Space ◽  
Space Time ◽  
Linear Wave ◽  
World Sheet ◽  
Superstring Theory ◽  
Curved Space ◽  
Consistent Solution ◽  
Left And Right

The heterotic superstring theory is formulated on the world sheet hab(τ,σ), in a flat space–time background [Formula: see text], by combining a 26-dimensional left-moving sector, consisting only of bosonic fields [Formula: see text], with a 10-dimensional right-moving sector, consisting of bosonic fields [Formula: see text] and Majorana–Weyl fermionic fields ψA(τ - σ), the string coordinates in 10-dimensional space–time being the sum [Formula: see text] and the right-moving sector supersymmetric due to the decoupling of left- and right-moving modes for closed strings. Here, we generalize the background to a curved space–time ĝAB(XC). The equation of motion for XA(τ,σ), given by Short, is then a non-linear modification of the linear wave equation □XA = 0 which yields decoupled left- and right-moving sectors in flat space–time. The linearity of the transverse modes can be maintained, however, if the metric, after reduction to four dimensions, is allowed to depend only upon co-moving time t ∝ X0, although it can be anisotropic, gij = gij(X0). Specializing to the isotropic, Friedmann cosmological space–time ds2 = dt2 - r2(t) dx2, where the radius function of the three-space dx2, assumed flat, is r(t), with d ln r/dt = 2/3γt for a perfect-fluid source whose pressure and energy-density are related by p = (γ - 1)ρ, we find that the string coordinates Xα(τ,σ)(α = 1,2,3), multiplied by the overall prefactor r ≡ t2/3γ, are purely oscillatory only in Minkowski space or for the value γ = 2/3. This result is equivalent to requiring that the Nordström energy-density ρ N ≡ (3γ - 2)ρ vanish. The case ρ = 2/3 corresponds to a space–time generated by an ensemble of cosmic strings, including the superstrings themselves in a self-consistent solution containing no other matter, and defines the Milne universe. The string is only world-sheet-supersymmetric if ρ N = 0, that is if ρ = 0 or γ = 2/3.

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.

scholarly journals Lorentz-violating effects in three-dimensional QED

2014 ◽  
Vol 29 (22) ◽  
pp. 1450112 ◽  
Author(s):  
R. Bufalo
Keyword(s):  
Quantum Electrodynamics ◽  
Lagrangian Density ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Space Time ◽  
Nonminimal Coupling ◽  
Coupling Term ◽  
Interaction Terms ◽  
Higher Derivative ◽  
Dimensional Space Time

Inspired in discussions presented lately regarding Lorentz-violating interaction terms in B. Charneski, M. Gomes, R. V. Maluf and A. J. da Silva, Phys. Rev. D86, 045003 (2012); R. Casana, M. M. Ferreira Jr., R. V. Maluf and F. E. P. dos Santos, Phys. Lett. B726, 815 (2013); R. Casana, M. M. Ferreira Jr., E. Passos, F. E. P. dos Santos and E. O. Silva, Phys. Rev. D87, 047701 (2013), we propose here a slightly different version for the coupling term. We will consider a modified quantum electrodynamics with violation of Lorentz symmetry defined in a (2+1)-dimensional space–time. We define the Lagrangian density with a Lorentz-violating interaction, where the space–time dimensionality is explicitly taken into account in its definition. The work encompasses an analysis of this model at both zero and finite-temperature, where very interesting features are known to occur due to the space–time dimensionality. With that in mind, we expect that the space–time dimensionality may provide new insights about the radiative generation of higher-derivative terms into the action, implying in a new Lorentz-violating electrodynamics, as well the nonminimal coupling may provide interesting implications on the thermodynamical quantities.

The Superstring Propagator and the Signature of Space-Time

1997 ◽  
Vol 12 (14) ◽  
pp. 987-998 ◽  
Author(s):  
M. D. Pollock
Keyword(s):  
Wave Function ◽  
Apparent Horizon ◽  
Scalar Particle ◽  
Space Time ◽  
Functional Integral ◽  
Global Symmetry ◽  
Hamiltonian Constraint ◽  
Superstring Theory ◽  
Acceptable Solution ◽  
Higher Derivative

The Faddeev (Newton–Wigner) propagator K for the heterotic superstring theory is derived from the Wheeler–DeWitt equation for the wave function of the Universe Ψ, obtained in the four-dimensional (mini-superspace) Friedmann space-time ds2=dt2-a2(t)dx2, after reduction from the ten-action. The effect of higher-derivative terms ℛ2 is to break the local invariance under time reparametrization to a global symmetry t→λt, and consequently there are no ghost or gauge-fixing contributions, a functional integral over the (constant) Lagrange multiplier λ being sufficient to enforce the Hamiltonian constraint implicitly. After Wick rotation of the time, [Formula: see text], the only physically acceptable solution for K decreases exponentially on the Planck time-scale ~ t P , explaining from the quantum cosmological viewpoint why the signature of space-time is Lorentzian rather than Euclidean. This is analogous to the case of the (two-dimensional) free relativistic scalar particle, discussed recently by Redmount and Suen, who found that the propagator decreases exponentially outside the light-cone on the scale of the Compton wavelength of the particle (in accordance with the Heisenberg indeterminacy principle). These two seemingly different forms of acausality are thus physically excluded in the same way. The propagator for the Schwarzschild black hole of mass M is also obtained from the Schrödinger equation for the wave function on the apparent horizon, due to Tomimatsu, and the Hawking temperature T H =(8π M)-1 is derived from the Euclidean form of this equation.

scholarly journals SPIN-3/2 FERMIONS IN TWISTOR FORMALISM

2001 ◽  
Vol 16 (32) ◽  
pp. 2103-2113 ◽  
Author(s):  
MITSUO J. HAYASHI
Keyword(s):  
Twistor Space ◽  
Integrability Condition ◽  
Local Existence ◽  
Flat Space ◽  
Space Time ◽  
Necessary Condition ◽  
Field Equations ◽  
Weyl Spinor ◽  
Cartan Theory ◽  
The Right

Consistency conditions for the local existence of massless spin-3/2 fields have been explored to find the facts that the field equations for massless helicity-3/2 particles are consistent if the space–time is Ricci-flat, and that in Minkowski space–time the space of conserved charges for the fields is its twistor space itself. After considering the twistorial methods to study such massless helicity-3/2 fields, we show in flat space–time that the charges of spin-3/2 fields, defined topologically by the first Chern number of their spin-lowered self-dual Maxwell fields, are given by their twistor space, and in curved space–time that the (anti-)self-duality of the space–time is the necessary condition. Since in N=1 supergravity torsions are the essential ingredients, we generalize our space–time to that with torsion (Einstein–Cartan theory), and investigate the consistency of existence of spin-3/2 fields in this theory. A simple solution to this consistency problem is found: The space–time has to be conformally (anti-)self-dual, left-(or right-) torsion-free. The integrability condition on α-surface shows that the (anti-)self-dual Weyl spinor can be described only by the covariant derivative of the right-(left-)handed torsion.

Newton, Einstein and Barth on time and eternity

2014 ◽  
Vol 67 (4) ◽  
pp. 436-449
Author(s):  
Li Qu
Keyword(s):  
Holy Spirit ◽  
Dimensional Space ◽  
Space Time ◽  
Einstein Equations ◽  
Historical Event ◽  
Theory Of Relativity ◽  
Newtonian Theory ◽  
Absolute Time ◽  
The Past ◽  
The Holy Spirit

AbstractFor two hundred years after 1687, Newton's notion of absolute time dominated the world of physics. However, Newtonian metaphysical absolute time is so ideal that it may only be realised and actualised by God. In the early twentieth century, Einstein breaks this dominant understanding of time fundamentally by his Special Theory of Relativity and General Theory of Relativity. In the Einsteinian paradigm, we are forced to think no longer of space and time but rather to look at a four-dimensional space-time continuum, in which time appears to be more space-like than temporal. The Newtonian theory implies that there is an absolute, dominant point from which the universe can be observed, whereas Einstein argues for the opposite: there can be no vantage perspective and no universal present by which God can divide past and future.Barth takes a trinitarian approach to interpret the concept of time. For Barth, the Father is coeternal with the Son and the Holy Spirit. The eternal immanent Trinity acts concretely as the temporal economic Trinity, thus the triune God is pre-, supra- and post- to us. In actual temporality, the Father, Son and Holy Spirit transcend time concretely in our history and penetrate time absolutely from divine eternity. God's eternity is both transcendent and immanent to human time.Such a trinitarian temporality might serve as a ‘dynamic privileged perspective’ since time, energy and movement are all created by God from eternity. On the one hand, the triune Creator transcends his creature and its creaturely form – time absolutely; on the other hand, even when God enters time and moves together with the time ‘uniformly’ in the Son and the Holy Spirit, he becomes concretely simultaneous with all time. Also the Barthian perspective might provide something which is lacking in Einstein's relative time, i.e. the direction of time from the past to the future. Since every historical event in Einsteinian four-dimensional continuum is posited as a static space-time slice and Einstein equations are time-reversible, there is no ontological difference between time dimensions at all. However, in Barth's trinitarian opinion, such extraordinary events as the creation, resurrection and Pentecost are ontologically superior to other events in human history because they do change our temporality in an absolute way. Penetrated by the trinitarian eternity, those discrete space-time slices also become communicable and hence take genuine temporal characteristics, i.e. the past, present and future.

A Class of Energetically Rigid Gravitational Fields

1974 ◽  
Vol 29 (11) ◽  
pp. 1527-1530 ◽  
Author(s):  
H. Goenner
Keyword(s):  
Flat Space ◽  
Second Fundamental Form ◽  
Momentum Tensor ◽  
Space Time ◽  
Curved Space ◽  
Geometrical Properties ◽  
Gravitational Fields ◽  
The Second Fundamental Form ◽  
Perfect Fluids ◽  
Higher Dimensional

In Einstein's theory, the physics of gravitational fields is reflected by the geometry of the curved space-time manifold. One of the methods for a study of the geometrical properties of space-time consists in regarding it, locally, as embedded in a higher-dimensional flat space. In this paper, metrics admitting a 3-parameter group of motion are considered which form a generalization of spherically symmetric gravitational fields. A subclass of such metrics can be embedded into a five- dimensional flat space. It is shown that the second fundamental form governing the embedding can be expressed entirely by the energy-momentum tensor of matter and the cosmological constant. Such gravitational fields are called energetically rigid. As an application gravitating perfect fluids are discussed.

scholarly journals Physical account of Weyl anomaly from Dirac Sea

2015 ◽  
Vol 30 (25) ◽  
pp. 1550147
Author(s):  
Yoshinobu Habara ◽  
Holger B. Nielsen ◽  
Masao Ninomiya
Keyword(s):  
Gravitational Field ◽  
Regularization Method ◽  
Energy Momentum Tensor ◽  
Dimensional Space ◽  
Momentum Tensor ◽  
Space Time ◽  
Two Dimensional ◽  
Dimensional Space Time ◽  
Dirac Sea ◽  
Mass Terms

We rederive in a physical manner the Weyl anomaly in two-dimensional space–time by considering the Dirac Sea. It is regularized by some bosonic extra species which are formally negatively counted. In fact, we calculate the trace of the energy–momentum tensor in the Dirac Sea in presence of background gravitational field. It has to be regularized, since the Dirac Sea is bottomless and thus causes divergence. The new regularization method consists in adding various massive bosonic species some of which are to be counted negative in the Dirac Sea. The mass terms in the Lagrangian of the regularization fields have a dependence on the background gravitational field.

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