Obtaining NRQM from QFT: Issues, Folklores and Facts by Padmanabhan

[gentzen]

Padmanabhan's paper "Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores and Facts" (https://arxiv.org/abs/1712.06605) is long (58 pages) and substantial, therefore it deserves its own thread.

Here is a recent discussion about Padmanabhan's paper in an unrelated thread:

Most think that QFT in the non-relativistic limit reduces to ordinary QM. A careful analysis in the paper I posted shows the limit is not ordinary QM.

The non-relativistic limit of relativistic QFT is non-relativistic QFT. Non-relativistic QFT, also known as "second quantization", is widely used in condensed matter physics. In general, states in non-relativistic QFT do not have a definite number of particles. However, when non-relativistic QFT is applied to states with a definite number of particles, the resulting theory is equivalent to non-relativistic QM.

The Padmanabhan's point is that the NR limit of (relativistic) QFT is not merely NRQM of particles, but NRQM of particles and antiparticles. While this is correct, I find it a bit trivial. For instance, it means that the NR limit of QFT based on the Dirac equation is not merely NRQM of electrons, but NRQM of electrons and positrons. True, but so what? It doesn't make the usual NRQM of electrons wrong, it only means that a similar theory can also be applied to positrons. As long as we study processes in which the number of electrons and positrons does not change, there is no much difference between NRQM and NR limit of relativistic QFT. Of course, when we study processes in which the number of particles and/or antiparticles changes, then we must take into account the full relativistic QFT, nobody denies that.

So loosely speaking, it is still true that NR limit of relativistic QFT is NRQM, with the only caveat that the latter describes particles and antiparticles.

He also points out that you get two Schrodinger's equations (one for the particle, and another for the antiparticle) in the Heisenberg picture, but not operating on the quantum state, but on the quantum field.

 

[Peter Morgan]

A quick note that the published version in Eur Phys J C 2018 is OA, https://doi.org/10.1140/epjc/s10052-018-6039-y

 

[pines-demon]

Is there a way to obtain RQM from QFT? If yes, then you can get NRQM from RQM.

 

[Peter Morgan]

There is no concept of a particle in axiomatic forms of QFT. We can add Wigner's definition of particles in terms of unitary irreducible representations of the Poincaré group, but that is far from the definition of a 'system' in von Neumann's axioms for NRQM, which are still closely followed in most modern presentations of QM. Padmanabhan, in his §8.1, puts that as (ii) in "note that: (i) QFT is more fundamental than NRQM, and (ii) we do not have a sensible notion of the single-particle wave function in RQM."
The more recent tradition of Algebraic QM is significantly closer to axiomatic QFT traditions insofar as we do not have to introduce an idea of particles and their properties in the axioms. For a textbook presentation, see, for example, François David's The Formalisms of Quantum Mechanics (arXiv preprint, derived from a course for undergraduates). In its rawest form, Algebraic QM is not more than the mathematics of generalized probability theory, which comes close to what Padmanabhan suggests is needed at the beginning of his §8.1, "One main conclusion – which we have reached from several different perspectives – is that, to make a seamless transition from QFT to NRQM, you need to describe NRQM in a language which is closer to that of QFT and not the other way around."
This slide, from a recent presentation of my research on YouTube (which is very long as a whole, at 3h26m20s), suggests that we can come close to unifying algebraic QM and QFT with this kind of approach,

In computing terms, we would call the description of a measurement that is embedded in the smearing of a measurement operator-valued distribution the metadata for the measurement. In QM, that metadata is given as something like "We measured this property of a system", which is OK when it's clear enough which system or particle is referred to, but falls apart when it is not clear enough, so that QFT relies on space-time coordinates as a way to share metadata about a measurement that is good enough for other physicists to reproduce the measurement in a similar experimental context.
I've always liked Padmanabhan's article insofar as he gives us a very helpful analysis, but I think he doesn't connect as many dots as can be connected. Before I discuss QFT in that talk, I suggest taking measurements to be directly connected to datasets, not to a system's properties, by equating the set of outcomes in a dataset to the spectrum of a measurement operator. This choice applies equally well to QFT and allows us to remove systems from the axioms of an algebraic QM: that is very deep in the overall structure of von Neumann's form of QM, so it of course results in a large network of new ideas that includes ways to rethink both the measurement problem and the renormalization 'problem'.

 

[Demystifier]

Is there a way to obtain RQM from QFT? If yes, then you can get NRQM from RQM.

It depends on what you mean by RQM. If by RQM you mean Bjorken & Drell 1, then yes, you can obtain it from Bjorken & Drell 2. But in Bjorken & Drell 1, there is no probability density of particle positions. So how do you obtain probability density of particle positions in NRQM?

My answer is the following. To define probability density of particle positions in RQM, or equivalently, the position operator in RQM, you must give up the requirement that all observables should be Lorentz covariant. Personally I think it's consistent, but that's not something you will find in Bjorken & Drell 1 or most other books on RQM.

 

[selfsimilar]

It depends on what you mean by RQM. If by RQM you mean Bjorken & Drell 1, then yes, you can obtain it from Bjorken & Drell 2. But in Bjorken & Drell 1, there is no probability density of particle positions. So how do you obtain probability density of particle positions in NRQM?

There has been some attempts but I don't know how the mainstream looks at it
you can also see all the references, there are many others that I have seen.

Relativistic particle in a box: Klein-Gordon vs Dirac Equations
https://arxiv.org/pdf/1711.06313

also
https://www.sciencedirect.com/science/article/abs/pii/S0375960105014726
https://arxiv.org/pdf/quant-ph/0701208v2

 

[bhobba]

There is no concept of a particle in axiomatic forms of QFT.

I am reading the paper's author's textbook on QFT. In it, he proves that if you start with the concept of a point particle and use RQM, you are inevitably led to the need to use a field as its position becomes smeared.

Thanks
Bill

 

[bhobba]

Ihere is no probability density of particle positions.

Exactly. See the above post.

Thanks
Bill

 

[Demystifier]

I am reading the paper's author's textbook on QFT. In it, he proves that if you start with the concept of a point particle and use RQM, you are inevitably led to the need to use a field as its position becomes smeared.

Yes, but the field cannot be something that removes all the conceptual problems of NRQM.

First, in NRQM the position is an observable, while in QFT the fermionic field cannot be an observable, because it's not a self-adjoint operator. One can argue that an observable does not need to be a fundamental entity, but can be something more complicated constructed from the fields. I'm fine with that, but then one can construct a position operator in relativistic QFT. Essentially, it's the Newton-Wigner operator expressed in terms of field operators. Sure, such a position operator is not Lorentz covariant, but since it is not a fundamental quantity, there is no need for it to be Lorentz covariant. Operationally, a measuring apparatus defines a preferred Lorentz frame in which the apparatus is at rest, so such an observable can be measured.

Second, there is the measurement problem in quantum theory. Saying that the fundamental entities are fields rather than particles does not make the problem go away. In fact, in some approaches to the problem, it makes the problem even a bit harder. Namely, if one argues that the problem goes away if one clearly identifies the objects which are "real" irrespective of measurements, then what these "real" objects are in the case of QFT? Perhaps the values of fundamental fields at all spacetime points? But what kind of "values" are they? Are they operator valued or c-number valued? If they are operator valued, then how does it solve the measurement problem, given that we actually measure values expressed as c-numbers? If they are c-number valued, then how does it work for fermionic fields?

 

[gentzen]

Yes, but the field cannot be something that removes all the conceptual problems of NRQM.

Padmanabhan doesn't claim this. Art Hobson is the one who believes this.

I guess Padmanabhan's textbook on QFT and his long paper (58 pages) are substantial and essentially "correct".

First, in NRQM the position is an observable, while in QFT the fermionic field cannot be an observable, because it's not a self-adjoint operator.

Yeah, it is an interesting question whether forming self-adjoint operators is still crucial in QFT. You could also just ensure that you get real numbers somehow, for example by taking squared absolute values, or by combining state and operators suitably. (Even so I could cite examples, I don't care about details here, and don't really understand what is OK and what is not OK. Maybe all cases which are OK can implicitly be reformulated to use self-adjoint operators instead.)
However, one important difference between QFT and NRQM is that QFT uses (anti-)commuting operators for modeling independence, while NRQM uses tensor products.

Second, there is the measurement problem in quantum theory. Saying that the fundamental entities are fields rather than particles does not make the problem go away.

Yeah, Art Hobson believes this, but bhobba already started to distance himself a bit from Art Hobson in this respect.