Shape of elementary particles in QFT, etc?

[A. Neumaier]

You missed my point. In a fixed gauge, this is equivalent to measurement of derivative of the potential.

And you missed my point. The potential is not the field in the accepted terminology. But you had stated that in classical field theory one can only measure changes of the field.

 

[A. Neumaier]

Sure, but how do you define "shape"? Note that it must be an intrinsic property, defined in the particle's frame (let's concentrate on massive particles which at least have a position in the literal sense). What you described is not the shape of a particle but the shape (or better said pattern) of probability (distributions) of observables. E.g., I'd not say an electron in the ground state of a hydrogen atom is a sphere with extension of one Bohr radius. When I measure, e.g., it's position I'll find a "pointlike" spot on the detector (thereby kicking it out of the bound state).

It is a matter of terminology, and different people use the terminology differently. You as a nuclear physicists express everything in terms of probabilities because you think in terms of properties obtained by actually doing statistics on many identically prepared particles; But quantum chemists talk about shapes of single atoms and molecules, and the borderline where one switsches from one language to the other is fluent. If you view the shape of a single atom in a high-resolution atom microscope you actually view the charge density which is the ensemble mean of the charge density operator. But one sees it as a definite shape. Now I know that people who talk about the shape of a photon just extrapolate this to the case where things are too small to be measured directly but where onecan still compute the quantities hence talk about them in a meaningful way.

 

[vanhees71]

Well, for atoms and molecules or other "composed objects" "shape" makes some sense. You can define it as some coarse-grained geometrical observable like some averaged one-particle density or a charge or energy density (as you write yourself). Of course, such "shape" definitions make a lot of sense in chemistry.

Even this can be tricky, as the example of the proton shows: Its charge radius depends on how you measure it. There seem to be different radii when using electrons vs. when using muons to determine it. It's not yet understood, how this result comes about.

The shape of a photon is even harder to define, and I don't think that it is very meaningful. Maybe you can define it as the (ensemble mean) of its energy-density, which at least is a well-defined although frame dependent quantity. Note that there's no preferred frame for a photon like for massive particles, where one chooses simply its rest frame. It's also highly state dependent. I guess you can "shape" photons in pretty arbitrary shapes (at least in principle) if you define it in that way. I'm however in doubt, whether this is a very helpful quantity to depict what a photon is beyond my more conservative attempt; it's just describes the detection probability anyway.

 

[Demystifier]

The potential is not the field in the accepted terminology.

Maybe in classical electrodynamics. But in QED it is certainly common to say that $A^{\mu}$ is a vector field with spin-1.

 

[A. Neumaier]

Maybe you can define it as the (ensemble mean) of its energy-density, which at least is a well-defined although frame dependent quantity.

I just asserted that this is the meaning with which the term is used by those who use it, by extrapolation form the interpretation that is valid for larger objects.

In the macroscopic and nonrelativistic limit it becomes the mass density which defines very obviously the shape, and since it can be defined for all objects no matter how large it is a meaningful physical concent, even though one can measure it only for sufficently robust objects such as single atoms on a surface or in a cavity, or mesoscopic or even macroscopic objects. (There is lots of nonmeasurable stuff that has a clear physical meaning, for example the detailed mass distribution in the interior of the sun or in the Andromeda galaxy.)

That shape is frame-dependent is already well-known form classical relativity (length contraction). It has to be so.

 

[A. Neumaier]

Maybe in classical electrodynamics. But in QED it is certainly common to say that $A^{\mu}$ is a vector field with spin-1.

But this is loose, formal but physically meaningless talk, since it is a gauge dependent quantity and gauge fixing is an arbitrary, unphysical procedure. Only gauge invariant quantities have a physical meaning.

 

[secur]

the borderline where one switsches from one language to the other is fluent.

That borderline may be fluid (or 'flexible', 'fuzzy', 'uncertain', 'indeterminate', 'ill-defined', etc), but I doubt it's fluent - in language or anything else. Unless you have a new theory of "conscious borderlines"? Given your belief that consciousness is a meaningless superstition, that would be quite a switsch!

 

[vanhees71]

Maybe in classical electrodynamics. But in QED it is certainly common to say that $A^{\mu}$ is a vector field with spin-1.

It is very important to say that it is a massless vector field, which together with locality implies that it must be a U(1) gauge field. See, e.g., Weinberg, Quantum Theory of Fields, vol. 1.

 

[vanhees71]

I just asserted that this is the meaning with which the term is used by those who use it, by extrapolation form the interpretation that is valid for larger objects.

In the macroscopic and nonrelativistic limit it becomes the mass density which defines very obviously the shape, and since it can be defined for all objects no matter how large it is a meaningful physical concent, even though one can measure it only for sufficently robust objects such as single atoms on a surface or in a cavity, or mesoscopic or even macroscopic objects. (There is lots of nonmeasurable stuff that has a clear physical meaning, for example the detailed mass distribution in the interior of the sun or in the Andromeda galaxy.)

True, but these quantities are measurable in principle. It's only not within our technical possibilities to do so.

That shape is frame-dependent is already well-known form classical relativity (length contraction). It has to be so.

This is a misconception you find astonishingly often. Of course, for a system with non-zero mass there's a preferred frame of reference, the center-momentum frame, and this frame is used to define intrinsic quantities of this system. If you want you can call the energy-density distribution in this frame the "shape" of the object, although I've never seen this anywhere in the literature. You can define this quantity in a frame-independent way as a scalar density. For a fluid it's given by $$\epsilon=u_{\mu} u_{\nu} \Theta^{\mu \nu},$$ where $u^{\mu}$ is the four-velocity field of the fluid (normalized such that $u_{\mu} u^{\mu}=1$) and $\Theta^{\mu \nu}$ is the energy-momentum tensor of the fluid.

 

[A. Neumaier]

If you want you can call the energy-density distribution in this frame the "shape" of the object,

This doesn't work for a photon, whereas the energy-density distribution in the lab frame always exists. This is the one of interest to the experimenter. http://arxiv.org/abs/1605.00023, http://arxiv.org/abs/1601.07142, http://arxiv.org/abs/1512.08213, http://link.aps.org/abs/10.1103/PhysRevAccelBeams.19.021304 are some recent references to photon shape.

 

[vanhees71]

Yes, that's because a single photon is a massless state, and there is no restframe for it. That's why, I don't like to call that quantity "shape". It's just proportional to the probability distribution for detecting a photon. From reading the abstracts of the cited papers, I think that's also the common understanding among quantum opticians.

 

[Arjun Wasan]

In string theory classic point like particles are in 0 dimensions.