Are tracks in collision experiments proof of particles?

  • #51
Yes, you either need wave packets or you can use a finite volume (cube) and impose periodic boundary conditions. Then your momentum eigenstates are welldefined (since the momentum operator is self-adjoint contrary to the case with rigid boundary conditions) square-integrable functions. At the end after (!) squaring the so regularized S-matrix elements, and dividing by the four-volume to get transition rates, you can take the infinite-volume limit getting a single energy-momentum conserving ##\delta## distribution as it should be.
 
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  • #52
atyy said:
I do think a proper derivation of LSZ needs wave packets
But this is unrelated to measurement/decoherence issues. It is just a choice of test functions.
 
  • #53
True, but what is it then what you are after concerning "measurement issues"? I always leave the measurements to the experimentalists, because they are much better in that as any theorist can be ;-)).
 
  • #54
A. Neumaier said:
But this is unrelated to measurement/decoherence issues. It is just a choice of test functions.

But if one has a wave packet, then we get to the point where we can use some variant of Mott's analysis, and decoherence etc. (In the Copenhagen view, a wave packet does not imply a particle with a trajectory, which is why one still needs additional argument to say why the wave packet leaves a track similar to a classical particle.)

Are you thinking that we cannot use Mott's analysis, or that we can, but he doesn't give enough detail of the decoherence process?
 
  • #56
vanhees71 said:
what is it then what you are after concerning "measurement issues"?
It was not me who tried to answer my quest with informal references to decoherence and measurement; so I pointed out that the information provided by a reference to LSZ does not give support to the idea that decoherence or measurement resolve the problem.

From my point of view it is a purely formal question how to reduce trains of particles in a beam (a prepared ensemble of single particles in a quantum mechanical textbook setting) to something interpretable in many-particle quantum field theory. Thus there should be a purely formal answer to my quest. Or at least an outline that looks like being formalizable.

The S-matrix from QFT doesn't do it, as it describes what happens to a single 2-particle system (1 beam particle plus 1 target particle) prepared at infinite distance upon collision, producing an additional outgoing spherical wave that extends to infinity.

But QFT is a description that gives correlation functions at all space-time positions, hence a complete history of everything.
In particular, QFT should be able to model the train of particles in a beam as a temporally resolved something in the QFT description, without having to resort to the interpretation of the single temporal train of ##N\gg 1## particles as an ensemble of 1-particle systems.

Treating the train as an ensemble of N-particle systems will not work since it will give the wrong statistics - at any time only one of the particles will collide with a target particle - so the cross section is that of a 2-particle collision and not one of ##N+1## particles!

All this is simply swept under the carpet if one says that for all practical purposes one may regard the system as decohered immediately after the collision (when exactly?? how??) into flying particles (where do they get their fairly well-determined momentum from if, as claimed, only a position measurement was made??) and that the measurement process (in its ill-defined quantum version!) takes care of it.

I know that this works in practice, but I am not satisfied with shut-up-and-calculate but want to look under the carpet!
 
  • #57
A. Neumaier said:
The S-matrix from QFT doesn't do it, as it describes what happens to a single 2-particle system (1 beam particle plus 1 target particle) prepared at infinite distance upon collision, producing an additional outgoing spherical wave that extends to infinity.

But QFT is a description that gives correlation functions at all space-time positions, hence a complete history of everything.
In particular, QFT should be able to model the train of particles in a beam as a temporally resolved something in the QFT description, without having to resort to the interpretation of the single temporal train of ##N\gg 1## particles as an ensemble of 1-particle systems.

Well, if that's your objection, I don't believe I have ever seen a treatment of this problem. Everyone treats the particles in a train as an independent preparation.

It's only when we try to apply quantum theory to the whole universe, eg. quantum fluctuations during inflation seeding structure formation that we try to do something like this, because we don't have multiple preparations of the universe.
 
  • #58
atyy said:
Are you thinking that we cannot use Mott's analysis, or that we can, but he doesn't give enough detail of the decoherence process?
''not enough detail'' is a strong exaggeration - he is completely silent about decoherence or measurement!

He just needs Born's rule for interpreting the final outcome. This makes it an exemplary contribution to the foundations. He explains without reference to anything outside the quantum formalism.

Moreover, there is no reference to the ##\alpha## particle! This makes his analysis very close to a field theoretical treatment. It is consistent with the possibility (implicitly indicated in the formulation of the thread title) that particles do not exist but are just a way of visualizing invisible happenings in the microscopic domain.

But we cannot use Mott's analysis directly in a QFT treatment since there is a mismatch between the statistical view of a train of many temporally separated particles in a beam (as an ensemble in the QM1 sense) and the temporally resolved view of many-particle QFT, where everything happening in space and time is described by correlation functions only.
 
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  • #59
Robert100 said:
Who the is secretly deleting comments in this thread, without even having the common courtesy to tell the original poster? That is rude behavior. Especially since I haven't been informed of even a single complaint or criticism.
No college or university science course I have been in has ever had an instructor act this way.
What do you mean? Your post https://www.physicsforums.com/posts/5379333/ is still here!
 
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  • #60
A. Neumaier said:
''not enough detail'' is a strong exaggeration - he is completely silent about decoherence or measurement!

He just needs Born's rule for interpreting the final outcome. This makes it an exemplary contribution to the foundations. He explains without reference to anything outside the quantum formalism.

Moreover, there is no reference to the ##\alpha## particle! This makes his analysis very close to a field theoretical treatment. It is consistent with the possibility (implicitly indicated in the formulation of the thread title) that particles do not exist but are just a way of visualizing invisible happenings in the microscopic domain.

I don't think anyone uses the word particle the way you use it. A particle and a field are the same in QFT, because of the Fock space. I think everyone would agree with you if they used your terminology. (And yes, Mott's analysis is severely lacking in detail, but he does enough that one can believe it ok to leave as a homework problem)

A. Neumaier said:
But we cannot use Mott's analysis directly in a QFT treatment since there is a mismatch between the statistical view of a train of many temporally separated particles in a beam (as an ensemble in the QM1 sense) and the temporally resolved view of many-particle QFT, where everything happening in space and time is described by correlation functions only.

Yeah, you are the only person who's ever asked this. Usually we only bother about such things if we really believe there is only one history, and not many independent preparations.
 
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  • #61
A. Neumaier said:
there is a mismatch between the statistical view of a train of many temporally separated particles in a beam (as an ensemble in the QM1 sense) and the temporally resolved view of many-particle QFT, where everything happening in space and time is described by correlation functions only.
atyy said:
Everyone treats the particles in a train as an independent preparation.
It's only when we try to apply quantum theory to the whole universe, e.g. quantum fluctuations during inflation seeding structure formation that we try to do something like this, because we don't have multiple preparations of the universe.
But there is an intermediate situation that doesn't need the whole universe. One can consider an ensemble of independently prepared trains of particles in a beam undergoing a collision with a target. Such an ensemble can be easily prepared in many labs around the world, or in the same lab on different says. Therefore there should be a QFT model where this ensemble is considered as a single system.
 
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  • #62
A. Neumaier said:
But there is an intermediate situation that doesn't need the whole universe. One can consider an ensemble of independently prepared trains of particles in a beam undergoing a collision with a target. Such an ensemble can be easily prepared in many labs around the world, or in the same lab on different says. Therefore there should be a QFT model where this ensemble is considered as a single system.

Agreed. I'm pretty sure this no one has done this. It's like saying the usual analysis of Bell experiments is inadequate, and that rather the Aspect experiment itself should be considered one member of an ensemble, and the Zeilinger experiment another member of the ensemble. I mean you are correct, but really, this seems masochistic.

I detect you think the standard should be higher for QFT, but it's ok if you just think that QFT is a particular type of QM.
 
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  • #63
atyy said:
I don't think anyone uses the word particle the way you use it. A particle and a field are the same in QFT, because of the Fock space.
No. Nobody uses the particle concept in the way you use it here.
  • First of all, Fock space describes free (i.e., asymptotic) particle states only.
  • Second, the (smeared) field is an operator, and cannot be associated in any way with one or more particles.
  • Third, even in a free quantum field theory, the complete information about the particle number is in the state of the system.
  • Fourth, Fock space contains states with an arbitrary particle number, and most states do not describe situations with a fixed particle number. They describe free particles only in a very loose sense.
  • Fifth, for an interacting QFT, the particle interpretation is completely lost in the renormalization process: Due to Haag's theorem there is no valid interaction picture, so particles exist only asymptotically.
This is why many textbooks only treat asymptotic theory, i.e., S-matrix computations.

In the books that don't restrict to asymptotics (books on nonequilibrium quantum field theory or statistical mechanics) one only recovers a modified, effective particle picture, given in terms of quasiparticles.
 
  • #64
A. Neumaier said:
What do you mean? Your post https://www.physicsforums.com/posts/5379333/ is still here!

Weird. I got notifications ("alerts") that multiple posts written by me were deleted, without explanation.
If I do not know why a post is deleted, then I cannot change how I write, to address the issue.
I'm glad to see then, that that particular post is still there.

Robert
 
  • #65
Robert100 said:
Weird. I got notifications ("alerts") that multiple posts written by me were deleted, without explanation.
If I do not know why a post is deleted, then I cannot change how I write, to address the issue.
I'm glad to see then, that that particular post is still there.

Robert
You can see onhttps://www.physicsforums.com/members/robert100.46911/ which posts are still there. Post can be deleted either if you violated the rules, or if you replied to a post that was deleted for this reason. Some posts are also just moved somewhere else if you replied off-topic.
 
  • #66
A. Neumaier said:
In the books that don't restrict to asymptotics (books on nonequilibrium quantum field theory or statistical mechanics) one only recovers a modified, effective particle picture, given in terms of quasiparticles.
What books would that be?
 
  • #67
  • #68
A. Neumaier said:
This would hold for a classical field but not for a quantum field. In quantum mechanics, discreteness is not rigidly associated with decomposability into pieces.

Orbital angular momentum is also quantized, but nobody deduces from the http://espace.library.uq.edu.au/view/UQ:161172/UQ161172.pdf the existence of angular momentum particles.

Indeed I meant classical field; I wouldn't expect any such thing from a quantum field, as it is an abstract concept devoid of simple visualization. Still, how do you explain with quantum field that charge on oil drops occurs in multiples of ##e##?
 
  • #69
A. Neumaier said:
No. Nobody uses the particle concept in the way you use it here.
  • First of all, Fock space describes free (i.e., asymptotic) particle states only.
  • Second, the (smeared) field is an operator, and cannot be associated in any way with one or more particles.
  • Third, even in a free quantum field theory, the complete information about the particle number is in the state of the system.
  • Fourth, Fock space contains states with an arbitrary particle number, and most states do not describe situations with a fixed particle number. They describe free particles only in a very loose sense.
  • Fifth, for an interacting QFT, the particle interpretation is completely lost in the renormalization process: Due to Haag's theorem there is no valid interaction picture, so particles exist only asymptotically.
This is why many textbooks only treat asymptotic theory, i.e., S-matrix computations.

In the books that don't restrict to asymptotics (books on nonequilibrium quantum field theory or statistical mechanics) one only recovers a modified, effective particle picture, given in terms of quasiparticles.

Sure. But it's just terminology. No physics disagreement, I think. I might say the condensed matter books do things a bit differently, so it depends on whether one's basic QFT book is say Peskin and Schroeder or Wen.
 
  • #70
A. Neumaier said:
Did you read the paper?? Mott nowhere mentions collapse. Instead he shows (quote from p.80) that [in the absence of a deflecting magnetic field]

This fully explains the tracks, without making any claims about collapse or a particle pointer state, or position measurement. The spherical wave is nowhere replaced by flying particles, as @mfb wants us make believe.

Yes, but the point of the Mott paper was to respond to a puzzle at the time. Namely why don't we see random ionization in a spherically symmetric pattern consistent with what a person might naively believe an S Wave is. The whole point (and the reason the paper is pretty much the grandfather of decoherence) was to note that there are two EQUIVALENT (at this level) descriptions of the phenomena. You can either treat the alpha nuclei as being the quantum object and the whole rest of the bubble chamber as a classical measuring device (in which case you need projection operators that selects out a particular pointer state) OR you treat the atoms in the chamber as a quantum mechanical system, (so you now have a composite system) and then you note that the probability for deviating from the path of the momentum of the 'particle' is negligable under standard Shroedinger evolution and you have the desired behaviour of a 'line' like track.. Later it was realized (but was probably obvious to Mott) that you might use exactly such a setup to 'explain' away the rather arbitrary divide between classical and quantum behaviour inherent in the first projection description.

Anyway, at the level of the nonrelativistic quantum mechanics being used here, its obvious that the behaviour of the 'particles' are absolutely no different in nature from say the two slit phenomenon. Something clicks in the detector with a probability pattern that looks like it follows a 'wave like' pattern, however the clicks are never a half click, and once you get a click you now have a thing that will reclick upon subsequent measurement. Further, this behaviour seems to be universal provided you give me a system with a Hamiltonian that has a spectrum with properly spaced eigenvalues. The rest is a matter of terminology, some people call it wave-particle duality, others just call it a quantum particle. I just don't think there is a distinction to be made here at all in what seems to be universal behaviour.

A. Neumaier said:
They refer to Mott on p.225 (top left) but don't take it up later. The present thread is about microscopic particles, but they consider (already visible from the title and the abstract) instead the problem of localization of macroscopic objects. Or did I miss something? Where do they discuss how a particle in a spherical state decoheres into a flying particle with well-defined momentum and reasonably well-defined position?
I don't see how their analysis applies to the case under discussion here.

The paper I linked was one of the first examples of the modern Einselection program, its not exactly the same setup as the one Mott considers in a bubble chamber, but the logic of what happens goes through in exactly the same fashion. The point is you have a 'quantum' environment that is repeatedly measuring and recording the behaviour of a moving object, and the preffered basis problem is 'solved' by the details of the system in consideration (is it a discrete or continuous variable being measured, how often is it being measured, is self interaction large or small, etc).

I don't know if this research ever reconsidered the exact same setup as the Mott paper, you are invited to do a literature search as I am unfortunately swamped for time..
 
  • #71
A. Neumaier said:
''not enough detail'' is a strong exaggeration - he is completely silent about decoherence or measurement!

He just needs Born's rule for interpreting the final outcome. This makes it an exemplary contribution to the foundations. He explains without reference to anything outside the quantum formalism.

Moreover, there is no reference to the ##\alpha## particle! This makes his analysis very close to a field theoretical treatment. It is consistent with the possibility (implicitly indicated in the formulation of the thread title) that particles do not exist but are just a way of visualizing invisible happenings in the microscopic domain.

But we cannot use Mott's analysis directly in a QFT treatment since there is a mismatch between the statistical view of a train of many temporally separated particles in a beam (as an ensemble in the QM1 sense) and the temporally resolved view of many-particle QFT, where everything happening in space and time is described by correlation functions only.
Of course he (and in my opinion all other theoreticians after him) only need Born's rule to interpret the meaning of the state. There's nothing else in the formalism. Sometimes you find attempts to derive Born's rule from the other postulates of quantum theory. I think that Weinberg has given a convincing line of arguments that this is not possible (in his newest textbook: Lectures on Quantum Mechanics, Cambridge University Press). Of course, to follow such (mostly mathematical) endeavers is very interesting and sometimes fruitful. A famous example is the attempt to derive the parallel postulate of Euclidean geometry from the other axioms, which lead Gauß et al to the discovery of non-Euclidean geometry.

Within quantum theory Mott's analysis is fully sufficient to explain the observation of tracks in matter from quantum theory. It's of course an interesting question to investigate, how to generalize the non-relativstic wave-function treatment in the "first-quantization formalism" to QFT.
 
  • #72
Jano L. said:
Indeed I meant classical field; I wouldn't expect any such thing from a quantum field, as it is an abstract concept devoid of simple visualization. Still, how do you explain with quantum field that charge on oil drops occurs in multiples of ##e##?
It is because in QED, the total charge operator has a discrete spectrum consisting of integral multiples of ##e##. Thus the analogy with angular momentum is complete; one doesn't need a particle concept.
 
  • #73
Haelfix said:
The rest is a matter of terminology, some people call it wave-particle duality, others just call it a quantum particle. I just don't think there is a distinction to be made here at all in what seems to be universal behaviour.
The behavior is indeed universal, but the terminology matters a lot for the intuition and the resulting apparent weirdness.

The tension between particles and waves goes back in the case of light to the times of Huygens (1690) and Newton (1704). Through the interference experiments of Young (1801) it was settled (conclusively for more than a century) in favor of waves. The particle picture was later explained through the paraxial approximation that leads to geometric optics. Thus one can get particles in some limit as an approximation of waves; but there is no way to get waves from particles. Thus the waves are fundamental. This is also the reason why elementary particle physics is based on quantum field theory and not on quantum particle theory! Nomen est omen.

There are only fields in Maxwell's equations but the particle picture is a useful approximation for many optical phenomena (not involving destructive interference). In the same way, there are only correlation functions in quantum field theory, the fundamental theory in modern physics, but the particle picture is a useful approximation for many microscopic phenomena, as long as one acknowledges its limitations and intrinsic approximate nature.
 
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  • #74
A. Neumaier said:
It is because in QED, the total charge operator has a discrete spectrum consisting of integral multiples of ##e##. [...]
Ouch! To my shame, I was not aware of this. Where can I find a derivation?
 
  • #75
For free electrons and positrons you have
$$\hat{Q}=-e \int \mathrm{d}^3 \vec{x} :\hat{\bar{\psi}}(t,\vec{x}) \gamma^0 \psi(t,\vec{x}): = -e \sum_{\sigma=\pm 1/2} \int \frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} [\hat{a}^{\dagger}(\vec{p},\sigma)\hat{a}(\vec{p},\sigma)-\hat{b}^{\dagger}(\vec{p},\sigma) \hat{b}(\vec{p},\sigma)].$$
A complete basis are the Fock states, which are the eigenstates of occupation numbers, which are either 0 or 1.
 
  • #76
strangerep said:
Ouch! To my shame, I was not aware of this. Where can I find a derivation?
In the free case, it is a consequence of the fact that the total charge operator is ##e## times the difference of the number operators for positrons and for electrons. Since the latter have a nonnegative integral spectrum and commute, the statement follows. In the interacting case, one has the same situation in naive perturbation theory. Then one has to argue that nothing bad happens to this fact during renormalization.
I don't think the limits involved change anything in the conclusion.
vanhees71 said:
A complete basis are the Fock states, which are the eigenstates of occupation numbers, which are either 0 or 1.
This is too sloppy. Occupation numbers are not operators, hence have no eigenvalues. The integrals in quastion have as eigenvalues a sum of occupation numbers, which is a nonnegative integer.
 
  • #77
For me the occupation numbers are represented by the number operators
$$\hat{N}_a(\vec{p},\sigma)=\hat{a}^{\dagger}(\vec{p},\sigma) \hat{a}(\vec{p},\sigma).$$
Strictly speaking, you have to first use some regularization (the most simple is to use the "finite-box quantization", i.e., put the system in a cubic box with periodic boundary conditions for the field modes) and afterwards take the infinite-volume limit. For the finite box the ##\hat{N}## are compatible observables and their eigenvectors span a basis of the Fock space. The eigenvalues for the fermion case are 0 and 1 for each occupation number due to the anti-commutation relations of the creation and annihilation operators. That's how we physicists construct the Fock space in our lectures. Maybe it's somehow not rigorous for mathematicians, but I guess that can be made rigorous (in fact the only thing that can be made rigorous in 1+3 dimensions seems to be construction of the free-particle Fock space).
 
  • #78
vanhees71 said:
For me the occupation numbers are represented by the number operators
$$\hat{N}_a(\vec{p},\sigma)=\hat{a}^{\dagger}(\vec{p},\sigma) \hat{a}(\vec{p},\sigma).$$
Strictly speaking, you have to first use some regularization (the most simple is to use the "finite-box quantization", i.e., put the system in a cubic box with periodic boundary conditions for the field modes) and afterwards take the infinite-volume limit. For the finite box the ##\hat{N}## are compatible observables and their eigenvectors span a basis of the Fock space. The eigenvalues for the fermion case are 0 and 1 for each occupation number due to the anti-commutation relations of the creation and annihilation operators. That's how we physicists construct the Fock space in our lectures. Maybe it's somehow not rigorous for mathematicians, but I guess that can be made rigorous (in fact the only thing that can be made rigorous in 1+3 dimensions seems to be construction of the free-particle Fock space).
But then you need extra explanatory work and an infinite volume limit to ensure that the integral over the operator-valued density with spectrum 0,1 is indeed an integer. (Classically, the integral over a characteristic function of a set has no reason to be an integer!)

Whereas if you take the integral (which is a standard 1-particle operator) as a whole and apply it to an N-particle state you immediately see that it gives N times the same state, revealing the spectrum. There is no need to invoke any particular basis in Fock space, and it works the same way for bosons and for fermions! The number operator is something more basic than the occupation number representation.
 
  • #79
vanhees71 said:
$$
\hat{Q}=-e \int \mathrm{d}^3 \vec{x} :\hat{\bar{\psi}}(t,\vec{x}) \gamma^0 \psi(t,\vec{x}): ~=~ -e \sum_{\sigma=\pm 1/2} \int \frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} [\hat{a}^{\dagger}(\vec{p},\sigma)\hat{a}(\vec{p},\sigma)-\hat{b}^{\dagger}(\vec{p},\sigma) \hat{b}(\vec{p},\sigma)] $$

A. Neumaier said:
In the free case, it is a consequence of the fact that the total charge operator is ##e## times the difference of the number operators for positrons and for electrons. Since the latter have a nonnegative integral spectrum and commute, the statement follows. [...]
Oh,... I was indeed aware of that much. I thought you were referring to something deeper.

This is then just a consequence of the tensor product constructions involved in building a multiparticle Fock space: the integer charge spectrum is "by construction". This is significantly different (imho) from the case of angular momentum, where one merely asks that ##SO(3)## be represented unitarily on Hilbert space, and derives the half-integral spectrum without further input. In the latter case, the half-integers were nowhere inserted by hand.

There is also no group theoretic analysis (afaik) that can simultaneously derive integer charges for leptons, and fractional charges for quarks (without putting it all in by hand at the start).

So I think it is not correct to say:
A. Neumaier said:
[...] Thus the analogy with angular momentum is complete
The analogy certainly is not complete since the angular momentum case does not involve inserting half-integers somewhere by hand.
A. Neumaier said:
one doesn't need a particle concept
But one does use a (field theoretic version of) a particle concept in that the Fock space is built up by tensoring elementary systems.
 
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  • #80
I am not a physicist, but what happens to the photon when the energy of light is spent and matter cannot be created nor destroyed?
 
  • #81
cave man said:
I am not a physicist, but what happens to the photon when the energy of light is spent and matter cannot be created nor destroyed?
Please ask your questions in a new thread and delete them here, where it is completely off-topic!
 
  • #82
strangerep said:
This is then just a consequence of the tensor product constructions involved in building a multiparticle Fock space: the integer charge spectrum is "by construction". This is significantly different (imho) from the case of angular momentum, where one merely asks that ##SO(3)## be represented unitarily on Hilbert space, and derives the half-integral spectrum without further input. In the latter case, the half-integers were nowhere inserted by hand.
Well, instead of ##SO(3)## you just need to consider a Heisenberg group and proceed in the same way. In the simplest case, where you have just one oscillator its representation theory gives you a unique regular unitary representation, e.g., realized on the dense subspace of Schwartz function of the Hilbert space ##L^2(R)##. Inside the algebra of linear operators on this dense subspace you can find ladder operators as linear combinations of ##q=x## and ##p=-i\partial_x## that generate the discrete spectrum of ##N=\frac12(p^2+q^2)-##const. It is not built in into the construction of ##L^2(R)##, unless you count everything as built-in that can be mathematically deduced! But the the discrete spectrum of ##SO(3)## is also built in!

Thus the analogy is really complete! One can even get the Heisenberg case as a limiting case of either ##SO(3)## or ##SO(1,2)##; see Section 22.2 of my online book Classical and Quantum Mechanics via Lie algebras.

For field theory one simply takes a much bigger Heisenberg group with an infinite number of independent oscillators, and pick the simplest of the now uncountably many inequivalent unitary representations. One never encounters particles unless one starts to look more closely at the eigenstates of ##N##.
 
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  • #83
Haelfix said:
the modern Einselection program [...]
I don't know if this research ever reconsidered the exact same setup as the Mott paper, you are invited to do a literature search as I am unfortunately swamped for time..
Figari and Teta recently wrote a book about Mott's setting and variations, Quantum Dynamics of a Particle in a Tracking Chamber, expanding on an earlier arXiv paper. Although they say that
Figari and Teta (p.19) said:
Mott’s analysis can be considered the original prototype of the
modern approach to the theory of environment-induced decoherence.
their ''environment'' consists (as in Mott's analysis) of 2 electrons only - far from the macroscopic heat bath needed to get irreversible amplification. True decoherence would presumably happen when a liquid drop condensates around the ions produced, but this step is only mentioned, not even superficially discussed. On the level of the formal discussion, one has a 3-particle system that ends up in a pure, entangled state, predicting by Born's rule probabilities consistent only with a straight path. Thus the relation to decoherence is superficial only. (They do, however, in Chapter 3 something with a macroscopic array of spins, which I haven't digested yet.)

But they make a remark that I found interesting:
Figari and Teta (p.89) said:
It is worth emphasizing that a modification of the environment is the only experimental output one can observe. Contrary to what is often stated, one should not “trace out” the environment degrees of freedom, but rather those of the particle.
It fits the theme of this thread in that particles appear to be ghostlike and only macroscopic (hence field-like) things can be observed.
 
  • #84
strangerep said:
The analogy certainly is not complete since the angular momentum case does not involve inserting half-integers somewhere by hand.
In the case of the Heisenberg group (or equivalently but better, the slightly bigger oscillator group which also contains ##N## as a generator) of a single oscillator, nothing is inserted by hand, except for the choice of the Hermitian operator. This is because there is more freedom than in the ##SO(3)## case, due to the non-compactness of the Heisenberg group. If you pick ##q## or ##p## to determine the spectrum you get all reals; if you pick ##N##, you get only the nonnegative integers.

From the interpretation in terms of a harmonic osciilator, the meaning of the eigenvalues of ##N## is the number of excitations of the eigenfunctions. Not the number of particles - after all, we have only a single oscillator, not enough to make up a particle. In a classical analogy, they count overtones - the number of zeros of a standing harmonic wave clamped at both ends.

When you increase the number of oscillators, the eigenvalues of ##N## (now summed over the oscillators) still count the number of excitations. Why should this interpretation suddenly change in the limit of infinitely many oscillators? It doesn't. Therefore the eigenvalues of the number operator in a free quantum field theory count the number of excitations, and nothing else.

In particular, they never count the number of particles, since so far, particles don't even make sense in our construction. To make sense of it we must impose a - somewhat weird and only historically justified - particle interpretation. In this particle interpretation, one says that an elementary excitation of the quantum field (i.e., a state in the ##N=1## eigenspace) constitutes an elementary particle, and defines the meaning of a single particle in this way! It is an arbitrary (only historically sanctioned) name for these states. It just amounts to using the word ''particle'' for ''elementary excitation'', thereby suggesting a sometimes appropriate, sometimes very misleading imagery.

Note that the particle interpretation is possible only when ##N## exists as an operator - i.e., in the free case, or, in the interacting case, asymptotically in the limit of infinite times for bound clusters in a scattering experiment! Therefore, in the real world, where one can never scatter in infinite time, the resulting particle picture is strictly speaking never appropriate - except in an approximate way!

Poincare invariance, Locality, and the uniqueness of the vacuum state now imply that the newly christened single particle space furnishes a causal unitary irreducible representation of the Poincare group, which were classified by Wigner in 1939. This is why particle theorists say that elementary particles are causal unitary irreducible representations of the Poincare group, Thus elementary particles are something exceedingly abstract, not tiny, fuzzy quantum balls!

For spin ##\le 1##, these representations happen to roughly match the solution space of certain wave equations for a single relativistic particle in the conventional sense of quantum mechanics, but only if one discards the contributions of all negative energy states of the latter. This already shows that there is something very unnatural about the relativistic particle picture. Problems abound if one tries to push the analogies further, and quantum field theorists in their right mind will never do so.

Thus from a quantum field perspective, particles are ghosts from the past still haunting us as long as we continue to believe in them. It is historical baggage that carries no real weight - except in the terminology, which grew historically and is difficult to change.

To be fair, the particle picture has a very practical use. But only as an approximate, semiclassical concept valid when the fields are concentrated along a single (possibly bent) ray and the resolution is coarse enough. But whenever these conditions apply, one is no longer in the quantum domain and can already describe everything classically, perhaps with small quantum corrections. Thus the particle concept is useful when and only when the semiclassical description is already adequate. Note that this domain of validity excludes experiments with beam-splitters, half-silvered mirrors, double slits, diffraction, long-distance entanglement, and the like. Thus it is no surprise that in the interpretation of experimens involving these, particle imagery leads to mind-boggling features otherwise only knowns from dreams and ghost stories. The latter they are!
 
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A.N, on the subject of buckyballs etc you mentioned once that we shouldn't limit our idea of fields. I am struggling with this a bit. Is there a buckyball field?
 
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Jilang said:
A.N, on the subject of buckyballs etc you mentioned once that we shouldn't limit our idea of fields. I am struggling with this a bit. Is there a buckyball field?
Yes. There is an effective field for every molecule. Technically, for every bound state of the fundamental fields. Molecules are such bound states.
 
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