How can point particles be Lorentz Contracted?

Question

We know that single electrons undergo Lorentz contraction as their fields change in different frames as if a spherical distribution of charge was contracted to be an ellipsoid. However, since electrons are point particles how can they be contracted? They wouldn't have any volume inside themselves to contract into. Is it perhaps the electrons wavefunction that contracts?

Answer 1

The electric field of the electron changes because it is moving

The difference in electric field between a stationary electron and a moving one has nothing to do with the length contraction of the electron. It's simply that Coulomb's Law holds only for a stationary charge; for a moving one you need to use the full Jefimenko Equations (check the Heaviside-Feynman Formula section for the point particle version). Plugging a moving point particle into those will give you the Lorentz-contracted electric field.

We know that single electrons undergo Lorentz contraction as their fields change in different frames as if a spherical distribution of charge was contracted to be an ellipsoid.

This is incorrect: an ellipsoid has almost the same charge distribution as a sphere. Steve is correct that technically a point charge can't really exist, but it doesn't matter if you model the electron as a point or a small ball. In the latter case, the length contraction of the ball will add a tiny correction to the electric field, but it's that: a tiny correction. The real difference still comes from the ball's velocity, not it's change in shape.

Answer 2

Points aren't contracted, only lengths are contracted. Point electron is point in all frames. But its EM field is not a point; it is present in all space, at points which have some distance between them. Thus it can be different in different frames.

Answer 3

The concept of the "point" is a mathematical fiction.

In the physical world, all things have some kind of volume, region, or extent, in both space and time.

So aside from the question about the nature of length contraction, the question simply proceeds from a mistake that "point particles" exist in the relevant sense.

Many physical things can be analysed and quantified adequately as points in cases where their volume would have no bearing on the calculation (given the accuracy requirements of a particular application), but once you're reasoning conceptually about the physics, you need to shift to recognising that everything has a non-zero volume, and a "point" is really associated with some volume which has a non-zero extent and a non-zero uncertainty about its boundaries.

I dare say this is one of the characteristic differences between how engineers and mathematicians tend to conceive of things differently, as engineers tend to be confronted with physical reality and human limitations often, and intrinsically think in terms of scales and tolerances and so on, whereas mathematicians can acquire and sustain conceptualisations that only need to make internal logical sense in their minds (and that system of sense doesn't have to closely correspond with any physical reality).

Answer 4

"Length contraction" means that an object of length $L$ in its own frame has length $\gamma L$ in another frame, where $\gamma$ is a constant depending on the relative velocity between the two frames (in the direction in which length is being measured).

When $L=0$, $\gamma L=0$.

Answer 5

How can point particles be Lorentz Contracted?

That can't happen.

We know that single electrons undergo Lorentz contraction as their fields change in different frames

The field can appear different in different frames, but the electron itself is still described as a point particle.

...However, since electrons are point particles how can they be contracted?

They can't.

There is a difference between the particle and the field it produces. A classical non-relativistic electron at a point $\vec a(t)$ produces a field: $$ \vec E(\vec r, t) = -|e|\frac{\vec r - \vec a(t)}{|\vec r - \vec a(t)|^3}\;. $$

The field is a vector function with values at all points in space $\vec r$ and time $t$. The relativistic generalization of this field $\vec E$ transforms under Lorentz transformations in well known ways. But, the particle itself is still located at a single point $\vec a$.

They wouldn't have any volume inside themselves to contract into.

Right.

Is it perhaps the electrons wavefunction that contracts?

No. I fear you are going to confuse yourself further if you start trying to bring in quantum mechanical aspects such as an "electrons wavefunction." The wave function is probabilistic, and does not describe the behavior of any specific single electron--just like how the electron double slit experiment can not produce a diffraction pattern with a single electron measurement, but requires many many repeated experiments with identically prepared systems.

Answer 6

The other answers present interesting conceptual discussions. Here is another approach to the problem, from the viewpoint of the Dirac distribution that describes the charge density of a point particle. In this discussion, I completely ignore the electromagnetic field, since the same argumentat below can be applied to other observables such as the mass density.

We first recall the coordinate transformation which describes a Lorentz boost, \begin{equation} x^\prime = \gamma (x -vt). \end{equation}

The charge distribution in the primed coordinates is thus \begin{equation} \delta(x^\prime) = \delta(\gamma (x - vt)) \end{equation}

Now we recall that the Dirac delta is only well defined as a distribution. Therefore, we should consider it together with a differential.

We note that the differential in the primed coordinates is stretched, \begin{equation} dx^\prime= \gamma dx. \end{equation} This is the Lorentz contraction, which is a property of the coordinates, and thus is always present.

We next recall a basic property of the Dirac delta distribution, \begin{equation} \delta(\lambda x - \lambda x_0) = \frac{1}{\lambda} \delta(x - x_0) \end{equation}

Putting all together, we get \begin{equation} \delta(x^\prime) dx^\prime = \delta(\gamma x - \gamma vt) \gamma dx = \delta(x - vt)dx, \end{equation} which integrated over the whole domain of $x$ still yields $Q = \int \delta(x-vt) dx=1$. This means that the total charge content of the point particle is still the same in every inertial reference frame.

The point particle is contracted only in the sense that a differential unit can be contracted or stretched by performing a change of variables.

Answer 7

In Classical Electrodynamics it's possible to consider macroscopic distribution and describe them in different reference of frame, this means that the values of the Electric and Magnetic fields and the charge density depends on the frame of reference. Saying "Lorentz contracting particles" doesn't have any sense because what you are really boosting are the fields and the volume in which the charge is distributed, but not the electron it self.

At the same time, in QFT, which is a different framework, you can always act with a Lorentz transformation on a field, a non observable object whose role is to generate a bunch of particles (in this case electrons or positrons) in some point of the space-time trough the creation and distruction operators. In QFT as in classical field theory, point are just labels and doesn't matter which one you use to describe the fields: this means that all the laws are equally formulated in any reference of frame,the only things that can change when you act with a generic Lorentz transformation on a field are the components of the momentum of the particle, the helicity or even the charge and spin if you act with charge conjucation and time reversal.

The value of these quantities depend on the observer in the same sense as the electric and magnetic field do in classical electrodynamics. In this sense you are describing the same object, the electron, from different frames of reference. This is one of the feature of QED.

Answer 8

The theory of point particles in classical electromagnetism is only an approximation. Maxwell's equations provide equations of motion for a field given the motion of charges and the initial conditions of the field. Then Lorentz force law gives the equation of motion for a charge in a given electromagnetic field. But the Lorentz force law on a point ignores the effect of the field of that point charge on the charge itself. The field tends to infinity as you get closer to the point charge and as a result there is no consistent theory of the self force of a classical point charged particle. This leads to problems with the consistency of the resulting theory, see Part One of "Inconsistency, Asymmetry, and Non-Locality: A philosophical investigation of classical electrodynamics" by Mathias Frisch. As such the classical theory of point charges is just an approximation and can't be treated consistently as a description of how the world works.

In quantum electrodynamics this self force problem doesn't occur:

https://arxiv.org/abs/2206.09472

Particles in quantum field theory are just an approximation for describing excitations of quantum fields in suitable limits in which an excitation is approximately localised. For example, if you try to probe a particle on a level smaller than the Compton wavelength additional particles are created and this vitiates the localisation of the particle. See Section 6.5 of "The conceptual framework of quantum field theory" by Anthony Duncan and Section 5 of this paper

https://arxiv.org/abs/quant-ph/0112148

So particles in QFT aren't point particles.

In the point particle approximation where you ignore the self force you can transform the field according to the standard equations for Lorentz transformations of electromagnetic fields that can be found in many textbooks:

https://www.feynmanlectures.caltech.edu/II_26.html