# What is the problem with point particles?

1. Aug 10, 2015

### nnerik

When I read about string theory, I sometimes get a feeling that the pointlike nature of particles in QFT is a problem that needs to be fixed, but I have never seen an explanation of the problem. Is this correct, and if so, why?

Point in case (from https://en.wikipedia.org/wiki/String_theory):

In physics, string theory is a theoretical framework in which thepoint-like particles of particle physics are replaced by one-dimensional objects called strings.​

And just to be sure, what does it mean that elementary particles are point-like in the first place? Does it mean anything besides that the particle has no internal structure? Because clearly the wave function is not point-like (although the position operator's eigen states are).

2. Aug 10, 2015

### Demystifier

As you said, the position eigenstates of particles are point-like. On the other hand, string theory does not have position eigenstates. It has string-configuration eigenstates, which are string-like rather than point-like.

3. Aug 10, 2015

### Demystifier

The problem is often explained, but under a different name. It is called the problem of UV divergences.

4. Aug 10, 2015

### nnerik

Thanks for the pointer, this Wikipedia article helped. See the following quote (attributed to Bjorken and Drell):

"The field functions are continuous functions of continuous parameters x and t, and the changes in the fields at a point x are determined by properties of the fields infinitesimally close to the point x. For most wave fields (for example, sound waves and the vibrations of strings and membranes) such a description is an idealization which is valid for distances larger than the characteristic length which measures the granularity of the medium. For smaller distances these theories are modified in a profound way. [...] it is a gross and profound extrapolation of present experimental knowledge to assume that a wave description successful at 'large' distances (that is, atomic lengths ≈10-8 cm) may be extended to distances an indefinite number of orders of magnitude smaller (for example, to less than nuclear lengths ≈ 10-13 cm)"
The image that appears in my mind, is that the continuous fields of QFT in ST are replaced by a liquid of stringy "atoms" through which the (quantum) waves propagate. Now, I am sure that this rather classical and ether-like description is wrong, but how else does the stringyness do any good, if it does not quantize space itself like molecules "quantize" the air? And by the way, do strings really get rid of the continuum and infinitessimal points? Would not the vibrations of the strings themselves be described by continuous waves shorter than the Planck length? Could not these waves be described as point-like particles on the strings themselves?

Any explanation or pointer will be appreciated.

5. Aug 10, 2015

### jerromyjon

The strings are one dimensional?

6. Aug 10, 2015

### Staff: Mentor

7. Aug 11, 2015

### Demystifier

Why the question mark?

8. Aug 11, 2015

### jerromyjon

Because it was a question.

9. Aug 11, 2015

### Demystifier

10. Aug 11, 2015

### jerromyjon

It may be "obvious" as far as words go but I was trying to imagine the mathematical implications, and it is not obvious to me how that works.

11. Aug 11, 2015

### Jano L.

The particle $\neq$ the $\psi$ function. The former is a point particle, the latter is a function whose support is never reduced to merely one point.

By being a point particle, we mean that particle has no internal structure. In non-relativistic theory of Schroedinger's equation, it also means the Hamiltonian we use is that of a point particle, not of an extended particle or a field.

Math note: the position operator has no eigenfunctions.

12. Aug 11, 2015

### Staff: Mentor

Hmmm. I think you should look into Rigged Hilbert Spaces.

Thanks
Bill