In
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 ...
, the Hamilton–Jacobi–Einstein equation (HJEE) or Einstein–Hamilton–Jacobi equation (EHJE) is an equation in the
Hamiltonian formulation of
geometrodynamics
In theoretical physics, geometrodynamics is an attempt to describe spacetime and associated phenomena completely in terms of geometry. Technically, its goal is to grand unification, unify the fundamental forces and reformulate general relativity ...
in
superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numb ...
, cast in the "geometrodynamics era" around the 1960s, by
Asher Peres
Asher Peres ( he, אשר פרס; January 30, 1934 – January 1, 2005) was an Israeli physicist. He is well known for his work relating quantum mechanics and information theory. He helped to develop the Peres–Horodecki criterion for quantum en ...
in 1962 and others. It is an attempt to reformulate general relativity in such a way that it resembles quantum theory within a
semiclassical approximation, much like the correspondence between
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 chemistry, ...
and
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
.
It is named for
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
,
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...
, and
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
. The EHJE contains as much information as all ten
Einstein field equation
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
s (EFEs). It is a modification of the
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechan ...
(HJE) from
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, and can be derived from the
Einstein–Hilbert action
The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the ac ...
using the
principle of least action
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states tha ...
in the
ADM formalism
The ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was first ...
.
Background and motivation
Correspondence between classical and quantum physics
In classical
analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...
, the dynamics of the system is summarized by the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
. In quantum theory, namely non-relativistic
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 chemistry, ...
(QM),
relativistic quantum mechanics
In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light ''c ...
(RQM), as well as
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
(QFT), with varying interpretations and mathematical formalisms in these theories, the behavior of a system is completely contained in a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued
probability amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the quan ...
(more formally as a
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
ket - an element of a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
). Using the polar form of the wave function, so making a Madelung transformation:
:
the
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
of is interpreted as the action, and the modulus is interpreted according to the
Copenhagen interpretation
The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as featu ...
as the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
. The
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
is the ''quantum of angular momentum''. Substitution of this into the quantum general
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
(SE):
:
and taking the limit yields the classical HJE:
:
which is one aspect of the
correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says t ...
.
Shortcomings of four-dimensional spacetime
On the other hand, the transition between quantum theory and ''general'' relativity (GR) is difficult to make; one reason is the treatment of space and time in these theories. In non-relativistic QM, space and time are not on equal footing; time is a parameter while
position is an operator. In RQM and QFT, position returns to the usual
spatial coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
alongside the time coordinate, although these theories are consistent only with SR in four-dimensional ''flat''
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
, and not
curved space
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
nor GR. It is possible to formulate
quantum field theory in curved spacetime
In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory treats spacetime as a fixed, classical background, while giving ...
, yet even this still cannot incorporate GR because gravity is not
renormalizable
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similarity, self-similar geometric structures, that are used to treat infinity, infinities arising in calculated ...
in QFT. Additionally, in GR particles move through curved spacetime with a deterministically known position and momentum at every instant, while in quantum theory, the position and momentum of a particle cannot be exactly known simultaneously; space and momentum , and energy and time , are pairwise subject to the
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
s
:
which imply that small intervals in space and time mean large fluctuations in energy and momentum are possible. Since in GR
mass–energy and
momentum–energy is the source of
spacetime curvature
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. G ...
, large fluctuations in energy and momentum mean the spacetime "fabric" could potentially become so distorted that it breaks up at sufficiently small scales.
There is theoretical and experimental evidence from QFT that vacuum does have energy since the motion of electrons in atoms is fluctuated, this is related to the
Lamb shift
In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which the ...
. For these reasons and others, at increasingly small scales, space and time are thought to be dynamical up to the
Planck length and
Planck time scales.
In any case, a four-dimensional
curved space
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
time continuum is a well-defined and central feature of general relativity, but not in quantum mechanics.
Equation
One attempt to find an equation governing the dynamics of a system, in as close a way as possible to QM and GR, is to reformulate the HJE in ''
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
curved space'' understood to be "dynamic" (changing with time), and ''not''
four-dimensional
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
spacetime dynamic in all four dimensions, as the EFEs are. The space has a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
(see
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
for details).
The
metric tensor in general relativity is an essential object, since
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
,
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
,
geodesic motion in
curved space
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
time, and other things, all depend on the metric. The HJE above is modified to include the metric, although it's only a function of the 3d spatial coordinates , (for example in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
) without the
coordinate time
In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial ...
:
:
In this context is referred to as the "metric field" or simply "field".
General equation (free curved space)
For a free particle in curved "
empty space" or "free space", i.e. in the absence of
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...
other than the particle itself, the equation can be written:
where is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the metric tensor and the
Ricci scalar curvature of the 3d geometry (not including time), and the "" instead of "" denotes the
variational derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on w ...
rather than the
ordinary derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
. These derivatives correspond to the field momenta "conjugate to the metric field":
:
the rate of change of action with respect to the field coordinates . The and here are analogous to and , respectively, in classical
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
. See
canonical coordinates
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of ...
for more background.
The equation describes how
wavefront
In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
s of constant action propagate in superspace - as the dynamics of
matter wave
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
s of a free particle unfolds in curved space. Additional source terms are needed to account for the presence of extra influences on the particle, which include the presence of other particles or distributions of matter (which contribute to space curvature), and sources of electromagnetic fields affecting particles with
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
or
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
. Like the Einstein field equations, it is
non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
in the metric because of the products of the metric components, and like the HJE it is non-linear in the action due to the product of variational derivatives in the action.
The quantum mechanical concept, that action is the phase of the wavefunction, can be interpreted from this equation as follows. The phase has to satisfy the principle of least action; it must be
stationary for a small change in the configuration of the system, in other words for a slight change in the position of the particle, which corresponds to a slight change in the metric components;
:
the slight change in phase is zero:
:
(where is the
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV = \ ...
of the
volume integral
In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ap ...
). So the constructive interference of the matter waves is a maximum. This can be expressed by the
superposition principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
; applied to many non-localized wavefunctions spread throughout the curved space to form a localized wavefunction:
:
for some coefficients , and additionally the action (phase) for each must satisfy:
:
for all ''n'', or equivalently,
:
Regions where is maximal or minimal occur at points where there is a probability of finding the particle there, and where the action (phase) change is zero. So in the EHJE above, each wavefront of constant action is where the particle ''could'' be found.
This equation still does not "unify" quantum mechanics and general relativity, because the semiclassical
Eikonal approximation In theoretical physics, the eikonal approximation (Greek εἰκών for likeness, icon or image) is an approximative method useful in wave scattering equations which occur in optics, seismology, quantum mechanics, quantum electrodynamics, and par ...
in the context of quantum theory and general relativity has been applied, to provide a transition between these theories.
Applications
The equation takes various complicated forms in:
*
Quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
*
Quantum cosmology
Quantum cosmology is the attempt in theoretical physics to develop a quantum theory of the universe. This approach attempts to answer open questions of classical physical cosmology, particularly those related to the first phases of the universe.
...
See also
*
Foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
*
Quantum geometry
In theoretical physics, quantum geometry is the set of mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at distance scales comparable to the Planck length. At these d ...
*
Quantum spacetime
In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebr ...
*
Calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
*The equation is also related to the
Wheeler–DeWitt equation
The Wheeler–DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and general r ...
.
*
Peres metric In mathematical physics, the Peres metric is defined by the proper time
:
^ = dt^2 - 2f(t+z, x, y) (dt+dz)^2-dx^2-dy^2-dz^2
for any arbitrary function ''f''. If ''f'' is a harmonic function with respect to ''x'' and ''y'', then the corresponding ...
References
Notes
Further reading
Books
*
*
*
*
*
Selected papers
* (''Equation A.3 in the appendix'').
*
*
*
*
{{DEFAULTSORT:Hamilton-Jacobi-Einstein equation
General relativity
Hamiltonian mechanics
Quantum gravity