Understanding Forces and Fields in Classical and Quantum Interactions

[Blue Scallop]

Classically... interactions are referred to as forces, then after it is quantum(ized) or quantized, it becomes fields... so strong force becomes strong field, weak force becomes weak field, and electromagnetic force becomes electromagnetic field, gravitational force becomes gravitational field.

but then isn't it electromagnetic field is classical? oh maybe that's why there is a label quantum electromagnetic field or quantum electrodynamics.. or second quantization...

but what happens to force = mass x acceleration or f = ma? What is the quantum version of it (since force are classical)?

 

[vanhees71]

First of all from a modern point of view "forces" are a somewhat difficult subject. Today theorists tend to think in terms of the action principle to derive equations of motion, on a fundamental level based on symmetry principles (a bit overexaggerating one could say that modern classical and quantum theory is applied mathematics of Lie-group theory ;-)).

Then one should realize that there are different levels of description in physics, and that's a gift of nature. You don't need the most sophisticated theory to describe everything. Often you can do a great job with the correct approximation and almost always you can solve a problem only by applying approximations. One only must take care about the range of validity of these approximations.

E.g., in atomic physics, thanks to the weakness of the electromagnetic interaction (which governs almost all aspects of atoms, molecules, and condensed matter) you can well do by approximating the electromagnetic field as classical and only quantize the atoms and atomic nuclei. Then you also come quite far with approximating the dynamics as non-relativistic. That's what's subject of the Quantum Mechanics 1.

Of course, to get the subtle details, like the Lambshifts of the hydrogen energy levels, you have to quantize the em. field and use the full scheme of relativistic QFT (in this case QED), but to get a pretty good (even quantitative) picture about atoms, molecules and solid-state physics the above described approximation does a very good job.

 

[Blue Scallop]

First of all from a modern point of view "forces" are a somewhat difficult subject. Today theorists tend to think in terms of the action principle to derive equations of motion, on a fundamental level based on symmetry principles (a bit overexaggerating one could say that modern classical and quantum theory is applied mathematics of Lie-group theory ;-)).

Then one should realize that there are different levels of description in physics, and that's a gift of nature. You don't need the most sophisticated theory to describe everything. Often you can do a great job with the correct approximation and almost always you can solve a problem only by applying approximations. One only must take care about the range of validity of these approximations.

E.g., in atomic physics, thanks to the weakness of the electromagnetic interaction (which governs almost all aspects of atoms, molecules, and condensed matter) you can well do by approximating the electromagnetic field as classical and only quantize the atoms and atomic nuclei. Then you also come quite far with approximating the dynamics as non-relativistic. That's what's subject of the Quantum Mechanics 1.

Of course, to get the subtle details, like the Lambshifts of the hydrogen energy levels, you have to quantize the em. field and use the full scheme of relativistic QFT (in this case QED), but to get a pretty good (even quantitative) picture about atoms, molecules and solid-state physics the above described approximation does a very good job.

Let's say you have a classical electron...

Apply Quantum Mechanics and you have the electron non-relativistic Schroedinger wave function.
Apply SR and you have the Dirac Equation and the electron has electron field and the electron the excitation of the electron field

What would happen if you apply General Relativity and QFT.. would you still have electron field and electron as excitation of the electron field or would you have something altogether different.. and does this depends on what is the final form of the quantum gravity (for example AdS/CFT or string theory, loop quantum gravity, etc.)?

Can others give feedback on this too. Thank you.

 

[PeterDonis]

What would happen if you apply General Relativity and QFT

You've already applied QFT; that's what applying SR gives you. The Dirac equation is a QFT equation.

Applying GR just let's spacetime be curved; QFT can be done in curved spacetime.

 

[Blue Scallop]

You've already applied QFT; that's what applying SR gives you. The Dirac equation is a QFT equation.

Applying GR just let's spacetime be curved; QFT can be done in curved spacetime.

But according to atyy:

"Well, the formalism of particles on a fixed curved background is only an approximation. We do expect the motion of all particles to modify the 4D spacetime. So Smolin is right to emphasize background independence.

However, I don't think background independence is what really distinguishes string theory and loop quantum gravity. I think the question the distinction is between emergent and non-emergent spacetime.

In string theory, spacetime is emergent ie. the fundamental dynamical entities are not related to spacetime in any simple way. In loop quantum gravity, the fundamental entities are essentially pieces of spacetime, so spacetime is not as drastically emergent.

The main problem in strings at the moment is that there isn't a non-perturbative formulation of the theory, ie. we don't know the full structure of the theory. There are non-perturbative formulations of some sectors due to AdS/CFT in which the boundary is fixed, but the bulk is fully emergent and dynamical.

The main problem in loop quantum gravity is that it hasn't been shown that all the little pieces of spacetime join up to make a classical spacetime that obeys Einstein's equations."

Won't the final quantum gravity whether it's strings or LQG or AdS/CFT affect the treatment of electron has having electron field and the electron the excitation of the electron field? For instance.. in AdS/CFT.. there may not even be electrons so, how could electrons be excitations of the electron field.. remember in AdS/CFT, only the boundary is fixed, while the entire bulk emerges, unless you mean the boundary also contain electron and electrons still being excitation of the electron field even at the boundary?

 

[PeterDonis]

the formalism of particles on a fixed curved background is only an approximation

Yes, but if we go beyond that approximation we are going beyond GR, and I was answering your question about what we get when we apply GR. GR has a fixed curved background--in the sense that when we do QFT in curved spacetime, we fix the stress-energy tensor everywhere so that we have a well-defined solution to the EFE, and that fixes the spacetime geometry everywhere. The stress-energy tensor we use can have "back reaction" terms in it which take into account the energy in the quantum fields, but it can only do so in an averaged sense. That's why this approach is only an approximation.

Won't the final quantum gravity whether it's strings or LQG or AdS/CFT affect the treatment of electron has having electron field and the electron the excitation of the electron field?

Only in the sense that we will know what more complete theory our current theories of GR and QFT are approximations to. But as approximations, GR and QFT will still be perfectly valid within their domains. They have to be, because we've tested them experimentally to many decimal places and they work.

 

[Blue Scallop]

Yes, but if we go beyond that approximation we are going beyond GR, and I was answering your question about what we get when we apply GR. GR has a fixed curved background--in the sense that when we do QFT in curved spacetime, we fix the stress-energy tensor everywhere so that we have a well-defined solution to the EFE, and that fixes the spacetime geometry everywhere. The stress-energy tensor we use can have "back reaction" terms in it which take into account the energy in the quantum fields, but it can only do so in an averaged sense. That's why this approach is only an approximation.
Only in the sense that we will know what more complete theory our current theories of GR and QFT are approximations to. But as approximations, GR and QFT will still be perfectly valid within their domains. They have to be, because we've tested them experimentally to many decimal places and they work.

You mean if String theory or LQG became true someday.. QFT are approximations to String Theory? But does the phrase makes sense.. String Theory is also a QFT.. so how can QFT be approximations to a QFT? Can you please give example how GR or QFT can be approximations to? in what sense?

 

[vanhees71]

Let's say you have a classical electron...

That's a very difficult challenge, because there's no consistent theory for the classical electron. The best we can do is to give approximations as a spinless charged point particle with an approximate equation of motion in external fields (the Landau-Lifshitz approximation of the Abraham-Lorentz-Dirac equation seems to be the most sensible one, as far as I know).

Apply Quantum Mechanics and you have the electron non-relativistic Schroedinger wave function.
Apply SR and you have the Dirac Equation and the electron has electron field and the electron the excitation of the electron field

The non-relativistic Schrödinger wave-function picture is consistent and has of course wide applications in atomic, molecular, and condensed-matter physics.

Apply SR, and you see that the Dirac Equation (as any other relativistic field equation of motion) has no consistent interpretation as a wave function for interacting particles (even particles interacting with classically approximated external background fields). The reason is that when interactions take place at relativistic energy transfers, it is possible to create and destroy particles (or photons), and that cries for using a many-body approach, of which relativistic local QFT is the most successful one (despite still unsolved mathematical problems).

What would happen if you apply General Relativity and QFT.. would you still have electron field and electron as excitation of the electron field or would you have something altogether different.. and does this depends on what is the final form of the quantum gravity (for example AdS/CFT or string theory, loop quantum gravity, etc.)?

Can others give feedback on this too. Thank you.

Now we enter the realm of open research topics. There is yet no consistent quantum theory of gravitation. Already to make sense of QFT with gravity treated classically is a challenge. E.g., to find a particle interpretation (i.e., asymptotic free states) in a non-Minkowskian classical (General Relativity) background space-time is pretty complicated. So concerning your last question, we cannot give a satisfactory answer. It's not even clear, what are the observational consequences of a quantization of gravity (i.e., according to GR the geometry of spacetime).

 

[PeterDonis]

if String theory or LQG became true someday

If those theories are correct, they're already true now. We just don't know it (yet). Changing our best current theory does not change reality. Electrons didn't work differently before QFT was discovered.

QFT are approximations to String Theory? But does the phrase makes sense.. String Theory is also a QFT.. so how can QFT be approximations to a QFT?

As I've cautioned you before in other threads, you need to stop focusing so much on the words and think about the actual models. "QFT" here just means our best current quantum field theory that we have good experimental evidence for: that would be the Standard Model. "String Theory" means a different quantum field theory, which might possibly turn out to be confirmed experimentally at some point.

Can you please give example how GR or QFT can be approximations to?

I can't because we don't know of any more comprehensive theory. But it's easy to give examples of theories being approximations to other theories. For example, Newtonian gravity is an approximation to GR when gravity is very weak and all velocities are very small compared to the speed of light. That means two things: (1) under the appropriate conditions (weak gravity, slow motion), Newtonian gravity gives answers that are accurate to an appropriate degree; and (2) using the GR equations, you can explain why the Newtonian equations are accurate to an appropriate degree under the appropriate conditions. Most GR textbooks discuss this.

 

[Blue Scallop]

If those theories are correct, they're already true now. We just don't know it (yet). Changing our best current theory does not change reality. Electrons didn't work differently before QFT was discovered.
As I've cautioned you before in other threads, you need to stop focusing so much on the words and think about the actual models. "QFT" here just means our best current quantum field theory that we have good experimental evidence for: that would be the Standard Model. "String Theory" means a different quantum field theory, which might possibly turn out to be confirmed experimentally at some point.

Standard model QFT uses folk space and idea of electron fields and electrons being excitations of the field. What I was asking earlier was whether String Theory also used folk space and similar idea of electron fields and electrons being excitations of the field mathematically.. so how can you say the latter is a different quantum field theory... in what sense different?

I can't because we don't know of any more comprehensive theory. But it's easy to give examples of theories being approximations to other theories. For example, Newtonian gravity is an approximation to GR when gravity is very weak and all velocities are very small compared to the speed of light. That means two things: (1) under the appropriate conditions (weak gravity, slow motion), Newtonian gravity gives answers that are accurate to an appropriate degree; and (2) using the GR equations, you can explain why the Newtonian equations are accurate to an appropriate degree under the appropriate conditions. Most GR textbooks discuss this.

 

[jtbell]

Standard model QFT uses folk space

Folk space?

 

[PeterDonis]

Standard model QFT uses folk space

Do you mean "Fock space"? If so, this concept is not at all particular to the Standard Model; it appears in just about any QFT, regardless of what the QFT is modeling.

how can you say the latter is a different quantum field theory

Because it's a quantum field theory of strings, not particles. Once again, if you would go look at the actual math instead of focusing on the ordinary language words, the differences would be obvious.

 

[Blue Scallop]

Do you mean "Fock space"? If so, this concept is not at all particular to the Standard Model; it appears in just about any QFT, regardless of what the QFT is modeling.
Because it's a quantum field theory of strings, not particles. Once again, if you would go look at the actual math instead of focusing on the ordinary language words, the differences would be obvious.

Thanks for that. Would read Brian Greene book sometimes (maybe next week). But first String Theory is not just a quantum gravity but it tries to explain all the fundamental forces. While Loop Quantum Gravity is only quantum gravity without talking of the other forces. Does this mean in LQG, it is say quantum field theory of loops? Or since it doesn't generalize QFT.. it is still quantum field theory of particles? and only the quantum gravity in the Planck scale differs in the math?

 

[PeterDonis]

Would read Brian Greene book sometimes (maybe next week).

I would not recommend reading anything by Brian Greene if you actually want to learn about the science. Find a good textbook.

String Theory is not just a quantum gravity but it tries to explain all the fundamental forces

Yes.

While Loop Quantum Gravity is only quantum gravity without talking of the other forces

More or less, yes; LQG tries to construct an underlying model from which spacetime geometry emerges. AFAIK it does not try to construct models from which fields on that spacetime geometry emerge.

Does this mean in LQG, it is say quantum field theory of loops?

AFAIK LQG is not a quantum field theory at all; it doesn't use any of the mathematical objects used in QFT.

 

[vanhees71]

Thanks for that. Would read Brian Greene book sometimes (maybe next week).

Peter recommended to read a science but not science-fiction book! SCNR.