# Relation between worldsheets and worldlines

If we want to find the propagator of say a zero-spin particle, the formal inverse can be computed by introducing $$-i\int_{0}^{\infty}e^{i(p^2 - m^2)s}\mathrm{d}s$$ We can rewrite this in path integral form as well. Is this somehow generalised by the introduction of a worldsheet action? If so, then is there an analogue of the Schwinger proper time formalism in string theory?

• Yes, the Schwinger proper time formalism applies to both point mechanics, field theory and string theory. – Qmechanic Nov 23 '17 at 9:46

There is a world-sheet formalism generalizing the worldline formalism. In this formalism, in the Feynman diagrams:

1. The integration over the light cone coordinate $p_{-} = \frac{1}{\sqrt{2}}(p_0 - p_3)$ is traded by an integration on $x^{+} = \frac{1}{\sqrt{2}}(x^0 + x^3)$ by means of a Fourier transform.

2. Then, the integration on the lightcone $p_{+} = \frac{1}{\sqrt{2}}(p_0 + p_3)$ and $x^{+}$ coordinates is separated from the other integration in the Feynman diagrams and treated as an integration over a world-sheet. Please, see the following works by Thorn and Bardakci first and second and references therein.

The idea can be illustrated on a scalar massless propagator , whose expression in the mixed lightcone coordinates is given by: $$\Delta(\mathbf{p}, p_{+}, x^{+}) = \langle \phi(\mathbb{p}, p_{+}, x^{+}) ) \phi(0)\rangle = \int \frac{dp_{-}}{2 \pi i } e^{-i p_{-}x^{+}} \frac{1}{p^2-i \epsilon} = \frac{\theta(x^{+})}{2 p_{-}} e^{\frac{x^{+} \mathbf{p}^2}{2 p_{+}}}$$ where $\mathbf{p}$ is the vector of the rest of the coordinates (of dimension $d-2$ where $d$ is the dimension of space time).

Bardakci and Thorn found a string worldsheet path integral representation of the above propagator as: $$\Delta(\mathbf{p}, p_{+}, x^{+}) = \int \mathcal{Dc}\mathcal{Dc}\mathcal{Dq} e^{-S}$$ with $$S = \int_0^{x^{+}} \int_0^{p_{+}} d\sigma \left ( \frac{1}{2}\mathbb{q}'^2 - b'c' \right)$$ Where $\mathbf{q}$ is a $d-2$ dimensional vector and $b$ and $c$ are Grasmann vector of half of the dimension of $\mathbf{q}$. The boundary condition for $b$ and $c$ are $0$ on all the worldsheet boundaries and for $\mathbf{q}$ , we have

$$\mathbb{q}(p_{+}, \tau) - \mathbb{q}(0, \tau) = \mathbf{p}$$

The above technique can be applied by means of inserting the above expressions for the propagators in the Feynman diagrams and use lightcone expressions for the vertices.

This formalism is useful in dealing with Yang-Mills theoies with large number colors, where the semiclassical contribution of the worldsheet integrals dominate, because the number of colors will appear multiplicatively in the worldsheet actions.

This formalism is a realization of 't Hooft idea of the dominance of the planar diagrams in the large color limit and provides in principle a method for summing these diagrams.