# What exactly is the difference between the position operator in non-relativistic QM and the Newton-Wigner operators in QFT?

I've read several threads over the past several days talking about how photons don't have wavefunctions in the same way as massive particles do because they don't have non-relativistic limits. If I understood correctly, that's because the usual position operator introduced in introductory QM courses really only applies to non-relativistic theories. The Newton-Wigner operators kept being mentioned as the closest QFT analog to the position operator from non-relativistic QM, so I've been trying to find information on them, but the relevant Wikipedia page is very sparse and vague and everything else I found was very long and technical. All I really want to know is:

1. What's the actual definition of the Newton-Wigner operators?

2. How does it differ from the definition of the position operator from non-relativistic QM?

From what little the Wikipedia article did say, I know the Newton-Wigner operators aren't Lorentz covariant, which, if I understand correctly, means they're reference-framed dependent w.r.t. the Lorentz transform. But is that the only difference between them and analogous operator from non-relativistic QM? If so, then why does position in particular often get singled out as being different in relativistic and non-relativistic QM and QFT, when other properties, such as energy and momentum, are also reference frame dependent?

First of all I suggest to have a look at my recent paper on this and related subjects where a detailed and technical review introduction takes place, for the massive KG particle ["On the Relativistic Spatial Localization for massive real scalar Klein-Gordon quantum particle][1]".

Given an elementary particle (referring to Wigner's definition), the Newton-Wigner observable is the unique projector-valued measure (i.e. triple of commuting self-adjoint operator) labelled by measurable sets in the rest 3-space of a Minkowskian reference frame, that is co-variant with respect to the group of spatial isometries of that Euclidean space.

Actually, this is true by assuming some further technical hypotheses on the regularity of the representation and it has been established by Wightman exploiting Mackey's results on imprimitivity systems.

The above-mentioned group of isometries of the spatial rest space is generated, in the Hilbert space of the particle, by momenta and angular momenta as usual. These observables are uniquely determined by the irreducible unitary representation of Poincaré group which defines the considered elementary particle in Wigner sense.

So, if one assumes that position is described by a (triple of commuting) self adjoint operator(s), there is only one way to define it, under the assumption of natural physical requirements.

Let $$L^2(\mathbb{R}^3, d\mu)$$ be the Hilbert space of a particle, for simplicity Klein-Gordon and massive. Above $$d\mu(p)$$ is the Lorentz invariant measure in momentum space. The Newton-Wigner position operator along the spatial k-th direction has this expression

$$(X^k\psi)(p) = i \left[\left(\frac{\partial}{\partial p^k} - \frac{p^k}{2p^2_0}\right)\psi\right](p)\:.$$ (Above, $$\psi$$ stays in the Schwartz space when viewed on the mass shell. However, there is only one selfadjoint extension of that operator.)
You see that at large mass, i.e., small 3-momenta the added term disappears and the covariant measure tends to the non relativistic one (up to a factor) $$d^3p$$. This proves that, in the non-relativistic limit we find the standard position operator for massive particles. (See also my old answer pointed out by @Gold in the comment under your post, but notice that there I employ the non-covariant measure $$d^3p$$ form scratch: I use a different but equivalent representation of the Hilbert space.)

The presence of spin makes even more complex the expression above.

In position representation (where the space is the joint spectrum of the three position operators), the expression $$X^k$$ is the standard one, but the Lorentz group acts non-locally. When representing in spacetime a completely localized state (a Dirac delta in position representation) one sees a wave spatially concentrated around a point in a region comparable with the Compton length of the particle.

Unfortunately the NW notion of position is plagued by a number of issues accumulated over the years. These are of two types.

1. Without considering the problem of the post-measurment state (after a position measurement). The NW notion of position gives rise to a supreluminal propagation of probability, as established by the Hegerfeldt theorem. In turn, this phenomenon implies a direct violation of (macroscopic) causality.

2. Considering also the post-measurment state. Here things become much more complicated. As the spectrum of the position operators is continuous, the standard L"uders projection postulate is untenable (Ozawa theorem) and one should choose a procedure leading to the post-measurment state in terms of a quantum operation (Kraus operators). Apparently natural choices are ruled out by the so called Malament theorem and more recent improvements which prove, once again, that causality violation takes place also along this route.

On the other hand, the expectation value of the Newton Wigner position operator enjoys quite nice properties. First of all, the "center of mass" of a free wavepacket describes a timelike worldline as expected. The Newton Wigner operators are Poincaré-covariant, but in a precise way as is well know (Wikipedia seems to be wrong on that).

The various issues arising from the analysis of NW notion of position, in my view, are just a symptom of the inadequacy of the description of observables only in terms of self-adjoint operators in QM (especially relativistic) and of the naivety of the projection postulate. In particular, except for a few cases, the projection postulate is physically untenable. The general way to formalize the quantum world is in terms of quantum operations, positive-operator valued measures, and all that.

[1]: https://arxiv.org/abs/2304.02133, Lett. Math. Phys. 113, 66 (2023). DOI:10.1007/s11005-023-01689-5

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