Loop Quantum Gravity Forum

[Loop]

I have created a new forum dedicated exclusively to LQG. You're welcome to post there. LQG will be probably the TOE that will give all the answers that humanity is searching
http://amazingforums.com/forum2/QGRAVITY/forum.html

[Xerxes314]

I hate to rain on your parade, but LQG is explicitly not a ToE. That\'s one of its major selling points: that it only solves the QM-GR problem and has nothing much to say about the standard model, which doesn\'t disagree with any experiments anyway.

Xerxes

[marlon]

LQG IS NOT A THEORY OF EVERYTHING. This is a big misconception. Let me give you this intro :

INTRODUCTION TO LOOP QUANTUM GRAVITY, everything you ever wanted to know...


In Loop Quantum Gravity, also referred to as LQG, the attempt is made to introduce the concept of quantum gravity. This is the unification of the General Theory of Relativity and the Quantummechanics. It is a very well established fact that gravitation and quantummechanics both have totally different foundations, which makes it very difficult to unify them at “first sight”. On the one hand position is uncertain in QM due to the Heisenberg-principle, while this is never the case in GTR. On the other hand, there is no metrical connection between space and time in QM, similar to the space-time-continuum of GTR. This leads to the fact that there is no curvature of space nor time in QM


In order to quantize the GTR we need gauge-fields, curved on a manifold just like in GTR. These gauge fields then need to be quantized just like other fermionic fields are quantized in QFT. When following this procedure, one needs to obey the following two laws at any point and time :

1) diffeomorfism invariance (this is the general covariance of GTR)
2) gauge invariance (like in QFT, invariance of gauge fields under local symmetries)


Basically these two laws ensure us that we have background independence so that we can choose any metric we want in order to describe the manifold. The different possible frames on that manifold must yield the same physical equations at any point on the manifold, that is the covariance (just like in GTR). These diffeomorfisms from one possible metric to another make sure that the physical laws remain the same when metrics are interchanged.


More specifically one needs to describe a manifold. Great mathematicians like Gauss and Riemann have taught us that this is done by the socalled connections. A metric describing a manifold is the most familiar example of a connection, i.e. the socalled metrical connection. There are other options though, like in GTR the connections are not metric-functions but they are gauge fields.


Next question is, how do we study some manifold ? What system can be followed in order to describe how objects behave on some chosen manifold ?
Well, we want background independence, so we must be able to chose any metric or connection we want in order to describe our manifold we are working on. In the early stages of LQG all possible metrics were used in order to implement this concept of back-ground-independence. A certain physical state was then represented as a probability-density containing all these metrics. This way of working was not very practical and in the mid-eighties it was even replaced by a description based upon the set of connections instead of all possible metrics.



Now, how does a connection work ? Well suppose you are on the manifold at a certain point A. Then you want to move in some direction on the manifold along a loop to that starts in A and comes back to A, like a circle. In order to describe this transition in mathematics, one uses the concept of parallel transport of tangent vectors. In order to be able to talk about such things as vectors, we need a reference frame that we can choose as we please because of the two laws mentioned above. Take a frame in A then make a very little step along the loop and look how this chosen frame has changed its position during the movement. Then complete the same procedure until you get back in A after completing the loop. Ofcourse it is not useful to look at the movement of the frame at every intermediate step along the loop. Actually one can integrate out the evolution of the frame over the entire trajectory that is followed from A to A.

When we start in A we actually take a tangent vector. This is an element of the tangent space of the manifold at point A. The transformation that is used to go from a point A on the manifold to the tangential space is called a projection. This tangent space can be turned into a socalled the Lie-Algebra, containing vectors written in terms of differentials, and provides the description for the movement from A to A along the loop. Now the operations that can be executed on the elements of a Lie-Algebra, like the identity or rotations, can be found in the Lie-Group.


As stated in the above paragraph it will be the intention to map elements from the Lie-Algebra to the Lie-Group. To be more specific : suppose we look at some vectors from the Lie-Algebra at A and we parallel transport them along the loop back to A. Now, we see for example that these vectors have rotated 45 degrees during their transport. This 45-degree-rotation is an element from the Lie-Group and the map between these two concepts gives us some idea on how vectors behave when replaced along some chosen loop on the manifold. Thus, yielding in a system to describe the manifold itself. It is proven that if you exponentiate Lie-Algebra-elements, you get the Lie-Group-elements.



More specifically, we take the frame around some loop and integrate all the differential motions of this frame during it’s transport. It is this integral that is exponentiated in order to get the corresponding group-element. In the Lie-Algebra, the group-element has a certain representation like a matrix. It is the trace if this matrix that is considered because the trace is a scalar and it will be the same for all reference frames. The map between the Lie-Algebra element and the Lie-Group element is called a Wilson Loop. Basically it “tries to feel” the metric by parallel transporting a Lie-Algebra element along a loop and “measuring” how this element changes it’s position with respect to the original position, after the loop is completed. Thus yielding a Lie-Group element.


The reason why we can ultimately speak about integrations and so on, is because initially everything is considered to be very very small. We work in terms of differential motions, which add up into the total motion between A and A. We use the Algebra’s in order to talk in terms of differentials d. As we move the frame along some \"d(loop)\" it experiences some \"d(rotation).\"


Now, once we have established such a relation, we can calculate the total movement by exponentiating the two differentials of the Lie-Algebra. The d(loop) ofcourse yields a transformation that describes the trajectory of the loop, while the d(rotation) yields the total rotation that has been undergone by the transported vector.









The main consequence of Loop Quantum Gravity is the fact that our space-time-continuum is no longer infinitely divisible. In LQG space has a “granular” structure that represents the fact that space is divided into elementary space-quanta of which the dimensions can be measured in LQG. The main problem of QFT is the fact that it relies on the existence of some physical background. As stated one of the main postulates of LQG is the fact that we need background independence. The diffeomorfisms give us the possibility to go from one metric to another and the physical laws must remain the same. Basically some physical state in LQG is a superposition over all possible backgrounds or in other words a physical state is a wavefunction over all geometries.



In String Theory, the main “competitor” when it comes to quantumgravity starts from the fact that there must be some kind of fictitious background space, thus actually undoing the aspects of general relativity. All calculations can then be made with respect to this background field and in the end the background independence must “somehow” be recovered. LQG starts from a totally different approach, though. We start from the knowledge we have from General Relativity, thus no background field, and we then try to rewrite the entire Quantum Field Theory in a certain way that no background-field is needed.



How to implement this nice background-independence in QFT has already shortly been introduced, i.e. The Wilson Loop and more generally the spin networks :

The map between the Lie-Algebra element and the Lie-Group element is called a Wilson Loop. Basically it “tries to feel” the metric by parallel transporting a Lie-Algebra element along a loop and “measuring” how this element changes it’s position with respect to the original position, after the loop is completed. Thus yielding a Lie-Group element.

The strategy is as follows : in stead of working with one specific metric like in “ordinary” QFT, just sum up over all possible metrics. So QFT should be redefined into somekind of pathintegral over all possible geometries. A wavefunction is then expressed in terms of all these geometries and one can calculate the probability of one specific metric over another. This special LQG-adapted wavefunction must obey the Wheeler-DeWitt equation, which can be viewed at as some kind of Schrödinger-equation for the gravitational field. So just like the dynamics of the EM-field is described by the Maxwell-equations, they dynamics of the gravitational-field are dedeterminedy the above mentioned equation. Now how can we describe the motion of some object or particle in this gravitational field. Or in other words, knowing the Maxwell equations, what will be the variant of the Lorentz-force ???


This is where the loops come in. First questions one must ask is :

Why exactly them loops ?

Well, let’s steal some ideas from particle physics... In QFT we have fermionic matter-fields and bosonic force-fields. The quanta of these force-fields or the socalled force-carrier-particles that mediate forces between matter-particles. Sometimes force-carriers can also interact with eachother, like strong-force-mediating gluons for example. These force carriers also have wavelike properties and in this view they are looked as excitations of the bosonic-forcefields. For example some line in a field can start to vibrate (think of a guitar-string) and in QFT one then says that this vibration is a particle. This may sound strange but what is really meant is that the vibration has the properties of some particle with energy, speed, and so on, corresponding to that of the vibration. These lines are also known as Faraday’s lines of force. Photons are \"generated\" this way in QFT, where they are excitations of the EM-field. Normally these lines go from one matter-particle to another and in the absence of particles or charges they form closed lines, aka loops. Loop Quantum Gravity is the mathematical description of quantum gravity in terms of loops on a manifold. We have already shown how we can work with loops on a manifold and still be assured of background-independence and gauge-invariance for QFT. So we want to quantize the gravitational field by expressing it in terms of loops. These loops are quantum excitations of the Faraday-lines that live in the field and who represent the gravitational force. Gravitons or closed loops that arise as low-energy-excitations of the gravitational field and these particles mediate the gravitational force between objects.


It is important to realize that these loops do not live on some space-time-continuum, they are space-time !!! The loops arise as excitations of the gravitational field, which on itself constitutes “space”. Now the problem is how to incorporate the concept of space or to put it more accurately : “how do we define all these different geometries in order to be able to work with a wave function ?”


The Wheeler-DeWitt equation has solutions describing excitations of the gravitational field in terms of loops. A great step was taken when Abhay Ashtekar rewrote the General Theory of Relativity in a similar form as the Yang-Mills-Theory of QFT. The main gauge-field was not the gravitational field. No, the gravitational field was replaced by the socalled connection-field that will then be used to work with different metrics. In this model space must be regarded as some kind of fabric weaved together by loops. This fabric contains finite small space-parts that reflect the quantization of space. It is easy to see that there are no infinite small space regions, thus no space-continuum. Quantummechanics teaches us that in order to look at very small distance-scales, an very big amount of energy is needed. But since we also work in General Relativity we must take into account the fact that great amounts of energy concentrated at a very small scale gives rise to black holes that make the space-region disappear. By making the Schwardzschildradius equal to the Comptonradius we can get a number expressing the minimum size of such a space-region. The result is a number that is in the order of the Planck Length.


Now how is space constructed in LQG ? Well, the above mentioned minimal space-regions are denoted by spheres called the nodes. Nodes are connected to eachother by lines called the links.



By quantizing a physical theory, operators that calculate physical quantities will acquire a certain set of possible outcomes or values. It can be proven that in our case the area of the surface between two nodes is quantized and the corresponding quantumnumbers can be denoted along a link. These surfaces I am referring are drawn as purple triangles. In this way a three-dimensional space can be constructed.







One can also assign a quantumnumber which each node, that corresponds to the volume of the grain. Finally, a physical state is now represented as a superposition of such spin-networks.

[Fernanda]

[Edited on 3-12-2005 by Fernanda]

[Motion+E]

If I were to create the universe...

It is essential that we understand the difference in energy...that there is INFLUENTIAL ENERGY and there is DISRUPTIVE ENERGY within the universe. To understand the differences, there is only one true form of Influential...all other forms are disruptive in nature. They are in fact complete opposites in that influential precedes matter and disruptive energy succeeds it.

For a model, think of a suspended particle. If we were to apply influential energy...the particle is pulled toward the energy. If we were to apply disruptive energy...the particle is pushed away from the energy. Essentially what this means is that there is truly only one form of a "forward" moving energy...which means energy would have to precede matter.

Another difference in influential and disruptive energies is how they interact with atoms. Disruptive interacts with the largest portion of the atom...the nucleus...and is short lived. Influential interacts with the smallest particle...the electron...and we will say that this is long lived to make things simple.

Time and energy exist simultaneously...or you can say that one does not exist without the other. However, if it were only energy that existed and nothing else, time would be suspended. It is there...it just would not be active. Time, as we know it, moves "forward" meaning that time is of influential energy.

It is important to determine TIME CONSTANT to understand anything else. This in essence is true time and is the constant within the entire universe. To picture it...think of a single electron...nothing else...no nucleus...no other particles of any kind. By applying the influential energy, the electron would move through the shortest span of space to the energy. If we started with the electron within a certain span of space to move through, the "distance" from its origin to the energy is now our measurement of time...this being a time that is constant throughout the universe. We would assign this measurement with the value of 0...meaning that it is unaffected...it is not negative nor is it positive...neutral throughout the universe...true time.

To make the alterations of time that take place throughout the universe less complex, we will use the model of having two sources of influential energy that are at exact 90 degrees from one and other within space. The two sources are not of the same power ratio, however, due to the "distance" of each from the origin of the electron...this is what we will use as an example. The electron would ultimately be more influenced by the closer and stronger energy source...to move through the shortest span of space. However, with the existence of a secondary source, it too would have an influence...therefore causing a "bend" or "arch" within the path of the electron...therefore affecting the amount of space or distance from the point of origin to the closer source of energy. This is how the measurement has been altered and the value is now positive depending on the arch.

[Motion+E]

If we were to apply my theory to the point of creation, the expansion of influential energy would have to precede anything else...for potential time and energy would have to exist within space before anything else. Applied to the "Big Bang Theory", think of it as the rolling out of a piece of carpet. As it unfolds, it distributes a uniform unaltered fabric in its path. Essentially this would mean that if there were an edge to "our" universe, there would be an excess of influential energy on the outskirts still expanding/unraveling...still pushing our boundaries further and further.

What would follow in sequence to influential energy would be light traveling faster than the rate at which "space and time" could unravel so therefore would be restricted to traveling with it...unable to surpass. Matter and disruptive energies are introduced...altering time and space...forming the influential energy based cores of planets...as at this stage, influential energy is still far more present than the disruptive, which would eventually change and cause "universal drag" on the expansion rate of the primary influential energy wave.

The universal drag could become so great that galaxies could potentially begin to catch up to the influential energy wave...where galaxies begin to accelerate in the expansion from one and other the closer they come to this wave that is now expanding at a slower rate.

[Motion+E]

There has been a great debate on what space is and its relation to time. You must have an understanding of space to understand gravity. There are two types of space...we will call one "inhabitable space" and the other as "habitable space" for now. It is important for you to understand the difference to comprehend the Big Bang when we cover it.

Essentially we start with inhabitable space...to make it as simple as possible...it is a place where you or the things around you can not exist. Let us say we add a two dimensional magnetic field to this space...now matter as you know it could exist in this two dimensional field and outside of that field it can not...this is only an example...and yes, matter as we know it is three dimensional.

In this two dimensional field, matter changes as the stability belt has changed...Iron becoming the most stable and is at the center of the belt...meaning that one end you have hydrogen and the other is uranium and they all are "magnetized" to become stable...migrating toward the state of iron...but matter can also be easily manipulated toward either direction...this is our quick version habitable space.

To understand inhabitable space in a few words...without these fields...matter splits in that the stability has changed...it goes in reverse from the center of the belt...meaning that matter is stripped to hydrogen...or clumped into uranium (which I am using due to it being the known natural element on the one extreme end)...there is no manipulation of matter in this space.

[Motion+E]

There has been a great debate on what space is and its relation to time. You must have an understanding of space to understand gravity. There are two types of space...we will call one "inverted space" and the other as "ideal space" for now. It is important for you to understand the difference to comprehend the Big Bang when we cover it.

Essentially we start with inverted space...to make it as simple as possible...it is a place where you or the things around you can not exist. Let us say we add a two dimensional magnetic field to this space...now matter as you know it could exist in this two dimensional field and outside of that field it can not...this is only an example...and yes, matter as we know it is three dimensional.

In this two dimensional field, matter changes as the stability belt has changed...Iron becoming the most stable and is at the center of the belt...meaning that one end you have hydrogen and the other is uranium and they all are "magnetized" to become stable...migrating toward the state of iron...but matter can also be easily manipulated toward either direction...we have a range...this is our quick version ideal space.

To understand inverted space in a few words...without these fields...matter splits in that the stability has changed...it goes in reverse from the center of the belt...meaning that matter is stripped to hydrogen...or clumped into uranium (which I am using due to it being the known natural element on the one extreme end)...there is no manipulation of matter in this space.

It is easy to predict what would happen in this situation with matter being imploded...we have the creation of ideal space.

Now you have a relative understanding of why space and time can not be separated...for they are the same energy.