ON THE DERIVATION OF THE WHEELER-DEWITT EQUATION IN THE HETEROTIC SUPERSTRING THEORY

1992 ◽  
Vol 07 (17) ◽  
pp. 4149-4165 ◽  
Author(s):  
M.D. POLLOCK
Keyword(s):  
Quantum Mechanics ◽  
Euler Number ◽  
Number Density ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Effective Lagrangian ◽  
Total Divergence ◽  
Set Up ◽  
The Universe ◽  
Non Locality

It has been shown by Pollock that the Wheeler-DeWitt equation for the wave function of the Universe Ψ cannot be derived for the D-dimensional, heterotic superstring theory, when higher-derivative terms [Formula: see text] are included in the effective Lagrangian [Formula: see text], because they occur as the Euler-number density [Formula: see text]. This means that [Formula: see text] cannot be written in the standard Hamiltonian form, and hence that macroscopic quantum mechanics does not exist at this level of approximation. It was further conjectured that the solution to this difficulty is to take into account the effect of the terms [Formula: see text], an expression for which has been obtained by Gross and Witten, and by Freeman et al. Here, this conjecture is proved, but it is pointed out that the theory must first be reduced to a lower dimensionality [Formula: see text]. When this is done, the reduced term R2 is no longer proportional to [Formula: see text], because of additional contributions arising from the dimensional reduction of [Formula: see text]. The Wheeler-DeWitt equation can now be derived in the form of a Schrödinger equation, in particular when [Formula: see text] (and [Formula: see text] is a total divergence which can be discarded), and quantum mechanics can be set up in the usual way. In the light of these results, it is argued that the non-locality of quantum mechanics is related to the cosmological horizon problem.

ON THE SEMI-CLASSICAL APPROXIMATION TO THE SUPERSTRING THEORY

1992 ◽  
Vol 07 (25) ◽  
pp. 6421-6430 ◽  
Author(s):  
M.D. POLLOCK
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Classical Limit ◽  
Schrodinger Equation ◽  
Cosmic Time ◽  
Canonical Coordinates ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Effective Lagrangian ◽  
Non Locality

The semi-classical limit of the compactified, heterotic superstring theory is examined, including the effects of higher-derivative terms [Formula: see text] in the effective Lagrangian. The total wave-function Ψ obeys a Schrödinger equation in the mini-superspace ds2=dt2−e2α(t)dx2, the canonical coordinates being the position α and the velocity (Hubble parameter) [Formula: see text], while the cosmic time coincides with the parameter introduced by Tomonaga, ∂/∂σ≡∂/∂t≡ξ∂/∂α. The wave function describing the matter, Ψ m , also obeys a linear Schrödinger equation. The relevance of this result to the problem of non-locality in quantum mechanics is discussed.

DIMENSIONAL REDUCTION AND THE VACUUM STATE IN THE SUPERSTRING THEORY

2002 ◽  
Vol 17 (30) ◽  
pp. 1965-1972 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Effective Action ◽  
Dimensional Reduction ◽  
Euler Number ◽  
Physical Space ◽  
Vacuum State ◽  
Number Density ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Derivative Term

The ten-dimensional effective action of the heterotic superstring theory contains a quadratic higher-derivative term in the form of the Euler-number density [Formula: see text]. We discuss the role of this term in determining the dimensionality [Formula: see text] of the physical space–time, obtained by dimensional reduction, applying the Feynman propagator and the zero-action hypothesis.

HOW DID THE UNIVERSE BEGIN?

1998 ◽  
Vol 07 (05) ◽  
pp. 727-735 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Time Interval ◽  
Field Equations ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Planck Time ◽  
Contracting Phase ◽  
Three Space ◽  
Previous Phase ◽  
The Universe ◽  
Radius Function

The question of the initial configuration of the Universe — did the expanding Friedmann space-time ds2 = dt2 - a2(t)dx2 tend to a singularity when extrapolated back in time, or was there a turning point, indicating a previous phase of contraction? — is re-examined in the context of the heterotic superstring theory of Gross et al. If the adiabatic index tends to the value γ = 1, then the higher-derivative terms ℛ2 in the Lagrangian L dominate the Einstein–Hilbert term R/16πG in the time interval t p ≲ t ≲ 4t p , during which the action is S ≈ 25ℏ, guaranteeing the approximate validity of the classical field equations (if the compactification process is ignored), where [Formula: see text] is the Newton gravitational constant and t p is the Planck time. Under these conditions, Ruzmaĭkina and Ruzmaĭkina have shown, for a flat three-space with K = 0, that the initial singularity can only be avoided at all if there is a spin-zero tachyon, a conclusion modified by Barrow and Ottewill if K = ±1. We have previously shown, however, that the theory is tachyon-free, and have argued that K has to vanish for the existence of a well-defined, quantum-mechanical ground state. Also, if there is no inflation, the radius function is always much too large for the terms in K to exert any effect, a(t) ≳ 5 × 1029t p . While if γ = 2, then ℛ2 never dominates R/16πG. Accordingly, we conjecture that the Universe did not bounce, irrespective of the value of γ, the absence of a prior contracting phase thus being an aspect of causality.

ON THE SUPERSTRING HAMILTONIAN IN THE FRIEDMANN SPACE–TIME

2006 ◽  
Vol 15 (06) ◽  
pp. 845-868 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Cosmological Solution ◽  
Vacuum State ◽  
Einstein Condensate ◽  
De Sitter ◽  
Superstring Theory ◽  
Bose Einstein Condensate ◽  
Higher Derivative ◽  
Effective Lagrangian ◽  
State Formula ◽  
De Sitter Vacuum

The ten-dimensional effective Lagrangian [Formula: see text] for the gravitational sector of the heterotic superstring theory is known up to quartic higher-derivative order [Formula: see text]. In cosmology, the reduced, four-dimensional line element assumes the Friedmann form ds2 = dt2 - a(t)2dx2, where t is comoving time and a(t) ≡ a0eα(t) is the radius function of the three-space dx2, whose curvature is k = 0, ± 1. The four-Lagrangian can then be expressed as the power-series [Formula: see text], where ˙ ≡ d/dt, from which the field equation can be derived by the method of Ostrogradsky. Here, we determine the coefficients Λ0, An, Bn, Cn, and Kn, which are all non-vanishing in general. We recover the previously obtained, high-curvature, anti-de Sitter vacuum state [Formula: see text] with effective cosmological constant Λ = {18/[175ζ(3) - 1/2]}1/3A r κ-2, whose existence makes it possible to envisage a singularity-free and horizon-free cosmological solution, stable to linear perturbations. It is interesting that all the coefficients of quartic origin arise from the near-cancellation of sums of opposite sign but magnitude f ≈ (28.6–369) times larger than the answer. They thus exhibit a slight asymmetry with regard to positive and negative energies, the anti-de Sitter vacuum being characterized by positive Nordström energy, and therefore only accessible at high curvatures. This vacuum state is a Bose–Einstein condensate of non-interacting gravitons at zero temperature, which, referred to comoving time, can only be formulated after the Wick rotation t → ±iτ, resulting in an imaginary horizon.

ON THE SPECTRUM OF PRIMORDIAL DENSITY FLUCTUATIONS

2006 ◽  
Vol 15 (09) ◽  
pp. 1487-1499 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Galaxy Formation ◽  
Gravitational Constant ◽  
Higher Dimensions ◽  
Density Fluctuations ◽  
Density Contrast ◽  
Superstring Theory ◽  
Dimensional Theory ◽  
Scale Invariant ◽  
Higher Derivative ◽  
The Universe

The problem of the origin and spectrum of cosmic density fluctuations is discussed, especially with reference to the heterotic superstring theory of Gross et al. It is shown that primordial variation of the gravitational constant, due to its renormalization by higher-derivative terms [Formula: see text] dependent on the moduli, or the Harrison mechanism applied at the Hagedorn temperature T H = 1.14 × 1017 GeV (where we have argued that the four-dimensional theory decompactifies to higher dimensions), both naturally give rise to the scale-invariant spectrum of Zeldovich with a density contrast δ ~ 10-4-10-3, as required by the indeterminacy principle and for galaxy formation in the Universe.

Cosmic Density Fluctuations, Magnetic Fields and Gravitational Waves

1997 ◽  
Vol 12 (15) ◽  
pp. 1069-1076 ◽  
Author(s):  
M. D. Pollock
Keyword(s):  
Galaxy Formation ◽  
Background Radiation ◽  
Density Fluctuations ◽  
Microwave Background ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Microwave Background Radiation ◽  
Dynamo Mechanism ◽  
Gravitational Wave Background ◽  
The Universe

It has previously been shown, for the heterotic superstring theory including higher-derivative terms ℛ2, how metric fluctuations, sufficient for galaxy formation in the Universe, arise as a consequence of the Heisenberg indeterminacy principle, applied to the dynamical auxiliary coordinate [Formula: see text] and its canonically conjugate momentum πξ, defined from the Friedmann space-time [Formula: see text]. This indeterminacy is distributed amongst the scalar, vector and tensor modes of the metric. Therefore, in addition to the fluctuations δρ/ρ in the matter, and in the cosmic microwave background radiation, there is a magnetic field, whose magnitude is estimated to agree approximately with the phenomenological value B c ~ 10-10 G required for the present-day intergalactic field (in the absence of a dynamo mechanism acting on a primordial field B s ≲ 10-17 G), and also a stochastic gravitational wave background, whose energy density must be bounded by the limit Ω gw ≲ 2.6×10-14h-2≈ 10-13 obtained by Krauss and White from the Sachs–Wolfe effect.

ON THE SUPERSTRING BLACK HOLE

2007 ◽  
Vol 16 (01) ◽  
pp. 123-140 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Black Hole ◽  
Energy Momentum Tensor ◽  
Energy Content ◽  
Momentum Tensor ◽  
Back Reaction ◽  
Effective Energy ◽  
De Sitter ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Effective Lagrangian

The effective Lagrangian for the heterotic superstring theory of Gross et al. contains higher-derivative gravitational terms [Formula: see text], n ≥ 2, which become important at large curvatures. This leads to a natural realization of the limiting-curvature hypothesis of Frolov et al., which was formulated to describe the interior of black holes. Assuming a purely geometrical, four-dimensional Schwarzschild black hole, for which all matter fields are zero, this interior consists of two regions: a shell of effective energy-density ρ immediately beyond the event horizon at r+ = 2M, due to the back reaction of the [Formula: see text] on the Schwarzschild metric, extending inward to a transition radius r0 ≈ M⅓, where the shell signature (- + - -) reverts to the exterior Lorentzian form (+ - - -), and an innermost core tending asymptotically to anti-de Sitter space as r → 0. The total mass-energy content of the hole M can be expressed in terms of the effective energy–momentum tensor Sij as the Nordström mass [Formula: see text], since the space–time is static and free of physical singularities. The conjecture that ρ N (r) is positive in the shell, which is necessary for the contribution to M N to be positive, is shown to be true for the term [Formula: see text], due to the unrenormalized [Formula: see text]. The corresponding "potential" energy–momentum tensor calculated in the Schwarzschild background is isotropic in the region r0 ≪ r ≪ r+, where [Formula: see text], while the dominant "kinetic" contribution is [Formula: see text], so that [Formula: see text].

INFLATION FROM THE SUPERSTRING VACUUM

2009 ◽  
Vol 24 (20n21) ◽  
pp. 4021-4037
Author(s):  
M. D. POLLOCK
Keyword(s):  
Background Radiation ◽  
Physical Space ◽  
Reduction Process ◽  
Tree Level ◽  
Internal Space ◽  
De Sitter ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Effective Lagrangian ◽  
Microwave Background Radiation

Quartic higher-derivative gravitational terms in the effective Lagrangian of the heterotic superstring theory renormalize the bare, four-dimensional gravitational coupling [Formula: see text], due to the reduction process [Formula: see text], according to the formula [Formula: see text], where A r and B r are the moduli for the physical space gij(xk) and internal space [Formula: see text], respectively. The Euler characteristic [Formula: see text] is negative for a three-generation Calabi–Yau manifold, and therefore both the additional terms, of tree-level and one-loop origin, produce a decrease in κ-2, which changes sign when κ-2 = 0. The corresponding tree-level critical point is [Formula: see text], if we set [Formula: see text] and λ = 15π2, for compactification onto a torus. Values [Formula: see text] yield the anti-gravity region κ-2 < 0, which is analytically accessible from the normal gravity region κ-2 > 0. The only non-singular, vacuum minimum of the potential [Formula: see text] is located at the point [Formula: see text], where [Formula: see text], the quadratic trace anomaly [Formula: see text] dominates over [Formula: see text], and a phase of de Sitter expansion may occur, as first envisaged by Starobinsky, in approximate agreement with the constraint due to the effect of gravitational waves upon the anisotropy of the cosmic microwave background radiation. There is no non-singular minimum of the potential [Formula: see text].

ON THE POSITIVITY OF THE GRAVITATIONAL POTENTIAL IN THE QUANTUM-COSMOLOGICAL SCHRÖDINGER EQUATION

1994 ◽  
Vol 03 (03) ◽  
pp. 569-578 ◽  
Author(s):  
M.D. POLLOCK
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Gravitational Theory ◽  
Space Time ◽  
Number Density ◽  
Anisotropic Space ◽  
Four Dimensions ◽  
Higher Derivative ◽  
Superstring Theories ◽  
The Universe

Quantization of the D(=M+1)-dimensional gravitational theory with higher-derivative terms [Formula: see text] in the Friedmann space-time ds2=dt2−e2α(t)dx2, where the M-space dx2 has curvature K, yields the Schrödinger equation (Wheeler-DeWitt equation) i∂Ψ/∂t=[−AMe−Mα∂2/∂ξ2+VM,K(α, ξ)]Ψ, where ξ≡dα/dt, provided that [Formula: see text] differs from the Euler-number density. The coefficient AM is positive if there is no spin-0 tachyon in [Formula: see text] and the potential VM,K is positive semi-definite if M=3 and K=0. The theory is classically stable if the spin-2 tachyon is also absent. All of these conditions are satisfied by the heterotic superstring, after reduction to four dimensions, but not by the bosonic string, which contains a spin-2 tachyon, nor by the type-II superstring, which contains a spin-0 tachyon. After generalization to the anisotropic space-time ds2= dt2−e2β(t)dy2−e2γ(t)dz2, where dy2 and dz2 have dimensions one and two, respectively, the Schrödinger equation becomes i∂Ψ/∂t=[–(4B)−1e−(β+2γ) (X∂2/∂ζ2+Y∂2+ 2Z∂2/∂ζ∂η)+…+V(β, γ; ζ, η)]Ψ, where ζ≡dβ/dt, η≡dγ/dt. The potential V is unbounded both from above and from below for all β≠γ, for all three superstring theories, and in fact for all dimensionalities. This explains why the Universe is isotropic, and why dimensional reduction and compactification occur.

Quantum Becoming?

2017 ◽  
Author(s):  
Craig Callender
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Time ◽  
Quantum Non Locality ◽  
Time Quantum ◽  
Temporal Becoming ◽  
Non Locality ◽  
Quantum Wavefunction

Two of quantum mechanics’ more famed and spooky features have been invoked in defending the idea that quantum time is congenial to manifest time. Quantum non-locality is said by some to make a preferred foliation of spacetime necessary, and the collapse of the quantum wavefunction is held to vindicate temporal becoming. Although many philosophers and physicists seek relief from relativity’s assault on time in quantum theory, assistance is not so easily found.

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