1st to 2nd quantization, potential approximates interactions?

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Discussion Overview

The discussion revolves around the nature of the potential term in the Lagrangian for a single particle, particularly in the context of first and second quantization. Participants explore whether the potential can be viewed as an approximation of interactions with virtual particles and the implications of quantum field theory on particle interactions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the potential term in the Lagrangian summarizes interactions of virtual particles at every point, suggesting that first quantization ignores these interactions.
  • Others argue that virtual particles are merely a visual representation of interactions encoded in the Lagrangian, not actual entities interacting with particles.
  • A participant questions whether any change in a particle's momentum can occur without interaction with another particle, suggesting that local interactions are necessary for such changes.
  • Some participants assert that quantum field theory describes particles as localized features of quantum fields, challenging the notion of particles as fundamental entities.
  • A participant raises the idea that a moving electron in a static EM field could experience changes in field strength, potentially linking this to photon interactions.
  • Another participant explores the mathematical representation of potentials, questioning whether a series expansion can represent any legitimate potential in certain approximations.
  • Concerns are raised about interpreting certain mathematical terms as plane waves, with a participant reflecting on their initial thoughts regarding the relationship between the Lagrangian and free particles.

Areas of Agreement / Disagreement

The discussion contains multiple competing views regarding the interpretation of potentials, the role of virtual particles, and the fundamental nature of particles versus fields. No consensus is reached on these topics.

Contextual Notes

Participants express uncertainty about the definitions and implications of various concepts, such as the nature of virtual particles and the mathematical representation of potentials. The discussion highlights the complexity of relating quantum field theory to classical mechanics.

friend
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I'm wondering if the potential term in the Lagrangian for a single particle is just an approximate way to summarize all the interactions of virtual particles created by the potential at every point. For example, the EM field is mediated by photons at every point in space. In reality an electron is interacting with the virtual photons created by the EM field at every point. Perhaps 1st quantization is just ignoring any other particles or effects and only concentrating on what happens to the electron as it pass through the sea of photons created by the field. If this is right, then all one has to do is average out 2nd quantization for a single particle to get to 1 quantization, right?
 
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friend said:
I'm wondering if the potential term in the Lagrangian for a single particle is just an approximate way to summarize all the interactions of virtual particles created by the potential at every point.
No. Rather virtual particles are a fancy way to visually illustrate interactions encoded in the Lagrangian.
friend said:
For example, the EM field is mediated by photons at every point in space. In reality an electron is interacting with the virtual photons created by the EM field at every point.
No. Only in virtuality, an electron is interacting with virtual photons. In reality, an electron interacts with the real electromagnetic field.

See Chapter A7 ''Virtual particles and vacuum fluctuations'' of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#A7
 
A. Neumaier said:
No. Rather virtual particles are a fancy way to visually illustrate interactions encoded in the Lagrangian.



Have we detected in any particle accelerator any change of momentum of a particle without an interaction with another particle? It seems to me that particles are defined with certain properties including momentum and that it would take some very local interaction (particle) to change those properties, right?
 
friend said:
Have we detected in any particle accelerator any change of momentum of a particle without an interaction with another particle? It seems to me that particles are defined with certain properties including momentum and that it would take some very local interaction (particle) to change those properties, right?

Yes. Local interactions are described both in classical and in quantum mechanics by means of fields. This is the reason why one needs quantum field theory to describe elementary particle interactions.

The potential-based approach of Hamiltonian classical or quantum mechanics corresponds instead to having _nonlocal_ interactions at a distance.
 
A. Neumaier said:
Yes. Local interactions are described both in classical and in quantum mechanics by means of fields. This is the reason why one needs quantum field theory to describe elementary particle interactions.

The potential-based approach of Hamiltonian classical or quantum mechanics corresponds instead to having _nonlocal_ interactions at a distance.

I'm not sure how to relate your answer. So let me ask the bottom line: Has anyone ever proven that ANY property of a particle can change that was not the result of an interaction with another particle? Thank you.
 
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friend said:
I'm not sure how to relate your answer. So let me ask the bottom line: Has anyone ever proven that ANY property of a particle can change that was not the result of an interaction with another particle? Thank you.

Particles are not fundamental.

In quantum field theory, which is the theory of elementary particles and their interactions, the basic entity is that of a quantum field, while particles are certain localized features of quantum fields. Thus everything that exists and behaves one way or another is defined and mediated by quantum fields. in a certain (only loosely defined, historically developed) way of speaking, one may interpret certain scattering diagrams as interactions mediated by particles, but this talk can be made rigorous only by eliminating the particles in terms of the fields.

See the entry ''Why are fields more fundamental than particles?'' in Chapter B3 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#complementarity
 
A. Neumaier said:
See the entry ''Why are fields more fundamental than particles?'' in Chapter B3 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#complementarity

You've mentioned in that FAQ that photons occur for varying EM fields but (virtual) photons do not enter the picture for static fields. However, if the electron moves at all through a static EM field (even by quantum fluctuations), then it would experience a change in EM field strength. And I wonder if a photon can be related to that change in field strength and thereby change the momentum of the electron.
 
friend said:
You've mentioned in that FAQ that photons occur for varying EM fields but (virtual) photons do not enter the picture for static fields. However, if the electron moves at all through a static EM field (even by quantum fluctuations), then it would experience a change in EM field strength. And I wonder if a photon can be related to that change in field strength and thereby change the momentum of the electron.

It depends on the scheme to compute the electron motion. In the Coloumb gauge, the static field is represented as a Potential in the Hamiltonian, and there are no associated quantum fluctuations. In a covariant gauge, you get Feynman diagrams with internal lines, which may loosely be interpreted as virtual particles. But ascribing reality to them is meaningless since their occurrence depends on the computational scheme used rather than on the underlying physics of quantum fields 9which is gauge independent).

See Chapter A7 ''Virtual particles and vacuum fluctuations'' of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#A7 , in particular the Sections ''Does the vacuum fluctuate?'' and ''Virtual particles and Coulomb interaction''.
 
  • #10
friend said:
I'm wondering if the potential term in the Lagrangian for a single particle is just an approximate way to summarize all the interactions of virtual particles created by the potential at every point.

How about

V(x)=m1(x-x1)2 + m2(x-x2)2 + m3(x-x3)2 + ...

Can this series be made to equal any legitimate potential at least in the linear or quadraic approximation for a correct choice of mi, and xi. After expanding each term this can be made to equal Ax2+Bx+C.

If so, then the mi(x-xi)2 might be seen as a plane wave expansion for a potential. Does such a thing exist in the literature?
 
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  • #11
friend said:
If so, then the mi(x-xi)2 might be seen as a plane wave expansion for a potential.
How can you see the terms as plane waves? Nothing wavy is going on here...
 
  • #12
A. Neumaier said:
How can you see the terms as plane waves? Nothing wavy is going on here...

Sorry, I was distracted by the TV. I was trying to relate the mi(x-xi)2 terms to free particles. I was thinking that since the lagrangian is in the exponential of the path integral, an exp(i*mi(x-xi)2) could be manipulated to a plane wave. Now that I think more about it, maybe not.

Anyway that was not really part of my question which was the potential term, V(x), in the lagrangian:

friend said:
How about

V(x)=m1(x-x1)2 + m2(x-x2)2 + m3(x-x3)2 + ...

Can this series be made to equal any legitimate potential at least in the linear or quadraic approximation for a correct choice of mi, and xi. After expanding each term this can be made to equal Ax2+Bx+C.
 
  • #13
friend said:
Sorry, I was distracted by the TV. I was trying to relate the mi(x-xi)2 terms to free particles. I was thinking that since the lagrangian is in the exponential of the path integral, an exp(i*mi(x-xi)2) could be manipulated to a plane wave. Now that I think more about it, maybe not.

Anyway that was not really part of my question which was the potential term, V(x), in the lagrangian:

What you get is simply a harmonic potential. th expansion in a sum of quadratic terms is completely irrelevant.
 

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