Is Energy conservation among sectors (EM,GR,QM) a principle?

[Roberto Pavani]

TL;DR

Energy conservation (exchange) among EM,GR,QM

Considering QM,EM,GR "sectors" we know that the energy conservation apply but this is a principle.

Noether theorem provides a justification of it for each sector but for the combined (intra-sector conservation) it seem a circular loop:
Energy is conserved through sectors -> I can write a combined action -> Noether proof the conservation -> then the Energy is conserved through sectors.

I'd like to have more comments if the Energy conservation when exchanged between sectors is just a principle or if it is a Theorem on modern physics.

Roberto Pavani
[Link to Orcid preprints redacted by the Mentors]

 

[Dale]

I'd like to have more comments if the Energy conservation when exchanged between sectors is just a principle or if it is a Theorem on modern physics

What is assumed and what is derived is something that individual authors can decide according to their own personal preferences.

 

[Roberto Pavani]

What is assumed and what is derived is something that individual authors can decide according to their own personal preferences.

true, but there is a big difference between:
I assume the Dirac Equation for my Theory
I derive the Dirac Equation from my Theory

 

[martinbn]

What do you mean by energy conservation when exchanged between sectors?

 

[Dale]

true, but there is a big difference between:
I assume the Dirac Equation for my Theory
I derive the Dirac Equation from my Theory

Right. But because different authors make different choices, there is no one single answer to the question.

 

[Roberto Pavani]

What do you mean by energy conservation when exchanged between sectors?

The conservation of energy between gravity and Electromagnetism and Quantum Mechanics (e.g annichilation of electron and positron)
We all know that the Energy is conserved, what I'm asking if for modern physics it is just a principle or a theorem.

 

[renormalize]

We all know that the Energy is conserved, what I'm asking if for modern physics it is just a principle or a theorem.

Can you define what what you mean by the word "principle"?
Or are you asking:
"We all know that the Energy is conserved, what I'm asking if for modern physics it is just a postulate or a theorem."
If that's your actual question, then the answer is: the conservation-of-energy is a theorem.

 

[javisot]

https://en.wikipedia.org/wiki/Noether's_theorem

 

[Dale]

I'm asking if for modern physics it is just a principle or a theorem.

Again, there is no single answer. Any author can choose whether they take it as an assumed principle or a theorem proven from other principles.

Explicitly, Noether’s theorem goes both ways. So either you can assume the symmetries and derive the conserved quantities or you can assume the conserved quantities and derive the symmetries.

 

[Roberto Pavani]

Noether’s theorem is ok, but can be used because we assume to write a combined lagrangian that is justified by the energy conservation among sectors, so it seems circular.
Was asking if there are any non-cicular explanation or if it is just an accepted principle.

 

[Roberto Pavani]

https://en.wikipedia.org/wiki/Noether's_theorem

Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law.

But it means that you can write a combined action of the physical system, then the energy is conserved.

But the reason you can write the combined action mus be different than "because the total energy is conserved", otherwise we are using the ouput of Noether's theorem as input on what we want to demonstrate.

We all know and agree to the principle of energy conservation.

So Noether as explanation is ok as long as there is a theorem that can write a combined action.

 

[renormalize]

We all know and agree to the principle of energy conservation.
So Noether as explanation is ok as long as there is a theorem that can write a combined action.

Our best current model of fundamental physics is the Standard Model of particle physics combined with General Relativity for gravitation. Both of those theories are based on Lagrangians. So the existence of a "combined action" is an empirical observation about nature.

 

[Roberto Pavani]

Our best current model of fundamental physics is the Standard Model of particle physics combined with General Relativity for gravitation. Both of those theories are based on Lagrangians. So the existence of a "combined action" is an empirical observation about nature.

Each sector separately has time-translation symmetry and its own conserved energy.
The fact that energy can flow between sectors while the *total* is conserved is what we use as a guiding principle to write the combined action.
Noether then formalizes it, but we have already used the conservation across sectors as input to write the combined action.
So the conservation is the input, not the output.
That's why I'd call it a principle.

 

[Dale]

Noether’s theorem is ok, but can be used because we assume to write a combined lagrangian that is justified by the energy conservation among sectors, so it seems circular.

This is incorrect. Not all Lagrangians have a conserved energy. That is kind of the point of Noether's theorem, the reason that she developed it. It lets you understand which Lagrangians will have which conserved quantities.

What you do is experimentally observe the equations of motion. From that you infer the Lagrangian. You can either do that by assuming a symmetry or by assuming a conserved quantity. From either you can derive the other and confirm that it leads to the observed equations of motion.

Was asking if there are any non-cicular explanation or if it is just an accepted principle.

As I have said now three times, every author can choose what they want to have as principles (postulates/axioms) and what they want as theorems.

That's why I'd call it a principle.

You are certainly free to make that choice.

 

[renormalize]

Each sector separately has time-translation symmetry and its own conserved energy.
The fact that energy can flow between sectors while the *total* is conserved is what we use as a guiding principle to write the combined action.
Noether then formalizes it, but we have already used the conservation across sectors as input to write the combined action.
So the conservation is the input, not the output.
That's why I'd call it a principle.

I think that's an eccentric view. What I would say instead is that, for each particle/field we observe in nature, we discover that the behavior of each individually derives from a Lagrangian and the concomitant principle of least action. We can add those individual Lagrangians, along with interaction terms coupling all those particles/fields with one another, to arrive at the overall Lagrangian that describes the experimentally-verified SM/GR model. To me, that means the "input" is: "Nature obeys the Principle of Least Action". From that single principle flows all the standard conservation laws (energy, linear-momentum, angular-momentum, charge, etc.) via Noether's theorem.

 

[Roberto Pavani]

I agree with you all.
Let me clarify what I mean: I'm not questioning that energy is conserved.
Noether's theorem guarantees that, once you have the total Lagrangian with time-translation invariance, total energy is conserved.
What I'm saying is: there is no theorem that tells us how to combine the individual sector Lagrangians into L_total.
The additivity of the action across sectors, the specific form of the interaction terms, the fact that the
"energy" defined by Noether for the combined system coincides with what we operationally measure;
these are empirical inputs, not derived results.
Noether tells you: "given L_total, here's your conserved quantity." It doesn't tell you how to construct L_total from the pieces.

Now, modern physics does provide partial answers:

The Standard Model
heavily constrains how sectors can be combined (gauge invariance + renormalizability + anomaly cancellation), though it still requires ~19 free parameters and the gauge group itself as empirical input.

String theory / supergravity
aims to derive the full L_total from a single underlying structure, potentially answering the question completely
(but remains unverified)

I was asking about the current state of the art on this specific point: how far are we from a principled derivation of L_total, rather than an empirically guided construction?

 

[martinbn]

I was asking about the current state of the art on this specific point: how far are we from a principled derivation of L_total, rather than an empirically guided construction?

Why do you think this question is related to conservation of energy?

 

[Roberto Pavani]

Why do you think this question is related to conservation of energy?

The connection is this: energy conservation was an empirical discovery long before we had Lagrangians.
When we formalized it via Noether's theorem, we didn't *explain* why energy is conserved.
We just *relocated* the mystery.
The question "why is energy conserved?" became "why does L_total have time-translation invariance?"
and, more fundamentally, "why does L_total take the form it does?"
So my point is: Noether's theorem doesn't close the question of energy conservation
it transforms it into a question about the structure of L_total.
And *that* question remains partially open.

 

[Roberto Pavani]

@renormalize I agree that "Nature obeys the Principle of Least Action" is a valid starting point.
But even granting that, the *form* of L_total (which sectors exist, how they couple, which gauge group) remains empirical input.
As you noted: "we can add those individual Lagrangians along with interaction terms."
My sense is that approaches deriving the Lorentzian structure itself from a deeper substrate (Girelli-Liberati-Sindoni 2008, Penrose's twistor programme, emergent spacetime models) may eventually dissolve this
question.
If the metric and the field content are *derived*, the inter-sector structure comes for free rather than being assembled by hand.

To make my question concrete: are there results in the literature that derive the interaction terms of L_total from more primitive assumptions, rather than fitting them to experiment?

 

[martinbn]

The connection is this: energy conservation was an empirical discovery long before we had Lagrangians.
When we formalized it via Noether's theorem, we didn't *explain* why energy is conserved.
We just *relocated* the mystery.
The question "why is energy conserved?" became "why does L_total have time-translation invariance?"
and, more fundamentally, "why does L_total take the form it does?"
So my point is: Noether's theorem doesn't close the question of energy conservation
it transforms it into a question about the structure of L_total.
And *that* question remains partially open.

My guess is that you and I have a different understanding of what energy is. It seems to me, that for you it is something that is there, that has some ontology, and then the question is if it is conserved or not. To me it is quite different. It is an abstract mathematical notion that may be defined or not for any given dynamical system depending on the symmetries of the mathematical description. Then the quesion whether energy is conserved is meaningless. By definition energy is the thing that is conserved due to a symmetry.

 

[A.T.]

we didn't *explain* why energy is conserved

It is conserved, because we humans define it such that it is conserved under certain conditions. Same for momentum.

If you ask why it is possible at all to define such conserved quantities, that's more of a philosophical question.

 

[Dale]

Let me clarify what I mean

I don’t think further clarification is needed. I understand your question. I have answered your question multiple times!

You seem to believe that there is only one answer and if people are not giving you the answer that you want then it must be because they don’t understand your question. I understand.

Again, this is a choice that can be made by each author. You want it to be an assumed principle, and you are free to do so. Other authors can choose differently.

The additivity of the action across sectors

Maybe you should think a little more deeply about how that works. Is it true that actions are simply added across sectors? If so, what is the meaning of the “interaction” terms? Is the form of the interaction terms an a priori assumption? If energy were not conserved “across sectors” then what would that look like in a Lagrangian?

how far are we from a principled derivation of L_total, rather than an empirically guided construction?

This is physics, not mathematics. The only justification for any Lagrangian that matters is that it reproduces the observed equations of motion.

The additivity of the action across sectors, the specific form of the interaction terms, the fact that the
"energy" defined by Noether for the combined system coincides with what we operationally measure;
these are empirical inputs, not derived results

I don’t think this is accurate. At best the experimental evidence gives you the equation of motion. Often not even that. Once you have the equation of motion then you construct a Lagrangian. Nature does not provide the Lagrangian. You can construct the Lagrangian by any means, such as an educated guess, inferring a conserved quantity, inferring a symmetry, etc. Neither the Lagrangian nor the means by which you arrive at the Lagrangian are given by nature. But the sole justification of the Lagrangian is that it produces the equations of motion that are given by nature.

 

[Roberto Pavani]

@Dale Your questions about the interaction terms are exactly my point:

1. "Is it true that actions are simply added across sectors?": No, and that's precisely the non-trivial step.
2. "Is the form of the interaction terms an a priori assumption?" In the SM, yes: the gauge group and the coupling structure are empirical inputs.
3. "If energy were not conserved across sectors, what would that look like in a Lagrangian?"
   It would look like a Lagrangian without time-translation invariance for the combined system, or one where the interaction terms break the symmetry. The fact that this doesn't happen is the empirical content I'm asking about.

So we agree: the construction of L_total from the sectors is non-trivial and contains empirical content.
My question is simply whether any framework has made that step *derived* rather than *assumed*.

As for "this is physics, not mathematics" — I agree. But physics has a history of turning empirical facts into derived results (Maxwell, Newton)

 

[Roberto Pavani]

For those interested, the programmes I had in mind are: emergent spacetime (Barceló, Liberati, Visser), twistor theory (Penrose), noncommutative geometry (Connes), thermodynamic gravity (Jacobson). None has fully derived L_total from first principles, but they constrain the problem from different angles.

 

[Dale]

My question is simply whether any framework has made that step *derived* rather than *assumed*.

And my answer to that question remains the same. It can be either depending on the choice of the author.

the construction of L_total from the sectors is non-trivial and contains empirical content

Of course! It would not be science otherwise. The point is that there is not a unique path from empirical content to law of physics. Different authors can justifiably take different paths. In any case each author is making some sort of assumption and some sort of derivation from that assumption. That part is an act of choice by the author.

None has fully derived L_total from first principles, but they constrain the problem from different angles.

So, why do you continue to think that I am not directly answering your question? Doesn't my answer explain the very thing you are asking about? Different authors can make different choices. As I said over and over and over.

 

[Roberto Pavani]

After some further research, I think I found one possible answer to my own question:
Chamseddine & Connes (2007/2008).
They derive the full SM Lagrangian, including the inter-sector couplings, from a spectral action principle on a noncommutative geometry.
In that framework, L_total is not assembled by hand; it follows from the choice of a finite-dimensional algebra and a Dirac operator.
Energy conservation across sectors is then a consequence, not an independent assumption, because the full Lagrangian is derived from a single action principle.

This is the kind of result I was asking about: a set of more primitive axioms from which the inter-sector structure is derived rather than postulated.

Thanks to all for the answers.

 

[Dale]

one possible answer to my own question:
Chamseddine & Connes

But that is not the only possible answer. Different authors will make different choices and they are all justifiable.