A Quantum field theory, spacetime, and coordinates

[Demystifier]

You may do all these transformations, but it won't lead to an equivalent formulation, because the equivalence proof requires that $S(J(x))$ be pulled through the derivative, but this does not work in general coordinates, hence the two formulations will be non-equivalent.

And I fully agree with you that it doesn't work in general coordinates. That's why I treat the hydrogen atom in spherical coordinates differently, by using the "dirty smart trick" I described before. And in genuinely curved spacetime, even that kind of trick will not work. I agree with all that. In general, I cannot avoid the explicit use of tetrads and covariant derivatives, which makes the computations rather complicated. But in some cases I can make things simpler, so this is what I do in my paper - I make things simpler in cases in which it is possible.

 

[vanhees71]

Or you can call it an element of a left module of the Dirac algebra, the latter being a Clifford algebra over the complex field using a (1,3) metric, which is isomorphic to a matrix algebra M₄(ℂ).
The left module structure is not reducible under the whole Lorentz group, which is, I venture to say, why mathematical physics more uses (or, rather, seems in my experience and in my preference more to use) Dirac spinors than Weyl spinors.
The Dirac spinor also supports a representation of the connected component of the conformal group in 1+3- or 3+1-dimensions, because (going a little too fast here, and it's surely not the only route) the Clifford algebras over ℝ using either a (2,3) or a (4,1) metric are both isomorphic to a matrix algebra M₄(ℂ). [I'm slightly correcting vanhees71 mentioning the Poincaré group instead of the Lorentz group, which I think he intended here; time now for someone to slightly correct me, I expect.]

That's an interesting different point of view, but I really mean the Poincare group, because it's not the Lorentz group alone that's important to construct the unitary representations of the Standard Model but indeed the Poincare group. Of course, all arguments about the Lorentz group also apply, because it's a subgroup of the Poincare group.

Also it is important to stress that in Nature the symmetry group is not the full Poincare group but only the proper orthchronous subgroup, because today we know by independent (!) experiments that the weak interaction does not obey all the various discrete symmetries, P, C, T, and CP (while CPT seems to be intact as predicted by local relativistic QFT).

 

[vanhees71]

No it isn't. In his terminology $A^\mu(x)$ is a coordinate vector. It is $^*\!A^\alpha(x)$ that is a coordinate scalar in his terminology.

I'm not familar with this terminology. In the way I know it, $A^{\mu}(x)$ are components of a four-vector field and transform as such:
$$A^{\prime \mu}(x')={\Lambda^{\mu}}_{\nu} A^{\nu}(\hat{\Lambda}^{-1} x'),$$
where $x$ and $x'=\Lambda x$ are the column vectors built by the vector components of the space-time vector, $x^{\mu}$ and $x^{\prime \mu}$ respectively; $\hat{\Lambda}=({\Lambda^{\mu}}_{\nu})$ is an $\mathbb{R}^{4 \times 4}$-Lorentz-transformation matrix.

Vectors and vector fields themselves are, of course, invariant under any kind of basis transformation by definition!

 

[rubi]

And I fully agree with you that it doesn't work in general coordinates. That's why I treat the hydrogen atom in spherical coordinates differently, by using the "dirty smart trick" I described before. And in genuinely curved spacetime, even that kind of trick will not work. I agree with all that. In general, I cannot avoid the explicit use of tetrads, which makes the computations rather complicated. But in some cases I can make things simpler, so this is what I do in my paper - I make things simpler in cases in which it is possible.

It's nice that we agree. Using the chain rule instead of transformation laws should still work in curved spacetime, although I don't know what the generalization of your equation to curved spacetime would look like. My issue was just that your equation spoils the covariance (under general transformations) of the Dirac equation, but of course you can still perform calculations with it if you keep track of all the transformations that are involved. The nice thing about covariant equations is that you can just take the equation in cartesian coordinates and perform the replacement $\partial_\mu\rightarrow\nabla_\mu$ to get the general version. When physicists spot an equation that involves indices, they often take it for granted that it's covariant, so I struggle a bit to introduce non-covariant equations. I don't think this is beneficial for students, because they don't know what a non-coordinate spin frame is and that's quite advanced stuff that can't be explained easily, but it would be really important to teach them, so they understand why certain things work differently with your equation. However, it's of course nice if there are situations, where your equation makes things simpler.

 

[vanhees71]

One must of course also be careful with the replacement rule mentioned. Sometimes you cannot simply make $\partial_{\mu} \rightarrow \nabla_{\mu}$, but you have to carefully use the proper means to determine the correct differential operator for what you want to model. A pretty nice example is the quantization of the spinning top in Kleinert's path-integral book. There the correct way is of course to use the Hamiltonian, properly formulated with angular-momentum operators, while a naive "canonical quantization prescription" leads to wrong equations.

As far as I can see in Bjorken Drell vol. 1 they simply got the correct Dirac equation for the hydrogen atom in terms of spherical coordinates, because they used the proper definitions of the operators involved (energy, total angular momentum, orbital angular momentum, and spin).

 

[rubi]

One must of course also be careful with the replacement rule mentioned. Sometimes you cannot simply make $\partial_{\mu} \rightarrow \nabla_{\mu}$, but you have to carefully use the proper means to determine the correct differential operator for what you want to model. A pretty nice example is the quantization of the spinning top in Kleinert's path-integral book. There the correct way is of course to use the Hamiltonian, properly formulated with angular-momentum operators, while a naive "canonical quantization prescription" leads to wrong equations.

Well, okay, it's still possible that $\partial_\mu$ is supposed to be a Lie derivative or something like that instead of a covariant derivative, which you can't see easily from the cartesian coordinates formulation. My point is probably that covariant equations have a fully coordinate-independent formulation, i.e. you can write them in a way that doesn't involve a choice of coordinates in the first place and thus emphasizes the physical content of the equation rather than effects that really may be artifacts of the coordinate choice. Quantization is yet a different problem though. Even for a coordinate-free equation, there may be many inequivalent ways to quantize it.

As far as I can see in Bjorken Drell vol. 1 they simply got the correct Dirac equation for the hydrogen atom in terms of spherical coordinates, because they used the proper definitions of the operators involved (energy, total angular momentum, orbital angular momentum, and spin).

Well, the calculation in Bjorken Drell is very brief anyway. They essentially skipped the angular part and just wrote down the result, which is quite sad, because most of the connection coefficients are in the angular part, so there was no real part in the book where I could point to in order to illustrate the presence of the coefficients.

 

[vanhees71]

It's not only in quantizing classical models. Of course, I agree with you, that one should formulate everything in a covariant way in terms of coordinate-independent equations.

I don't remember where I've found the example (maybe in Misner Thorne, Wheeler, Gravitation?) that it's not clear how in a naive way you make the Maxwell Equations generally covariant by naively using the equations within SR in Cartesian coordinates and substituting the partial derivatives by covariant derivatives. It's of course unique when using the action principle, which has the great advantage that everything is written down in a manifestly covariant way as a scalar Lagrangian.

 

[rubi]

I don't remember where I've found the example (maybe in Misner Thorne, Wheeler, Gravitation?) that it's not clear how in a naive way you make the Maxwell Equations generally covariant by naively using the equations within SR in Cartesian coordinates and substituting the partial derivatives by covariant derivatives. It's of course unique when using the action principle, which has the great advantage that everything is written down in a manifestly covariant way as a scalar Lagrangian.

What's true is that there are in general many ways to generalize Maxwell's equation to curved spacetime, because you could for instance add a coupling to the Ricci scalar or other curvature invariants (any many more things) that vanish in the flat limit. But if you stay in flat spacetime, an equation will always have a unique coordinate-free formulation (of course up to equivalence, e.g. you can multiply both sides by 5). But if the equation is not covariant, the coordinate-free formulation will look require the introduction of unnatural background structure.

 

[Peter Morgan]

That's an interesting different point of view, but I really mean the Poincare group, because it's not the Lorentz group alone that's important to construct the unitary representations of the Standard Model but indeed the Poincare group.

Right. I wondered whether that was what you really intended. I was specifically channeling the Wightman axioms, per Haag, Local Quantum Physics, p. 57:

.
So the components of the field in that context are subject to transformations taken from a representation of the Lorentz group. The standard model of particle physics is the same, I think. I'd be interested in your choice of an example of a theory in which components of the field transform according to a representation of the Poincaré group?

 

[George Jones]

Or you can call it an element of a left module of the Dirac algebra, the latter being a Clifford algebra over the complex field using a (1,3) metric, which is isomorphic to a matrix algebra M₄(ℂ).

Representations of the Poincare group often are constructed by using infinite-dimensional function spaces that map into the above space, i.e., infinite-dimensional representations of the Lorentz group are used to construct infinite-dimensional representations of the Poincare group, where the action of the Poincare group is on the function space, and the action of the Lorentz group is on the codomain of the functions in the function space. Wave equations (e.g., the Dirac equations) are used to project onto subspaces of the function spaces that are relevant to a particular representation of the Poincare group.

 

[PeterDonis]

I don't think there are any relativists who use a different definition.

Just to note, this thread is in the quantum forum, not the relativity forum, so the question is really what definition quantum field theorists use.