How it's known that the variational principle works for relativity

[kent davidge]

This question is actually about relativity and quantum field theories. I have the impression that we just use the variational principle, and given the right lagrangian, they lead to equations that we know, are correct. That seems to me a good reason for "believing" that the variationa principle works.

But then in quantum field theories, they start from a variational principle and it leads to equations. Then the only way to see if the equations are right is making experiments, right? But if we knew in advance that the variational principle necessarely needs to the correct equations, then we would not need to test the resultant equations?

 

[Dale]

You always need to test your equations against experiments. That is inherent in science.

 

[kent davidge]

You always need to test your equations against experiments. That is inherent in science.

I agree. Then my guess is correct?

 

[haushofer]

I agree. Then my guess is correct?

You need experiments. Whether you want to test your equations of motion, or, from an variational principle, you want to test your action. You would call it "testing your action", if I understand you correctly.

 

[Vanadium 50]

Then my guess is correct?

No, Kent, your guess is totally wrong.

You should know by now that "I have the impression" is not a valid source. I would have hoped that after N threads, it would have dawned on you that your "impressions" usually don't lead you to understanding. They lead you to lots of running around with the rest of us trying to get you to let go of your "impressions" and you clutching to them for dear life,

The relationship between Lagrangian, Hamiltonian and equations of motion is just math. You can't have one right and the others wrong any more than you can have 2+2 sometimes 4 and sometimes 5. The question of whether this set of mathematical objects is correct or not is, as Dale and haushofer have pointed out, depends on experiment.

 

[kent davidge]

You need experiments. Whether you want to test your equations of motion, or, from an variational principle, you want to test your action. You would call it "testing your action", if I understand you correctly.

thanks
is there a way to show that the theories (relativity, qft) must satisfy a variational principle?

 

[Vanadium 50]

Who cares if they must? Fact is they do.

 

[Dale]

I agree. Then my guess is correct?

You made several guesses which seem mutually contradictory. I cannot tell which guess you mean and not all of your guesses were correct, so I cannot say yes or no here.

 

[PeterDonis]

is there a way to show that the theories (relativity, qft) must satisfy a variational principle?

What do you mean by "must"?

If you mean, we have these theories which have equations of motion, and we want to know if those equations of motion can be derived from a variational principle, then you show that by finding such a principle and deriving the equations of motion from it.

If you mean, we want to prove that any theory that matches experiments must satisfy a variational principle, I have no idea how you would show that since we have no way of ever knowing that we have found all possible theories that could match experiments.

 

[vanhees71]

I think the variational principle is just the best language we have to express the fundamental laws of physics that are found by observation, including high-precision measurements.

The variational principle is so suited to do so, because it admits a simple realization of the symmetry principles underlying the fundamental laws of physics, which themselves are a discovery based on observations. It all starts with the spacetime model and its symmetries. Fortunately we can neglect gravity as far as elementary-particle physics is concerned. Thus we can use the Einstein-Minkowski spacetime with its high symmetry. Just as a spacetime it has the proper orthochronous Poincare group as the part of the symmetry group that forms a Lie group of transformations continuously connected with the identity operation. From this already follow all possible kinds of fields by analyzing the unitary ray representations of the corresponding Lie algebra, leading to representations of the central extension of the covering group by exponentiation. In the semidirect product of spatio-temporal translations and Lorentz transformations, you simply have to substitute ##\mathrm{SL}(2,\mathbb{C})## for the proper orthochronous Lorentz group. As it turns out the proper orthochrnous Poincare group has no non-trivial central charges und thus all the ray representations are liftable to proper unitary representations (in contradistinction to the Gailei Lie algebra whose non-trivial central charge is mass, while the proper unitary representations of the classical Gailei group do not lead to physically sensible dynamics, i.e., the notion of "massless particles" doesn't make sense within non-relativistic physics).

The further analysis leads to the conclusion that among the possible field equations only those with positive or vanishing four-momentum square (which defines the mass of the particle related to the corresponding non-interacting field via ##p_{\mu} p^{\mu}=m^2 c^2##). In addition an irreducible representation is characterized by the sign of the eigenvalues of ##p^0##, and the spin (i.e., the representation of SU(2) in the "little group", which describes the transformation of the field under rotations, for spin-##s## fields implying ##(2s+1)## polarization degrees of freedom). Massless fields for ##s=0## and ##s=1/2## are simply described by the limit ##m \rightarrow 0## of the corresponding massive cases, while for ##s \geq 1## you need to represent the null-rotations within the little group trivially, leading to a gauge symmetry, which leaves only 2 out of the ##(2s+1)## polarizations of a massive particle, the states with helicity ##h=\pm s##. This defines the free particles as represented within a irreducible unitary representation of the proper orthochronous Lorentz group.

Then there are also the discrete space-time symmetries, i.e., space-reflections and "time reversal". Together with the assumption that any physical theory should have a Hamiltonian bounded from below, time reversal is necessarily realized by an antiunitary transformation.

As it also turned out the only successful relativistic quantum theories of interacting particles so far are all local quantum field theories, where the Hamiltonian is expressed by quantum fields transforming locally under Poincare transformations (as their classical pendants) and obeys the principle of microcausality (i.e., local observables commute at space-like separations of their space-time arguments). This together with the demand that the Hamiltonian is bounded from below leads to the necessity of antiparticles (with the possibility that particles and antiparticles might be identical for the socalled strictly neutral particles), the spin-statistics theorem (half-integer-spin particles are fermions, integer-spin particles are bosons), and the PCT theorem, according to which the "great reflection" consisting of space reflections (parity, ##\hat{P}##), charge-conjugation (exchanging particles with their antiparticles, ##\hat{C}##), and time-reversal (##\hat{T}##) must necessarily be a symmetry.

Given that mathematical framework, which of course has developed in a fascinating interplay between theory and experiment in the years from about 1928 ("Dirac equation") to the mid 1970ies (completion of the standard model by the discovery of asymptotic freedom of QCD), it is clear how to build the right Lagrangian in the action principle given the empirical input. The result is the standard model of elementary particle physics, which is the most successful theory ever, though it's considered incomplete since we don't know how to finally also incorporate the gravitational interaction within the framework in a satisfactory way and because we also don't know yet how to describe "dark matter" (who knows, whether it's really "particles" in the same way as the known ones).

 

[haushofer]

thanks
is there a way to show that the theories (relativity, qft) must satisfy a variational principle?

No.

It so happens to work. There is no "need", just like there's no "need" for nature to be described by mathematics in the first place.

But who knows, maybe we find a deeper reason for the action principle.

 

[kent davidge]

maybe we find a deeper reason for the action principle

The variational principle can be derived for some classical systems. I mean, it is possible to show that some types of classical systems satisfy it. So I thought that perhaps there were similar derivations for all theories.

 

[kent davidge]

Given that mathematical framework [...] it is clear how to build the right Lagrangian in the action principle given the empirical input

Oh, I see now. Thanks!

 

[haushofer]

The variational principle can be derived for some classical systems. I mean, it is possible to show that some types of classical systems satisfy it. So I thought that perhaps there were similar derivations for all theories.

Do you have some references? I'm not sure I understand your statement and your use of "derive".

 

[Ibix]

The variational principle can be derived for some classical systems. I mean, it is possible to show that some types of classical systems satisfy it. So I thought that perhaps there were similar derivations for all theories.

Is your point that we have other ways (e.g. school-level Newtonian physics) to describe classical systems, but only the variational approach for more advanced physics? I think this is just an accident of history. It certainly doesn't mean that we trust Lagrangian approaches more for classical physics. It just means that we can infer that Lagrangian methods in classical physics match experiment if they reproduce other approaches that we already know match experiment.

 

[PeterDonis]

The variational principle can be derived for some classical systems. I mean, it is possible to show that some types of classical systems satisfy it.

Showing that a system satisfies a variational principle is not the same as "deriving" a variational principle. "Deriving" would mean proving that the variational principle follows as a mathematical theorem from some set of axioms that are considered "more fundamental" than the variational principle. I'm not aware of any physical system for which that is true; variational principles are typically taken as the most fundamental principles involved whenever they are used.

 

[strangerep]

The variational principle can be derived for some classical systems. I mean, it is possible to show that some types of classical systems satisfy it. So I thought that perhaps there were similar derivations for all theories.

If by the word "theory" you mean a set of equations, then what you're looking for is called the "Inverse Problem in the Calculus of Variations", also known as the "Inverse Problem for Lagrangian Mechanics".

As you'll see on that Wiki page, a Lagrangian function can be found for a given set of equations provided certain so-called "Helmholtz conditions" hold.

 

[vanhees71]

Is your point that we have other ways (e.g. school-level Newtonian physics) to describe classical systems, but only the variational approach for more advanced physics? I think this is just an accident of history. It certainly doesn't mean that we trust Lagrangian approaches more for classical physics. It just means that we can infer that Lagrangian methods in classical physics match experiment if they reproduce other approaches that we already know match experiment.

I trust Lagrangian approaches more, because it provides a tool to formulate dynamical laws in accordance with the fundamental symmetry principles. Particularly in relativistic physics the Lagrangian approach is more simple and thus helps to find the correct covariant description.

 

[Ibix]

I trust Lagrangian approaches more, because it provides a tool to formulate dynamical laws in accordance with the fundamental symmetry principles. Particularly in relativistic physics the Lagrangian approach is more simple and thus helps to find the correct covariant description.

Fair enough - but you are essentially talking about quality assurance here (aside: I am amused by the fact I had two typos in "quality assurance", somehow). The point I was trying to make is that a correct non-variational approach (where available) gives identical predictions to a correct variational one. You can use either. But variational approaches generalise better to more complex physics, as you say.

Whatever approach you use, the only way you know it's accurate is to make a prediction and do an experiment.

However, variational methods are often introduced by showing that they reproduce Newton's Newtonian mechanics. You then don't need to do more experiments - the ones done before also count as tests of the variational approach. My suspicion is that the OP thinks we think variational approaches work because they reproduce older mathematics - missing the transitive properties of experimental results in this context.

 

[vanhees71]

Sure. Any approach to derive dynamical laws is as good as any other, and after all everything is subject to experimental test.

 

[kent davidge]

Do you have some references?

yes, I first saw the derivation in Townsend, Mechanics. I can't give you the exact pages, but I am sure it is right in the beggining, as the book is mostly about problems solved with variational principle, so the derivation is certainly in the first pages.

variational principles are typically taken as the most fundamental principles involved whenever they are used

as you said, typically, not always, as I said, a derivation for some systems can be found in Townsend, Mechanics.

 

[PeterDonis]

a derivation for some systems can be found in Townsend, Mechanics.

A derivation from what other principles or axioms?

 

[kent davidge]

A derivation from what other principles or axioms?

D'Alembert's principle. However, the author argues why the principle holds, that is, he does not simply state it and derive the Euler-Lagrange equations from it (which would already be a nice thing).

 

[vanhees71]

That's the usual way: You start with D'Alembert's principle of virtual displacements to derive the EL equations, from which you immediately see that they can be deduced from a variational principle, Hamilton's stationary-action principle. I never understood, why one needs D'Alembert's principle. It's anyway equivalent to Hamilton's action principle, and it can as well be shown that the "naive" Newtonian equations of motion can be derived from it (provided the forces have a potential or extensions of this concept like a four-vector potential for the em. interaction).

 

[kent davidge]

You start with D'Alembert's principle of virtual displacements to derive the EL equations, from which you immediately see that they can be deduced from a variational principle

However that does not work for Relativity or QFT, correct?

 

[vanhees71]

Indeed, I'd say relativistic laws of motion are best derived using the action principle, because it's pretty simple to write down relativistically invariant actions, given the transformation laws for fields, which themselves can be found by analyzing the representations of the Lorentz group.