Wick rotation, scalar field and invariants

In summary, the conversation discusses the concept of Wick rotation and its application to the Minkowski and Euclidean metrics. It also addresses the implications of this rotation on the mass-shell condition and the Green function, as well as the behavior of plane-waves in euclidean space. The main questions raised are whether the mass-shell condition still holds after Wick rotation, what happens to invariants constructed from scalar product, and if there is a relationship between Wick rotation and the non-eigenfunction behavior of plane-waves.
  • #1
Melsophos
6
0
One point about Wick rotation is puzzling me and I can not find explanations in books. It concerns the invariants formed from scalar product and solutions to equation. So I will expose my way of reasoning to let you see if it is correct and at the end ask more specific questions.

Let's start with the Minkoswki metric
[tex]ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu = - dt^2 + d\vec x^2[/tex]

Now if we go to euclidean time
[tex]t = -i\tau[/tex]
(the sign being chosen to have an exponential decay: [itex]e^{-iHt}\to e^{-H\tau}[/itex]) then the metric becomes euclidean
[tex]ds^2 = \delta_{\mu\nu} dx^\mu dx^\nu = dt^2 + d\vec x^2[/tex]
and
[tex]S = \int d^d x \; L = -i \int d^d x_E \; L = i \int d^d x_E \; L_E = i S_E[/tex]
so that
[tex]S = i S_E, \quad
L = - L_E, \quad
Z = \int d\phi\; e^{iS} = \int d\phi\; e^{-S_E}[/tex]

Now take the scalar field lagrangian
[tex]L = -\frac{1}{2} \Big((\partial^\mu \phi)^2 + m^2 \phi^2 \Big) - V(\phi)[/tex]
This gives the usual Klein-Gordon equation with potential:
[tex](-\Delta + m^2) \phi = V'(\phi)[/tex]
Plugging plane-waves into the free equation ([itex]V(\phi) = 0[/itex]) gives the mass-shell condition
[tex]p^2 = -m^2[/tex]
The Green function is (without care about epsilon factors)
[tex]G(p) = \frac{1}{p^2 + m^2}[/tex]

Applying the Wick rotation gives
[tex]L_E = \frac{1}{2} \Big((\partial_E^\mu \phi)^2 + m^2 \phi^2 \Big) + V(\phi)[/tex]
and the equation
[tex](-\Delta_E + m^2) \phi = -V'(\phi)[/tex]
Fourier transform gives the Green function
[tex]G_E(p) = \frac{1}{p_E^2 + m^2}[/tex]
which have no poles since [itex]p_E^2 \ge 0[/itex]. This just says that plane-waves are not eigenfunctions of the Laplacian (but instead exponentials; which seems to agree with the fact that euclidean has a better behavior – and so there is no waves propagating in euclidean space).

Now my problem is with the previous mass-shell condition: if we Wick rotate it, then it becomes
[tex]p_E^2 = -m^2[/tex]
which seems to be in disagreement with the fact that scalar product in euclidean space is positive-definite. So does this mean that we should give up this relation and totally forget about it? And in this case, does this happen also to all invariant constructed from spacelike vectors? At a first glance I would have change the sign for timelike scalars, but then this gives the Helmholtz equation with plane-wave solutions and this reintroduces the pole in Green function, so it does not seem to be a good solution.
The fact that the mass-shell does not hold anymore seems in agreement that plane-waves are not eigenfunctions.
 
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  • #2
So my questions are: Does the mass-shell relation still hold (in some form) after Wick rotation?What happens to all invariant constructed from scalar product?Is there a relation between Wick rotation and the fact that plane-waves are not eigenfunctions of the Laplacian?
 

1. What is Wick rotation and how is it used in physics?

Wick rotation is a mathematical technique used in theoretical physics to relate different theories in different dimensions. It involves rotating the time coordinate in a theory with a Euclidean metric, allowing for the use of techniques from Euclidean geometry to solve problems in a Lorentzian space-time. This technique is particularly useful in quantum field theory and general relativity.

2. What is a scalar field and how does it differ from a vector field?

A scalar field is a mathematical function that assigns a scalar value (such as temperature or pressure) to every point in space. In contrast, a vector field assigns a vector (such as velocity or force) to every point in space. Scalar fields are used to describe quantities that only have magnitude, while vector fields describe quantities that have both magnitude and direction.

3. What are invariants and why are they important in physics?

Invariants are quantities that do not change under a certain transformation or operation. In physics, they are important because they allow us to simplify and solve complex problems by identifying quantities that are conserved or remain the same in different situations. For example, energy and momentum are invariants in many physical systems, which allows us to use conservation laws to solve for unknown quantities.

4. How are Wick rotation, scalar fields, and invariants related?

Wick rotation is often used to simplify calculations involving scalar fields in quantum field theory. By rotating the time coordinate, we can use techniques from Euclidean geometry to solve problems that would be much more complicated in Lorentzian space-time. Additionally, invariants can be used to identify symmetries in the theories involving scalar fields, which can help us understand the underlying physical principles at play.

5. Can Wick rotation be applied to all physical theories?

In theory, Wick rotation can be applied to any theory that involves a Minkowski space-time, as long as the rotation does not introduce any singularities or inconsistencies. However, it is most commonly used in quantum field theory and general relativity, as these theories involve calculations that are simplified by the use of Euclidean techniques. In other theories, the use of Wick rotation may not be necessary or useful.

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