Path Intergral quantization for Relativistic Point like particle?

[aspidistra]

Dear all,
Since standard QM textbook Sakurai or Shankar only mention Non-relativistic path
integral and QFT text deal with path integral for field theory, I want to ask whether
there is a subject like "Path Intergral quantization for Relativistic point like Particles"?
If so, is this subject has been well done and is there any readable textbook
or review articles? thanks in advance. :)

 

[Halcyon-on]

In quantum theory a relativistic particle is not point like, it is a field. This is the reason why one should use field theory in the relativistic limit. The field can be thought of as a multiparticle state due to pair creation at high energies.

 

[aspidistra]

So...Is that means since in real world particles can be created and annihilated,
and a field theory can describe a such process a point like particle model can not,
therefore a project which use point like particle quantization scheme is dead?

 

[Avodyne]

No, it can be done, but it's surprisingly tricky. See string theory books such as Polchinski for how to do it.

 

[jostpuur]

In this thread (Dirac-Feynman-action principle and pseudo-differential operators) I was asking about pretty much the same problem. Instead of focusing on the Lagrangian

$$L = -mc^2\sqrt{1 - |v|^2/c^2}$$

and Hamiltonian

$$H = \sqrt{(mc^2)^2 + |p|^2c^2}$$

I was interested in a simpler (and more hypothetical) system defined by Lagrangian

$$L \propto |v|^{\alpha}$$

and

$$H \propto |p|^{\frac{\alpha}{\alpha - 1}}.$$

When $\alpha=2$ this is the usual non-relativistic system, but with $\alpha\neq 2$ this system leads to the same mathematical problems as the relativistic point particle system.

My short thread did not settle the problem, but I got an idea which could be the key to the solution. I only have not had enough energy and motivation to start working on it. It could be that the key is to use Hamiltonian formulation of the path integral, and then pay special attention to the order of integration. It seems clear that the Lagrangian formulation does not work, because it cannot produce the Fourier transforms needed to express the pseudo differential operators. Consequently, one cannot perform the calculation in such manner that one would derive Lagrangian path integral from the Hamiltonian path integral. The propagator must be derived from the Hamiltonian path integral via some other way.

 

[Demystifier]

No, it can be done, but it's surprisingly tricky. See string theory books such as Polchinski for how to do it.

But string-theory textbooks, including the one by Polchinski, do not discuss path-integral quantization of point particles. Or if I am wrong, can you specify the exact place (page number, equation number, ...) where it is discussed?
Still, you are making a good point. I am sure that a stringy type of reasoning can be used to formulate path-integral quantization of relativistic point particles. The question is whether someone has already done it?
(I am not talking about describing particle creation and destruction. I am talking about path-integral formulation of relativistic quantum mechanics (Part I of Bjorken-Drell), irrespective on whether such theory is satisfying physically.)

 

[Avodyne]

But string-theory textbooks, including the one by Polchinski, do not discuss path-integral quantization of point particles. Or if I am wrong, can you specify the exact place (page number, equation number, ...) where it is discussed?

Section 3.2, but there's not a lot of detail. But a quick google search turned up this nice discussion by Dan Kabat:

http://www.phys.columbia.edu/~kabat/strings/Spring08/handout2.pdf

 

[jostpuur]

a quick google search turned up this nice discussion by Dan Kabat:

http://www.phys.columbia.edu/~kabat/strings/Spring08/handout2.pdf

Then adopting the path integral prescription we have (at least naively) an expression for the relativistic propagator
$$\langle x_f, t_f | x_i, t_i\rangle = \int\mathcal{D}x(\cdot) e^{iS}$$
$$S = -mc\int\limits_{t_i}^{t_f} \sqrt{1 - |\dot{x}|^2/c^2}$$
I'm not sure to what extent this path integral makes sense.

Yeah... that's the problem. It does not make sense.

However, I don't think that the string guys' solution to the problem makes sense either.

Recall that in non-relativistic QM the path integral time evolution is equivalent to the Schrödinger equation time evolution. Doesn't it then seem natural to assume in relativistic QM the relativistic path integral is going to be equivalent with some relativistic Schrödinger equation too? If not, then what is the relativistic path integral supposed to give then? If the path integral gives some propagator, surely it will give some Schrödinger equation too? Simply substitute infinitesimal time to the time evolution.

When some theoreticians explain that they have a relativistic path integral, but are unable to give the corresponding equivalent Schrödinger equation, it does not look like that the theoreticians know what they are doing.

It would of course be lot easier to show that some relativistic path integral is equivalent with some relativistic Schrödinger equation, if one first knew what the relativistic Schrödinger equation is.

 

[Demystifier]

Section 3.2, but there's not a lot of detail. But a quick google search turned up this nice discussion by Dan Kabat:

http://www.phys.columbia.edu/~kabat/strings/Spring08/handout2.pdf

Thanks!

 

[haushofer]

Good question, which I also had during my first time exposure to path integral quantization. I never really could find out what the precise problem was.

In quantum theory a relativistic particle is not point like, it is a field. This is the reason why one should use field theory in the relativistic limit. The field can be thought of as a multiparticle state due to pair creation at high energies.

Maybe I'm nitpicking, but doesn't QFT still speak about "relativistic pointlike particles", but which are excitations of a field? If so, I can't understand your reasoning :)

 

[Demystifier]

Doesn't it then seem natural to assume in relativistic QM the relativistic path integral is going to be equivalent with some relativistic Schrödinger equation too?

Yes it does, but in the sense that I have already explained to you in
https://www.physicsforums.com/showthread.php?t=248653

 

[jostpuur]

In quantum theory a relativistic particle is not point like, it is a field. This is the reason why one should use field theory in the relativistic limit. The field can be thought of as a multiparticle state due to pair creation at high energies.

Maybe I'm nitpicking, but doesn't QFT still speak about "relativistic pointlike particles", but which are excitations of a field? If so, I can't understand your reasoning :)

Furthermore, also non-relativistic particles are excitations of fields, and still they can be thought to be point particles. (In general, particles are excitations of fields!) It would be interesting to see how precisely is the point particle picture derived out from the field excitation picture, but this seems to belong to those topics that all authors prefer avoiding.

Yes it does, but in the sense that I have already explained to you in
https://www.physicsforums.com/showthread.php?t=248653

I understood nothing out of that back then

If I recall correctly, there were two thoughts oscillating in my mind. Other one was that string theorists are mad, and the second one was that I should try reading Zwiebach more carefully from the beginning, because the light cone stuff seemed unfamiliar.

 

[Demystifier]

If I recall correctly, there were two thoughts oscillating in my mind. Other one was that string theorists are mad, and the second one was that I should try reading Zwiebach more carefully from the beginning, because the light cone stuff seemed unfamiliar.

Well, even if strings do not exist, I am convinced that string theorists are not mad. So the second option seems more viable.

 

[jostpuur]

Notice that the two options are not mutually exclusive.

 

[Demystifier]

Notice that the two options are not mutually exclusive.

Do you know an example when it is worthwhile to read something written by a mad person?

 

[Avodyne]

Here's another explanation:

http://www.physics.thetangentbundle.net/wiki/String_theory/bosonic_string/relativistic_point_particle

When you work out the hamiltonian, it turns out to be zero! So the Schrödinger equation simply says that the wave function does not depend on proper time. On the other hand, there is a constraint that the 4-momentum be on-shell. This requires the wave function to obey the Klein-Gordon equation.

Thus the theory is very clever in evading the problem that the Schrödinger equation is first-order in time, but there is no relativistic equation of this form for a spin-zero particle.