I Who is Ballentine and why is he important in the world of quantum mechanics?

[bhobba]

Why don't you start by asking them back, "What is a field?"

A theorem called the no interaction theorem says that in relativity, particles left to themselves will never interact.

But we know they often do. To get around this, fields are postulated to exist. Wigner showed they all must be tensors. In fact, by starting with a tensor type, it is often possible to write down the equations of the field with just a little bit of other knowledge, e.g.,

https://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

We know from Noether's Theorem they have momentum and energy - properties associated usually with real things. Because of this, we think fields are real but mathematically described by tensors. What they are is anyone's guess.

Regarding Quantum Field Theory, Weinberg often mentioned what he called a folk theorem. There is no rigorous proof, but physicists generally believe it is true. It says any theory that includes Special Relativity and Quantum Mechanics, plus the cluster decomposition property, will look like a Quantum Field theory at large enough distances.

https://www.arxiv-vanity.com/papers/hep-th/9702027/

Particles (very real things) are like knots in these fields. Therefore, we think of them as real and know how to describe them mathematically, but just like classical fields, what they are is anyone's guess.

Thanks
Bill

 

[bhobba]

Is Ballentine really good at explaining this sort of thing? Thanks for all your very helpful replies.

Ballentine, in my opinion, is THE book on quantum mechanics.

But you have to build up to it:


Thanks
Bill

 

[apostolosdt]

l’ll be naively honest: Until I became a member of these Forums, l had never heard of Ballentine, or his RMP article or his QM book for that matter. Of course that has no weight on his work or on other members’ opinion. But it’s kind of strange. One possible explanation might be that I never worked with people who were interested in interpretations of quantum mechanics—I was more of the sort “Shut up and do your Feynman diagrams.”

 

[vanhees71]

A theorem called the no interaction theorem says that in relativity, particles left to themselves will never interact.

But we know they often do. To get around this, fields are postulated to exist. Wigner showed they all must be tensors. In fact, by starting with a tensor type, it is often possible to write down the equations of the field with just a little bit of other knowledge, e.g.,

https://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

We know from Noether's Theorem they have momentum and energy - properties associated usually with real things. Because of this, we think fields are real but mathematically described by tensors. What they are is anyone's guess.

The "real things" are observables, in relativistic QFTs local ones, represented by corresponding local operators, built with help of the fields. In the case of gauge theories, of course only gauge-covariant local field operators can represent such observables.

That's often not clearly mentioned, particularly not in non-relativistic QM in the semiclassical approximation, i.e., with the electrons (or other charged particles) "quantized" and the em. field kept classical. There's a lot of confusion in the textbook literature about this. Even recently there was a paper about the confusion related to the Landau problem (charged particle in a homogeneous magnetic field). All this is, of course, clarified. A good treatment can be found in the textbook by Cohen-Tannoudji (Complement H III in Vol. 1 of Cohen-Tannoudji, Diu, Laloe, Wiley 2020).

Regarding Quantum Field Theory, Weinberg often mentioned what he called a folk theorem. There is no rigorous proof, but physicists generally believe it is true. It says any theory that includes Special Relativity and Quantum Mechanics, plus the cluster decomposition property, will look like a Quantum Field theory at large enough distances.

https://www.arxiv-vanity.com/papers/hep-th/9702027/

Particles (very real things) are like knots in these fields. Therefore, we think of them as real and know how to describe them mathematically, but just like classical fields, what they are is anyone's guess.

"Particles" are asymptotic free Fock states with "particle number" 1. Particles are not an easy concept in QFT! Of course Weinberg is always the right address to clarify conceptional questions (at least concerning QFT ;-)).

 

[bhobba]

l had never heard of Ballentine, or his RMP article or his QM book for that matter.

Ballentine is famous for years ago publishing a paper on the statistical interpretation:

https://www.unicamp.br/~chibeni/textosdidaticos/ballentine-1970.pdf

His book is well respected here because it integrates graduate-level QM with his favoured statistical interpretation.

As a personal comment, I learned QM from Dirac and von Neumann. I got sidetracked into investigating Rigged Hilbert Spaces. Then I read Ballentine, and it was a revelation - all was much clearer. I have since read many other books, some of which are very good, like Weinberg, but others were not to my taste.

You are correct. Ballentine is well thought of by many people here (not all, though).

Thanks
Bill

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