2004 Nobel Laureates Lecture Stockholm - Question on Beta function!

[AndersHermansson]

I attended a public lecture on QCD with David J. Gross H. David Politzer Frank Wilczek this morning here in Stockholm! =) I very much enjoyed it. There was one thing that especially caught my attention. They mentioned that one step in their work was calculating the Beta function for some configuration and they showed their result, and that it was important for it to be negative. Does anybody know what the physical relevance of that function and their calculation of it is?

[lethe]

I attended a public lecture on QCD with David J. Gross H. David Politzer Frank Wilczek this morning here in Stockholm! =) I very much enjoyed it. There was one thing that especially caught my attention. They mentioned that one step in their work was calculating the Beta function for some configuration and they showed their result, and that it was important for it to be negative. Does anybody know what the physical relevance of that function and their calculation of it is?
in a renormalizable quantum field theory, the physical constants like the coupling constant depends on the energy scale. for example, the charge of an electron grows at higher energies. sometimes a physical picture of this is given as follows: the electron surrounds itself with virtual pairs of electrons and positrons, polarized so that the positrons are situated closer to to the electron. this shields the charge of the electron. at higher energies, the shielding effect gets weaker and weaker, so the charge gets larger and larger.

the beta function describes how this coupling constant depends on energy. for electromagnetism, the coupling constant grows with energy (and even goes to infinity at the Landau pole. The unshielded electron has infinite charge!). Perturbation theory only works if the coupling constant is small, so one of the nice things about electromagnetism is that the coupling constant is very small (1/137) meaning that perturbation theory works fantastically well at attainable energy levels. (and at higher energy levels, electromagnetism unifies with weak nuclear, so we don't have to worry about things like perturbation failing at high energy in QED or blowing up at the Landau pole, we only have to worry about what happens in the electroweak theory)

in QCD, however, the coupling constant turned out to be about 1, meaning that perturbation theory simply wouldn't work for QCD. what Gross discovered is that the beta function of QCD is negative, meaning that the coupling constant gets smaller at larger energies, as compared with QED where it gets larger, which means that perturbation theory can work for QCD if you crank up the energy beams high enough. that's what asymptotic freedom means; at high enough energy, the coupling constant gets small enough that hte theory is essentially a free theory (i.e. no coupling). perturbation theory works great, and we don't have to worry about things like the Landau pole.

[AndersHermansson]

I guess Perturbation theory is a way to approximate? It makes sense having to be able to that also for QCD.

Thanks alot!

[lethe]

I guess Perturbation theory is a way to approximate? It makes sense having to be able to that also for QCD.

Thanks alot!
yeah, perturbation theory is a way to approximate solutions to a theory when you have exact solutions to a nearby problem. you view the solutions as perturbed versions of the exact solutions. but it only works if the exact theory is somehow nearby. It's pretty standard in physics.

It's like this, all the observables in the quantum theory are built out of correlation functions, which can be defined as functional derivatives of the partition function Z. the partition function is defined to be ∫d[f]e^-S, where S is the action, and f stands for your fields. For example, for a free theory, the action goes like S=∫dx(∂f)²-m²f². This is just a sort of sketch of a typical action, but the important thing about it is that it is quadratic in the field, which means that the classical equation of motion will be linear, and that the partition function integral is essentially a Gaussian integral like ∫dxe^{-x^2} which can be evaluated exactly (do you know how to evaluate that integral?). It is equal to √π. But the free theory is not too interesting, because the particles cannot interact with each other, and in the real world, we know that particles do interact. basically, linear equations obey the superposition principle, meaning that if you have two solutions to a linear equation, they can pass each other without effecting each other at all.

So for an interacting theory, the action will have at least an additional term that is higher order than quadratic, a simple example would be something like S_int=S_free+gf³, and g is the coupling constant. This coupling term makes the classical equations of motion nonlinear, but of course we're interested in the quantum theory. We don't know how to evaluate integrals that look like e^{x^2+gx^3}, but we can approximate the answer as a Taylor series: e^{-x^2+gx^3}=e^{-x^2}(1+gx³/2+g²x⁶/6+...) and each of the infinite terms in this series can be integrated exactly.

to first order, we approximate the interacting theory as a free theory, and then add a first order correction of order g, and a second order correction of order g² and keep going until you get as much accuracy as you want. You are perturbing the theory away from a free theory. each term in the series is represented by a Feynman diagram. The Feynman diagrams are a shorthand mnemonic for calculating those integrals.

And here's the real point: if g is much less than 1, then each successive order term is smaller then the previous. for example, in QED, the coupling is 1/137. so fourth order terms are very small, and can be neglected. but if g is greater than 1, then the higher order terms are <i>larger</i> than the lower order terms, and cannot be neglected. and calculating terms of higher order is <i>hard</i>. calculating tree level terms takes about a page. calculating 1 loop terms takes a few pages, and calculating third order terms takes, i dunno, maybe a hundred pages, depending on the theory, i've never done one.

so unless the coupling is small, you simply cannot use this method to calculate anything; the interacting theory is not "nearby" enough to the free theory. before Gross discovered this result about the beta function, a lot of people thought that quantum field theory was not the right tool for particle physics, and had to be replaced. some people (Dirac, maybe?) thought that physicists should be looking for another theory. After the result by Gross, people started taking QCD seriously, people started believing in quarks, and people started believing that quantum field theory could be the fundamental theory. so give that man a Nobel!

[AndersHermansson]

yeah, perturbation theory is a way to approximate solutions to a theory when you have exact solutions to a nearby problem. you view the solutions as perturbed versions of the exact solutions. but it only works if the exact theory is somehow nearby. It's pretty standard in physics.

It's like this, all the observables in the quantum theory are built out of correlation functions, which can be defined as functional derivatives of the partition function Z. the partition function is defined to be &int;d[f]e^-S, where S is the action, and f stands for your fields. For example, for a free theory, the action goes like S=&int;dx(&part;f)²-m²f². This is just a sort of sketch of a typical action, but the important thing about it is that it is quadratic in the field, which means that the classical equation of motion will be linear, and that the partition function integral is essentially a Gaussian integral like &int;dxe^{-x^2} which can be evaluated exactly (do you know how to evaluate that integral?). It is equal to &radic;&pi;. But the free theory is not too interesting, because the particles cannot interact with each other, and in the real world, we know that particles do interact. basically, linear equations obey the superposition principle, meaning that if you have two solutions to a linear equation, they can pass each other without effecting each other at all.


Most of it way over my head, I'm afraid. It's my first year as undergraduate =)
I know the superpositionprinciple and that the integral you asked about can be evaluate by partial integration tho.

Maybe an example would help. Take momentum, how is it built from which correlation functions? (Maybe it's too naive of a question).



So for an interacting theory, the action will have at least an additional term that is higher order than quadratic, a simple example would be something like S_int=S_free+gf³, and g is the coupling constant. This coupling term makes the classical equations of motion nonlinear, but of course we're interested in the quantum theory. We don't know how to evaluate integrals that look like e^{x^2+gx^3}, but we can approximate the answer as a Taylor series: e^{-x^2+gx^3}=e^{-x^2}(1+gx³/2+g²x⁶/6+...) and each of the infinite terms in this series can be integrated exactly.

to first order, we approximate the interacting theory as a free theory, and then add a first order correction of order g, and a second order correction of order g² and keep going until you get as much accuracy as you want. You are perturbing the theory away from a free theory. each term in the series is represented by a Feynman diagram. The Feynman diagrams are a shorthand mnemonic for calculating those integrals.

And here's the real point: if g is much less than 1, then each successive order term is smaller then the previous. for example, in QED, the coupling is 1/137. so fourth order terms are very small, and can be neglected. but if g is greater than 1, then the higher order terms are <i>larger</i> than the lower order terms, and cannot be neglected. and calculating terms of higher order is <i>hard</i>. calculating tree level terms takes about a page. calculating 1 loop terms takes a few pages, and calculating third order terms takes, i dunno, maybe a hundred pages, depending on the theory, i've never done one.

so unless the coupling is small, you simply cannot use this method to calculate anything; the interacting theory is not "nearby" enough to the free theory. before Gross discovered this result about the beta function, a lot of people thought that quantum field theory was not the right tool for particle physics, and had to be replaced. some people (Dirac, maybe?) thought that physicists should be looking for another theory. After the result by Gross, people started taking QCD seriously, people started believing in quarks, and people started believing that quantum field theory could be the fundamental theory. so give that man a Nobel!

Hmm. So at higher energies, g is sufficiently small for the series to converge, and perturbation theory becomes useful?

Thanks again!

[lethe]

Most of it way over my head, I'm afraid. It's my first year as undergraduate =)
The action is a functional, this means for every possible path through phase space that a system can take, it assigns a real number (phase space is the space of all possible coordinates and their derivatives. one of Newton's postulates of classical mechanics is that a trajectory is determined solely by the values of the initial positions and their derivatives. the system evolves according to a 2nd order differential equation). Hamilton's least action principle then states that the system will choose the path which minimizes the action. The action is something that has to look at the whole path through phase space, so it needs to add up information about the positions and velocities at every point along the path. the way you add up points that span a continuous path is to integrate (you can't add all the real numbers from 0 to 1, you can only integrate along that interval). If you parameterize the path by an independent variable t, then we want the action to look like S=&int;dt L, where L is some function of the coordinates and derivatives. L is called the Lagrangian.

So to find which path the system takes, we need only minimize the derivative of the action. This is done more or less the same as in first semester calculus: take the derivative and set it equal to zero. The condition to extremize the action turns out to be d/dt(&part;L/&part;v)-&part;L/&part;x=0. This important equation is known as the Euler-Lagrange equation, and gives the equation of motion of the system, which tells how the system evolves in time. If the Lagrangian happens to be equal to kinetic energy minus potential energy (T-V), as it usually is, then this simply reduces to Newton's second law: dp/dt=-dV/dx, where p=dT/dv is the momentum.

So I can specify a theory completely by specifying the action. Now let's see what I was saying before about when the Lagrangian is quadratic. The kinetic energy T should be something like T=1/2mv², and then dT/dv=mv. The potential energy V should be something like V=1/2kx², with dV/dx=kx, and so the Euler-Lagrange equation gives a=-&omega;²x, where the frequency &omega; is just &radic;(k/m). The solutions to this linear equation are easily seen to be like x=e^i&omega;t. This is of course the Lagrangian for the simple harmonic oscillator, and dV/dx=kx is simply Hooke's law. A free field theory is simply an infinite collection of coupled harmonic oscillators.

So what about when I include a higher order term in the potential? Try V(x)=1/2kx²+1/3gx³, then dV/dx=kx+mgx², and the equation of motion is no longer linear: a=-&omega;²x+gx². In this simple case, you could actually solve this nonlinear differential equation in terms of elliptic functions, but for more complicated potentials, it would be impossible. So one also obtains the solution in a power series in g in terms of the solution to the harmonic oscillator above.

Finally, I have to mention that only classical systems (where Planck's constant h=0) obey the Euler-Lagrange equation. Quantum systems have to be summed over all paths through phase space, each path is assigned a phase (complex number with |z|=1) by the formula e^iS. When you sum this phase over all paths, the sum is dominated by the stable value of S (the minimum), but other values contribute as well.

I know the superpositionprinciple and that the integral you asked about can be evaluate by partial integration tho.
The Gaussian integral is usually evaluated this way: let I=&int;e^{-x^2}dx. Then I²=(&int;e^{-x^2}dx)(&int;e^{-y^2}dy), where I have changed the name of the dummy variable in the second integral. This can be rewritten I²=&int;&int;e^{-x^2-y^2}dxdy, and now change to polar coordinates where you have dA=dxdy=rdrd&theta; and r²=x²+y² so I²=&int;&int;re^{-r^2}drd&theta; which is an elementary double integral.


Maybe an example would help. Take momentum, how is it built from which correlation functions? (Maybe it's too naive of a question).

Quantum field theory is a quantum many particle theory. In quantum theories, the things you measure are matrix elements between states. So like for example, an initial state might be an electron and a positron, and the corresponding final state might be two photons. You would want to measure the transition amplitude between these two states.

I suppose that you could measure the matrix elements of the momentum operator, it wouldn't be much different. But this isn't done in practice; you don't measure the momentum usually, you input it as a parameter. What you usually do is take those transition amplitudes between states and turn them into cross sections or decay rates. These are the things that particle physicists measure. The decay rates and amplitudes depend on the matrix elements in rather simple ways (eg d&sigma;/dp=|M|² (times some kinematical factors)) And the LSZ reduction formula tells you how to get those transition amplitudes from the correlation functions.


Hmm. So at higher energies, g is sufficiently small for the series to converge, and perturbation theory becomes useful?

um well, we should be so lucky. The series probably never converges, actually. It's probably what's known as an asymptotic series, where the first couple of terms start to approximate the answer, but after enough terms starts to diverge again. It'll start to diverge after 1/g terms, so for QED, it gets very close to the right value until you go out to the 137th order. considering that people only calculate up to about 3rd or 4th order, that's pretty good! but for QCD, with g=1, the series starts to diverge already with the first term, so you never get close to the right answer, unless you move to an energy scale where g is much smaller.

also, notice that the Taylor series for e^x converges for all x, no matter how large, but 1+x+x²/2 is not a good approximation for e^x if x is large. What's important is not whether the series converges, but rather whether the first couple of terms give you a close answer to the true value or not.

[AndersHermansson]

The action is a functional, this means for every possible path through phase space that a system can take, it assigns a real number (phase space is the space of all possible coordinates and their derivatives. one of Newton's postulates of classical mechanics is that a trajectory is determined solely by the values of the initial positions and their derivatives. the system evolves according to a 2nd order differential equation). Hamilton's least action principle then states that the system will choose the path which minimizes the action. The action is something that has to look at the whole path through phase space, so it needs to add up information about the positions and velocities at every point along the path. the way you add up points that span a continuous path is to integrate (you can't add all the real numbers from 0 to 1, you can only integrate along that interval). If you parameterize the path by an independent variable t, then we want the action to look like S=&int;dt L, where L is some function of the coordinates and derivatives. L is called the Lagrangian.

So to find which path the system takes, we need only minimize the derivative of the action. This is done more or less the same as in first semester calculus: take the derivative and set it equal to zero. The condition to extremize the action turns out to be d/dt(&part;L/&part;v)-&part;L/&part;x=0. This important equation is known as the Euler-Lagrange equation, and gives the equation of motion of the system, which tells how the system evolves in time. If the Lagrangian happens to be equal to kinetic energy minus potential energy (T-V), as it usually is, then this simply reduces to Newton's second law: dp/dt=-dV/dx, where p=dT/dv is the momentum.

So I can specify a theory completely by specifying the action. Now let's see what I was saying before about when the Lagrangian is quadratic. The kinetic energy T should be something like T=1/2mv², and then dT/dv=mv. The potential energy V should be something like V=1/2kx², with dV/dx=kx, and so the Euler-Lagrange equation gives a=-&omega;²x, where the frequency &omega; is just &radic;(k/m). The solutions to this linear equation are easily seen to be like x=e^i&omega;t. This is of course the Lagrangian for the simple harmonic oscillator, and dV/dx=kx is simply Hooke's law. A free field theory is simply an infinite collection of coupled harmonic oscillators.


I had to reread this a couple of times. =)

I was going to ask you if the Lagrangian depended on all of the paths through phase space, or if there was lots of Lagrangians, one for each path. But now i see there is only one Lagrangian, and it describes the path of least action?


So what about when I include a higher order term in the potential? Try V(x)=1/2kx²+1/3gx³, then dV/dx=kx+mgx², and the equation of motion is no longer linear: a=-&omega;²x+gx². In this simple case, you could actually solve this nonlinear differential equation in terms of elliptic functions, but for more complicated potentials, it would be impossible. So one also obtains the solution in a power series in g in terms of the solution to the harmonic oscillator above.

Finally, I have to mention that only classical systems (where Planck's constant h=0) obey the Euler-Lagrange equation. Quantum systems have to be summed over all paths through phase space, each path is assigned a phase (complex number with |z|=1) by the formula e^iS. When you sum this phase over all paths, the sum is dominated by the stable value of S (the minimum), but other values contribute as well.


The Gaussian integral is usually evaluated this way: let I=&int;e^{-x^2}dx. Then I²=(&int;e^{-x^2}dx)(&int;e^{-y^2}dy), where I have changed the name of the dummy variable in the second integral. This can be rewritten I²=&int;&int;e^{-x^2-y^2}dxdy, and now change to polar coordinates where you have dA=dxdy=rdrd&theta; and r²=x²+y² so I²=&int;&int;re^{-r^2}drd&theta; which is an elementary double integral.




Quantum field theory is a quantum many particle theory. In quantum theories, the things you measure are matrix elements between states. So like for example, an initial state might be an electron and a positron, and the corresponding final state might be two photons. You would want to measure the transition amplitude between these two states.

I suppose that you could measure the matrix elements of the momentum operator, it wouldn't be much different. But this isn't done in practice; you don't measure the momentum usually, you input it as a parameter. What you usually do is take those transition amplitudes between states and turn them into cross sections or decay rates. These are the things that particle physicists measure. The decay rates and amplitudes depend on the matrix elements in rather simple ways (eg d&sigma;/dp=|M|² (times some kinematical factors)) And the LSZ reduction formula tells you how to get those transition amplitudes from the correlation functions.



um well, we should be so lucky. The series probably never converges, actually. It's probably what's known as an asymptotic series, where the first couple of terms start to approximate the answer, but after enough terms starts to diverge again. It'll start to diverge after 1/g terms, so for QED, it gets very close to the right value until you go out to the 137th order. considering that people only calculate up to about 3rd or 4th order, that's pretty good! but for QCD, with g=1, the series starts to diverge already with the first term, so you never get close to the right answer, unless you move to an energy scale where g is much smaller.

also, notice that the Taylor series for e^x converges for all x, no matter how large, but 1+x+x²/2 is not a good approximation for e^x if x is large. What's important is not whether the series converges, but rather whether the first couple of terms give you a close answer to the true value or not.

Awesome. Thanks!

[lethe]

I was going to ask you if the Lagrangian depended on all of the paths through phase space, or if there was lots of Lagrangians, one for each path. But now i see there is only one Lagrangian, and it describes the path of least action?

There is only one Lagrangian for a given theory. It is a function from phase space to the real line. Its shape determines the path of least action, yep.

[Yuriy]

AndersHermansson,
You got an excellent response on your question: I got a pleasure, reading lethe's posts...
But there should be done one very significant addition, which I do not want to be missed from this wonderful song.
The problem is that Lagrangian defines a lot, but it does not describs what we see and have in Nature ... alone. The second part of any QFT is the vacuum. The Physical vacuum of our Nature. All observable characteristics of any field are defined by the structure of Lagrangian and ... vacuum. Only together they define observable quantities. And here we come to need to know what the physical vacuum is, what symmetries it has, which symmetries of Lagrangian it "supports" and which not. Do you recall the old "classic" Dirac's theory of electron-positrons field? Even there without imagination of our vacuum as a huge "see" of "already taken energetical levels" he could not make his theory physically meaningful...
I hope, lethe will tell you with the same excellence what is the role that vacuum plays in QFT today, why some of his asymmetries are very important, and what they give for reconstruction of exact knowledge of Quantified Fields in the contemporary Physics...