# What do we learn from quantizing the relativistic point particle?

In many textbooks on string theory, some time is spend on quantizing the relativistic point particle as a warming-up for quantizing the Nambu-Goto action for relativistic strings.

However, I have not found a clear explanation of why the theory of the quantum relativistic point particle is insufficient. What does this quantum theory tell us, and why is the upgrade to strings needed?

• Commented Feb 28 at 12:40
• If you cannot effectively quantize a relativistic point particle then you have no hope of quantizing a world sheet. In some sense quantizing point particles is as fundamental to string theory as point mechanics is to continuum mechanics. Commented Feb 28 at 13:16
• some worldline aspects of the distinction between particles and strings, and why the former is not unique, is explained in doi.org/10.1016/0370-2693(90)90792-5 Commented Feb 29 at 5:51

## 2 Answers

The theory of a relativistic point particle is insufficient for the same reason that the theory of a relativistic string is insufficient! Specifically, the former can compute the amplitude for a particle interaction when "intermediate processes" are specified. But it does not explain where these come from or offer a window onto non-perturbative physics. Think of evaluating a Feynman diagram by stringing together propagators which (in the scalar case) are written as $$$$D_{yz} = \int_0^\infty dT \int_{x(0)=y}^{x(T)=z}Dx e^{-S} = \int_0^\infty dT (4\pi T)^{-d/2} e^{-m^2 T - (y - z)^2 / 4T}. \quad (1)$$$$ This is the amplitude for a particle to propagate from position $$y$$ to position $$z$$, integrated over all possible amounts of time that the propagation can take. This is enough to compute all our favourite perturbative cross sections without referring to fields.

However, it is still better to use fields because this makes the process less ad hoc. Computing matrix elements of a time evolution operator with respect to in and out states is already a natural thing before you perturb in any small coupling. And one can constrain the interaction vertices which appear in this operator using ideas of effective field theory. In this language, (1) is just one of the propagators arising from Wick's theorem when written in terms of Schwinger parameters.

So this "wordline theory" became more satisfying with the introduction of QFT. We would also like to make "worldsheet theory" (string theory) more satisfying in the same way but this has not been done yet. The basic framework of string theory, which was introduced at around the same time as QFT, is unfortunately still perturbative by its very nature. As one of the comments seems to suggest, I would use the quantization of a point particle as a tool to help learn string theory but not to motivate it.

The motivation should really just be the theory's surprising properties. In roughly historical order, some are:

1. It gives a plausible explanation for why the resonances of hadrons form Regge trajectories even if precise predictions have not materialized.
2. It lets us see what a UV complete theory of quantum gravity looks like whether or not you believe that it's the UV completion appearing in nature.
3. It is constrained enough to have a possibly finite number of solutions which is not the case in QFT without gravity.
4. Because of AdS/CFT, it can be helpful even if your ultimate interest still is QFT without gravity.

I will give a different perspective on the worldline formalism. Yes, the worldline formalism is perturbative, where the worldline proper time is the Schwinger proper time. However, once we use BRST quantization for the worldline, which is required if one wants Poincaré covariance to be manifest, there is a natural way of constructing field theories out of worldline formalism, which then can be used for non-perturbative computations. In fact, this construction of field theories, which starts from the first quantization, usually gives not just the possibly degenerated gauge invariant action but also its ghosts, anti-fields and anti-ghosts, i.e. one lands itself in a BV action.

Instead of explaining what I meant by natural way, I will refer to the following papers, and lecture notes, that do this very well: