Relationships between QM and QFT Particles

[Buzz Bloom]

What are the mathematical relationships (if any) between the particles as described by Quantum Mechanics and the particles described by Quantum Field Theory?

A specific question related to the general question above arose in post #14 of the thread: How can a particle be a combination of other particles? The two more specific questions paraphrased below were asked there, but failed to attract any answers. I am hoping that starting a separate thread on this question might attract some answers.

1. Is there a QFT about EM and photons?
2. If so, is there a theoretical or mathematical connection between such a QFT and the relativistic Maxwell equations that define the behavior of the EM fields?

I found the following in post #2 on the thread: Relation between QM and QFT.

QFT is the unification of special relativity and QM.
In QM the basic ingredient are wavefunctions.
In QFT the basic ingredient are fields of which the fluctuations correspond to particles.​

I also found the following in post #2 on the thread: Question on particles/fields in QFT

You can consider each excitation of the field is a particle since each excitation is discrete and obeys the energy momentum relation E^2=p^2+m^2 if the field obeys the Klein-Gordon equation. But … In the end, QFT is a theory of fields and not particles.

Specifically, for example, a^†_→p |0> creates a "particle" in a specific momentum eigenstate, and so this "particle" is not localized over any region of spacetime. So this may notion of particles is not quite in resonance with the normal notion of a particle as a corpuscular entity localized in space (to a point, or w/e).​

Note: The appearance of the notation for the operation above is not quite right as a copy from the thread. It's the best I can do.

Superficially, these two quotes above appear to be quite contradictory. I suppose that one can resolve this apparent contradiction simply by saying that these two concepts of "particle" are related to each other by both using the word "particle". ;)

3. Since the concept of a "particle" in QM and in QFT are apparently so different, would the following be a reasonable suggestion: In order to minimize confusion and/or misunderstanding it would be useful to use different words/phrases for these two concepts, like e.g., "particle" (for QM) and "QFT particle"? If not, why not?

 

[jimgraber]

What are the mathematical relationships (if any) between the particles as described by Quantum Mechanics and the particles described by Quantum Field Theory?

A specific question related to the general question above arose in post #14 of the thread: How can a particle be a combination of other particles? The two more specific questions paraphrased below were asked there, but failed to attract any answers. I am hoping that starting a separate thread on this question might attract some answers.

1. Is there a QFT about EM and photons?

Yes. This theory is usually called QED or Quantum Electro Dynamics and credited to Feynman, Schwinger and Tomonaga.
Google for a good start.

 

[atyy]

A particle in non-relativistic QM and a particle in relativistic QFT are "essentially" the same. The way one can see this is by reformulating non-relativistic QM of many identical particles as a non-relativistic QFT. This reformulation is (misleadingly) called "second quantization".

http://hitoshi.berkeley.edu/221b/QFT.pdf

http://web.physics.ucsb.edu/~mark/qft.html (Eq 1.30 and surrounding text)

This language is used a lot in condensed matter physics, since it deals with the quantum mechanics of many identical particles.

 

[Buzz Bloom]

Hi jim and atty:

Thank you both for your posts. Frankly I am quite confused by what seems to be contradictions between your posts and the material I quoted from two other threads whch seems to contradict each other. I suppose it might be that both of those two quotes are just incorrect. I will need a bit of time to think about what specifically is confusing me before trying to phrase some useful followup questions.

I have watched the series of QED lectures Feynman gave in New Zealand. I am now surprised that QED is considered to be a QFT. Feynman diagrams look nothing like what I would expect from (1) "In QFT the basic ingredient are fields of which the fluctuations correspond to particles" and (2) "You can consider each excitation of the field is a particle since each excitation is discrete and obeys the energy momentum relation" and "this 'particle' is not localized over any region of spacetime".

Regards,
Buzz

 

[HomogenousCow]

A particle in non-relativistic QM and a particle in relativistic QFT are "essentially" the same. The way one can see this is by reformulating non-relativistic QM of many identical particles as a non-relativistic QFT. This reformulation is (misleadingly) called "second quantization".

http://hitoshi.berkeley.edu/221b/QFT.pdf

http://web.physics.ucsb.edu/~mark/qft.html (Eq 1.30 and surrounding text)

This language is used a lot in condensed matter physics, since it deals with the quantum mechanics of many identical particles.

I read the first link and now I'm a little confused. Why are there no anti-particles in non-relativistic QFT? He's quantizing a complex scalar field which in RQFT gives two species of particles. Is this because the Lagrangian in the NRQFT model does not have a quadratic mass term?

 

[atyy]

I have watched the series of QED lectures Feynman gave in New Zealand. I am now surprised that QED is considered to be a QFT. Feynman diagrams look nothing like what I would expect from (1) "In QFT the basic ingredient are fields of which the fluctuations correspond to particles" and (2) "You can consider each excitation of the field is a particle since each excitation is discrete and obeys the energy momentum relation" and "this 'particle' is not localized over any region of spacetime".

Let's just define QFT as the quantum mechanics of many identical particles (in second quantized language). The essential formalism here is Hilbert spaces, operators, commutation relations.

There are two ideas in Feynman's approach.

(1) Quantum mechanics can be reformulated as a particle taking all paths, so that when one calculates the amplitude to start in one state and end in another, one gives each path a weight and sums over the paths. Because there are many paths, the non-classical oaths can be considered "fluctuations" from the classical path. This is known as Feynman's "path integral formulatin of quantum mechanics". The quantum mechanics of a single non-relativistic particle can be formulated using the path integral, which should allow you to see how the idea that QFT is the quantum mechanics of many identical particles can also be reformulated using the path integral.

(2) Once we have the path integral formulation, we may wish to use approximations such as Taylor series to calculate various quantities. Bear in mind that algebraic expressions like the binomial coefficients have diagrammatic counterparts such as the number of ways of choosing some subset of things https://en.wikipedia.org/wiki/Binomial_coefficient. The Feynman diagrams are analogous, and are very convenient as a visual mnemonic for some algebra.

You can find those ideas explained in http://muchomas.lassp.cornell.edu/8.04/Lecs/lec_FeynmanDiagrams/notes.html.

For reference, these ideas can be made mathematically sound, eg. via the Osterwalder-Schrader conditions: http://www.einstein-online.info/spotlights/path_integrals.

 

[bhobba]

What are the mathematical relationships (if any) between the particles as described by Quantum Mechanics and the particles described by Quantum Field Theory?

The following examines it in detail:
http://www.worldscientific.com/worldscibooks/10.1142/5111

Briefly QM is the 'dilute' weak field limit of QFT. There are a number of equivalent formulations of QM:
http://susanka.org/HSforQM/[Styer]_Nine_Formulations_of_Quantum_Mechanics.pdf

The mathematics of QFT naturally leads to interpretation F, the second quantisation formulation. In the dilute limit (or weak field approximation) you have one or no particles which is bog standard QM.

It is the view of the above book, and my view as well, that many of the issues of QM are rendered trivial by considering QFT from the start. At the lay level the following explains that approach:
https://www.amazon.com/dp/0473179768/?tag=pfamazon01-20

Thanks
Bill

 

[atyy]

I read the first link and now I'm a little confused. Why are there no anti-particles in non-relativistic QFT? He's quantizing a complex scalar field which in RQFT gives two species of particles. Is this because the Lagrangian in the NRQFT model does not have a quadratic mass term?

I'm not sure what the exact condition for having anti-particles is. David Tong's notes http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf have some comments on this issue in section 2.8, after Eq 2.113.

 

[Nugatory]

Since the concept of a "particle" in QM and in QFT are apparently so different, would the following be a reasonable suggestion: In order to minimize confusion and/or misunderstanding it would be useful to use different words/phrases for these two concepts, like e.g., "particle" (for QM) and "QFT particle"? If not, why not?

The two concepts are not as different as you're thinking. What's going on here is that non-relativistic QM works with energies low enough that the particle number is fixed and with an assumed absolute time that allows us to treat position as an observable so that we can have a position basis. These properties allow us to write wave functions that look like $\psi(x,t)$ and then interpret them as a probabilistic description of the position of a classical particle. I'd wager that that's the picture most students form when they first encounter $\psi(x,t)$; I'm also hearing that picture when you say "the normal notion of a particle as a corpuscular entity localized in space (to a point, or w/e)" above.

However, that semi-classical model is not required by the formalism of even non-relativistic QM (all that $\psi(x,t)$ is telling you is the probability of finding a particle at a particular place and time if you look) and it starts to fall apart as soon as you encounter the double-slit experiment and realize that the particle could not have had a definite trajectory or been a "corpuscular entity" between source and screen.

Thus, I prefer to deal with the terminology problem (which is completely the result of trying to attach English words to the mathematical formalism) by saying that in quantum mechanics (both non-relativistic and QFT) the word "particle" doesn't mean what it does in ordinary English, namely some form of "corpuscular entity". It's just that non-relativistic QM doesn't immediately punish you for thinking that it might, whereas QFT forces you to check the notion at the door.

 

[Buzz Bloom]

A particle in non-relativistic QM and a particle in relativistic QFT are "essentially" the same. The way one can see this is by reformulating non-relativistic QM of many identical particles as a non-relativistic QFT.

Hi @atyy:

Thanks for your post.

Q4. By "non-relativistic QM" I assume you mean that "particles" with non-zero rest mass are moving at a speed which is sufficiently less than c so that

1/sqrt(1-(v^2/c^2)) -1 - (1/2) v^2 << 1.​

Is that correct?

Q5. If so, wouldn't "non-relativistic QM" be able to make good predictions of the behavior of photons?

Regards,
Buzz

 

[Buzz Bloom]

Quantum mechanics can be reformulated as a particle taking all paths, so that when one calculates the amplitude to start in one state and end in another, one gives each path a weight and sums over the paths.

Hi @atyy:

Again, thanks for your post.

Q6. If the answer to Q5 in my previous post is "YES", wouldn't "non-relativistic QM" be able to make good predictions about the behavior of photons (or other zero rest mass particles) with respect to possible multiple paths, e.g. of photons with respect to double split behavior?

Q7. I interpret the quote to mean that the "reformulated" math of "non-relativistic QM" would be able to predict the behavior of a non-relativistic particle (e.g., an electron) with respect to a double split. Is this correct?

Q8. If so, why couldn't the "reformulated" math of "non-relativistic QM" be extended by replacing the non-relativistic expressions mv and (1/2)mv^2 for momentum and kinetic energy with their relativistic forms, without using the math procedures of QFT?

Regards,
Buzz

 

[Buzz Bloom]

The mathematics of QFT naturally leads to interpretation F, the second quantisation formulation. In the dilute limit (or weak field approximation) you have one or no particles which is bog standard QM.

Hi Bill:

Thanks for your citations.

Q9.
I underlined "bog" as a context for this question. I assume "bog" is a typo, but since I can not guess the word was intended, what word did you intend?

Regards,
Buzz

 

[Buzz Bloom]

What's going on here is that non-relativistic QM works with energies low enough that the particle number is fixed and with an assumed absolute time that allows us to treat position as an observable so that we can have a position basis.

However, that semi-classical model ... starts to fall apart as soon as you encounter the double-slit experiment and realize that the particle could not have had a definite trajectory or been a "corpuscular entity" between source and screen.

Thus, I prefer to deal with the terminology problem ... by saying that in quantum mechanics ... the word "particle" doesn't mean what it does in ordinary English, namely some form of "corpuscular entity". It's just that non-relativistic QM doesn't immediately punish you for thinking that it might, whereas QFT forces you to check the notion at the door.

Hi @Nugatory:

Thank you very much for your helpful post. Please see my next post.

Regards,
Buzz

 

[bhobba]

I underlined "bog" as a context for this question. I assume "bog" is a typo, but since I can not guess the word was intended, what word did you intend?

It simply means the usual QM taught in introductory QM courses - not relativistic QM or QFT.

Thanks
Bill

 

[Buzz Bloom]

Hi to all who posted to this thread:

I want to thank all of you for your posts. I feel I now have a much better understand of the quantum/particle concept. I also now feel able to express some further questions.

I have had for sometime an idea about an approach to understanding quantum phenomena (see below) that seems to have some similarities to the descriptions of QFT in this thread.

Q10. In what ways does the idea described below differ from QFT concepts?

Q11. In spite if any such differences, do you think it is possible for the idea described below to be developed into mathematical tools that would calculate the same predictive results as a QFT?

Q12. If not, why not?

A brief summary of the intent of the idea is:

There are no force particles, only continuous physical dynamic energy fields, e.g., an EM field. The energy exchanged by an interaction between a material particle (i.e., a non-force particle with a non-zero rest mass) and a force field is limited to a set of discrete values. Also, the energy exchanged by an interaction between two force fields is similarly limited to a set of discrete values.

There are no direct interation between material particles. All such interactions are mediated by force fields.​

The following summarizes the characteristics of the math for the above summary:

A quantized force can be represented as a continuous physical dynamic energy field, e.g., an EM field, with one or more energy scalars or vectors or tensors at each point, e.g., electric vectors and magnetic vectors. Such an energy field fills the ordinary 3+1D space-time.

There is a corresponding amplitude function which provides a set of weighted amplitude values for every possible path the field energy might take from a source to a destination. The set of amplitudes include a single amplitude for each discrete energy value that could be exchanged with a target at the destination. These amplitudes would be calculated from the equations defining the distribution of the field energy throughout space-time.

For each discrete energy value, a composite amplitude could be calculated as a sum or integral over the possible paths between source and destination.​

Regards to all,
Buzz

 

[Nugatory]

I assume "bog" is a typo,

Not a typo, but sometimes it's hard to tell with these Australians

 

[bhobba]

The following summarizes the characteristics of the math for the above summary.

I can't say I follow what you are saying.

I will simply reiterate the fundamental fact of QFT. It can only be described in mathematics. Attempts do convey it in words will fail. I did my best in my reply - but even that falls well short.

Thanks
Bill

 

[bhobba]

Not a typo, but sometimes it's hard to tell with these Australians

https://en.wiktionary.org/wiki/bog_standard

Its common here in Aus. That it has its origins in toilets probably says something about our view of the world. Its the same quirk that made the Rats Of Tobruk revel in what was meant to be derogatory:
http://www.convictcreations.com/history/tobruk.htm

Thanks
Bill

 

[Buzz Bloom]

I can't say I follow what you are saying.

I will simply reiterate the fundamental fact of QFT. It can only be described in mathematics.

Hi Bill:

Thanks for your post. I apologize for my inadequate skills to express my thoughts clearly. I was trying to present what I hoped might be a coherent description of what such a mathematical form of my idea might include. I don't have the skills needed to develop such a mathematical theory.

(Q10) I was also hoping that my characterization of the mathematical relationships would be clear enough so that differences between these relationships and those of QFTs might be recognized and characterized by experts like yourself.

(Q11,12) I also hoped that that any serious flaws in my idea that would prevent its being developed into useful math might be identified.

Regards,
Buzz

 

[bhobba]

I also hoped that that any serious flaws in my idea that would prevent its being developed into useful math might be identified.

The issue for me is I can't quite follow it. That may be my my fault - I don't know. Maybe someone else can help.

But I have to say once you know field theory and QM, QFT more or less follows automatically. It can't be expressed linguistically - but the math is very clear.

Added Later
Maybe spelling out how its done will help. The most advanced form of particle mechanics is the Lagrangian formalism. To apply it to a field you break the field into small blobs and treat those blobs as particles. You then take the size of the blobs to zero (technically we say it becomes a continuum) and you get the equations of a field. In QFT you use QM to treat those blobs as quantum particles and take the limit - you then get QFT. Its pretty much unavoidable.

Just as an aside it is also thought to be the origin of the need for renormalisation - but that is another story.

Thanks
Bill

 

[jtbell]

I assume "bog" is a typo,

No, it's part of the British idiom "bog standard."

https://britishisms.wordpress.com/2015/03/27/bog-standard/

http://www.bbc.co.uk/worldservice/learningenglish/radio/specials/1728_uptodate/page25.shtml

http://english.stackexchange.com/questions/66102/american-equivalent-of-bog-standard

Now, if y'all will excuse me for a minute, I need to go to the bog...

 

[Buzz Bloom]

Hi jtbell:

Thanks for the citations. It always feels good to learn a new word or phrase, even a slang one.

Regards,
Buzz

 

[vanhees71]

Hi @atyy:

Thanks for your post.

Q4. By "non-relativistic QM" I assume you mean that "particles" with non-zero rest mass are moving at a speed which is sufficiently less than c so that

1/sqrt(1-(v^2/c^2)) -1 - (1/2) v^2 << 1.​

Is that correct?

Q5. If so, wouldn't "non-relativistic QM" be able to make good predictions of the behavior of photons?

Regards,
Buzz

No! A photon is massless, i.e., the dispersion relation is $E=|\vec{p}|$, and there is no non-relativistic limit for those quanta (NOT PARTICLES!).

Saying it in the other way around: For very deep Galileo-group-theoretical reasons, while in relativistic QT you can describe massless quanta, there is no way to make sense of non-relativistic quanta with vanishing mass.

 

[Buzz Bloom]

there is no way to make sense of non-relativistic quanta with vanishing mass.

Hi Vanhees:

Thanks for your post.

I interpret your answer to my questions to mean (1) non-relativistic QM is restriced to only non-zero mass particles with sufficeiently slow speeds that relatavistic effects are small enough to be ignored, and (2) that this is understood to exclude particles which have zero mass, e.g. photons.

Based on my interpretation of the relationship of reIativity to QM and QFT in atty's posts #3 and #6, I find the exclusion of photons to be a bit odd. I need to do some investigation about the history of QM. My recollections are admittedly unreliably vague, but I seem to remember that earlier than the relativistic QFT of QED had been developed, non-relativistic QM had produced an explanation of the double slit phenomenon with respect to photons.

Regards,
Buzz

 

[Nugatory]

I interpret your answer to my questions to mean (1) non-relativistic QM is restriced to only non-zero mass particles with sufficeiently slow speeds that relatavistic effects are small enough to be ignored, and (2) that this is understood to exclude particles which have zero mass, e.g. photons.

It is restricted to particles with sufficiently slow speeds that relativistic effects can be neglected. That pretty much has to exclude photons - they travel at the speed of light, which is not such a speed.

 

[Buzz Bloom]

It is restricted to particles with sufficiently slow speeds that relativistic effects can be neglected.

Hi Nugatory:

Thnaks for your post.

OK. I accept as a definition that that "non-relativistic (NR) QM" excludes photons. I also accept that the currently best understanding of the photon double slit phenomena is based on QED, the QFT relevant to EM.

I am now thinking of a double slit experiment with non-relativistic electrons (NREs). Given that the results of such an experiment involves EM and no relativity, would it be possible to calculate the results based on the Feynman concept being applied to an NRE which travels through all possible paths, i.e., goes through both slits, in reaching its destination? If so, would the use of the Feynman concept by definition make this approach an NR QFT and/or an NR QED? Or are QFTs and QED necessarily by definition relativistic?

ADDED
Below is a quote from atyy's post #3,

A particle in non-relativistic QM and a particle in relativistic QFT are "essentially" the same. The way one can see this is by reformulating non-relativistic QM of many identical particles as a non-relativistic QFT.​

This seems to imply that a QFT can be either relativistic or NR. It also suggests that using the Feynman concept is also excluded from QM.

What I am trying to understand here is a definitional difference between QM and QFTs. Is it by definition so that QM (1) excludes all but NR particles, and (2) excludes the Feynman concept of a particlal traveling all possible paths? If so, are there any additional definitional distinctions?

Regards,
Buzz

 

[bhobba]

What I am trying to understand here is a definitional difference between QM and QFTs.

Standard QM is the principles of QM applied to things where position is an observable. If that's the case and you assume the Galilean transformations what you get is the usual QM talked about in books. The detail of this can be found in Chapter 3 of Ballentine - QM - A Modern Development - but it uses advanced math so can't be explained in words.

There is a branch of physics called field theory that deals with fields like EM fields. To handle that with the methods of mechanics ie of particles, you break the field into a large number of blobs and treat each of those blobs like a particle. Then you let the blob size go to zero to get a continuum.

Quantum Field Theory does exactly the same thing. You have a field, break it into blobs, apply the methods of standard QM to those blobs, then take the blob size to zero. That way you get a Quantum Field Theory. Its exactly analogous to classical field theory.

That's the definitional difference. How does all these weird things like 'knots' in the field being particles etc come about? That requires a lot of study and deep math and can't be explained linguistically.

Thanks
Bill

 

[Buzz Bloom]

Hi Bill:

Thank you very much for your post #26. It is is very helpful.

Next I would like to understand how the definitional differences between QM and QFT in your post relate to (1) relativity and (2) Feyman's concept of multiple paths.

I conclude from earlier discussion in this thread together with your post that only non-relativestic particles have observable positions. I am guessing that I have misunderstood the concept of "observable position". Isn't the position of a photon when it is detected by a detector observed to be at the time of detection at the poistion of the sensitve part of the detector? I am now guessing that having an observable position means that the position is in principle observable at all times, not only at a detector. Is that correct?

I am also guessing that it is possible to use Feyman's concept of multiple paths with NR particles, and doing this could be in the context of QM rather than QFT. Is this correct?

Regards,
Buzz

 

[bhobba]

I conclude from earlier discussion in this thread together with your post that only non-relativestic particles have observable positions.

They can have observable positions in QFT, but the situation is very complex
http://arnold-neumaier.at/physfaq/topics/position.html

In standard QM it is always assumed to have position - in QFT it may or may not.

In books on QFT I have read its not a usual thing that's worried about ie its not important - at least at the level I have investigated - which is not advanced.

Thanks
Bill

 

[vanhees71]

Hi Nugatory:

Thnaks for your post.

OK. I accept as a definition that that "non-relativistic (NR) QM" excludes photons. I also accept that the currently best understanding of the photon double slit phenomena is based on QED, the QFT relevant to EM.

I am now thinking of a double slit experiment with non-relativistic electrons (NREs). Given that the results of such an experiment involves EM and no relativity, would it be possible to calculate the results based on the Feynman concept being applied to an NRE which travels through all possible paths, i.e., goes through both slits, in reaching its destination? If so, would the use of the Feynman concept by definition make this approach an NR QFT and/or an NR QED? Or are QFTs and QED necessarily by definition relativistic?

ADDED
Below is a quote from atyy's post #3,

A particle in non-relativistic QM and a particle in relativistic QFT are "essentially" the same. The way one can see this is by reformulating non-relativistic QM of many identical particles as a non-relativistic QFT.​

This seems to imply that a QFT can be either relativistic or NR. It also suggests that using the Feynman concept is also excluded from QM.

What I am trying to understand here is a definitional difference between QM and QFTs. Is it by definition so that QM (1) excludes all but NR particles, and (2) excludes the Feynman concept of a particlal traveling all possible paths? If so, are there any additional definitional distinctions?

Regards,
Buzz

Here a lot is very confused. Let's try to sort things out separately.

(a) Quantum field theory vs. quantum mechanics

Quantum field theory is a formulation of quantum theory of many-body systems, for which the particle number is not necessarily conserved. In this sense it is more general than quantum mechanics, which is always about quantum systems with a fixed number of particles. Any problem of quantum mechanics (in this sense) can be formulated in terms of quantum field theory and both theories are completely equivalent in these cases. Quantum field theory can be used as a formulation non-relativistic quantum theory as well as for relativistic. In the relativistic case quantum mechanics is not so well suited, because in relativistic quantum theory you always have some probability to create and destroy particles (or quanta).

The big arena of non-relativistic QFT is condensed-matter physics, where you often use the concept of "quasi-particles". This technique has been invented by Landau in low-temperature physics of liquid helium. The point is that formally you can describe collective excitations of a many-body system in an analogous way as dilute gases. The "particle-like excitations" are then described as an almost ideal gas of quasi-particles. This is in some way a misnomer, because it's just the mathematical analogy which coined this name quasi-particles. An example are phonons in solid-state physics. They are quantized collective vibrations (like sound waves propagating in the classical picture). The point however is that you can describe bulk properties like the specific heat of the crystal using these quantized collective vibrations, i.e., a set of independent harmonic oscillators (which are mathematically precisely analogous to the description of non-interacting particles in quantum field theory). Now the lattice vibrations are perturbations of the lattice and these perturbations can interact with the electrons and also among themselves, and these "interactions" between electrons and quasiparticles or among the quasiparticles themselves can be treated with perturbation theory. You have corresponding Feynman diagrams and everything very much like in relativistic quantum field theory used in particle physics.

(b) Feynman's path-integral formulation of quantum theory

Feynman's path-integral formulation is just another way to express the same quantum theories discussed above. In quantum mechanics you have functional integrals over trajectories in phase space to begin with. It can be derived from the usual operator formulation of quantum theory. In non-relativistic QM often you have a Hamiltonian which looks like $H=p^2/(2m)+V(x)$ (single relativistic particle moving in a potential force field, e.g., an electrostatic field). Now, the only path integrals you can really solve analytically are path-integrals where the integrand is a Gaussian, and this is what happens in this case: You can integrate analytically over the momentum part of the phase-space trajectory and are left with a path-integral over configuration space, and this is what Feynman figured out in his PhD thesis (published in Review of Modern Physics).

The path-integral technique is also applicable in quantum-field theory, but there you integrate over field configurations not particle trajectories. At the end it's just another formulation of quantum-field theory in the operator language.

Which method to use, depends on the problem you working on. Often the path-integral formalism can be simpler than the operator formulation or the other way around.

 

[Buzz Bloom]

Hi Bill and vanhees:

Thank you both very much for your very helpful posts. I think I now have an understanding about the differences and similaities regarding the concept of "particle" between QM and QFT at a sufficiently "intermediate" level appropriate to my current interest and mental capabilities.

Regards,
Buzz

 

[DrDu]

The main difference between non-relativistic and relativistic QFT is that the former can be shown to exist while the latter doesn't.
Galilean covariance guarantees the conservation of mass and particle number while in relativistic QFT's even the slightest interaction leads to the creation of an infinity of particles which then live in a completely different Hilbert space. Up to now, this spoils any consistent QFT. All we have are some perturbation series aka Feynman diagrams.

 

[Buzz Bloom]

in relativistic QFT's even the slightest interaction leads to the creation of an infinity of particles which then live in a completely different Hilbert space. Up to now, this spoils any consistent QFT.

Hi @DrDu:

Thanks for you post. It makes a correction to what I previously thought I understood.

As I understand the quote above, you are saying that while non-relativistic QM and QFT are consistent, relativistic QFT is not. I conclude from this, and from the earlier discussion in this trhread, that relativity doesn't work well regarding both QM and QFT. For QM, relativity doesn't apply at all, even for photons. Does your post imply that QFT also doesn't deal with photons adequately, ir is it just relativistic non-zero mass particles that cause QFT problerms.

Regards,
Buzz

 

[DrDu]

The problem is with interacting particles. Hence, it also applies to photons, as photons also interact with electrons and other charged particles. From what I remember, the problems are even more severe for massless particles than for massive ones. The problem of showing that the Yang-Mills theory - which is thought to be maybe the most well behaved QFT - actually exists as a well defined mathematical theory is one of the millenium problems of the Clay institute:

https://en.wikipedia.org/wiki/Yang–Mills_existence_and_mass_gap

 

[atyy]

There are rigourously constructed interacting relativistic QFTs in less than 1+1D and 2+1D. The question is open in 3+1D.

In rigourous interacting relativistic QFT, there are not particles in any fundamental sense. But in non-rigourous language that physicist use for interacting relativistic QFT, there are particles.