Posted:

**Jul 14, 2010 8:26 am**OK, so as I mentioned previously, I’ve decided to try to learn what loop quantum gravity is, and I thought that it might be useful to post my learning process as I go along, so that people can chip in with extra insights/questions/objections and hopefully we’ll all get to know a bit more than we did before. At this stage I know virtually nothing about what loop quantum gravity, so I can’t plan a coherent and logically ordered set of posts on the subject, so please expect a random mishmash of half understood ideas and concepts. I don’t know who will read this as it’s not exactly mainstream interest – maybe only people who are fairly high on the geekometer spectrum .

Some abbreviations:

LQG = Loop Quantum Gravity

GR = General Relativity

QED = Quantum Electrodynamics

QFT = Quantum Field Theory

So, since I don’t yet know much about LQG and my copy of Rovelli’s monograph hasn’t arrived from Amazon yet, I thought I’d start with:

Post1: WHY DO WE WANT A QUANTUM THEORY OF GRAVITY ANYWAY?

Good question. In fact I remember SkinnyPuppy started a thread over on RDF with precisely this question. Gravity is expressed in terms of the spacetime in which everything lives. The stuff of our universe (fields/particles) consists of objects (particles and fields) which live “in” or “on” this spacetime. Why not just leave the spacetime as it is (“classical” and described by general relativity), and keep the quantum nonsense just for the fields which live on it ?

1 Singularities

Well, the first thing that comes to mind is that even if we treat the spacetime purely classically, as dictated by general relativity, we find, under a set of conditions which are still pretty general, Hawking and Penrose proved several decades ago that the spacetime will contain singularities. These are regions where general relativity is unable to describe what the spacetime is like. At a singularity, a key variable which GR uses to describe the spacetime - the curvature - becomes infinite. Infinities happen in other theories too - for a point charge, the electric field becomes infinite at the origin, but this is due to the pointlike nature of the source. A GR singularity is not like this - there doesn't have to be a source in the region where the curvature diverges. The gravitational field manages to screw itself up without any outside help !

It is hoped that a quantum theory of gravity will somehow avoid this singular behaviour.

2 All the other Force Fields are Quantised, so why not Gravity ?

The other 3 forces of nature - strong, weak and electromagnetic forces all interact with "matter" in a manner described by quantum laws. The forces themselves are described heuristically, in quantum field theoretic terms, by interaction of an "exchange particle" with the matter in question.

Electromagnetic Force = Exchange of (virtual) photons between electrically charged particles

Weak Force = Exchange of (virtual) electroweak gauge bosons between weakly interacting particles

Strong Force = Exchange of (virtual) gluons between quarks

Since we humans don't have much imagination, we expect gravity to be described in a similar way – gravitational force = exchange of gravitons etc.

3 Planck Scale / Cutoffs

Firstly, Quantum Field Theory in a few paragraphs:

QFT is conventionally built up in Minkowski space (the flat spacetime of special relativity): The basis for building up a quantum field theory is a function called the Lagrangian, this tells you which fields (electrons, photons, quarks etc) you’re dealing with, their symmetry properties, which of the fields interact with each other and how strongly they interact etc. You can then take the Fourier transform of the fields. This will split them into “positive frequency” and “negative frequency” components. After introducing (anti)commutation relations, the Fourier coefficients of the positive frequency components are then treated as annihilation operators and the Fourier coefficients of the negative frequency components as creation operators. This process is known as second quantization. Being operators, the operators need something to operate on. They operate on vectors in a potentially infinite dimensional space called a Hilbert space. The starting point for this process is a state vector called the vacuum, usually denoted as |0>.

With this structure you can start to calculate probability amplitudes using perturbation theory. In QFT perturbation theory works like this: in the Lagrangian, the interaction between fields is defined by creating terms with a “product” of the fields which interact with each other. For example in QED, the Lagrangian would contain terms like

eψ-ψA

Here e is the electron charge (you could replace it with the fine structure constant if you want). ψ denotes the electron/positron field, and A denotes the electromagnetic field(ψ with a minus superscript is meant to be “psi bar”, but I don’t know how to type this, also there should be a gamma matrix in the formula and a Lorentz index on the A, but I’m just trying to convey the concept and skip the technicalities).

Suppose I want to compute the probability amplitude for some process – let’s take a trivial process – an electron goes from point A to point B. If it doesn’t interact along the way, there is a simple formula for this probability amplitude ( the electron propagator). However, suppose along the way it emits a virtual photon and then reabsorbs this photon. Drawing this on a diagram, we see the diagram has a loop. Following the rules and computing the probability amplitude in this case, we unfortunately get the answer infinity !

This useless result comes from the rule which says that we have to integrate, from zero to infinity, over the momentum of the virtual photon. We do this since the photon could have any momentum, and we have to add the probability amplitudes for each possibility. There is a very cunning way around this problem, called renormalisation which was far from obvious – QFT had been around for a long time before Feynman/Schwinger/Tomonaga invented the renormalisation procedure in the 50s (?).

I won’t go into renormalisation in detail, but basically one way to do it involves first “taming” the divergent integral by integrating not up to infinity, but to some arbitrary cutoff momentum value Λ. Then the integral is split into a convergent and divergent part and the divergent part subtracted by means of adding “counterterms” to the Lagrangian. After a bit of jiggery pokery replacing the bare coupling constants by their physical values and applying a renormalisation prescription, finite answers which are independent of the cutoff can be extracted from the integral. For a nice not-too-technical presentation of renormalisation, see here.

This whole process seems rather contrived and inelegant (clever yes, but elegant – no). The only reason I mention all this stuff is that one hope for a full quantum gravity theory is that the need for this messy process may vanish. The reason for this hope is that integrals to infinity in the loop momentum will no longer be needed, since in the new theory quantum gravity will provide a natural cut off for the integrals (or whatever the equivalents of the integrals are in the new theory). By combining the fundamental constants G, h and c in various dimensionally consistent ways, “natural” values of energy, mass, length, time etc can be derived. These are the values at which quantum gravity is expected to become relevant. The divergent integrals giving so much trouble in QFT would somehow be cutoff naturally at momenta below the Planck energies.

4 Time in Quantum theory vs Time in Relativity

I mentioned in 3 that a crucial part of QFT in Minkowski space was the separation of the field into positive and negative frequency components. We needed this in order to have the concept of creation and annihilation operators which we needed to describe it as a particle theory. To be able to define positive and negative frequency components, we need a time coordinate. This is fine in Minkowski space - if I make a Lorentz transformation to a new time coordinate I still have the same definition of positive and negative frequencies.

However, in general relativity,

no unique way to choose a time coordinate

=> different choices of time coordinate give different definitions positive/negative frequencies

=> different definitions of creation and annihilation operators

=> in a given state, different definitions of the presence or absence of particles.

(In fact, you don't even need curved spacetime for this - a non inertial frame will do - this is the root of the Unruh effect).

A key aspect of quantum theory is that physical observables are modelled as operators. When the system is in a certain state, the value of the observable is one of the eigenvalues resulting when that operator is applied to the state vector (see http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics). In QFT, time is treated in a different way to the other parameters of the system. It is not an observable. There is no operator I can apply whose eigenvalues are the time. It is merely a parameter. This feels a little unnatural - the fact that there is this classical background clock ticking, driving the quantum system.

Hope has been expressed that in a quantum gravity theory, time will somehow emerge FROM the theory rather than being part of the background on which the theory is formulated.

5 Why Not Stick with Quantum Field Theory on a Curved Background ?

We know how to do QFT on Minkowski space. Much work has been done to investigate what happens to QFT when you replace the Minkowski metric with a general spacetime. I said in (4) that the presence of quantum particles depends upon spacetime curvature (or acceleration, according to the principle of equivalence). There is a now well-developed theory which treats this and many fascinating results have been obtained (Unruh effect, Hawking radiation, black hole entropy). Maybe that's as far as it goes ?

Well if it stops here, we do not yet have a complete theory of gravitons. Classical GR admits gravitational wave solutions, so we would expect there to at least be a quantised way of treating these. Most other wave phenomena in physics seem to have a quantum treatment. Gravitons should be able to interact with each other as well as with “source” matter, so the theory which models them must be able to handle this. If this is tried, using the conventional renormalisation scheme of (3), the prescription fails – renormalisation does not work. What happens is that more and more counter terms are required at successively higher order approximations of the theory.

Well, that’s just a bit of woffle about some of the motivations for quantizing gravity. It’s not an exhaustive list.

LQG is based around the technique of canonical quantization of GR, so in my next post I’ll try to say something about the canonical formulation of GR.

Some abbreviations:

LQG = Loop Quantum Gravity

GR = General Relativity

QED = Quantum Electrodynamics

QFT = Quantum Field Theory

So, since I don’t yet know much about LQG and my copy of Rovelli’s monograph hasn’t arrived from Amazon yet, I thought I’d start with:

Post1: WHY DO WE WANT A QUANTUM THEORY OF GRAVITY ANYWAY?

Good question. In fact I remember SkinnyPuppy started a thread over on RDF with precisely this question. Gravity is expressed in terms of the spacetime in which everything lives. The stuff of our universe (fields/particles) consists of objects (particles and fields) which live “in” or “on” this spacetime. Why not just leave the spacetime as it is (“classical” and described by general relativity), and keep the quantum nonsense just for the fields which live on it ?

1 Singularities

Well, the first thing that comes to mind is that even if we treat the spacetime purely classically, as dictated by general relativity, we find, under a set of conditions which are still pretty general, Hawking and Penrose proved several decades ago that the spacetime will contain singularities. These are regions where general relativity is unable to describe what the spacetime is like. At a singularity, a key variable which GR uses to describe the spacetime - the curvature - becomes infinite. Infinities happen in other theories too - for a point charge, the electric field becomes infinite at the origin, but this is due to the pointlike nature of the source. A GR singularity is not like this - there doesn't have to be a source in the region where the curvature diverges. The gravitational field manages to screw itself up without any outside help !

It is hoped that a quantum theory of gravity will somehow avoid this singular behaviour.

2 All the other Force Fields are Quantised, so why not Gravity ?

The other 3 forces of nature - strong, weak and electromagnetic forces all interact with "matter" in a manner described by quantum laws. The forces themselves are described heuristically, in quantum field theoretic terms, by interaction of an "exchange particle" with the matter in question.

Electromagnetic Force = Exchange of (virtual) photons between electrically charged particles

Weak Force = Exchange of (virtual) electroweak gauge bosons between weakly interacting particles

Strong Force = Exchange of (virtual) gluons between quarks

Since we humans don't have much imagination, we expect gravity to be described in a similar way – gravitational force = exchange of gravitons etc.

3 Planck Scale / Cutoffs

Firstly, Quantum Field Theory in a few paragraphs:

QFT is conventionally built up in Minkowski space (the flat spacetime of special relativity): The basis for building up a quantum field theory is a function called the Lagrangian, this tells you which fields (electrons, photons, quarks etc) you’re dealing with, their symmetry properties, which of the fields interact with each other and how strongly they interact etc. You can then take the Fourier transform of the fields. This will split them into “positive frequency” and “negative frequency” components. After introducing (anti)commutation relations, the Fourier coefficients of the positive frequency components are then treated as annihilation operators and the Fourier coefficients of the negative frequency components as creation operators. This process is known as second quantization. Being operators, the operators need something to operate on. They operate on vectors in a potentially infinite dimensional space called a Hilbert space. The starting point for this process is a state vector called the vacuum, usually denoted as |0>.

With this structure you can start to calculate probability amplitudes using perturbation theory. In QFT perturbation theory works like this: in the Lagrangian, the interaction between fields is defined by creating terms with a “product” of the fields which interact with each other. For example in QED, the Lagrangian would contain terms like

eψ-ψA

Here e is the electron charge (you could replace it with the fine structure constant if you want). ψ denotes the electron/positron field, and A denotes the electromagnetic field(ψ with a minus superscript is meant to be “psi bar”, but I don’t know how to type this, also there should be a gamma matrix in the formula and a Lorentz index on the A, but I’m just trying to convey the concept and skip the technicalities).

Suppose I want to compute the probability amplitude for some process – let’s take a trivial process – an electron goes from point A to point B. If it doesn’t interact along the way, there is a simple formula for this probability amplitude ( the electron propagator). However, suppose along the way it emits a virtual photon and then reabsorbs this photon. Drawing this on a diagram, we see the diagram has a loop. Following the rules and computing the probability amplitude in this case, we unfortunately get the answer infinity !

This useless result comes from the rule which says that we have to integrate, from zero to infinity, over the momentum of the virtual photon. We do this since the photon could have any momentum, and we have to add the probability amplitudes for each possibility. There is a very cunning way around this problem, called renormalisation which was far from obvious – QFT had been around for a long time before Feynman/Schwinger/Tomonaga invented the renormalisation procedure in the 50s (?).

I won’t go into renormalisation in detail, but basically one way to do it involves first “taming” the divergent integral by integrating not up to infinity, but to some arbitrary cutoff momentum value Λ. Then the integral is split into a convergent and divergent part and the divergent part subtracted by means of adding “counterterms” to the Lagrangian. After a bit of jiggery pokery replacing the bare coupling constants by their physical values and applying a renormalisation prescription, finite answers which are independent of the cutoff can be extracted from the integral. For a nice not-too-technical presentation of renormalisation, see here.

This whole process seems rather contrived and inelegant (clever yes, but elegant – no). The only reason I mention all this stuff is that one hope for a full quantum gravity theory is that the need for this messy process may vanish. The reason for this hope is that integrals to infinity in the loop momentum will no longer be needed, since in the new theory quantum gravity will provide a natural cut off for the integrals (or whatever the equivalents of the integrals are in the new theory). By combining the fundamental constants G, h and c in various dimensionally consistent ways, “natural” values of energy, mass, length, time etc can be derived. These are the values at which quantum gravity is expected to become relevant. The divergent integrals giving so much trouble in QFT would somehow be cutoff naturally at momenta below the Planck energies.

4 Time in Quantum theory vs Time in Relativity

I mentioned in 3 that a crucial part of QFT in Minkowski space was the separation of the field into positive and negative frequency components. We needed this in order to have the concept of creation and annihilation operators which we needed to describe it as a particle theory. To be able to define positive and negative frequency components, we need a time coordinate. This is fine in Minkowski space - if I make a Lorentz transformation to a new time coordinate I still have the same definition of positive and negative frequencies.

However, in general relativity,

no unique way to choose a time coordinate

=> different choices of time coordinate give different definitions positive/negative frequencies

=> different definitions of creation and annihilation operators

=> in a given state, different definitions of the presence or absence of particles.

(In fact, you don't even need curved spacetime for this - a non inertial frame will do - this is the root of the Unruh effect).

A key aspect of quantum theory is that physical observables are modelled as operators. When the system is in a certain state, the value of the observable is one of the eigenvalues resulting when that operator is applied to the state vector (see http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics). In QFT, time is treated in a different way to the other parameters of the system. It is not an observable. There is no operator I can apply whose eigenvalues are the time. It is merely a parameter. This feels a little unnatural - the fact that there is this classical background clock ticking, driving the quantum system.

Hope has been expressed that in a quantum gravity theory, time will somehow emerge FROM the theory rather than being part of the background on which the theory is formulated.

5 Why Not Stick with Quantum Field Theory on a Curved Background ?

We know how to do QFT on Minkowski space. Much work has been done to investigate what happens to QFT when you replace the Minkowski metric with a general spacetime. I said in (4) that the presence of quantum particles depends upon spacetime curvature (or acceleration, according to the principle of equivalence). There is a now well-developed theory which treats this and many fascinating results have been obtained (Unruh effect, Hawking radiation, black hole entropy). Maybe that's as far as it goes ?

Well if it stops here, we do not yet have a complete theory of gravitons. Classical GR admits gravitational wave solutions, so we would expect there to at least be a quantised way of treating these. Most other wave phenomena in physics seem to have a quantum treatment. Gravitons should be able to interact with each other as well as with “source” matter, so the theory which models them must be able to handle this. If this is tried, using the conventional renormalisation scheme of (3), the prescription fails – renormalisation does not work. What happens is that more and more counter terms are required at successively higher order approximations of the theory.

Well, that’s just a bit of woffle about some of the motivations for quantizing gravity. It’s not an exhaustive list.

LQG is based around the technique of canonical quantization of GR, so in my next post I’ll try to say something about the canonical formulation of GR.