Exploring the Ambiguity of Wick Rotation in Time-Ordered Correlators

In summary, "Exploring the Ambiguity of Wick Rotation in Time-Ordered Correlators" delves into the complexities and potential ambiguities associated with the Wick rotation—a mathematical transformation that shifts time from real to imaginary values in quantum field theory. The paper examines how this rotation affects time-ordered correlators, which are essential for calculating physical observables. It highlights the nuances involved in defining these correlators and the implications of different approaches to Wick rotation, emphasizing the need for careful consideration of the underlying assumptions and physical contexts. Overall, the work aims to clarify the role of Wick rotation in theoretical frameworks and its impact on quantum field calculations.
  • #1
thatboi
133
18
Hi all,
I am reading the following post: https://physics.stackexchange.com/q...o-time-ordered-correlator-in-real-time-how-do
and was confused regarding the portion in the accepted answer where they made a distinction for the ##t>0## and ##t<0## case. I believe the ##e^{\pm i\pi}## factor comes from the negative sign in front of ##t^{2}## but cannot understand why there is an ambiguity in the final result.
 

FAQ: Exploring the Ambiguity of Wick Rotation in Time-Ordered Correlators

What is Wick rotation and why is it important in quantum field theory?

Wick rotation is a mathematical technique used in quantum field theory to transform integrals in Minkowski space (which has a time coordinate with a different sign in the metric) into integrals in Euclidean space (where all coordinates are treated similarly). This is done by rotating the time coordinate into the complex plane. It simplifies calculations, particularly in the context of path integrals, by converting oscillatory integrals into exponentially decaying ones, making them easier to evaluate.

What are time-ordered correlators and their significance in quantum field theory?

Time-ordered correlators are functions that describe the correlations between quantum fields at different points in spacetime, with the time coordinates ordered in a specific sequence. They are crucial for understanding the behavior of quantum fields and particles, as they provide information about the propagation and interaction of particles. In particular, they are used to calculate physical observables such as scattering amplitudes and response functions.

How does ambiguity arise in Wick rotation when dealing with time-ordered correlators?

Ambiguity in Wick rotation for time-ordered correlators arises because the process of rotating the time coordinate into the complex plane can lead to different paths of integration, depending on the ordering of time coordinates. This can result in different, sometimes conflicting, expressions for the correlators. The ambiguity is particularly pronounced when dealing with interacting field theories or when the correlators involve complex time dependencies.

What are the common methods to resolve the ambiguity in Wick rotation?

Common methods to resolve the ambiguity in Wick rotation include carefully defining the contour of integration in the complex time plane, using analytic continuation, and employing boundary conditions that ensure consistency. Another approach is to use regularization techniques to handle divergences and ambiguities systematically. In some cases, physical principles such as causality and unitarity can be used to guide the correct choice of contour and resolve ambiguities.

What are the implications of resolving the ambiguity in Wick rotation for practical calculations in quantum field theory?

Resolving the ambiguity in Wick rotation has significant implications for practical calculations in quantum field theory. It ensures that the computed time-ordered correlators are consistent and physically meaningful, leading to reliable predictions for particle interactions and other phenomena. Accurate time-ordered correlators are essential for computing scattering amplitudes, decay rates, and other observables that can be compared with experimental data, thereby validating or refining theoretical models.

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