Quantum Field Theory Made Easy
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29-04-2014, 05:51 AM
Quantum Field Theory Made Easy
Just posted another blog, The Essential Quantum Field Theory. Needless to say, QFT is perhaps the greatest accomplishment of the human mind. Just about everything that smells electronics owes its existence to this theory.

Any comment would be greatly appreciated.

Enjoy.
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29-04-2014, 08:46 AM
RE: Quantum Field Theory Made Easy
I'm not sure what level of audience you are targeting with this work. I consider myself an interested armature who is light on the maths and pretty rusty with physics maths in general. My notes are taken from that perspective.

My notes:
1. In the introduction you pose the question of whether to interpret the particle as fundamental or the field as fundamental, but you never name either concept. When your first heading says "So why quantum field theory?" you haven't yet stated that QFT is the interpretation that fields are fundamental, nor have you named the particle-centric interpretation.
2. Protons are a collection of up and down quarks, so is there a proton field pervading the universe or a an up quark field and a down quark field. Perhaps sticking to photon and electron would be clearer unless you want to head into the territory of compound particles at this point in the article. If so, could a Hafnof in another part of the universe be created that is identical to me... and would that suggest a Hafnof field pervades the universe? The proton example gets even more complex when you consider that the up and down quarks are constantly swapping with one another through colour force exchanges.
3. You use the abbreviation SR without expanding it first, and for that matter QM is also not expanded on or before first use.
4. As a lay person I'm not finding the explanation of point ii under "So why QFT" very clear. What about QM+SR implies that the number of particles is not conserved. Wouldn't it be clearer to state that QM+SR say that energy is conserved but not the number of particles? You could the go on to say that we routinely create and destroy particles in colliders while energy is conserved in these collisions. The Klein-Gordon and Dirac equations are not explained in this paragraph, and in the absence of explanation they don't themselves have any explanatory power regarding why QFT should be preferred.
5. Under the "What is QFT" heading you seem to be contrasting QFT and QM, suggesting that QM is a particle-centric understanding of quantum behaviour. But under "Why QFT" point ii you seem to refer to QM as a foundation for understanding QFT. So again the connection and distinction being drawn seems unclear.
6. Under Units and scales you seem to be introducing terms here that are not explained. L (length?), M (mass? momentum?), T (time?), c (speed of light?), G (?), and even ℏ are all introduced without introductory text. You then go on to say that ℏ=c=1 by which you mean that you are trying to sync up the units, but again it does not make for a particularly clear expression I think. My understanding of what you are saying here is that we are setting length per time such that the speed of light is one, and ℏ is expressed in electron volt seconds... right? I'm not right, am I? Smile
7. Should ℏ = L2M2T-1 actually be L2MT-1 (lose the square off the mass) or am I mistaken?
8. From CP to QM reuses L for the Lagrangian. It is explained, though, so I can't be too critical about that.
9. We didn't cover classical mechanics in terms of Lagrangians back in my day, so I'm finding that although I have rough understanding of most of this section there are parts where I get lost... especially when terms are not defined. S = action, L = lagrangian. Ok. x = distance, fine. v = velocity. q. Is that momentum? No, that should be p here, I think. Ok. An Poisson brackets? Do I need to look that up? That's about the point where I start getting lost Wink There are a number of concepts applied in here in quick succession that should be a breeze to someone who is actively studying the field but might need an explanation or at least a quick reference for someone who is not quite so engaged with the material.
10. Just as a matter of formatting I'm finding it incongruous that we have all of the regular mathematical symbols going on here except for the vinculum. Maybe I'm old fashioned, but seeing fractions expressed side by side doesn't help my comprehension. Perhaps a bit of mathml would help here?
11. It's not clear what "just a label" means here under the CFT heading. The labels are clearly still indicating something useful... so what is it? Smile
12. Another place I feel like I have missed something here is the prefixing of S with "δ" when we move to field theory. Later on δ is used for a seemingly different purpose on its own with subscripts and seemingly as a function... or has it been a constant all along? I know what a partial differential equation is, but I'm not recalling that particular notation or where it fits into whatever preexisting knowledge I assume I can drag up when needed. I'm seeing L and ℒ also and not having a proper understanding as to their relationship. π is also introduced later on without a clear explanation, and as much as I love ψ it too seems to be introduced with no explanation

I'll stop there for tonight, but in summary I guess it seems like the article is written with the assumption that I know QM and/or QFT pretty well before I start. If that's that audience you are after then you might be hitting your mark. If you are trying to touch base with the interested amateur however you should pause for each new mathematician's name, each new physicist's name, and for each new symbol to briefly (one sentence, possibly with a reference elsewhere for further reading) point aside so that that the reader can either read the background information required to understand the material or can put a box around it such that they understand what they don't understand and can get on with working through the rest of the material. That is - see if you can explain each concept sufficiently to let the reader understand whether they need to understand it, or can just ignore it as a solved mathematical nicety.

Give me your argument in the form of a published paper, and then we can start to talk.
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30-04-2014, 06:41 AM
RE: Quantum Field Theory Made Easy
(29-04-2014 08:46 AM)Hafnof Wrote:  I'm not sure what level of audience you are targeting with this work. I consider myself an interested armature who is light on the maths and pretty rusty with physics maths in general. My notes are taken from that perspective.

My notes:
1. In the introduction you pose the question of whether to interpret the particle as fundamental or the field as fundamental, but you never name either concept. When your first heading says "So why quantum field theory?" you haven't yet stated that QFT is the interpretation that fields are fundamental, nor have you named the particle-centric interpretation.

Thanks for that point. The theory is Quantum Field Theory (QFT), and NOT Quantum Particle Theory (QPT). So it should be obvious that historically, QFT was developed not QPT. So the prevailing paradigm is that we have fields as the fundamental building blocks, and particles are products of these fields.


Quote:2. Protons are a collection of up and down quarks, so is there a proton field pervading the universe or a an up quark field and a down quark field. Perhaps sticking to photon and electron would be clearer unless you want to head into the territory of compound particles at this point in the article. If so, could a Hafnof in another part of the universe be created that is identical to me... and would that suggest a Hafnof field pervades the universe? The proton example gets even more complex when you consider that the up and down quarks are constantly swapping with one another through colour force exchanges.

Quarks are the fundamental particles. So Quark-field rules. The forces are treated in the interaction picture as exchanges of particles: for the emf, the photons; for the strong force, the gluons; and for the weak force, the W+,W-, and Z bosons. Now this would be covered in the Standard Model (SM). My blog does not cover this topic as it would require to go over such topics as Renormalization, Gauge Theory - topics that would bring us far from that blog. Perhaps one day, I will post on those topics but not right now.

Quote:3. You use the abbreviation SR without expanding it first, and for that matter QM is also not expanded on or before first use.

For those familiar with my blog, they would know that QM= Quantum Mechanics, and SR+ Special Relativity. These abbreviations are standard practice in physics.


Quote:4. As a lay person I'm not finding the explanation of point ii under "So why QFT" very clear. What about QM+SR implies that the number of particles is not conserved. Wouldn't it be clearer to state that QM+SR say that energy is conserved but not the number of particles? You could the go on to say that we routinely create and destroy particles in colliders while energy is conserved in these collisions. The Klein-Gordon and Dirac equations are not explained in this paragraph, and in the absence of explanation they don't themselves have any explanatory power regarding why QFT should be preferred.

This bit was in line with a previous blog called The Essential Quantum Mechanics, in which you will see there is no place for the creation and destruction of particles. QM was not developed with that intention. So the reason why QFT had to be developed.

Quote:5. Under the "What is QFT" heading you seem to be contrasting QFT and QM, suggesting that QM is a particle-centric understanding of quantum behaviour. But under "Why QFT" point ii you seem to refer to QM as a foundation for understanding QFT. So again the connection and distinction being drawn seems unclear.
Ordinary QM was developed for a one-particle states, and does an adequate job. But when the pioneers of QM turned their attention to many-particles systems, it failed miserably. However, the basis of QM remained and was transferred to QFT. So I would suggest to read up on QM, my blog would one starting point, but there are other blogs you can google.

Quote:6. Under Units and scales you seem to be introducing terms here that are not explained. L (length?), M (mass? momentum?), T (time?), c (speed of light?), G (?), and even ℏ are all introduced without introductory text. You then go on to say that ℏ=c=1 by which you mean that you are trying to sync up the units, but again it does not make for a particularly clear expression I think. My understanding of what you are saying here is that we are setting length per time such that the speed of light is one, and ℏ is expressed in electron volt seconds... right? I'm not right, am I? Smile

In the mks system, L is measured in meters, M in kilograms, and T in seconds. But at sub-atomic scale, that mks system is inadequate as you would carry very large numbers for c, the speed of light, and very small numbers for ℏ. Since any measurement system is arbitrary, then we can use a simple one. So ℏ=c=1 makes the equations a lot simpler. One can always go back to the original system by dimensional analysis.

Code:
7. Should ℏ = L2M2T-1 actually be L2MT-1 (lose the square off the mass) or am I mistaken?

ℏ = L2M2T-1 is in the mks system. In the ℏ=c=1 system, it is dimentionless ( L2M2T-1 = 1)


Quote:8. From CP to QM reuses L for the Lagrangian. It is explained, though, so I can't be too critical about that.

Not clear to me which part of the blog you are referring to.

Quote:9. We didn't cover classical mechanics in terms of Lagrangians back in my day, so I'm finding that although I have rough understanding of most of this section there are parts where I get lost... especially when terms are not defined. S = action, L = lagrangian. Ok. x = distance, fine. v = velocity. q. Is that momentum? No, that should be p here, I think. Ok. An Poisson brackets? Do I need to look that up? That's about the point where I start getting lost Wink There are a number of concepts applied in here in quick succession that should be a breeze to someone who is actively studying the field but might need an explanation or at least a quick reference for someone who is not quite so engaged with the material.

One note: when we deal with one dimension, we usually use x for the position. But when we deal with a generalized coordinate, the standard practice is to use q. And p is always the momentum, unless stated otherwise.

The Poisson brackets are used to go from classical physics to quantum physics. It's the pivotal point when we go from particles to fields.

Quote:10. Just as a matter of formatting I'm finding it incongruous that we have all of the regular mathematical symbols going on here except for the vinculum. Maybe I'm old fashioned, but seeing fractions expressed side by side doesn't help my comprehension. Perhaps a bit of mathml would help here?
Well, some of these fractions are built in the math, especially the 2 pi, and different powers of it, and these come from Fourier transforms. They're a bit of a nuisance, but that's part of the territory.

Quote:11. It's not clear what "just a label" means here under the CFT heading. The labels are clearly still indicating something useful... so what is it? Smile

You are probably referring to position. In QM, position is an operator, while in QFT, it is a label. That distinction is very important.

Quote:12. Another place I feel like I have missed something here is the prefixing of S with "δ" when we move to field theory. Later on δ is used for a seemingly different purpose on its own with subscripts and seemingly as a function... or has it been a constant all along? I know what a partial differential equation is, but I'm not recalling that particular notation or where it fits into whatever preexisting knowledge I assume I can drag up when needed. I'm seeing L and ℒ also and not having a proper understanding as to their relationship. π is also introduced later on without a clear explanation, and as much as I love ψ it too seems to be introduced with no explanation

"δ"is sometimes used to mean a " small difference", and sometimes as a "function" as in the Dirac Delta function. Context will usually clear that up, but I can see it could be confusing to the uninitiated.

Quote:I'll stop there for tonight, but in summary I guess it seems like the article is written with the assumption that I know QM and/or QFT pretty well before I start. If that's that audience you are after then you might be hitting your mark. If you are trying to touch base with the interested amateur however you should pause for each new mathematician's name, each new physicist's name, and for each new symbol to briefly (one sentence, possibly with a reference elsewhere for further reading) point aside so that that the reader can either read the background information required to understand the material or can put a box around it such that they understand what they don't understand and can get on with working through the rest of the material. That is - see if you can explain each concept sufficiently to let the reader understand whether they need to understand it, or can just ignore it as a solved mathematical nicety.

As a suggestion, review QM. It would greatly help.
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