Variational Monte Carlo

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Variational_Monte_Carlo 

In computational physics, variational Monte Carlo (VMC) is a quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system.

The basic building block is a generic wave function ${\displaystyle |\mathrm{\Psi }(a)⟩}$ depending on some parameters ${\displaystyle a}$. The optimal values of the parameters ${\displaystyle a}$ is then found upon minimizing the total energy of the system.

In particular, given the Hamiltonian ${\displaystyle \mathcal{H}}$, and denoting with ${\displaystyle X}$ a many-body configuration, the expectation value of the energy can be written as:

$${\displaystyle E(a)=\frac{⟨\mathrm{\Psi }(a)|\mathcal{H}|\mathrm{\Psi }(a)⟩}{⟨\mathrm{\Psi }(a)|\mathrm{\Psi }(a)⟩}=\frac{\int |\mathrm{\Psi }(X,a){|}^2\frac{\mathcal{H}\mathrm{\Psi }(X,a)}{\mathrm{\Psi }(X,a)}dX}{\int |\mathrm{\Psi }(X,a){|}^{2}dX}.}$$

Following the Monte Carlo method for evaluating integrals, we can interpret ${\displaystyle \frac{|\mathrm{\Psi }(X,a){|}^2}{\int |\mathrm{\Psi }(X,a){|}^{2}dX}}$ as a probability distribution function, sample it, and evaluate the energy expectation value ${\displaystyle E(a)}$ as the average of the so-called local energy ${\displaystyle E_{\text{loc}}(X)=\frac{\mathcal{H}\mathrm{\Psi }(X,a)}{\mathrm{\Psi }(X,a)}}$. Once ${\displaystyle E(a)}$ is known for a given set of variational parameters ${\displaystyle a}$, then optimization is performed in order to minimize the energy and obtain the best possible representation of the ground-state wave-function.

VMC is no different from any other variational method, except that the many-dimensional integrals are evaluated numerically. Monte Carlo integration is particularly crucial in this problem since the dimension of the many-body Hilbert space, comprising all the possible values of the configurations ${\displaystyle X}$, typically grows exponentially with the size of the physical system. Other approaches to the numerical evaluation of the energy expectation values would therefore, in general, limit applications to much smaller systems than those analyzable thanks to the Monte Carlo approach.

The accuracy of the method then largely depends on the choice of the variational state. The simplest choice typically corresponds to a mean-field form, where the state ${\displaystyle \mathrm{\Psi }}$ is written as a factorization over the Hilbert space. This particularly simple form is typically not very accurate since it neglects many-body effects. One of the largest gains in accuracy over writing the wave function separably comes from the introduction of the so-called Jastrow factor. In this case the wave function is written as $\mathrm{\Psi }(X)=\exp (\sum u(r_{ij}))$, where ${\displaystyle r_{ij}}$ is the distance between a pair of quantum particles and ${\displaystyle u(r)}$ is a variational function to be determined. With this factor, we can explicitly account for particle-particle correlation, but the many-body integral becomes unseparable, so Monte Carlo is the only way to evaluate it efficiently. In chemical systems, slightly more sophisticated versions of this factor can obtain 80–90% of the correlation energy (see electronic correlation) with less than 30 parameters. In comparison, a configuration interaction calculation may require around 50,000 parameters to reach that accuracy, although it depends greatly on the particular case being considered. In addition, VMC usually scales as a small power of the number of particles in the simulation, usually something like N²⁻⁴ for calculation of the energy expectation value, depending on the form of the wave function.

Wave function optimization in VMC

QMC calculations crucially depend on the quality of the trial-function, and so it is essential to have an optimized wave-function as close as possible to the ground state. The problem of function optimization is a very important research topic in numerical simulation. In QMC, in addition to the usual difficulties to find the minimum of multidimensional parametric function, the statistical noise is present in the estimate of the cost function (usually the energy), and its derivatives, required for an efficient optimization.

Different cost functions and different strategies were used to optimize a many-body trial-function. Usually three cost functions were used in QMC optimization energy, variance or a linear combination of them. The variance optimization method has the advantage that the exact wavefunction's variance is known. (Because the exact wavefunction is an eigenfunction of the Hamiltonian, the variance of the local energy is zero). This means that variance optimization is ideal in that it is bounded from below, it is positive defined and its minimum is known. Energy minimization may ultimately prove more effective, however, as different authors recently showed that the energy optimization is more effective than the variance one.

There are different motivations for this: first, usually one is interested in the lowest energy rather than in the lowest variance in both variational and diffusion Monte Carlo; second, variance optimization takes many iterations to optimize determinant parameters and often the optimization can get stuck in multiple local minimum and it suffers of the "false convergence" problem; third energy-minimized wave functions on average yield more accurate values of other expectation values than variance minimized wave functions do.

The optimization strategies can be divided into three categories. The first strategy is based on correlated sampling together with deterministic optimization methods. Even if this idea yielded very accurate results for the first-row atoms, this procedure can have problems if parameters affect the nodes, and moreover density ratio of the current and initial trial-function increases exponentially with the size of the system. In the second strategy one use a large bin to evaluate the cost function and its derivatives in such way that the noise can be neglected and deterministic methods can be used.

The third approach, is based on an iterative technique to handle directly with noise functions. The first example of these methods is the so-called Stochastic Gradient Approximation (SGA), that was used also for structure optimization. Recently an improved and faster approach of this kind was proposed the so-called Stochastic Reconfiguration (SR) method.

VMC and deep learning

In 2017, Giuseppe Carleo and Matthias Troyer used a VMC objective function to train an artificial neural network to find the ground state of a quantum mechanical system. More generally, artificial neural networks are being used as a wave function ansatz (known as neural network quantum states) in VMC frameworks for finding ground states of quantum mechanical systems. The use of neural network ansatzes for VMC has been extended to fermions, enabling electronic structure calculations that are significantly more accurate than VMC calculations which do not use neural networks.

Alcubierre drive

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Alcubierre_drive 

Two-dimensional visualization of an Alcubierre drive, showing the opposing regions of expanding and contracting spacetime that displace the central region

The Alcubierre drive ([alkuˈβjere]) is a speculative warp drive idea according to which a spacecraft could achieve apparent faster-than-light travel by contracting space in front of it and expanding space behind it, under the assumption that a configurable energy-density field lower than that of vacuum (that is, negative mass) could be created. Proposed by theoretical physicist Miguel Alcubierre in 1994, the Alcubierre drive is based on a solution of Einstein's field equations. Since those solutions are metric tensors, the Alcubierre drive is also referred to as Alcubierre metric.

Objects cannot accelerate to the speed of light within normal spacetime; instead, the Alcubierre drive shifts space around an object so that the object would arrive at its destination more quickly than light would in normal space without breaking any physical laws.

Although the metric proposed by Alcubierre is consistent with the Einstein field equations, construction of such a drive is not necessarily possible. The proposed mechanism of the Alcubierre drive implies a negative energy density and therefore requires exotic matter or manipulation of dark energy. If exotic matter with the correct properties does not exist, then the drive cannot be constructed. At the close of his original article, however, Alcubierre argued (following an argument developed by physicists analyzing traversable wormholes) that the Casimir vacuum between parallel plates could fulfill the negative-energy requirement for the Alcubierre drive.

Another possible issue is that, although the Alcubierre metric is consistent with Einstein's equations, general relativity does not incorporate quantum mechanics. Some physicists have presented arguments to suggest that a theory of quantum gravity (which would incorporate both theories) would eliminate those solutions in general relativity that allow for backward time travel (see the chronology protection conjecture) and thus make the Alcubierre drive invalid.

History

In 1994, Miguel Alcubierre proposed a method for changing the geometry of space by creating a wave that would cause the fabric of space ahead of a spacecraft to contract and the space behind it to expand. The ship would then ride this wave inside a region of flat space, known as a warp bubble, and would not move within this bubble but instead be carried along as the region itself moves due to the actions of the drive. The local velocity relative to the deformed spacetime would be subluminal, but the speed at which a spacecraft could move would be superluminal, thereby rendering possible interstellar flight, such as a visit to Proxima Centauri within a few days.

Alcubierre metric

The Alcubierre metric defines the warp-drive spacetime. It is a Lorentzian manifold that, if interpreted in the context of general relativity, allows a warp bubble to appear in previously flat spacetime and move away at effectively faster-than-light speed. The interior of the bubble is an inertial reference frame and inhabitants experience no proper acceleration. This method of transport does not involve objects in motion at faster-than-light speeds with respect to the contents of the warp bubble; that is, a light beam within the warp bubble would still always move more quickly than the ship. Because objects within the bubble are not moving (locally) more quickly than light, the mathematical formulation of the Alcubierre metric is consistent with the conventional claims of the laws of relativity (namely, that an object with mass cannot attain or exceed the speed of light) and conventional relativistic effects such as time dilation would not apply as they would with conventional motion at near-light speeds.

An extension of the Alcubierre metric that eliminates the expansion of the volume elements and instead relies on the change in distances along the direction of travel is that of mathematician José Natário. In his metric, spacetime contracts towards the prow of the ship and expands in the direction perpendicular to the motion, meaning that the bubble actually "slides" through space, roughly speaking by "pushing space aside".

The Alcubierre drive remains a hypothetical concept with seemingly difficult problems, although the amount of energy required is no longer thought to be unobtainably large. However, Alexey Bobrick and Gianni Martire claim that, in principle, a class of subluminal, spherically symmetric warp drive spacetimes can be constructed based on physical principles presently known to humanity, such as positive energy. Furthermore, a study by Remo Garattini and Kirill Zatrimaylov shows that the amount of negative energy density required to sustain a warp bubble can in principle be reduced if the bubble is moving in an external gravitational field, such as that of a black hole.

Mathematics

Using the ADM formalism of general relativity, the spacetime is described by a foliation of space-like hypersurfaces of constant coordinate time t, with the metric taking the following general form:

${\displaystyle d{s}^2=-\left({\alpha }^2-{\beta }_i{\beta }^i\right)d{t}^2+2{\beta }_{i}d{x}^{i}dt+{\gamma }_{ij}d{x}^{i}d{x}^j,}$

where

• α is the lapse function that gives the interval of proper time between nearby hypersurfaces,
• βⁱ is the shift vector that relates the spatial coordinate systems on different hypersurfaces,
• γ_ij is a positive-definite metric on each of the hypersurfaces.

The particular form that Alcubierre studied is defined by:

${\displaystyle \begin{array}{rl}\alpha & =1,\\ {\beta }^x& =-v_s(t)f(r_s(t)),\\ {\beta }^y& ={\beta }^z=0,\\ {\gamma }_{ij}& ={\delta }_{ij},\end{array}}$

where

${\displaystyle \begin{array}{rl}{v}_s(t)& =\frac{d{x}_s(t)}{dt},\\ r_s(t)& =\sqrt{(x-x_s(t){)}^2+y^2+z^2},\\ f(r_s)& =\frac{\tanh (\sigma (r_s+R))-\tanh (\sigma (r_s-R))}{2\tanh (\sigma R)},\end{array}}$

with arbitrary parameters R > 0 and σ > 0. Alcubierre's specific form of the metric can thus be written:

${\displaystyle d{s}^2=\left(v_s(t{)}^{2}f(r_s(t){)}^2-1\right)d{t}^2-2v_s(t)f(r_s(t))dxdt+d{x}^2+d{y}^2+d{z}^2.}$

With this particular form of the metric, it can be shown that the energy density measured by observers whose 4-velocity is normal to the hypersurfaces is given by:

${\displaystyle -\frac{c^4}{8\pi G}\frac{v_s^2\left(y^2+z^2\right)}{4g^2{r}_s^2}{\left(\frac{df}{d{r}_s}\right)}^2,}$

where g is the determinant of the metric tensor.

Thus, because the energy density is negative, one needs exotic matter to travel more quickly than the speed of light. The existence of exotic matter is not theoretically ruled out; however, generating and sustaining enough exotic matter to perform feats such as faster-than-light travel (and to keep open the "throat" of a wormhole) is thought to be impractical. According to writer Robert Low, within the context of general relativity it is impossible to construct a warp drive in the absence of exotic matter.

Connection to dark energy and dark matter

See also: Negative mass

Astrophysicist Jamie Farnes from the University of Oxford has proposed a theory, published in the peer-reviewed scientific journal Astronomy & Astrophysics, that unifies dark energy and dark matter into a single dark fluid, and which is expected to be testable by the Square Kilometre Array around 2030. Farnes found that Albert Einstein had explored the idea of gravitationally repulsive negative masses while developing the equations of general relativity, an idea which leads to a "beautiful" hypothesis where the cosmos has equal amounts of positive and negative qualities. Farnes' theory relies on negative masses that behave identically to the physics of the Alcubierre drive, providing a natural solution for the current "crisis in cosmology" due to a time-variable Hubble parameter.

As Farnes' theory allows a positive mass (i.e. a ship) to reach a speed equal to the speed of light, it has been dubbed "controversial". If the theory is correct, which has been highly debated in the scientific literature, it would explain dark energy, dark matter, allow closed timelike curves (see time travel), and suggest that an Alcubierre drive is physically possible with exotic matter.

Physics

With regard to certain specific effects of special relativity, such as Lorentz contraction and time dilation, the Alcubierre metric has some apparently peculiar aspects. In particular, Alcubierre has shown that a ship using an Alcubierre drive travels on a free-fall geodesic even while the warp bubble is accelerating: its crew would be in free fall while accelerating without experiencing accelerational g-forces. Enormous tidal forces, however, would be present near the edges of the flat-space volume because of the large space curvature there, but a suitable specification of the metric would keep the tidal forces very small within the volume occupied by the ship.

The original warp-drive metric and simple variants of it happen to have the ADM form, which is often used in discussing the initial-value formulation of general relativity. This might explain the widespread misconception that this spacetime is a solution of the field equation of general relativity. Metrics in ADM form are adapted to a certain family of inertial observers, but these observers are not really physically distinguished from other such families. Alcubierre interpreted his "warp bubble" in terms of a contraction of space ahead of the bubble and an expansion behind, but this interpretation could be misleading, since the contraction and expansion actually refer to the relative motion of nearby members of the family of ADM observers.

In general relativity, one often first specifies a plausible distribution of matter and energy, and then finds the geometry of the spacetime associated with it; but it is also possible to run the Einstein field equations in the other direction, first specifying a metric and then finding the energy–momentum tensor associated with it, and this is what Alcubierre did in building his metric. This practice means that the solution can violate various energy conditions and require exotic matter. The need for exotic matter raises questions about whether one can distribute the matter in an initial spacetime that lacks a warp bubble in such a way that the bubble is created at a later time, although some physicists have proposed models of dynamical warp-drive spacetimes in which a warp bubble is formed in a previously flat space. Moreover, according to Serguei Krasnikov, generating a bubble in a previously flat space for a one-way faster-than-light trip requires forcing the exotic matter to move at local faster-than-light speeds, something that would require the existence of tachyons, although Krasnikov also notes that when the spacetime is not flat from the outset, a similar result could be achieved without tachyons by placing in advance some devices along the travel path and programming them to come into operation at preassigned moments and to operate in a preassigned manner. Some suggested methods avoid the problem of tachyonic motion, but would probably generate a naked singularity at the front of the bubble. Allen Everett and Thomas Roman comment on Krasnikov's finding (Krasnikov tube):

[The finding] does not mean that Alcubierre bubbles, if it were possible to create them, could not be used as a means of superluminal travel. It only means that the actions required to change the metric and create the bubble must be taken beforehand by some observer whose forward light cone contains the entire trajectory of the bubble.

For example, if one wanted to travel to Deneb (2,600 light-years away) and arrive less than 2,600 years in the future according to external clocks, it would be required that someone had already begun work on warping the space from Earth to Deneb at least 2,600 years ago:

A spaceship appropriately located with respect to the bubble trajectory could then choose to enter the bubble, rather like a passenger catching a passing trolley car, and thus make the superluminal journey ... as Krasnikov points out, causality considerations do not prevent the crew of a spaceship from arranging, by their own actions, to complete a round trip from Earth to a distant star and back in an arbitrarily short time, as measured by clocks on Earth, by altering the metric along the path of their outbound trip.

Difficulties

Mass–energy requirement

The metric of this form has significant difficulties because all known warp-drive spacetime theories violate various energy conditions. Nevertheless, an Alcubierre-type warp drive might be realized by exploiting certain experimentally verified quantum phenomena, such as the Casimir effect, that lead to stress–energy tensors that also violate the energy conditions, such as negative mass–energy, when described in the context of the quantum field theories.

If certain quantum inequalities conjectured by Ford and Roman hold, the energy requirements for some warp drives may be unfeasibly large as well as negative. For example, the energy equivalent of −10⁶⁴ kg might be required to transport a small spaceship across the Milky Way—an amount orders of magnitude greater than the estimated mass of the observable universe. Counterarguments to these apparent problems have also been offered, although the energy requirements still generally require a Type III civilization on the Kardashev scale.

Chris Van Den Broeck of the Katholieke Universiteit Leuven in Belgium, in 1999, tried to address the potential issues. By contracting the 3+1-dimensional surface area of the bubble being transported by the drive, while at the same time expanding the three-dimensional volume contained inside, Van Den Broeck was able to reduce the total energy needed to transport small atoms to less than three solar masses. Later in 2003, by slightly modifying the Van den Broeck metric, Serguei Krasnikov reduced the necessary total amount of negative mass to a few milligrams. Van Den Broeck detailed this by saying that the total energy can be reduced dramatically by keeping the surface area of the warp bubble itself microscopically small, while at the same time expanding the spatial volume inside the bubble. However, Van Den Broeck concludes that the energy densities required are still unachievable, as are the small size (a few orders of magnitude above the Planck scale) of the spacetime structures needed.

In 2012, physicist Harold White and collaborators announced that modifying the geometry of exotic matter could reduce the mass–energy requirements for a macroscopic space ship from the equivalent of the planet Jupiter to that of the Voyager 1 spacecraft (c. 700 kg) or less, and stated their intent to perform small-scale experiments in constructing warp fields. White proposed to thicken the extremely thin wall of the warp bubble, so the energy is focused in a larger volume, but the overall peak energy density is actually smaller. In a flat 2D representation, the ring of positive and negative energy, initially very thin, becomes a larger, fuzzy torus (donut shape). However, as this less energetic warp bubble also thickens toward the interior region, it leaves less flat space to house the spacecraft, which has to be smaller. Furthermore, if the intensity of the space warp can be oscillated over time, the energy required is reduced even more. According to White, a modified Michelson–Morley interferometer could test the idea: one of the legs of the interferometer would appear to have a slightly different length when the test devices were energised. Alcubierre has expressed skepticism about the experiment, saying: "from my understanding there is no way it can be done, probably not for centuries if at all".

In 2021, physicist Erik Lentz described a way warp drives sourced from known and familiar purely positive energy could exist—warp bubbles based on superluminal self-reinforcing "soliton" waves. The claim is controversial, with other physicists arguing that all physically reasonable warp drives violate the weak energy condition, as well as both the strong and dominant energy conditions.

Placement of matter

Krasnikov proposed that if tachyonic matter cannot be found or used, then a solution might be to arrange for masses along the path of the vessel to be set in motion in such a way that the required field was produced. But in this case, the Alcubierre drive vessel can only travel routes that, like a railroad, have first been equipped with the necessary infrastructure. The pilot inside the bubble is causally disconnected from its walls and cannot carry out any action outside the bubble: the bubble cannot be used for the first trip to a distant star because the pilot cannot place infrastructure ahead of the bubble while "in transit". For example, traveling to Vega (which is 25 light-years from Earth) requires arranging everything so that the bubble moving toward Vega with a superluminal velocity would appear; such arrangements will always take more than 25 years.

Coule has argued that schemes, such as the one proposed by Alcubierre, are infeasible because matter placed en route of the intended path of a craft must be placed at superluminal speed—that constructing an Alcubierre drive requires an Alcubierre drive even if the metric that allows it is physically meaningful. Coule further argues that an analogous objection will apply to any proposed method of constructing an Alcubierre drive.

Survivability inside the bubble

An article by José Natário (2002) argues that crew members could not control, steer or stop the ship in its warp bubble because the ship could not send signals to the front of the bubble.

A 2009 article by Carlos Barceló, Stefano Finazzi, and Stefano Liberati uses quantum theory to argue that the Alcubierre drive at faster-than-light velocities is impossible mostly because extremely high temperatures caused by Hawking radiation would destroy anything inside the bubble at superluminal velocities and destabilize the bubble itself; the article also argues that these problems are absent if the bubble velocity is subluminal, although the drive still requires exotic matter.

Damaging effect on destination

Brendan McMonigal, Geraint F. Lewis, and Philip O'Byrne have argued that were an Alcubierre-driven ship to decelerate from superluminal speed, the particles that its bubble had gathered in transit would be released in energetic outbursts akin to the infinitely-blueshifted radiation hypothesized to occur at the inner event horizon of a Kerr black hole; forward-facing particles would thereby be energetic enough to destroy anything at the destination directly in front of the ship.

Wall thickness

The amount of negative energy required for such a propulsion is not yet known. Pfenning and Allen Everett of Tufts hold that a warp bubble traveling at 10-times the speed of light must have a wall thickness of no more than 10⁻³² meters—close to the limiting Planck length, 1.6 × 10⁻³⁵ meters. In Alcubierre's original calculations, a bubble macroscopically large enough to enclose a ship of 200 meters would require a total amount of exotic matter greater than the mass of the observable universe, and straining the exotic matter to an extremely thin band of 10⁻³² meters is considered impractical. Similar constraints apply to Krasnikov's superluminal subway. Chris Van den Broeck constructed a modification of Alcubierre's model that requires much less exotic matter but places the ship in a curved spacetime "bottle" whose neck is about 10⁻³² meters.

Causality violation and semiclassical instability

Calculations by physicist Allen Everett show that warp bubbles could be used to create closed timelike curves in general relativity, meaning that the theory predicts that they could be used for backwards time travel. While it is possible that the fundamental laws of physics might allow closed timelike curves, the chronology protection conjecture hypothesizes that in all cases where the classical theory of general relativity allows them, quantum effects would intervene to eliminate the possibility, making these spacetimes impossible to realize. A possible type of effect that would accomplish this is a buildup of vacuum fluctuations on the border of the region of spacetime where time travel would first become possible, causing the energy density to become high enough to destroy the system that would otherwise become a time machine. Some results in semiclassical gravity appear to support the conjecture, including a calculation dealing specifically with quantum effects in warp-drive spacetimes that suggested that warp bubbles would be semiclassically unstable, but ultimately the conjecture can only be decided by a full theory of quantum gravity.

Alcubierre briefly discusses some of these issues in a series of lecture slides posted online, where he writes: "beware: in relativity, any method to travel faster than light can in principle be used to travel back in time (a time machine)". In the next slide, he brings up the chronology protection conjecture and writes: "The conjecture has not been proven (it wouldn't be a conjecture if it had), but there are good arguments in its favor based on quantum field theory. The conjecture does not prohibit faster-than-light travel. It just states that if a method to travel faster than light exists, and one tries to use it to build a time machine, something will go wrong: the energy accumulated will explode, or it will create a black hole."

Relation to Star Trek warp drive

The Star Trek television series and films use the term "warp drive" to describe their method of faster-than-light travel. Neither the Alcubierre theory, nor anything similar, existed when the series was conceived—the term "warp drive" and general concept originated with John W. Campbell's 1931 science fiction novel Islands of Space. Alcubierre stated in an email to William Shatner that his theory was directly inspired by the term used in the show and cites the "'warp drive' of science fiction" in his 1994 article. A USS Alcubierre appears in the Star Trek tabletop RPG Star Trek Adventures. Since the release of Star Trek: The Original Series, more recent Star Trek spin-off series have made closer use of the theory behind the Alcubierre Drive incorporating warp bubbles/fields into the in-universe science.

Kaluza–Klein theory

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory

 

Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons

In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to string theory. In their setup, the vacuum has the usual 3 dimensions of space and one dimension of time but with another microscopic extra spatial dimension in the shape of a tiny circle. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions.

The five-dimensional (5D) theory developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Albert Einstein in 1919 and published them in 1921. Kaluza presented a purely classical extension of general relativity to 5D, with a metric tensor of 15 components. Ten components are identified with the 4D spacetime metric, four components with the electromagnetic vector potential, and one component with an unidentified scalar field sometimes called the "radion" or the "dilaton". Correspondingly, the 5D Einstein equations yield the 4D Einstein field equations, the Maxwell equations for the electromagnetic field, and an equation for the scalar field. Kaluza also introduced the "cylinder condition" hypothesis, that no component of the five-dimensional metric depends on the fifth dimension. Without this restriction, terms are introduced that involve derivatives of the fields with respect to the fifth coordinate, and this extra degree of freedom makes the mathematics of the fully variable 5D relativity enormously complex. Standard 4D physics seems to manifest this "cylinder condition" and, along with it, simpler mathematics.

In 1926, Oskar Klein gave Kaluza's classical five-dimensional theory a quantum interpretation, to accord with the then-recent discoveries of Werner Heisenberg and Erwin Schrödinger. Klein introduced the hypothesis that the fifth dimension was curled up and microscopic, to explain the cylinder condition. Klein suggested that the geometry of the extra fifth dimension could take the form of a circle, with the radius of 10⁻³⁰ cm. More precisely, the radius of the circular dimension is 23 times the Planck length, which in turn is of the order of 10⁻³³ cm. Klein also made a contribution to the classical theory by providing a properly normalized 5D metric. Work continued on the Kaluza field theory during the 1930s by Einstein and colleagues at Princeton University.

In the 1940s, the classical theory was completed, and the full field equations including the scalar field were obtained by three independent research groups: Yves Thiry, working in France on his dissertation under André Lichnerowicz; Pascual Jordan, Günther Ludwig, and Claus Müller in Germany, with critical input from Wolfgang Pauli and Markus Fierz; and Paul Scherrer working alone in Switzerland. Jordan's work led to the scalar–tensor theory of Brans–Dicke; Carl H. Brans and Robert H. Dicke were apparently unaware of Thiry or Scherrer. The full Kaluza equations under the cylinder condition are quite complex, and most English-language reviews, as well as the English translations of Thiry, contain some errors. The curvature tensors for the complete Kaluza equations were evaluated using tensor-algebra software in 2015, verifying results of J. A. Ferrari and R. Coquereaux & G. Esposito-Farese. The 5D covariant form of the energy–momentum source terms is treated by L. L. Williams.

Kaluza hypothesis

In his 1921 article, Kaluza established all the elements of the classical five-dimensional theory: the Kaluza–Klein metric, the Kaluza–Klein–Einstein field equations, the equations of motion, the stress–energy tensor, and the cylinder condition. With no free parameters, it merely extends general relativity to five dimensions. One starts by hypothesizing a form of the five-dimensional Kaluza–Klein metric ${\displaystyle {\tilde{g}}_{ab}}$, where Latin indices span five dimensions. Let one also introduce the four-dimensional spacetime metric ${\displaystyle g_{\mu \nu }}$, where Greek indices span the usual four dimensions of space and time; a 4-vector ${\displaystyle A^{\mu }}$ identified with the electromagnetic vector potential; and a scalar field ${\displaystyle \varphi }$. Then decompose the 5D metric so that the 4D metric is framed by the electromagnetic vector potential, with the scalar field at the fifth diagonal. This can be visualized as

${\displaystyle {\tilde{g}}_{ab}\equiv \left[\begin{array}{cc}{g}_{\mu \nu }+{\varphi }^2{A}_{\mu }{A}_{\nu }& {\varphi }^2{A}_{\mu }\\ {\varphi }^2{A}_{\nu }& {\varphi }^2\end{array}\right].}$

One can write more precisely

${\displaystyle {\tilde{g}}_{\mu \nu }\equiv g_{\mu \nu }+{\varphi }^2{A}_{\mu }{A}_{\nu },{\tilde{g}}_{5\nu }\equiv {\tilde{g}}_{\nu 5}\equiv {\varphi }^2{A}_{\nu },{\tilde{g}}_{55}\equiv {\varphi }^2,}$

where the index ${\displaystyle 5}$ indicates the fifth coordinate by convention, even though the first four coordinates are indexed with 0, 1, 2, and 3. The associated inverse metric is

${\displaystyle {\tilde{g}}^{ab}\equiv \left[\begin{array}{cc}{g}^{\mu \nu }& -A^{\mu }\\ -A^{\nu }& g_{\alpha \beta }{A}^{\alpha }{A}^{\beta }+\frac{1}{{\varphi }^2}\end{array}\right].}$

This decomposition is quite general, and all terms are dimensionless. Kaluza then applies the machinery of standard general relativity to this metric. The field equations are obtained from five-dimensional Einstein equations, and the equations of motion from the five-dimensional geodesic hypothesis. The resulting field equations provide both the equations of general relativity and of electrodynamics; the equations of motion provide the four-dimensional geodesic equation and the Lorentz force law, and one finds that electric charge is identified with motion in the fifth dimension.

The hypothesis for the metric implies an invariant five-dimensional length element ${\displaystyle ds}$:

${\displaystyle d{s}^2\equiv {\tilde{g}}_{ab}d{x}^{a}d{x}^b=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+{\varphi }^2(A_{\nu }d{x}^{\nu }+d{x}^5{)}^2.}$

Field equations from the Kaluza hypothesis

The Kaluza–Klein–Einstein field equations of the five-dimensional theory were never adequately provided by Kaluza or Klein because they ignored the scalar field. The full Kaluza field equations are generally attributed to Thiry, who obtained vacuum field equations, although Kaluza originally provided a stress–energy tensor for his theory, and Thiry included a stress–energy tensor in his thesis. But as described by Gonner, several independent groups worked on the field equations in the 1940s and earlier. Thiry is perhaps best known only because an English translation was provided by Applequist, Chodos, & Freund in their review book. Applequist et al. also provided an English translation of Kaluza's article. Translations of the three (1946, 1947, 1948) Jordan articles can be found on the ResearchGate and Academia.edu archives. The first correct English-language Kaluza field equations, including the scalar field, were provided by Williams.

To obtain the 5D Kaluza–Klein–Einstein field equations, the 5D Kaluza–Klein–Christoffel symbols ${\displaystyle {\tilde{\mathrm{\Gamma }}}_{bc}^a}$ are calculated from the 5D Kaluza–Klein metric ${\displaystyle {\tilde{g}}_{ab}}$, and the 5D Kaluza–Klein–Ricci tensor ${\displaystyle {\tilde{R}}_{ab}}$ is calculated from the 5D connections.

The classic results of Thiry and other authors presume the cylinder condition:

${\displaystyle \frac{\mathrm{\partial }{\tilde{g}}_{ab}}{\mathrm{\partial }{x}^5}=0.}$

Without this assumption, the field equations become much more complex, providing many more degrees of freedom that can be identified with various new fields. Paul Wesson and colleagues have pursued relaxation of the cylinder condition to gain extra terms that can be identified with the matter fields, for which Kaluza otherwise inserted a stress–energy tensor by hand.

It has been an objection to the original Kaluza hypothesis to invoke the fifth dimension only to negate its dynamics. But Thiry argued that the interpretation of the Lorentz force law in terms of a five-dimensional geodesic militates strongly for a fifth dimension irrespective of the cylinder condition. Most authors have therefore employed the cylinder condition in deriving the field equations. Furthermore, vacuum equations are typically assumed for which

${\displaystyle {\tilde{R}}_{ab}=0,}$

where

${\displaystyle {\tilde{R}}_{ab}\equiv {\mathrm{\partial }}_c{\tilde{\mathrm{\Gamma }}}_{ab}^c-{\mathrm{\partial }}_b{\tilde{\mathrm{\Gamma }}}_{ca}^c+{\tilde{\mathrm{\Gamma }}}_{cd}^c{\tilde{\mathrm{\Gamma }}}_{ab}^d-{\tilde{\mathrm{\Gamma }}}_{bd}^c{\tilde{\mathrm{\Gamma }}}_{ac}^d}$

and

${\displaystyle {\tilde{\mathrm{\Gamma }}}_{bc}^a\equiv \frac{1}{2}{\tilde{g}}^{ad}({\mathrm{\partial }}_b{\tilde{g}}_{dc}+{\mathrm{\partial }}_c{\tilde{g}}_{db}-{\mathrm{\partial }}_d{\tilde{g}}_{bc}).}$

The vacuum field equations obtained in this way by Thiry and Jordan's group are as follows.

The field equation for ${\displaystyle \varphi }$ is obtained from

${\displaystyle {\tilde{R}}_{55}=0\Rightarrow ◻\varphi =\frac{1}{4}{\varphi }^3{F}^{\alpha \beta }{F}_{\alpha \beta },}$

where ${\displaystyle F_{\alpha \beta }\equiv {\mathrm{\partial }}_{\alpha }{A}_{\beta }-{\mathrm{\partial }}_{\beta }{A}_{\alpha },}$ ${\displaystyle ◻\equiv g^{\mu \nu }{\mathrm{\nabla }}_{\mu }{\mathrm{\nabla }}_{\nu },}$ and ${\displaystyle {\mathrm{\nabla }}_{\mu }}$ is a standard, 4D covariant derivative. It shows that the electromagnetic field is a source for the scalar field. Note that the scalar field cannot be set to a constant without constraining the electromagnetic field. The earlier treatments by Kaluza and Klein did not have an adequate description of the scalar field and did not realize the implied constraint on the electromagnetic field by assuming the scalar field to be constant.

The field equation for ${\displaystyle A^{\nu }}$ is obtained from

${\displaystyle {\tilde{R}}_{5\alpha }=0=\frac{1}{2\varphi }{g}^{\beta \mu }{\mathrm{\nabla }}_{\mu }({\varphi }^3{F}_{\alpha \beta })-A_{\alpha }\varphi ◻\varphi .}$

It has the form of the vacuum Maxwell equations if the scalar field is constant.

The field equation for the 4D Ricci tensor ${\displaystyle R_{\mu \nu }}$ is obtained from

${\displaystyle \begin{array}{rl}{\tilde{R}}_{\mu \nu }-\frac{1}{2}{\tilde{g}}_{\mu \nu }\tilde{R}& =0\Rightarrow \\ R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R& =\frac{1}{2}{\varphi }^2\left(g^{\alpha \beta }{F}_{\mu \alpha }{F}_{\nu \beta }-\frac{1}{4}{g}_{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }\right)+\frac{1}{\varphi }({\mathrm{\nabla }}_{\mu }{\mathrm{\nabla }}_{\nu }\varphi -g_{\mu \nu }◻\varphi ),\end{array}}$

where ${\displaystyle R}$ is the standard 4D Ricci scalar.

This equation shows the remarkable result, called the "Kaluza miracle", that the precise form for the electromagnetic stress–energy tensor emerges from the 5D vacuum equations as a source in the 4D equations: field from the vacuum. This relation allows the definitive identification of ${\displaystyle A^{\mu }}$ with the electromagnetic vector potential. Therefore, the field needs to be rescaled with a conversion constant ${\displaystyle k}$ such that ${\displaystyle A^{\mu }\to k{A}^{\mu }}$.

The relation above shows that we must have

${\displaystyle \frac{k^2}{2}=\frac{8\pi G}{c^4}\frac{1}{{\mu }_0}=\frac{2G}{c^2}4\pi {ϵ}_0,}$

where ${\displaystyle G}$ is the gravitational constant, and ${\displaystyle {\mu }_0}$ is the permeability of free space. In the Kaluza theory, the gravitational constant can be understood as an electromagnetic coupling constant in the metric. There is also a stress–energy tensor for the scalar field. The scalar field behaves like a variable gravitational constant, in terms of modulating the coupling of electromagnetic stress–energy to spacetime curvature. The sign of ${\displaystyle {\varphi }^2}$ in the metric is fixed by correspondence with 4D theory so that electromagnetic energy densities are positive. It is often assumed that the fifth coordinate is spacelike in its signature in the metric.

In the presence of matter, the 5D vacuum condition cannot be assumed. Indeed, Kaluza did not assume it. The full field equations require evaluation of the 5D Kaluza–Klein–Einstein tensor

${\displaystyle {\tilde{G}}_{ab}\equiv {\tilde{R}}_{ab}-\frac{1}{2}{\tilde{g}}_{ab}\tilde{R},}$

as seen in the recovery of the electromagnetic stress–energy tensor above. The 5D curvature tensors are complex, and most English-language reviews contain errors in either ${\displaystyle {\tilde{G}}_{ab}}$ or ${\displaystyle {\tilde{R}}_{ab}}$, as does the English translation of Thiry. In 2015, a complete set of 5D curvature tensors under the cylinder condition, evaluated using tensor-algebra software, was produced.

Equations of motion from the Kaluza hypothesis

The equations of motion are obtained from the five-dimensional geodesic hypothesis in terms of a 5-velocity ${\displaystyle {\tilde{U}}^a\equiv d{x}^a/ds}$:

${\displaystyle {\tilde{U}}^b{\tilde{\mathrm{\nabla }}}_b{\tilde{U}}^a=\frac{d{\tilde{U}}^a}{ds}+{\tilde{\mathrm{\Gamma }}}_{bc}^a{\tilde{U}}^b{\tilde{U}}^c=0.}$

This equation can be recast in several ways, and it has been studied in various forms by authors including Kaluza, Pauli, Gross & Perry, Gegenberg & Kunstatter, and Wesson & Ponce de Leon, but it is instructive to convert it back to the usual 4-dimensional length element ${\displaystyle c^{2}d{\tau }^2\equiv g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}$, which is related to the 5-dimensional length element ${\displaystyle ds}$ as given above:

${\displaystyle d{s}^2=c^{2}d{\tau }^2+{\varphi }^2(k{A}_{\nu }d{x}^{\nu }+d{x}^5{)}^2.}$

Then the 5D geodesic equation can be written for the spacetime components of the 4-velocity:

${\displaystyle U^{\nu }\equiv \frac{d{x}^{\nu }}{d\tau },}$

${\displaystyle \frac{d{U}^{\nu }}{d\tau }+{\tilde{\mathrm{\Gamma }}}_{\alpha \beta }^{\mu }{U}^{\alpha }{U}^{\beta }+2{\tilde{\mathrm{\Gamma }}}_{5\alpha }^{\mu }{U}^{\alpha }{U}^5+{\tilde{\mathrm{\Gamma }}}_{55}^{\mu }(U^5{)}^2+U^{\mu }\frac{d}{d\tau }\ln \frac{cd\tau }{ds}=0.}$

The term quadratic in ${\displaystyle U^{\nu }}$ provides the 4D geodesic equation plus some electromagnetic terms:

${\displaystyle {\tilde{\mathrm{\Gamma }}}_{\alpha \beta }^{\mu }={\mathrm{\Gamma }}_{\alpha \beta }^{\mu }+\frac{1}{2}{g}^{\mu \nu }{k}^2{\varphi }^2(A_{\alpha }{F}_{\beta \nu }+A_{\beta }{F}_{\alpha \nu }-A_{\alpha }{A}_{\beta }{\mathrm{\partial }}_{\nu }\ln {\varphi }^2).}$

The term linear in ${\displaystyle U^{\nu }}$ provides the Lorentz force law:

${\displaystyle {\tilde{\mathrm{\Gamma }}}_{5\alpha }^{\mu }=\frac{1}{2}{g}^{\mu \nu }k{\varphi }^2(F_{\alpha \nu }-A_{\alpha }{\mathrm{\partial }}_{\nu }\ln {\varphi }^2).}$

This is another expression of the "Kaluza miracle". The same hypothesis for the 5D metric that provides electromagnetic stress–energy in the Einstein equations, also provides the Lorentz force law in the equation of motions along with the 4D geodesic equation. Yet correspondence with the Lorentz force law requires that we identify the component of 5-velocity along the fifth dimension with electric charge:

${\displaystyle k{U}^5=k\frac{d{x}^5}{d\tau }\to \frac{q}{mc},}$

where ${\displaystyle m}$ is particle mass, and ${\displaystyle q}$ is particle electric charge. Thus electric charge is understood as motion along the fifth dimension. The fact that the Lorentz force law could be understood as a geodesic in five dimensions was to Kaluza a primary motivation for considering the five-dimensional hypothesis, even in the presence of the aesthetically unpleasing cylinder condition.

Yet there is a problem: the term quadratic in ${\displaystyle U^5}$,

${\displaystyle {\tilde{\mathrm{\Gamma }}}_{55}^{\mu }=-\frac{1}{2}{g}^{\mu \alpha }{\mathrm{\partial }}_{\alpha }{\varphi }^2.}$

If there is no gradient in the scalar field, the term quadratic in ${\displaystyle U^5}$ vanishes. But otherwise the expression above implies

${\displaystyle U^5\sim c\frac{q/m}{G^{1/2}}.}$

For elementary particles, ${\displaystyle U^5>{10}^{20}c}$. The term quadratic in ${\displaystyle U^5}$ should dominate the equation, perhaps in contradiction to experience. This was the main shortfall of the five-dimensional theory as Kaluza saw it, and he gives it some discussion in his original article.

The equation of motion for ${\displaystyle U^5}$ is particularly simple under the cylinder condition. Start with the alternate form of the geodesic equation, written for the covariant 5-velocity:

${\displaystyle \frac{d{\tilde{U}}_a}{ds}=\frac{1}{2}{\tilde{U}}^b{\tilde{U}}^c\frac{\mathrm{\partial }{\tilde{g}}_{bc}}{\mathrm{\partial }{x}^a}.}$

This means that under the cylinder condition, ${\displaystyle {\tilde{U}}_5}$ is a constant of the five-dimensional motion:

${\displaystyle {\tilde{U}}_5={\tilde{g}}_{5a}{\tilde{U}}^a={\varphi }^2\frac{cd\tau }{ds}(k{A}_{\nu }{U}^{\nu }+U^5)=\text{constant}.}$

Kaluza's hypothesis for the matter stress–energy tensor

Kaluza proposed a five-dimensional matter stress tensor ${\displaystyle {\tilde{T}}_M^{ab}}$ of the form

${\displaystyle {\tilde{T}}_M^{ab}=\rho \frac{d{x}^a}{ds}\frac{d{x}^b}{ds},}$

where ${\displaystyle \rho }$ is a density, and the length element ${\displaystyle ds}$ is as defined above.

Then the spacetime component gives a typical "dust" stress–energy tensor:

${\displaystyle {\tilde{T}}_M^{\mu \nu }=\rho \frac{d{x}^{\mu }}{ds}\frac{d{x}^{\nu }}{ds}.}$

The mixed component provides a 4-current source for the Maxwell equations:

${\displaystyle {\tilde{T}}_M^{5\mu }=\rho \frac{d{x}^{\mu }}{ds}\frac{d{x}^5}{ds}=\rho U^{\mu }\frac{q}{kmc}.}$

Just as the five-dimensional metric comprises the four-dimensional metric framed by the electromagnetic vector potential, the five-dimensional stress–energy tensor comprises the four-dimensional stress–energy tensor framed by the vector 4-current.

Quantum interpretation of Klein

Kaluza's original hypothesis was purely classical and extended discoveries of general relativity. By the time of Klein's contribution, the discoveries of Heisenberg, Schrödinger, and Louis de Broglie were receiving a lot of attention. Klein's Nature article suggested that the fifth dimension is closed and periodic, and that the identification of electric charge with motion in the fifth dimension can be interpreted as standing waves of wavelength ${\displaystyle {\lambda }^5}$, much like the electrons around a nucleus in the Bohr model of the atom. The quantization of electric charge could then be nicely understood in terms of integer multiples of fifth-dimensional momentum. Combining the previous Kaluza result for ${\displaystyle U^5}$ in terms of electric charge, and a de Broglie relation for momentum ${\displaystyle p^5=h/{\lambda }^5}$, Klein obtained an expression for the 0th mode of such waves:

${\displaystyle m{U}^5=\frac{cq}{G^{1/2}}=\frac{h}{{\lambda }^5}\Rightarrow {\lambda }^5\sim \frac{h{G}^{1/2}}{cq},}$

where ${\displaystyle h}$ is the Planck constant. Klein found that ${\displaystyle {\lambda }^5\sim {10}^{-30}}$ cm, and thereby an explanation for the cylinder condition in this small value.

Klein's Zeitschrift für Physik article of the same year, gave a more detailed treatment that explicitly invoked the techniques of Schrödinger and de Broglie. It recapitulated much of the classical theory of Kaluza described above, and then departed into Klein's quantum interpretation. Klein solved a Schrödinger-like wave equation using an expansion in terms of fifth-dimensional waves resonating in the closed, compact fifth dimension.

Group theory interpretation

The space M × C is compactified over the compact set C, and after Kaluza–Klein decomposition one has an effective field theory over M.

In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set, and construction of this compact dimension is referred to as compactification.

In modern geometry, the extra fifth dimension can be understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U(1). In Kaluza–Klein theory this group suggests that gauge symmetry is the symmetry of circular compact dimensions. Once this geometrical interpretation is understood, it is relatively straightforward to replace U(1) by a general Lie group. Such generalizations are often called Yang–Mills theories. If a distinction is drawn, then it is that Yang–Mills theories occur on a flat spacetime, whereas Kaluza–Klein treats the more general case of curved spacetime. The base space of Kaluza–Klein theory need not be four-dimensional spacetime; it can be any (pseudo-)Riemannian manifold, or even a supersymmetric manifold or orbifold or even a noncommutative space.

The construction can be outlined, roughly, as follows. One starts by considering a principal fiber bundle P with gauge group G over a manifold M. Given a connection on the bundle, and a metric on the base manifold, and a gauge invariant metric on the tangent of each fiber, one can construct a bundle metric defined on the entire bundle. Computing the scalar curvature of this bundle metric, one finds that it is constant on each fiber: this is the "Kaluza miracle". One did not have to explicitly impose a cylinder condition, or to compactify: by assumption, the gauge group is already compact. Next, one takes this scalar curvature as the Lagrangian density, and, from this, constructs the Einstein–Hilbert action for the bundle, as a whole. The equations of motion, the Euler–Lagrange equations, can be then obtained by considering where the action is stationary with respect to variations of either the metric on the base manifold, or of the gauge connection. Variations with respect to the base metric gives the Einstein field equations on the base manifold, with the energy–momentum tensor given by the curvature (field strength) of the gauge connection. On the flip side, the action is stationary against variations of the gauge connection precisely when the gauge connection solves the Yang–Mills equations. Thus, by applying a single idea: the principle of least action, to a single quantity: the scalar curvature on the bundle (as a whole), one obtains simultaneously all of the needed field equations, for both the spacetime and the gauge field.

As an approach to the unification of the forces, it is straightforward to apply the Kaluza–Klein theory in an attempt to unify gravity with the strong and electroweak forces by using the symmetry group of the Standard Model, SU(3) × SU(2) × U(1). However, an attempt to convert this interesting geometrical construction into a bona-fide model of reality flounders on a number of issues, including the fact that the fermions must be introduced in an artificial way (in nonsupersymmetric models). Nonetheless, KK remains an important touchstone in theoretical physics and is often embedded in more sophisticated theories. It is studied in its own right as an object of geometric interest in K-theory.

Even in the absence of a completely satisfying theoretical physics framework, the idea of exploring extra, compactified, dimensions is of considerable interest in the experimental physics and astrophysics communities. A variety of predictions, with real experimental consequences, can be made (in the case of large extra dimensions and warped models). For example, on the simplest of principles, one might expect to have standing waves in the extra compactified dimension(s). If a spatial extra dimension is of radius R, the invariant mass of such standing waves would be M_n = nh/Rc with n an integer, h being the Planck constant and c the speed of light. This set of possible mass values is often called the Kaluza–Klein tower. Similarly, in Thermal quantum field theory a compactification of the euclidean time dimension leads to the Matsubara frequencies and thus to a discretized thermal energy spectrum.

However, Klein's approach to a quantum theory is flawed and, for example, leads to a calculated electron mass in the order of magnitude of the Planck mass.

Examples of experimental pursuits include work by the CDF collaboration, which has re-analyzed particle collider data for the signature of effects associated with large extra dimensions/warped models.

Robert Brandenberger and Cumrun Vafa have speculated that in the early universe, cosmic inflation causes three of the space dimensions to expand to cosmological size while the remaining dimensions of space remained microscopic.

Space–time–matter theory

One particular variant of Kaluza–Klein theory is space–time–matter theory or induced matter theory, chiefly promulgated by Paul Wesson and other members of the Space–Time–Matter Consortium. In this version of the theory, it is noted that solutions to the equation

${\displaystyle {\tilde{R}}_{ab}=0}$

may be re-expressed so that in four dimensions, these solutions satisfy Einstein's equations

${\displaystyle G_{\mu \nu }=8\pi T_{\mu \nu }}$

with the precise form of the T_μν following from the Ricci-flat condition on the five-dimensional space. In other words, the cylinder condition of the previous development is dropped, and the stress–energy now comes from the derivatives of the 5D metric with respect to the fifth coordinate. Because the energy–momentum tensor is normally understood to be due to concentrations of matter in four-dimensional space, the above result is interpreted as saying that four-dimensional matter is induced from geometry in five-dimensional space.

In particular, the soliton solutions of ${\displaystyle {\tilde{R}}_{ab}=0}$ can be shown to contain the Friedmann–Lemaître–Robertson–Walker metric in both radiation-dominated (early universe) and matter-dominated (later universe) forms. The general equations can be shown to be sufficiently consistent with classical tests of general relativity to be acceptable on physical principles, while still leaving considerable freedom to also provide interesting cosmological models.

Geometric interpretation

The Kaluza–Klein theory has a particularly elegant presentation in terms of geometry. In a certain sense, it looks just like ordinary gravity in free space, except that it is phrased in five dimensions instead of four.

Einstein equations

The equations governing ordinary gravity in free space can be obtained from an action, by applying the variational principle to a certain action. Let M be a (pseudo-)Riemannian manifold, which may be taken as the spacetime of general relativity. If g is the metric on this manifold, one defines the action S(g) as

${\displaystyle S(g)={\int }_{M}R(g)\mathrm{vol}(g),}$

where R(g) is the scalar curvature, and vol(g) is the volume element. By applying the variational principle to the action

${\displaystyle \frac{\delta S(g)}{\delta g}=0,}$

one obtains precisely the Einstein equations for free space:

${\displaystyle R_{ij}-\frac{1}{2}{g}_{ij}R=0,}$

where R_ij is the Ricci tensor.

Maxwell equations

By contrast, the Maxwell equations describing electromagnetism can be understood to be the Hodge equations of a principal ${\displaystyle \mathrm{U}(1)}$-bundle or circle bundle ${\displaystyle \pi :P\to M}$ with fiber ${\displaystyle \mathrm{U}(1)}$. That is, the electromagnetic field ${\displaystyle F}$ is a harmonic 2-form in the space ${\displaystyle {\mathrm{\Omega }}^2(M)}$ of differentiable 2-forms on the manifold ${\displaystyle M}$. In the absence of charges and currents, the free-field Maxwell equations are

${\displaystyle \mathrm{d}F=0\text{and}\mathrm{d}\star F=0,}$

where ${\displaystyle \star }$ is the Hodge star operator.

Kaluza–Klein geometry

To build the Kaluza–Klein theory, one picks an invariant metric on the circle ${\displaystyle S^1}$ that is the fiber of the U(1)-bundle of electromagnetism. In this discussion, an invariant metric is simply one that is invariant under rotations of the circle. Suppose that this metric gives the circle a total length ${\displaystyle \mathrm{\Lambda }}$. One then considers metrics ${\displaystyle \hat{g}}$ on the bundle ${\displaystyle P}$ that are consistent with both the fiber metric, and the metric on the underlying manifold ${\displaystyle M}$. The consistency conditions are:

• The projection of ${\displaystyle \hat{g}}$ to the vertical subspace ${\displaystyle {\mathrm{Vert}}_{p}P\subset T_{p}P}$ needs to agree with metric on the fiber over a point in the manifold ${\displaystyle M}$.
• The projection of ${\displaystyle \hat{g}}$ to the horizontal subspace ${\displaystyle {\mathrm{Hor}}_{p}P\subset T_{p}P}$ of the tangent space at point ${\displaystyle p\in P}$ must be isomorphic to the metric ${\displaystyle g}$ on ${\displaystyle M}$ at ${\displaystyle \pi (P)}$.

The Kaluza–Klein action for such a metric is given by

${\displaystyle S(\hat{g})={\int }_{P}R(\hat{g})\mathrm{vol}(\hat{g}).}$

The scalar curvature, written in components, then expands to

${\displaystyle R(\hat{g})={\pi }^{\ast }\left(R(g)-\frac{{\mathrm{\Lambda }}^2}{2}|F{|}^2\right),}$

where ${\displaystyle {\pi }^{\ast }}$ is the pullback of the fiber bundle projection ${\displaystyle \pi :P\to M}$. The connection ${\displaystyle A}$ on the fiber bundle is related to the electromagnetic field strength as

${\displaystyle {\pi }^{\ast }F=dA.}$

That there always exists such a connection, even for fiber bundles of arbitrarily complex topology, is a result from homology and specifically, K-theory. Applying Fubini's theorem and integrating on the fiber, one gets

${\displaystyle S(\hat{g})=\mathrm{\Lambda }{\int }_M\left(R(g)-\frac{1}{{\mathrm{\Lambda }}^2}|F{|}^2\right)\mathrm{vol}(g).}$

Varying the action with respect to the component ${\displaystyle A}$, one regains the Maxwell equations. Applying the variational principle to the base metric ${\displaystyle g}$, one gets the Einstein equations

${\displaystyle R_{ij}-\frac{1}{2}{g}_{ij}R=\frac{1}{{\mathrm{\Lambda }}^2}{T}_{ij}}$

with the electromagnetic stress–energy tensor being given by

${\displaystyle T^{ij}=F^{ik}{F}^{jl}{g}_{kl}-\frac{1}{4}{g}^{ij}|F{|}^2.}$

The original theory identifies ${\displaystyle \mathrm{\Lambda }}$ with the fiber metric ${\displaystyle g_{55}}$ and allows ${\displaystyle \mathrm{\Lambda }}$ to vary from fiber to fiber. In this case, the coupling between gravity and the electromagnetic field is not constant, but has its own dynamical field, the radion.

Generalizations

In the above, the size of the loop ${\displaystyle \mathrm{\Lambda }}$ acts as a coupling constant between the gravitational field and the electromagnetic field. If the base manifold is four-dimensional, the Kaluza–Klein manifold P is five-dimensional. The fifth dimension is a compact space and is called the compact dimension. The technique of introducing compact dimensions to obtain a higher-dimensional manifold is referred to as compactification. Compactification does not produce group actions on chiral fermions except in very specific cases: the dimension of the total space must be 2 mod 8, and the G-index of the Dirac operator of the compact space must be nonzero.

The above development generalizes in a more-or-less straightforward fashion to general principal G-bundles for some arbitrary Lie group G taking the place of U(1). In such a case, the theory is often referred to as a Yang–Mills theory and is sometimes taken to be synonymous. If the underlying manifold is supersymmetric, the resulting theory is a super-symmetric Yang–Mills theory.

Empirical tests

No experimental or observational signs of extra dimensions have been officially reported. Many theoretical search techniques for detecting Kaluza–Klein resonances have been proposed using the mass couplings of such resonances with the top quark. An analysis of results from the Large Hadron Collider (LHC) in December 2010 severely constrains theories with large extra dimensions.

The observation of a Higgs-like boson at the LHC establishes a new empirical test which can be applied to the search for Kaluza–Klein resonances and supersymmetric particles. The loop Feynman diagrams that exist in the Higgs interactions allow any particle with electric charge and mass to run in such a loop. Standard Model particles besides the top quark and W boson do not make big contributions to the cross-section observed in the H → γγ decay, but if there are new particles beyond the Standard Model, they could potentially change the ratio of the predicted Standard Model H → γγ cross-section to the experimentally observed cross-section. Hence a measurement of any dramatic change to the H → γγ cross-section predicted by the Standard Model is crucial in probing the physics beyond it.

An article from July 2018 gives some hope for this theory; in the article they dispute that gravity is leaking into higher dimensions as in brane theory. However, the article does demonstrate that electromagnetism and gravity share the same number of dimensions, and this fact lends support to Kaluza–Klein theory; whether the number of dimensions is really 3 + 1 or in fact 4 + 1 is the subject of further debate.