- Wheeler–deWitt equation
theoretical physics, the Wheeler-DeWitt equation [DeWitt, B.S., “Quantum Theory of Gravity. I. The Canonical Theory”, Phys. Rev., 160, 1113-1148, (1967).] is a functional differential equation. It is ill defined in the general case, but very important in theoretical physics, especially in quantum gravity. It is a functional differential equation on the space of three dimensional spatial metrics. The Wheeler-deWitt equation has the form of an operator acting on a wave functional, the functional reduce to a function in cosmology. Contrary to the general case, the Wheeler-deWitt equation is well defined in mini-superspaces like the configuration space of cosmological theories. An example of such a wave functionis the Hartle-Hawking state.
Simply speaking, the Wheeler-DeWitt equation says:where is the total
Hamiltonian constraintin quantized general relativity.
Although the symbols and may appear familiar, their interpretation in the Wheeler-deWitt equation is substantially different from non-relativistic quantum mechanics. is no longer a spatial wave function in the traditional sense of a complex-valued function that is defined on a 3-dimensional space-like surface and normalized to unity. Instead it is a functional of field configurations on all of spacetime. This wave function contains all of the information about the geometry and matter content of the universe. is still an operator that acts on the
Hilbert spaceof wave functions, but it is not the same Hilbert space as in the nonrelativistic case, and the Hamiltonian no longer determines evolution of the system, so the Schrödinger equationno longer applies.
In fact, the principle of
general covariancein general relativity implies that global evolution per se does not exist; is just a label we assign to one of the coordinate axes. Thus, what we think about as time evolution of any physical system is just a gauge transformation, similar to that of QED induced by U(1) local gauge transformation where plays the role of local time. The role of a Hamiltonian is simply to restrict the space of the "kinematic" states of the Universe to that of "physical" states - the ones that follow gauge orbits. For this reason we call it a "Hamiltonian constraint." Upon quantization, physical states become wave functions that lie in the kernelof the Hamiltonian operator.
In general, the Hamiltonian vanishes for a theory with general covariance or time-scaling invariance.
eigenstateof the Hamiltonian usually depends on "n""x" , "n""y", "n""z", ..., for the continuous case we have the form of the energy in terms of a functional:
Hence the ground state satisfies
Wikimedia Foundation. 2010.
Look at other dictionaries:
Dirac equation — Quantum field theory (Feynman diagram) … Wikipedia
Bryce DeWitt — Infobox Scientist name = Bryce Seligman DeWitt birth date = January 8 1923 death date = death date and age|2004|9|23|1923|1|8 residence = United States of America nationality = American field = Theoretical physicist work institution = Institute… … Wikipedia
Propagateur de l'équation de Schrödinger — Le terme propagateur a été introduit en physique par Feynman en 1948 pour sa formulation de la mécanique quantique en intégrales de chemin, une nouvelle approche de la quantification centrée sur le Lagrangien, contrairement à la procédure… … Wikipédia en Français
Bryce DeWitt — Pour les articles homonymes, voir DeWitt. Bryce Seligman DeWitt Bryce DeWitt, Vilkovisky et Barvinsky à Moscou en 1990 … Wikipédia en Français
Contributors to general relativity — General relativity Introduction Mathematical formulation Resources Fundamental concepts … Wikipedia
General relativity — For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. General relativity Introduction Mathematical formulation Resources … Wikipedia
Introduction to gauge theory — This article is an accessible, non technical introduction to the subject. For the main encyclopedia article, see Gauge theory. Quantum field theory … Wikipedia
Propagator — This article is about Quantum field theory. For plant propagation, see Plant propagation. Quantum field theory … Wikipedia
Loop quantum gravity — Not to be confused with the path integral formulation of LQG, see spin foam. This article is about LQG in its Canonical formulation.. Beyond the Standard Model … Wikipedia
Universal wavefunction — The Universal Wavefunction or Universal Wave Function is a term introduced by Hugh Everett in his Princeton PhD Thesis, [Bryce Seligman DeWitt, R. Neill Graham, eds, The Many Worlds Interpretation of Quantum Mechanics , Princeton Series in… … Wikipedia