Defeated by quantum field theory yet again

[HomogenousCow]

I haven't taken a course in qft yet, just looking ahead to see what's to come, and so far things are not looking good, I read the firet few chapters of qft in a nutshell, and jesus christ what is this stuff, where are the postulates? The equations of motion? How do I even do these crazy path integrals??What is this business about everything being harmonic oscillators??
*crawls back into my nonrelavistic cave where things are nice and cozy

 

[Jorriss]

What is your background in quantum mechanics?

 

[dextercioby]

We have axiomatical quantum field theory, don't worry. But one should first start with the basics and have a firm grip on non-relativistic quantum mechanics, including the path integral formulation by Feynman.

 

[jfy4]

Check out chapter two in Sakurai's quantum book "modern quantum mechanics." This outlines dynamics, and has a good part on the path integral in non-relativistic quantum mechanics. Also, DON'T GIVE UP! YOU CAN DO IT! :) Start over in Zee's book! and do it again until it works.

 

[tom.stoer]

One can do quantum field theory in the canonical i.e. Hamiltonian approach (it's not popular in most textbooks b/c all scattering and Feynman stuff becomes rather complex, but there are applications where this is advantageous). In this approach one can make QFT look like QM with (countable) infinitly many d.o.f. and one does not need to introduce any path integral.

 

[atyy]

*crawls back into my nonrelavistic cave where things are nice and cozy

There's a way to make non-relativistic QM for many identical particles into QFT called second quantization.
http://www.mit.edu/~levitov/8514/#lecturenotes (Lecture 3)
http://research.physics.illinois.edu/ElectronicStructure/598SCM-F04/lecture_notes/lect18a-2ndQ.pdf
http://arxiv.org/abs/hep-th/0409035 (p13-15).

where are the postulates?

Postulates for reloativistic QFT are in section 3.5 of http://uqu.edu.sa/files2/tiny_mce/plugins/filemanager/files/4282179/non11.pdf

 

[HomogenousCow]

Could I get some help tho, because I have no where Zee is getting this stuff from:
1. Why is the klein gordon equation relavistic??
2.Why do the number of indicies on the stress energy tensor tell us that the spin of the gravitons is 2?
3. How do I interperet particle fields? Are they still probability amplitudes?

 

[tom.stoer]

1.: b/c it is formulated in terms of scalars and 4-vectors
2.: it's the metric tensor which tell's us that it's spin 2

The metric is symmetric, so it has 4+3+2+1 = 10 indep. components; one can identify a gauge invariance allowing us to gauge away 8 components, so we are left with 2 d.o.f. which indicates that we are talking about a massless spin 1,2,... field. The reason why it's spin 2 (not spin 1) can be seen by looking at its multipole expansion: it starts with the quadrupole instead of a dipole

 

[Sonderval]

"How do I interperet particle fields? Are they still probability amplitudes?"
That's my favorite question ;-) (talked about that a lot in the past few weeks)
The problem you have with QFT may be at least in part due to the fact that nobody ever tells you what exactly a QFT field and its state is.
In QFT, the fields themselves are (in path integral formulation) classical fields, not prob. amplitudes. Consider an elastic membrane - the field is the displacement at each point.
In QFT, you now have a probability amplitude for each possible field configuration - the state of a quantum field is a superposition of all possible field configurations, each with its own probability amplitude.
Since it is very difficult to calculate with this kind of object (its a wave functional - a wave function with functions as arguments), people usually don't use it, but conceptionally I think it is important to understand this.

If you do a Fourier analysis, each Fourier component of your field behaves like a harmonic oscillator - so in the ground state the probability of finding a value a_k of the k'th Fourier mode is given by a gaussian centered at zero. (And that is why people will tell you that the expectation value of the field is zero for a vacuum state - but there is still a probability of measuring a non-zero state, exactly as for the position of a particle in a QHO).

If you have a 1-particle state in mode k, this means that the prob. amplitude for the Fourier coefficient of a_k looks like the first excited function of the QHO. It still has a zero expectation value, but now it has a different prob. amplitude. (And this is why you can read that a state with a definite particle number has a vanishing expectation value of the corresponding classical field.)

Hope this helps.

 

[HomogenousCow]

"How do I interperet particle fields? Are they still probability amplitudes?"
That's my favorite question ;-) (talked about that a lot in the past few weeks)
The problem you have with QFT may be at least in part due to the fact that nobody ever tells you what exactly a QFT field and its state is.
In QFT, the fields themselves are (in path integral formulation) classical fields, not prob. amplitudes. Consider an elastic membrane - the field is the displacement at each point.
In QFT, you now have a probability amplitude for each possible field configuration - the state of a quantum field is a superposition of all possible field configurations, each with its own probability amplitude.
Since it is very difficult to calculate with this kind of object (its a wave functional - a wave function with functions as arguments), people usually don't use it, but conceptionally I think it is important to understand this.

If you do a Fourier analysis, each Fourier component of your field behaves like a harmonic oscillator - so in the ground state the probability of finding a value a_k of the k'th Fourier mode is given by a gaussian centered at zero. (And that is why people will tell you that the expectation value of the field is zero for a vacuum state - but there is still a probability of measuring a non-zero state, exactly as for the position of a particle in a QHO).

If you have a 1-particle state in mode k, this means that the prob. amplitude for the Fourier coefficient of a_k looks like the first excited function of the QHO. It still has a zero expectation value, but now it has a different prob. amplitude. (And this is why you can read that a state with a definite particle number has a vanishing expectation value of the corresponding classical field.)

Hope this helps.

Yet, I am still not told what these particle fields represent physically
also I have no idea how to calculate these runctional integrals, no where have I ever seen them explained, how is it even defined? What does the infitesimal represent now?

 

[Sonderval]

"what these particle fields represent physically"
To make it concrete, think of the electromagnetic field (slightly simplified): The classical field is nothing but the four-potential. This is quantised, so in a quantum state, you have different amplitudes for different values of the four potential.

If the functional integral (I assume you mean the path integral?) seems complicated, just discretise it in your mind. Think of a space-time grid with a discrete number of points. Then the iegral becomes a sum over all possible field configurations, i.e., for each field configuration, you calculate exp(iS/hbar) and then you sum up all these to get the total value of the functional integral. It is completely analoguous to the path integral formulation of quantum mechanics (which is explained by Zee):
Instead of summing over all possible paths, you sum over all possible field configurations.

As so often in physics, QFT may seem so difficult because you have to learn new physics and new mathematical tricks, and many books don't make it clear whether something is just a mathematical trick or "true" physics...

Hope this helps a bit more - if not, it would be nice if you could make your questions more specific.

 

[dextercioby]

[...]
2.Why do the number of indicies on the. stress energy tensor tell us that the spin of the gravitons is 2?
[...]

The energy-momentum 4 tensor has 2 spacetime indices irrespective of the spin content of the field. The linearized metric tensor has spin content 0 + 2. Removing the spin 0 content is done for example by requiring it to be traceless.

 

[HomogenousCow]

Well I never had a firm grasp on the path integral formalism even in quantum mechanics, and most of the texts I've read are difficult to understand for me, I don't understand how the integral is defined. It clearly is no longer some multiple Reinman sum, so what is it?

 

[George Jones]

Well I never had a firm grasp on the path integral formalism even in quantum mechanics, and most of the texts I've read are difficult to understand for me, I don't understand how the integral is defined. It clearly is no longer some multiple Reinman sum, so what is it?

It doesn't exist; it is very useful.

Form "Quantum Field Theory: A Tourist Guide for Mathematicians" by Gerald Folland

... some new insights into the material we have developed and some tools for extending it further, at the price of working with some mathematically ill-defined integrals over infinite-dimensional spaces. The attractive thing about this offer is that these intregrals have an intuitive foundation, lead to meaningful calculations, and are tantalizingly close to being mathematically respectable. ... Our main objective in this chapter, however, is to describe functional integrals as used by physicists, so the reader must be prepared to exercise a certain amount of suspension of disbelief. There is some solid mathematics here and there, but much of the purported mathematics is fiction. Like good literary fiction, however, it contains a lot of truth.

 

[HomogenousCow]

I really dislike this kind of ill-defined procedures...what is rhe best text on path integrals?

 

[tom.stoer]

I really dislike this kind of ill-defined procedures...what is rhe best text on path integrals?

I would recommend either Feynman's lectures or Sakurai. The PIs can be defined rigorously in QM, but not in QFT. Anyway, I think it's better to get an idea what they are about, what they mean and how they are related to ordinary quantum mechanics.

 

[HomogenousCow]

so..
1. what do particle fields represent
2. why don't we just use good old kets and bras for QFT?
3. why can't we just solve for a bunch of eigen fields from the sources and then just superimpose them

 

[tom.stoer]

2. why don't we just use good old kets and bras for QFT?

as I said in post #5: we can, but you don't find it in many textbooks (unfortunately)

3. why can't we just solve for a bunch of eigen fields from the sources and then just superimpose them

b/c it's too complicated

 

[Sonderval]

"what do particle fields represent"
As I told you - for the photon, the field A is the vector potential.

"why don't we just use good old kets and bras for QFT?"
because - again as I told you - the state in QFT is a wave functional (a wave function of functions), making dealing with it awkward. You can do it - I explained in my post before how for a vacuum and 1-particle state. See the book by Hatfield "QFT oof point particles and strings" (I seem to recommend that a lot...)

"why can't we just solve for a bunch of eigen fields from the sources and then just superimpose them"
Because in the end what we are most interested in are scattering processes and for those that would be difficult to do.

 

[tom.stoer]

Because in the end what we are most interested in are scattering processes and for those that would be difficult to do.

Sorry to say that but scattering processes are only one sector of QFT; non-perturbative effects like, chiral symmetry breaking, quark condensates, color confinement, QCD bound states, their masses, their form factors etc. are likewise interesting - and there you will a lot of work based on the canonical formalism not using path integrals.

 

[Sonderval]

tom is of course right - I was just thinking about path integrals because this is where we started.
Stupid side question, though: Things like QCD bound states can be calculated with lattice gauge theory, which in essence is nothing but numerically solving a path integral, isn't it?

 

[tom.stoer]

Stupid side question, though: Things like QCD bound states can be calculated with lattice gauge theory, which in essence is nothing but numerically solving a path integral, isn't it?

Not a stupid question. Yes, you are right, lattice gauge theory is based on path integrals; but people use canonical formulations as well, especially to address the more funmdamental questions like e.g. the mechanism behind confinement

 

[HomogenousCow]

What do non force mediator fields represent? like the electron field in The QED lagrangian

 

[Sonderval]

They represent the electron field (you probably guessed *that*...)

The most intuitive way to understand them may be by their relation to the charge density or current. See for example this page, section 3.7:
http://www.quantumfieldtheory.info/Chap03.pdf
(Be warned though that Bob Klauber has an idosyncratic way of looking at many things which is not standard - nevertheless this text helped me a lot in understanding parts of QFT)

@Tom
Thanks - I was just confused because you seemed to imply that path integral always implies perturation theory.

 

[Sonderval]

They represent the electron field (you probably guessed *that*...)

The most intuitive way to understand them may be by their relation to the charge density or current. See for example this page, section 3.7:
http://www.quantumfieldtheory.info/Chap03.pdf
(Be warned though that Bob Klauber has an idosyncratic way of looking at many things which is not standard - nevertheless this text helped me a lot in understanding parts of QFT)

@Tom
Thanks - I was just confused because you seemed to imply that path integral always implies perturation theory.

 

[A. Neumaier]

I really dislike this kind of ill-defined procedures...what is the best text on path integrals?

The path integral in qwuantum mechanics is (unlike that in quantum field theory) perfectly well-defined mathematically.

The Feynman integral book by Johnson and Lapidus
http://tocs.ulb.tu-darmstadt.de/110841727.pdf
tries to do everything in QM with the path integral!

The book ''Feynman integral calculus'' by Smirnov
https://www.amazon.com/gp/search?index=books&linkCode=qs&keywords=3540306102&tag=pfamazon01-20
is a textbook, and has problems and solutions.

 

[HomogenousCow]

I followed Sondervals link and found it to be a better text, until the third chapter that is
I find the section very confusing, I accepted that the solutions to the Klein Gordon equation are operators ( even though I had no idea how they would operate on states), he then goes on in length about the a and b operators ( and their conjugates), I felt fine with that, a killed off a particle and b killed off an antiparticle, but then he doesn't actually explain why we had to solve for them in the first place, he does not actually make use of the original solutions at any point.
Futhermore how do I actually get results that I used to get in quantum mechanics, for example for non interacting particles how do I calculate the position probability density for a state etc etc?

 

[A. Neumaier]

I followed Sondervals link and found it to be a better text, until the third chapter that is
I find the section very confusing, I accepted that the solutions to the Klein Gordon equation are operators ( even though I had no idea how they would operate on states), he then goes on in length about the a and b operators ( and their conjugates), I felt fine with that, a killed off a particle and b killed off an antiparticle, but then he doesn't actually explain why we had to solve for them in the first place, he does not actually make use of the original solutions at any point.
Futhermore how do I actually get results that I used to get in quantum mechanics, for example for non interacting particles how do I calculate the position probability density for a state etc etc?

The position representation is very nonrelativistic, as it singles out a time coordinate.
Almost nobody works in the relativisitc domain in a position representation.

Relativistic multiparticle theory is almost always done as field theory, with the free case treated as a warm-up. So one starts with what is familiar from the nonrelativistic regime, to build up motivation for the relativistic field treatment. Afterwards forget the Klein Gordon equation - it is useless and superseded.

Indeed, if you look at Weinberg's QFT book, he builds the fields from scratch, using unitary irreducible representations of the Poincare group as a start rather than free classical fields. This also has the advantage that it works for any spin.

 

[meopemuk]

Futhermore how do I actually get results that I used to get in quantum mechanics, for example for non interacting particles how do I calculate the position probability density for a state etc etc?

Hi HomogeneousCow,

your questions are very important and relevant. You can find some of them asked and answered in http://arxiv.org/abs/physics/0504062

Cheers.
Eugene.