M-theory is a solution proposed for the unknown theory of everything

M-theory is a solution proposed for the unknown which would combine all five and 11-dimensional together. According to Dr. , who proposed the theory, mathematical tools which have yet to be invented are needed in order to fully understand it.

M-theory's relation to superstrings and supergravity

M-Theory in various geometric backgrounds is associated with the different superstring theories (in different geometric backgrounds), and these limits are related to each other by the principle of . Two physical theories are dual to each other if they have identical physics after a certain mathematical transformation.

and are related by , as are the two theories. and Heterotic are related by the . Type IIB is also S-dual with itself.

The type II theories have two supersymmetries in the ten-dimensional sense, the rest just one.
The type I theory is special in that it is based on unoriented open and closed strings.
The other four are based on oriented closed strings.
The IIA theory is special because it is non-chiral (parity conserving).
The other four are chiral (parity violating).

In each of these cases there is an 11th dimension that becomes large at strong coupling. In the IIA case the 11th dimension is a circle. In the HE case it is a line interval, which makes eleven-dimensional display two ten-dimensional boundaries. The strong coupling limit of either theory produces an 11-dimensional space-time. This eleven-dimensional description of the underlying theory is called "M- theory". A string's space-time history can be viewed mathematically by functions like

Xμ (σ,τ)

that describe how the string's two-dimensional sheet coordinates (σ,τ) map into space-time Xμ.

One interpretation of this result is that the 11th dimension was always present but invisible because the radius of the 11th dimension is proportional to the string coupling constant and the traditional perturbative string theory presumes it to be infinitesimal. Another interpretation is that is not a fundamental concept of M-theory at all.

Characteristics of M-theory

M-theory contains much more than just strings. It contains both higher and lower dimensional objects. These objects are called where p denotes their spatial dimensionality (thus, 1-brane for a string and 2-brane for a membrane). Higher dimensional objects were always present in superstring theory but could never be studied before the because of their nature.

Insights into non-perturbative properties of p-branes stem from a special class of p-branes called Dirichlet p-branes (Dp-branes). This name results from the assigned to the ends of open strings in type I superstrings.

Open strings of the type I theory can have endpoints which satisfy the . Under this condition, the endpoints of strings are free to move about but no momentum can flow into or out of the end of a string. The T duality infers the existence of open strings with positions fixed in the dimensions that are T-transformed. Generally, in type II theories, we can imagine open strings with specific positions for the end-points in some of the dimensions. This lends an inference that they must end on a preferred surface. Superficially, this notion seems to break the relativistic invariance of the theory, possibly leading to a paradox. The resolution of this paradox is that strings end on a p-dimensional dynamic object, the .

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The importance of stems from the fact that they make it possible to study the excitations of the brane using the renormalizable 2D quantum field theory of the open string instead of the non-renormalizable world-volume theory of the D-brane itself. In this way it becomes possible to compute non-perturbative phenomena using perturbative methods. Many of the previously identified p-branes are D-branes ! Others are related to D-branes by duality symmetries, so that they can also be brought under mathematical control. D-branes have found many useful applications, the most remarkable being the study of . and have shown that D-brane techniques can be used to count the quantum microstates associated to classical black hole configurations. The simplest case first explored was static extremal charged black holes in five dimensions. Strominger and Vafa proved for large values of the charges the entropy S = logN, where N is equal to the number of quantum states that system can be in, agrees with the prediction (1/4 the area of the event horizon).

This result has been generalized to black holes in 4D as well as to ones that are near extremal (and radiate correctly) or rotating, a remarkable advance. It has not yet been proven that there is any problematic breakdown of quantum mechanics due to black holes.

Loop quantum gravity (LQG), also known as loop gravity, and canonical quantum general relativity, is a proposed quantum theory of which attempts to blend together the seemingly incompatible theories of and . This theory is one of a family of theories called canonical quantum gravity. It was developed in parallel with , a rigorous framework for nonperturbative quantization of -invariant . In plain English this is a quantum theory of gravity in which the very space that all other physics occurs in is quantized.

Loop quantum gravity (LQG) is a proposed theory of spacetime which is built from the ground up with the idea of spacetime quantization via the mathematically rigorous theory of . It preserves many of the important features of general relativity, such as local Lorentz invariance, while at the same time employing quantization of both space and time at the in the tradition of . In this sense, both general relativity and quantum mechanics can be thought of as approximations to LQG in their respective domains; thus, LQG is one of the several competing theories that attempts to combine the two into a . However, both the mathematics and physics behind LQG are controversial and it is not clear whether LQG truly unifies the two theories, or if this unification is more "forced" than would be hoped.

This is not the most popular theory of quantum gravity; many physicists have philosophical problems with it. For one thing, the critics of this theory cite that it does not predict the existence of extra dimensions, and does not predict the masses or charges of particles, such as in . The rebuttal in general boils down to LQG being a theory of gravity and nothing more. In the view of those scientists who agree with LQG, the fact that it does not predict any of those properties of particles is not a problem. There are many other theories of quantum gravity, and a list of them can be found on the page.

Loop quantum gravity in general, and its ambitions

LQG in itself was initially less ambitious than string theory, purporting only to be a . String theory, on the other hand, appears to predict not only gravity but also various kinds of matter and energy that lie inside spacetime. Many string theorists believe that it is not possible to quantize gravity in 3+1 dimensions without creating these artifacts. But this is not proven, and it is also not proven that the matter artifacts of string theory are exactly the same as observed matter. Should LQG succeed as a quantum theory of gravity, the known matter fields would have to be incorporated into the theory a posteriori. , one of the fathers of LQG, has explored the possibility that string theory and LQG are two different approximations to the same ultimate theory.

The main claimed successes of loop quantum gravity are: (1) that it is a of 3-space geometry, with quantized area and volume operators; (2) that it includes a calculation of the of ; and (3) that it is a viable gravity-only alternative to string theory. However, these claims are not universally accepted. While many of the core results are rigorous , their physical interpretations are speculative. LQG may or may not be viable as a refinement of either gravity or geometry; entropy is calculated for a kind of hole which may or may not be a black hole.

It should be noted that several quantum gravity alternatives, including and , are also sometimes called "loop quantum gravity".

The incompatibility between quantum mechanics and general relativity

Quantum field theory studied on curved (non-Minkowskian) backgrounds has shown that some of the core assumptions of quantum field theory cannot be carried over. In particular, the vacuum, when it exists, is shown to depend on the path of the observer through space-time (see ).

Historically, there have been two reactions to the apparent inconsistency of quantum theories with the necessary background-independence of general relativity. The first is that the geometric interpretation of general relativity is not fundamental, but . The other view is that background-independence is fundamental, and quantum mechanics needs to be generalized to settings where there is no a priori specified time.

Loop quantum gravity is an effort to formulate a background-independent quantum theory. is a background-independent quantum theory, but it lacks -propagating local degrees of freedom needed for 3 + 1 dimensional gravity.

Being closely related to and theory, LQG is mostly established at the level of rigour of .

The ingredients of loop quantum gravity

Loop quantization

At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeomorphism-invariant gauge theories, which one might call loop quantization. While originally developed in order to quantize vacuum general relativity in 3+1 dimensions, the formalism can accommodate arbitrary spacetime dimensionalities, fermions (Baez and Krasnov), an arbitrary (or even ), and supersymmetry (Smolin), and results in a quantization of the kinematics of the corresponding diffeomorphism-invariant gauge theory. Much work remains to be done on the dynamics, the classical limit and the correspondence principle, all of which are necessary in one way or another to make contact with experiment.

In a nutshell, loop quantization is the result of applying quantization to a algebra of gauge-invariant classical observables. Non-canonical means that the basic observables quantized are not generalized coordinates and their conjugate momenta. Instead, the algebra generated by spin network observables (built from holonomies) and field strength fluxes is used.

Loop quantization techniques are particularly successful in dealing with topological quantum field theories, where they give rise to state-sum/spin-foam models such as the Turaev-Viro model of 2+1 dimensional general relativity. A much studied topological quantum field theory is the so-called BF theory in 3+1 dimensions, because classical general relativity can be formulated as a BF theory with constraints, and it is hoped that a consistent quantization of gravity may arise from perturbation theory of BF spin-foam models.

Lorentz invariance

For detailed discussion see the page.

LQG is a of a classical which is equivalent to the usual in that it leads to the same describing with . As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just as in . A positive can be realized in LQG by replacing the with the corresponding .

Diffeomorphism invariance and background independence

(also known as diffeomorphism invariance) is the invariance of physical laws (for example, the equations of general relativity) under arbitrary coordinate transformations. This symmetry is one of the defining features of general relativity. LQG preserves this symmetry by requiring that the physical states must be invariant under the generators of diffeomorphisms. The interpretation of this condition is well understood for purely spatial diffemorphisms; however the understanding of diffeomorphisms involving time (the ) is more subtle because it is related to dynamics and the so-called in general relativity, and a generally accepted calculational framework to account for this constraint is yet to be found.

Whether or not Lorentz invariance is broken in the low-energy limit of LQG, the theory is formally . The equations of LQG are not embedded in or presuppose space and time (except for its topology that cannot be changed), but rather they are expected to give rise to space and time at large distances compared to the Planck length. It has not been yet shown that LQG's description of spacetime at the Planckian scale has the right continuum limit described by general relativity with possible quantum corrections.

Problems

As of now there is not one experiment which verifies or refutes any aspect of LQG. This is a problem which plagues many current theories of quantum gravity. This problem is so persistent because LQG applies on a small scale to the weakest of the forces of nature. This problem however cannot be minimized as it is the biggest problem any scientific theory can have; theory without experiment is just faith. The second problem is that a crucial free parameter in the theory known as the is a logarithm of a . This has negative implications for the computation of the entropy of a black hole using LQG (although to be fair the transcendental number is just the result of a calculation and not an experiment, which would be the true test of scientific reality). Since Bekenstein and Hawking computed the entropy of a black hole this computation has been a crucial litmus test for any theory of quantum gravity. Last and most profoundly LQG has failed to gain support in the physics community at large mainly because of its limited scope. An observation is that many scientist believe that we could formulate a theory of quantum gravity which is just for four dimensions and is unconcerned with other forces but why? Why do that when via String theory or M theory we are so close to a theory that takes account of everything we know and predicts so very much that we do not know? At this point Loop theorists disagree. They feel that a proper theory of quantum gravity is a prerequisite for any theory of everything. This philosophical problem could be the most fatal problem that LQG faces in the future. Only time and experimentation will tell.