Question

When it comes to construct a well-defined path integral for the Polyakov action in Bosonic String Theory, most authors assume that the world sheet metric g is Riemannian (i.e. it has Euclidean signature). However, the target space is generally taken to be a Lorentzian manifold, such as Minkowski space. The constraint imposed over g is that it must be conformal to the metric induced both by the target space (Lorentzian) metric and the embedding X. This is not forbidden, but if the pullback happens to be Riemannian, we lose the usual view of a string propagating in a timelike direction in spacetime. How to solve this conceptual problem?

Answer 1

In Bosonic String Theory, the world sheet metric g is usually Riemannian, while the target space is Lorentzian. The constraint on g is that it must be conformal to the metrics induced by the target space and the embedding. One possible solution to the conceptual problem of losing the timelike interpretation is to consider a world sheet metric with Lorentzian signature.

Step-by-step explanation:

In Bosonic String Theory, the world sheet metric g is typically assumed to be Riemannian (Euclidean signature), while the target space is generally taken to be a Lorentzian manifold such as Minkowski space. The constraint on g is that it must be conformal to the metric induced by both the target space metric and the embedding X. This means that the pullback of the metric may be Riemannian, which can lead to the loss of the usual view of a string propagating in a timelike direction in spacetime.

One possible solution to this conceptual problem is to consider a world sheet metric that is not restricted to be Riemannian. By allowing the world sheet metric to have Lorentzian signature, we can maintain the interpretation of the string propagating in a timelike direction in spacetime. However, this leads to additional complications in the formulation of the path integral and requires new mathematical techniques.

Overall, the choice of the world sheet metric in Bosonic String Theory is a topic of ongoing research and different approaches may be suitable depending on the specific context and goals of the study.