# Unitarity and locality on patgh integrals

• melthengylf
In summary, the conversation discusses the use of Feynman path integrals to integrate eiS/hbar along all fields, generating a unitary and local theory. The question is whether the locality is due to the action being an integral of local functions and whether the unitarity is related to the purely imaginary exponent. The answer lies in the Osterwalder-Schrader theorem, which allows for the recovery of a relativistic local QFT from Euclidean path integrals. Additional resources on this topic include the book "Quantum Fields and Strings: A Course for Mathematicians" and the paper "The Osterwalder-Schrader Theorem in a Rigorous Perspective".
melthengylf
my question is this: you know than in feynman path integra, you integrate eiS/hbar along all the fields. you also know that S is real and that it is the integral of local functions (fields and derivatives of fields). you also know that path integral generates an unitary and local theory. the question is, it generates a local theory because the action is an integral of local functions? it generates unitary theory because the exponent involved is purely imaginary? if not, are these facts anyhow related?? thank you very much. I'm sorry for the untidiness.

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## 1. What is unitarity in path integrals?

Unitarity in path integrals refers to the principle that the total probability of all possible outcomes of a quantum system must add up to 1. This means that the wavefunction must be normalized and all probabilities must be positive. Unitarity ensures that the laws of quantum mechanics are preserved in the path integral formalism.

## 2. How does unitarity relate to the conservation of probability?

Unitarity is closely related to the conservation of probability. In quantum mechanics, the total probability of all possible outcomes must remain constant over time. This is known as the conservation of probability. In path integrals, unitarity ensures that the conservation of probability is maintained by requiring the total probability to be equal to 1.

## 3. What is locality in path integrals?

Locality in path integrals refers to the principle that the action of a quantum system only depends on its local environment. This means that the probability of a particle's path is only affected by the immediate surrounding fields and not by distant fields. Locality is a fundamental principle in quantum field theory and is necessary for the consistency of path integrals.

## 4. How does locality affect the calculation of path integrals?

Locality plays an important role in the calculation of path integrals. It allows us to break down the calculation into smaller, local pieces and then combine them to get the total probability. This makes the calculation more manageable and allows us to use perturbation theory for more complex systems. Without locality, the calculation of path integrals would be much more difficult.

## 5. What are some examples of systems where unitarity and locality are violated in path integrals?

There are some systems where unitarity and locality are violated in path integrals, such as non-local quantum field theories or theories with non-unitary time evolution. These systems are still under active research and their implications are not fully understood. Violations of unitarity and locality can also arise in certain quantum gravity theories or at extreme conditions, such as black holes or the early universe.

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