Signature change
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Signature change is concerned with when the signature of spacetime can change from positive definite (++++) to Lorentzian (-+++).
There are several reasons for interest in signature change including:
1) In the early universe there might be a transition from spacetime which has three space dimensions and one time dimension to four space dimensions.
This side-steps the question of what happens at ever earlier times: instead one now has an origin of time, when the fourth space dimension becomes time, and now the problem is to produce a theory to explain this.
2) In the path integral approach to quantum field theory many more systems are soluble when the signature is taken to be positive definite, essentially one has exponential functions rather than trigometrical functions in the integral which allows the integral to be solved. One then has to access the value of these solutions by applying some form of signature change to recover Lorentzian signature.
In practise most people just change the signature by fiat whenever calculations seem to require it; however there is a more systematic method using the line element field (sometimes called the direction field). This expresses a Lorentzian metric as a kind of sum of a positive definite metric and the line element field, and this separation can be carried through to the Lagrangian.
Then the line element field can be treated as just another field. The classical theory is well-defined, the simplest way of treating the quantum theory gives a type of Klein-Gordon equation governing signature change.
