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.

On the Vanishing of the Cosmological Vacuum Energy

1997 ◽  
Vol 12 (16) ◽  
pp. 1127-1130 ◽  
Author(s):  
M. D. Pollock
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Vacuum Energy ◽  
Ground State Solution ◽  
Space Time ◽  
State Solution ◽  
Superstring Theory ◽  
The Universe

By demanding the existence of a globally invariant ground-state solution of the Wheeler–De Witt equation (Schrödinger equation) for the wave function of the Universe Ψ, obtained from the heterotic superstring theory, in the four-dimensional Friedmann space-time, we prove that the cosmological vacuum energy has to be zero.

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.

Solusi Persamaan Schrödinger Berbasis Panjang Minimal untuk Potensial Coulomb Termodifikasi

2018 ◽  
Vol 2 (2) ◽  
pp. 43-47
Author(s):  
A. Suparmi, C. Cari, Ina Nurhidayati
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
Minimal Length ◽  
Energy Spectra ◽  
Atomic Particle ◽  
Approximate Analytical Solutions ◽  
Radial Schrödinger Equation

Abstrak – Persamaan Schrödinger adalah salah satu topik penelitian yang yang paling sering diteliti dalam mekanika kuantum. Pada jurnal ini persamaan Schrödinger berbasis panjang minimal diaplikasikan untuk potensial Coulomb Termodifikasi. Fungsi gelombang dan spektrum energi yang dihasilkan menunjukkan kharakteristik atau tingkah laku dari partikel sub atom. Dengan menggunakan metode pendekatan hipergeometri, diperoleh solusi analitis untuk bagian radial persamaan Schrödinger berbasis panjang minimal diaplikasikan untuk potensial Coulomb Termodifikasi. Hasil yang diperoleh menunjukkan terjadi peningkatan energi yang sebanding dengan meningkatnya parameter panjang minimal dan parameter potensial Coulomb Termodifikasi. Kata kunci: persamaan Schrödinger, panjang minimal, fungsi gelombang, energi, potensial Coulomb Termodifikasi Abstract – The Schrödinger equation is the most popular topic research at quantum mechanics. The  Schrödinger equation based on the concept of minimal length formalism has been obtained for modified Coulomb potential. The wave function and energy spectra were used to describe the characteristic of sub-atomic particle. By using hypergeometry method, we obtained the approximate analytical solutions of the radial Schrödinger equation based on the concept of minimal length formalism for the modified Coulomb potential. The wave function and energy spectra was solved. The result showed that the value of energy increased by the increasing both of minimal length parameter and the potential parameter. Key words: Schrödinger equation, minimal length formalism (MLF), wave function, energy spectra, Modified Coulomb potential

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.

COLLECTIVE SPIN BY LINEARIZATION OF THE SCHRÖDINGER EQUATION FOR NUCLEAR COLLECTIVE MOTION

1988 ◽  
Vol 03 (09) ◽  
pp. 859-866 ◽  
Author(s):  
MARTIN GREINER ◽  
WERNER SCHEID ◽  
RICHARD HERRMANN
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Degrees Of Freedom ◽  
Collective Motion ◽  
First Order ◽  
Energy And Momentum

The free Schrödinger equation for multipole degrees of freedom is linearized so that energy and momentum operators appear only in first order. As an example, we demonstrate the linearization procedure for quadrupole degrees of freedom. The wave function solving this equation carries a spin. We derive the operator of the collective spin and its eigenvalues depending on multipolarity.

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