A The g_ij as potentials for the gravitational field

[haushofer]

If he's agreeing that, in standard Newtonian mechanics, Newton's laws, including the law of gravity, are only invariant under Galilean transformations, then I think there is no dispute. I was not expressing any opinion about what was in the thesis he referenced except that it isn't standard Newtonian mechanics.

Indeed. I fully agree with your earlier statements and I was expressing myself not clearly. I'm also aware that the view I represent here on "Newtonian physics" is not standard.

Having said that, I'm still curious how you (or Vanhees or someone else) see the transformation of the wave function under Galilei boosts of the Schrodinger equation as transforming with an extra phase factor. It probably has to do something with projective representations of the Bargmann algebra. But somehow, that example always reminded me a bit of this discussion of the Newton potential.

 

[vanhees71]

Sure, you can write all equations in a general covariant way. E.g., for Newtonian mechanics and many other theories you can just use the action principle, leading to a formalism that is forminvariant under arbitrary diffeomorphisms in configuration space (in all kinds of generalized variables the EoM are given by the Euler-Lagrange equation) or general canonical transformations in the Hamiltonian formulation.

Still, the symmetry group of standard Newtonian mechanics is the Galilei group, because for a symmetry all the transformations making up (a representation of) the symmetry group the 1st variation of the action must be invariant. You know better than me that you can extend Galilean mechanics to a kind of Newton Cartan theory:

https://en.wikipedia.org/wiki/Newton–Cartan_theory

The realization of the Galilei group in non-relativistic QT is subtle. The proper unitary transformations of the classical (10-dim) Galilei group does not lead to a useful quantum theory, at least nothing that would apply to the real world (Inönü and Wigner). Rather you need the ray representation of a non-trivial central extension with the mass as an additional non-trivial "central charge" of the Lie algebra. In any case there's no reason not to also extend the ray representations of the symmetry group to its covering group. So in non-relativistic QT the classical Galilei group is realized as a unitary representation of the central extension of the covering group (the latter just implies that instead of the rotation subgroup SO(3) you use its covering group SU(2), allowing for half-integer spin, which obviously is needed to describe everything we call "matter", i.e., the leptons and baryons which are all fermions with half-integer spin). The central extension with the mass as an additional independent observable leads to a superselection rule forbidding to superimpose state vectors from representations of different mass eigenvalues, and this establishes an 11th independent conservation law from the Galilei symmetry, the conservation of mass.

E. In¨on¨u and E. P. Wigner, Representations of the Galilei group, Il Nuovo Cimento 9, 705 (1952),
https://doi.org/10.1007/BF02782239.

 

[PeterDonis]

one can easily deduce that under a transformation

No, you can't deduce this. And you admit that you're not:

This transformation of $\partial^i \phi$ is not "induced by the coordinate transformation". But there is a formalism in which we can interpret this kind of transformations: we call it the formalism of pseudosymmetries, which has its origin in sigma-models. The transformation of $x^i(t)$ does not induce the transformation on $\partial^i \phi$; instead the transformation of $\partial^i \phi$ can be regarded as a redefinition of $\partial^i \phi$. This "redefinition" (which for PeterDonis was the "magic"-part) is still a symmetry of the action, but because this redefinition is not induced by the transformation of $x^i(t)$, the corresponding symmetry does not have an accompanying Noether charge. That's why we call it a "pseudosymmetry". It's the price to pay if you want to connect the Newtonian limit of GR to the usual Newton theory on the level of symmetries.

Personally, I'm not familiar with this notion of "pseudosymmetries" or with sigma-models, so I can't comment on them. But you admit that whatever these notions involve, it is not "deducing" anything; it's just declaring by fiat that $\partial^i \phi$ works the way you want it to. That's not deducing. It's assuming.

 

[haushofer]

No, you can't deduce this. And you admit that you're not:

Of course you can. We're doing the limit of GR, and you know how the Christoffel connection transforms. That's the whole point: in good old Newtonian gravity, $\partial^i \phi$ is "just a gradient of a scalar"; in GR it turns out to be $\Gamma^i_{00}$. And we know how $\Gamma^i_{00}$ transforms under a gct, so we also know how it transforms under the proposed accelerations (copy from Landau&Lifshitz):

Naming i=i, k=l=0 gives you the transformation of $\Gamma^i_{00}$; the $\ddot{\xi}^i$ which pops up is just the inhomogenous term of the transformation. On top of that, $\Gamma^i_{00}$ transforms as a vector under constant rotations, it gives you the Coriolis and centrifugal force under rotations which depend on time, it transforms as a scalar under Galilei boosts, etc.

But if you don't believe me, you can check the textbook Newtonian limit of GR yourself and check which coordinate transformations you're left with after taking the limit. The covariance under gct's is broken down to covariance under the Galilei-group plus accelerations.

I admit that in the good-old Newtonian theory, we can't deduce this transformation of $\Gamma^i_{00}$ from its tensorial properties.

Personally, I'm not familiar with this notion of "pseudosymmetries" or with sigma-models, so I can't comment on them. But you admit that whatever these notions involve, it is not "deducing" anything; it's just declaring by fiat that $\partial^i \phi$ works the way you want it to. That's not deducing. It's assuming.

Well, I'm OK with that naming. My fiat is the correspondence principle. It depends on how comfortable you are with bending the rules.

 

[haushofer]

Sure, you can write all equations in a general covariant way. E.g., for Newtonian mechanics and many other theories you can just use the action principle, leading to a formalism that is forminvariant under arbitrary diffeomorphisms in configuration space (in all kinds of generalized variables the EoM are given by the Euler-Lagrange equation) or general canonical transformations in the Hamiltonian formulation.

Still, the symmetry group of standard Newtonian mechanics is the Galilei group, because for a symmetry all the transformations making up (a representation of) the symmetry group the 1st variation of the action must be invariant. You know better than me that you can extend Galilean mechanics to a kind of Newton Cartan theory:

https://en.wikipedia.org/wiki/Newton–Cartan_theory

The realization of the Galilei group in non-relativistic QT is subtle. The proper unitary transformations of the classical (10-dim) Galilei group does not lead to a useful quantum theory, at least nothing that would apply to the real world (Inönü and Wigner). Rather you need the ray representation of a non-trivial central extension with the mass as an additional non-trivial "central charge" of the Lie algebra. In any case there's no reason not to also extend the ray representations of the symmetry group to its covering group. So in non-relativistic QT the classical Galilei group is realized as a unitary representation of the central extension of the covering group (the latter just implies that instead of the rotation subgroup SO(3) you use its covering group SU(2), allowing for half-integer spin, which obviously is needed to describe everything we call "matter", i.e., the leptons and baryons which are all fermions with half-integer spin). The central extension with the mass as an additional independent observable leads to a superselection rule forbidding to superimpose state vectors from representations of different mass eigenvalues, and this establishes an 11th independent conservation law from the Galilei symmetry, the conservation of mass.

E. In¨on¨u and E. P. Wigner, Representations of the Galilei group, Il Nuovo Cimento 9, 705 (1952),
https://doi.org/10.1007/BF02782239.

Ok. But how does this relate exactly to the fact that the Schrodinger equation is only covariant with respect to boosts if the wave function transforms under it with an extra phase factor? Is that somehow a hint at the level of equations of motion of the group-theoretical result from the Inönü and Wigner paper?

 

[vanhees71]

It follows from the representation of the central extension of the covering group of the classical Galilei group how the wave function transforms. I have worked out once the representation theory for the Galilei group as lecture notes for a QM2 lecture. It's, however, in German (the quantum theoretical part starts with Sect. 2.5):

https://itp.uni-frankfurt.de/~hees/publ/hqm.pdf

 

[haushofer]

It follows from the representation of the central extension of the covering group of the classical Galilei group how the wave function transforms. I have worked out once the representation theory for the Galilei group as lecture notes for a QM2 lecture. It's, however, in German (the quantum theoretical part starts with Sect. 2.5):

https://itp.uni-frankfurt.de/~hees/publ/hqm.pdf

Kein problem, toll ja! Vielen dank! Ich werde es bald mal ansehen! :P

(That's more or less all the German that's left from attending 6 years German as a high-school subject)

Can you point to the page where you transform the Schrodinger equation under boosts and derive the transformation of the wave function?

 

[vanhees71]

This is on p. 87-88.

 

[PeterDonis]

Of course you can.

We've been round and round about this before, and you admit your viewpoint is not mainstream, and it's off topic for this thread anyway, and the OP is long gone.

I think this thread can be closed.