Quantum Field Theory: An Introduction to Physics Beyond the Standard Model

[jeff]

Originally posted by Si
...He doesn't say that the causality condition follows from something to do with measurements at spacelike separation not affecting each other...

He does mention this in the second last full paragraph on page 198.

Originally posted by Si
He doesn't quantize fields of classical field theories such as electromagnetism, as there is no real physical reason to do so. Rather he starts with "particles" (defined to be eigenstates of the generators of the Poincare group), then shows how fields arise from the need to satisfy the very obvious cluster decompostion principle.

Particle states arise since it's their masses and spins that label the irreducible representations of SO(3,1) under which they transform. The cluster decomposition principle is invoked to explain why and how the hamiltionian must be constructed from creation and annhilation operators acting on these states. But it's lorentz invariance that requires these operators be grouped together to form quantum fields that satisfy causality.

You're right that weinberg doesn't construct QED by quantizing maxwell, but he deduces it first from the gauge-invariance principle he shows any quantum theory of massless particles with spin must satisfy.

Here's a question for you. Can you verify the expression on page 548 in section 13.4?

Originally posted by Si
By the way, don't buy all three volumes in one go! Volume one will teach you a lot about the basics, and can be read without the other two, which cover more advanced topics which no-one here (including myself) is interested in yet.

Actually, I have read all three.

 

[Si]

Originally posted by jeff
He does mention this in the second last full paragraph on page 198.

Although this is not really his reason for making it so. He gives a more formal argument - that it's needed for Lorentz invariance.

Particle states arise since it's their masses and spins that label the irreducible representations of SO(3,1) under which they transform. The cluster decomposition principle is invoked to explain why and how the hamiltionian must be constructed from creation and annhilation operators acting on these states. But it's lorentz invariance that requires these operators be grouped together to form quantum fields that satisfy causality.

Yes, I didn't mention LI + causality for brevity. CDP + LI + causality (+ anything else?) leads to fields.

You're right that weinberg doesn't construct QED by quantizing maxwell, but he deduces it first from the gauge-invariance principle he shows any quantum theory of massless particles with spin must satisfy.

Actually here I felt was one of Weinberg's weaker points. I arrived at the same question I do from other QFT books when the author tries to derive the QED Lagrangian from the gauge invariance principle: Is the QED Lagrangian the only possibility (for Abelian fields)? Perhaps I missed something in Weinberg's argument.

Here's a question for you. Can you verify the expression on page 548 in section 13.4?

Probably not, as I am only on Chapter 12! However, I will try to look at it tonight.

Actually, I have read all three.

Sorry, I was referring only to those people who had never read Weinberg, and who want to learn / re-learn the basics. I think you're the first person I've met who has! What did you think, was it the best approach for you or is there another author you prefer?

By the way, Weinberg discusses his approach in Volume 1 in hep-th/9702027 (http://xxx.soton.ac.uk/abs/hep-th/9702027 ), with some nice caveats added.

 

[jeff]

Originally posted by Si
What did you think, was it the best approach for you or is there another author you prefer?

Weinberg's are the only QFT texts I studied systematically, and were used in the course I took in my final year as an undergraduate. Since string theory is my primary interest, the effective field theory approach was useful.

Originally posted by Si
Although this is not really his reason for making it so. He gives a more formal argument - that it's needed for Lorentz invariance.

You probably noticed this, but just in case; in the same paragraph on p198, weinberg explains that it's because of the difficulty of defining measurability for dirac fields that he avoided invoking causality.

Originally posted by Si
...here I felt was one of Weinberg's weaker points...: Is the QED Lagrangian the only possibility (for Abelian fields)?

Weinberg argues that QED is the most general possible lorentz-invariant QFT coupling a massless particle of helicity ±1.

Originally posted by Si
By the way, Weinberg discusses his approach in Volume 1 in hep-th/9702027 (http://xxx.soton.ac.uk/abs/hep-th/9702027), with some nice caveats added.

This appears as one article in a collection entitled "conceptual foundations of quantum field theory" based on a symposium on foundational aspects of QFT. The book includes responses to each lecture (including weinberg's). The link to amazon.com is

 

[planetology]

Well, I would certainly be impressed if Weinberg has a good explanation of how to arrive at the ground state propagator. That is one thing that all authors seem to deal with in the most annoyingly fast and loose terms - never bothering to justify the Wick rotation, and sliding from initial and final position states to the ground state by a complete slight of hand - Jeeze! Even if they would just acknowlege the fact they are cheating with the math, at least the poor students would not be left wondering what the hell we missed.

 

[Si]

I'm not sure I understand, but I will have a go.

If your problem is the apparent arbitrary insertion of the iε in the denominator: This correctly reproduces the position space propagator.

If your problem is a deeper one concerning scattering theory, I had the same problem when learning QFT. Indeed, Weinberg's section 3.1 was for me the most clear physical and mathematical justification of the relationship between interacting and free states, but I still feel I've missed something and would like to discuss this more.

 

[jeff]

Originally posted by planetology
I would certainly be impressed if Weinberg has a good explanation of how to arrive at the ground state propagator.

All results in weinberg are explained in the sense that they're carefully presented in the context of his view of QFT as the unique consequence of reconciling quantum mechanics with special relativity, and that any quantum theory - even if it isn't a field theory (like string theory for example) - will at sufficiently low energies look like one.

Originally posted by planetology
...authors seem to deal with in the most annoyingly fast and loose terms

Can you be more specific about authors or methods?

 

[planetology]

Originally posted by Si

If your problem is the apparent arbitrary insertion of the iε in the denominator: This correctly reproduces the position space propagator.

Are you saying the key is knowing the right answer ahead of time? My complaint is with authors who claim to be doing a derivation - meaning the Wick rotation should stand convincingly on its own logic. I don't necessarily doubt that it does, I just want to see it spelled out so I can understand it, too. (I've seen one other justification for simplifying the integral, which is that the oscillatory terms cancel out; I know of a theorem to that effect - but not due to any QFT author bothering to cite it.)

The other issue I mentioned is the derivation of the propagator from path integrals. What one gets directly is the integral representation for the position state transition <qf|exp(-iHt)|qi>. But what is really of interest is the transition of energy states, not position states. The ground state transition amplitude <0|exp(-iHt)|0> invariably magically appears, using the exact same integral representation that was derived for the postition state transition with no mention of the fact that energy states are superpositions of position states. The Wick rotation to simplify the integral is sometimes there in the mix. Which authors? Can't recall them all off my head... Ryder and Mandl & Shaw come to mind; but I've never seen it done in significantly more detail really.

 

[jeff]

Originally posted by planetology
...the Wick rotation should stand convincingly on its own logic.

The wick rotation is an example of "analytic continuation" in which functions analytic on some domain are extended to functions analytic on some larger domain. The physical justification of wick rotations lies in a theorem due to riemann saying that analytic continuations are unique. So the result of "repackaging" amplitudes by wick rotating them to a domain on which they converge is uniquely determined by the original oscillatory expression, i.e. no information has been added or removed, it's just been reexpressed in a form congenial to explicit calculation.

 

[Si]

Originally posted by planetology
Are you saying the key is knowing the right answer ahead of time?

If I understand you correctly, yes. If you have Weinberg(?), look at equation 6.2.1 for the propagator. This definition is unique, and comes from the commutation relations of the creation and annihilation operators. The i&epsilon; and the choice of sign of &epsilon; is introduced in the definition of the step function used for time ordering, 6.2.15. That it is correct follows from Cauchy's Theorem, closing the contour at &pm; &infinity; and taking the residue. The opposite sign would simply give the wrong result. Thus one gets the usual definition of the propagator, 6.2.18. In the S-matrix, this is integrated over x to produce a momentum space delta function. The momentum integral(s) are done again be using by rotating in the p0 plane, as allowed by Cauchy's Theorem, and this is called Wick rotation. The direction of rotation is determined by the sign of epsilon;, since we cannot rotate across the pole.

Following section 9.2, the PI applies to any QM with 'coordinates' q_i and 'momenta' p_j satisfying the canonical commutation relations. In old QM, q_i is the 3 spatial-position coordinates of a particle. In QFT, q_i is the field at each point in space. The PI gives the matrix element of operators sandwiched between two q-eigenstates (in QFT, eigenstates of the field at a given spatial point). To get the same matrix element with the field eigenstates replaced with the vacuum requires calculating the scalar product of the vacuum with the field eigenstates. If the magnitude of time is large, this change is equivalent to subtracting an i&epsilon; from the Hamiltonian in the PI, as well as an irrelevant normalization of the matrix element. This is shown to be equivalent to subtracting an i&epsilon; in the denominator of the propagators.

The S-matrix may then be calculated from the vacuum-vacuum matrix element of operators using 6.4.3 and the discussion before it.

This all rests on the axioms for how states behave when the magnitude of time is large. These are developed in chapter 3, although I still have problems with it.

 

[jeff]

Originally posted by Si
The i&epsilon; and the choice of sign of &epsilon; is introduced in the definition of the step function used for time ordering...

To get the same matrix element with the field eigenstates replaced with the vacuum requires calculating the scalar product of the vacuum with the field eigenstates. If the magnitude of time is large, this change is equivalent to subtracting an i&epsilon; from the Hamiltonian in the PI...

Just so there's no confusion, in both these examples, the i&epsilon; implements boundary conditions by closing the contour of integration in the upper or lower half-plane. Wick rotations on the other hand change oscillatory to convergent amplitudes by rotating contours of integration along the real axis so that they lie along the imaginary axis, an example of which is given in Weinberg I p475.

Originally posted by Si
This all rests on the axioms for how states behave when the magnitude of time is large. These are developed in chapter 3, although I still have problems with it.

What is it that's bugging you about this?

 

[Si]

Originally posted by jeff
What is it that's bugging you about this?

One of my problems is I'm not sure! Let me just state what I understand: We introduce the principle that there exist "free" states, where "free" means that acting on them with &exp;[-iHt] gives the same result as acting on them with &exp;[iH0 t], where H0 is called the "free" particle Hamiltonian. An example of a physical free state: Consider a large box with a proton and an electron in it, a 2 particle state, with both particles in near momentum eigenstates. However, at any point x, if the probability to find one particle is finite, the probability to find the other at x will be small. So they must not be in perfect momentum eigenstates. This is why Weinberg gives a little "spread" (using g(&alpha;)) to these states, so they don't overlap. Here we assume that Fourier transform of p-states gives the spacetime locations. This free state must be a (near) eigenstate of H. Thus, such a state will change by a small amount after a finite time, but will change by a finite amount after a large time. 3.1.11 refers to (in the Schroedinger picture) states in the process of interaction at finite &tau;, which means the state on the LHS of 3.1.12 must be (for finite &tau;) heavily overlapping, and changing by a finite amount for a finite change in &tau;. Correct? I would like to regularize Weinberg's argument by making everything finite, and then let the size of the box + time magnitude go to infinity, the spread to zero (more slowly) etc., but couldn't find an obvious way to do it.

Do we define H0 to have the same spectrum as H because it is the only way to get 3.1.12? Or is it an axiom? And is there some physical reason for it? What is the physical interpretation for the eigenstates of H0?

The fact that H has two sets of eigenstates even though a Hermitian operator should only have one means that 3.1.11 is not quite correct - there is a discontinuity in the theory (which I guess shows up in 3.1.17), so 3.1.11 should be spread with g(&alpha;)?

 

[planetology]

I just realized I said "propagator," when what I really meant was vacuum-vacuum transition amplitude. Sorry for the confusion.

Originally posted by Si

Following section 9.2, the PI applies to any QM with 'coordinates' q_i and 'momenta' p_j satisfying the canonical commutation relations. In old QM, q_i is the 3 spatial-position coordinates of a particle. In QFT, q_i is the field at each point in space. The PI gives the matrix element of operators sandwiched between two q-eigenstates (in QFT, eigenstates of the field at a given spatial point). To get the same matrix element with the field eigenstates replaced with the vacuum requires calculating the scalar product of the vacuum with the field eigenstates. If the magnitude of time is large, this change is equivalent to subtracting an i&epsilon; from the Hamiltonian in the PI, as well as an irrelevant normalization of the matrix element. This is shown to be equivalent to subtracting an i&epsilon; in the denominator of the propagators.

It makes sense to me what you are saying, but looking at Weinberg (I don't own it), his notation is odd to me, and his mathematical treatment so abbreviated it's a bit difficult for me to see his full justification of the equivalence between the inner product and the subtraction of i&epsilon;. I will study that section more carefully to see what more I can get out of it.

Originally posted by Jeff

The physical justification of wick rotations lies in a theorem due to riemann saying that analytic continuations are unique. So the result of "repackaging" amplitudes by wick rotating them to a domain on which they converge is uniquely determined by the original oscillatory expression, i.e. no information has been added or removed, it's just been reexpressed in a form congenial to explicit calculation.

That seems like it would be helpful to see. Know a good reference?

 

[jeff]

Originally posted by planetology
That [principle of analytic continuation] seems like it would be helpful to see. Know a good reference?

It's a basic result covered in every introductory course in complex analysis. Just look under analytic continuation in any complex analysis text.

 

[jeff]

Originally posted by Si
Do we define H0 to have the same spectrum as H because it is the only way to get 3.1.12? Or is it an axiom? And is there some physical reason for it? What is the physical interpretation for the eigenstates of H0?

Intuitively, since &Psi;_&alpha;^&plusmn; are states of non-interacting particles, we should be able to define them in terms of some corresponding set of free particle states &Phi;_&alpha; of a free particle hamiltonian H₀ in a way that they have the same appearance as the &Psi;_&alpha;^&plusmn;. This means that if we write H = H₀+V, then since the &Psi;_&alpha;^&plusmn; are eigenstates of the full physical hamiltonian H, V must be chosen so that the masses appearing in H₀ are the physical masses etc.

Originally posted by Si
This is why Weinberg gives a little "spread" (using g(?)) to these states...

In exp(-iH&tau;)&Psi; on p109, &Psi; describes a state seen by an observer at some point during a collision process. Now, the whole idea of defining scattering amplitudes in terms of in and out states depends on the assumption that the collision process occurs over some finite interval of time. If &Psi; is an energy eigenstate &Psi;_&alpha; so that we know it's exact energy, by the time-energy uncertainty principle the collision process is spread out across all time, in which case the whole idea of in and out states goes down the toilet. We see this mathematically by noting that in that case exp(-iH&tau;)&Psi; = exp(-iE_&alpha;&tau;)&Psi;_&alpha;, so that taking &tau; &rarr; &plusmn; &infin; achieves nothing since exp(-iE_&alpha;&tau;) is purely oscillatory and so has no limit. Thus, since &Psi;_&alpha;^&plusmn; are effectively states of non-interacting particles so that 3.1.1 requires they be energy eigenstates of the hamiltonian H, we must consider &int;d&alpha;exp(-iE_&alpha;&tau;)g(&alpha;)&Psi;_&alpha;^&plusmn; rather than just individual energy eigenstates &Psi;_&alpha;^&plusmn;. Therefore, the correspondence between the &Psi;_&alpha;^&plusmn; and the &Phi;_&alpha; must be given in terms of wave packets: &int;d&alpha;exp(-iE_&alpha;&tau;)g(&alpha;)&Psi;_&alpha;^&plusmn; &rarr; &int;d&alpha;exp(-iE_&alpha;&tau;)g(&alpha;)&Phi;_&alpha; for &tau; &rarr; -&infin; or &tau; &rarr; +&infin; respectively.

Originally posted by Si
The fact that H has two sets of eigenstates even though a Hermitian operator should only have one means that 3.1.11 is not quite correct...

&Psi;_&alpha;^&plusmn; are states in the same hilbert space (see 1st full paragraph after 3.2.1) and by energy conservation their energy eigenvalues must be equal.

 

[Si]

Originally posted by planetology
It makes sense to me what you are saying, but looking at Weinberg (I don't own it), his notation is odd to me, and his mathematical treatment so abbreviated it's a bit difficult for me to see his full justification of the equivalence between the inner product and the subtraction of i&epsilon;. I will study that section more carefully to see what more I can get out of it.

I was put off Weinberg for a long time because his notation was different to others. But once I got familiar with it, I found in fact that his notation was simpler and more general.

 

[planetology]

Originally posted by jeff
It's a basic result covered in every introductory course in complex analysis. Just look under analytic continuation in any complex analysis text.

Ah, I should have known Churchill could rescue me. Back in the day it always did.

Jeff and Si - thanks for all the helpful posts. Looking back at that stuff, I realize the Wick rotation was the thing that really was bugging me, and that all I needed was a slight clue. And keep lobbying for Weinberg - I may buy it yet :)

 

[gnl]

Fields

Thanks for the link to "Fields". I just downloaded it. It's huge! Never seen anything like that.

Another good set of notes is the book by Carroll on General Relativity, also on the archive.

Giuseppe

 

[tiny-tim]

Weinberg QTF Volume I free online

… not everyone has Weinberg's books. Siegel's book is online and free, so I thought it would be good …

As for a Weinberg workshop, be my guest. The only problem is that the number of people who can participate is limited by the number of people who have the book.

My reason for liking Weinberg: His approach feels much more "pure" than other books, particularly in that all axioms are simple and physically inuitive, and are introduced only when needed. Thus your knowledge doesn't get entangled, the various theorems become more powerful since they can then be extended to other theories, and theorems can be obtained more completely and generally yet made simpler. His brief yet comprehensive style helps one avoid getting confused, although sometimes he is a bit too brief!
…
By the way, don't buy all three volumes in one go! Volume one will teach you a lot about the basics, and can be read without the other two, which cover more advanced topics which no-one here (including myself) is interested in yet.

… By the way, Weinberg discusses his approach in Volume 1 in hep-th/9702027 (http://xxx.soton.ac.uk/abs/hep-th/9702027 ), with some nice caveats added.

Volume I of Weinberg's QTF is also available free online (most pages), at http://books.google.com/books?id=h9...nother book on quantum field theory"&f=false".

(It says "Volume 2" at the top, but it is Volume I! … if the above link doesn't work, do a google "Books" search for Volume 2 … that's why I've publicised the link here, since a google search for Volume I doesn't find it )