# Wick rotation and interference

• causalset
You Wick rotate the Schroedinger equation and its complex conjugate, and you get a diffusion equation. You then have to choose a time-symmetric ordering for the derivatives in the diffusion equation. One choice (the one I understand the least) is to choose the so-called "Moyal" ordering. This leads to diffusion processes in the complex plane, which in turn lead to complex-valued amplitudes for transition probabilities. When you calculate the expectation value of the transition probability amplitude, you have to integrate over all time variables in the complex plane, but there is a contour integral you can do which makes the integral "simply" converge. This is the analogue of the oscillatory integral that I was talking about in my other post. I

#### causalset

In light of the fact that Wick rotation can replace imaginary exponent wtih the real one, how does it reproduce interference? For example, in double slit experiment, how would we reproduce the fact that probability of reaching certain spots on the screen is zero? After all, the exponent of something "real" is always positive, so there is no room for cancelation.

causalset said:
In light of the fact that Wick rotation can replace imaginary exponent wtih the real one, how does it reproduce interference? For example, in double slit experiment, how would we reproduce the fact that probability of reaching certain spots on the screen is zero? After all, the exponent of something "real" is always positive, so there is no room for cancelation.

The replacement is only valid for certain kinds of calculations, namely when you can undo the Wick rotation at the end of your calculation to get back a result in Minkowski space.
Then the oscillatory nature reappears.

Don't worry about the rotation will kill the imaginary part...Though the action S in the exponent seems become "real",but the integral interval of time shifts from real zone into imaginary zone, which will guarantee that the amplitude still is complex value instead of a pure real number...Hence,the distribution of the pole in the complex plane will not change after the Wick rotation and it will give the same results.
Wick rotation is just more like a math skill...so a math skill won't offer any new physical facts but give a fresh viewpoint of the equation..

Causalset,

Wick rotating the Schroedinger equation and its complex conjugate leads to a theoretical framework called Euclidean Quantum Mechanics, developed by Jean-Claude Zambrini. In this framework, one can have a linear superposition of transition probability solutions to the Wick rotated Schroedinger equations, and hence have interference of transition probabilities. See these papers:

Stochastic mechanics according to E. Schrödinger
Phys. Rev. A 33, 1532–1548 (1986)
http://pra.aps.org/abstract/PRA/v33/i3/p1532_1

Euclidean quantum mechanics
Phys. Rev. A 35, 3631–3649 (1987)
http://pra.aps.org/abstract/PRA/v35/i9/p3631_1

The first paper explicitly gives the formula (equation 4.13, section C) for the interference of transition probabilities under two-slit boundary conditions.

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A. Neumaier said:
The replacement is only valid for certain kinds of calculations, namely when you can undo the Wick rotation at the end of your calculation to get back a result in Minkowski space.
Then the oscillatory nature reappears.

This doesn't answer the OP's question. The Wick rotation is perfectly valid for wavefunctions used to describe two-slit interference. See the papers I cited.

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Maaneli said:
This doesn't answer the OP's question. The Wick rotation is perfectly valid for wavefunctions used to describe two-slit interference. See the papers I cited.

But how do you transform equation 4.13, section C back to Minkowski space and recover the interference pattern? Zambrini is apparently silent about this!

A. Neumaier said:
But how do you transform equation 4.13, section C back to Minkowski space and recover the interference pattern? Zambrini is apparently silent about this!

This is a different question, and it doesn't invalidate the answer to the OP's question.

Also, Zambrini does actually show how one can transform between the Euclidean and real-time diffusions (the real-time diffusions also properly describe two-slit interference). What one has to do is change the time-symmetric ordering of the stochastic derivatives in the definition of the mean acceleration.

Maaneli said:
This is a different question, and it doesn't invalidate the answer to the OP's question.

The question was:
''how would we reproduce the fact that probability of reaching certain spots on the screen is zero? After all, the exponent of something "real" is always positive, so there is no room for cancellation.''
and therefore requires an answer for how you can get back the Minkowski result from the Euclidean calculation. Zambrini is of no help here, it seems.

A. Neumaier said:
The question was:
''how would we reproduce the fact that probability of reaching certain spots on the screen is zero? After all, the exponent of something "real" is always positive, so there is no room for cancellation.''

I'm point out to the OP that Zambrini shows that, contrary to his assumption that there is no room for cancellation, there is in fact an interference of transition probabilities in the Euclidean case.

A. Neumaier said:
and therefore requires an answer for how you can get back the Minkowski result from the Euclidean calculation. Zambrini is of no help here, it seems.

No. Once again, his question is about how one would reproduce the prediction of interference in the context of a Wick rotated version of QM. If you read the Zambrini paper, you'll see how this is done.