The Wheeler–DeWitt equation
for
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
and
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
, is a
field equation attributed to
John Archibald Wheeler
John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in ...
and
Bryce DeWitt
Bryce Seligman DeWitt (January 8, 1923 – September 23, 2004), was an American theoretical physicist noted for his work in gravitation and quantum field theory.
Life
He was born Carl Bryce Seligman, but he and his three brothers, including th ...
. The equation attempts to mathematically combine the ideas of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
and
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, a step towards a theory of
quantum gravity.
In this approach,
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called '
problem of time'. More specifically, the equation describes the quantum version of the
Hamiltonian constraint using metric variables. Its commutation relations with the
diffeomorphism constraint
In theoretical physics, it is often important to study theories with the diffeomorphism symmetry such as general relativity. These theories are invariant under arbitrary coordinate transformations. Equations of motion are generally derived from th ...
s generate the Bergman–Komar "group" (which ''is'' the
diffeomorphism group
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
on-shell
In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called "on the mass shell" or simply more often on shell; while those that do not are called "off the mass shell", ...
).
Quantum gravity
All defined and understood descriptions of string/M-theory deal with fixed asymptotic conditions on the background spacetime. At infinity, the "right" choice of the time coordinate "t" is determined (because the space-time is asymptotic to some fixed space-time) in every description, so there is a preferred definition of the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
(with nonzero eigenvalues) to evolve states of the system forwards in time. This avoids all the need to dynamically generate a time dimension using the Wheeler–DeWitt equation. Thus, the equation has not played a role in string theory thus far.
There could exist a Wheeler–DeWitt-style manner to describe the bulk dynamics of quantum theory of gravity. Some experts believe that this equation still holds the potential for understanding quantum gravity; however, decades after the equation was published, completely different approaches, such as string theory, have brought physicists as clear results about quantum gravity.
Motivation and background
In
canonical gravity, spacetime is
foliated into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is
and given by
:
In that equation the Latin indices run over the values 1, 2, 3 and the Greek indices run over the values 1, 2, 3, 4. The three-metric
is the field, and we denote its conjugate momenta as
. The Hamiltonian is a constraint (characteristic of most relativistic systems)
:
where
and
is the Wheeler–DeWitt metric.
Quantization "puts hats" on the momenta and field variables; that is, the functions of numbers in the classical case become operators that modify the state function in the quantum case. Thus we obtain the operator
:
Working in "position space", these operators are
:
:
One can apply the operator to a general wave functional of the metric
where:
:
which would give a set of constraints amongst the coefficients
. This means the amplitudes for
gravitons at certain positions is related to the amplitudes for a different number of gravitons at different positions. Or, one could use the two-field formalism, treating
as an independent field so that the wave function is