Ads/cft correspondence and quantum / classical correspondance

[Heidi]

Hi Pf
I do not know if it is only an analogy but i consider these two correspondences:
I) the transformation of a quantum spin into a statistical problem by a Wick rotation.
we start with a qubit evolving in time between 0 and t from S0 to St. we use a exp(iHt) operator to describe its evolution. we can use a Wick rotation to map each Feynman path to [0 t] on the x line such than the amplitudes are transformed in probabilities exp(-H).
Here we hace a correspondance between a 0 dimension system evolving in time to a 1 dimension statistical system.
In the statistical system the interferences have disapeared but the notion of temperature appears.
II) in the Ads/cft formalism we have a 2 dimensional quantum system without
gravity and we consider the 3 dimension bulk in which gravity appears.

I wonder if in the point of view of information it is only an analogy.

 

[Demystifier]

In the statistical system the interferences have disapeared but the notion of temperature appears.

No. What disappears in statistical mechanics at a fixed temperature is a change with time.

 

[Heidi]

When we compute the probability in qm we add the amplitudes a+b and we multiply it by its conjugare getting aa* + bb* + ab* + ba*
if you Wick rotate that in a statistical problem you add the probabilities aa* + bb*

 

[Heidi]

If you Wick rotate exp(-i Ht) into exp(- H tau) you get non négative real numbers that cannot cancel the other terms. that is why interferences disappear.

 

[Demystifier]

If you Wick rotate exp(-i Ht) into exp(- H tau) you get non négative real numbers that cannot cancel the other terms. that is why interferences disappear.

Three points:
1) By this argument, perhaps you can elliminate destructive interference, but not constructive interference.
2) Even destructive interference cannot be elliminated in some cases, for example when the wave function is stationary, so that interference pattern does not change with time.
3) By the Wick rotation a quantum system remains quantum and does not turn into a classical system. For instance, quantum indistinguashibility of identical particles is not affected by the Wick rotation.

 

[Heidi]

i only say that there is a correspondence between a 0 dimensional problem and a 1 dimensional problem. it is well explained ln Le Bellac book.

 

[Demystifier]

i only say that there is a correspondence between a 0 dimensional problem and a 1 dimensional problem. it is well explained ln Le Bellac book.

I agree with that. But it has nothing to do with AdS/CFT, because in AdS/CFT one assumes that both times are real (or both imaginary if one considers AdS/CFT in thermal equilibrium).

 

[Heidi]

Michel Le Bellac is the author of this book in 1998:
https://www.amazon.com/dp/2868833594/?tag=pfamazon01-20

I quote several translated lines

In quantum mechanics, an important role is often played by the
classic path going from A to B, that is to say by the path which makes
stationary action. In the corresponding physics problem
classical statistic, the configuration corresponding to the classical path
is the "configuration of Landau": indeed the configuration of Landau
is that which makes the Hamiltonian stationary (cf. II.B). To fluctuations
quantum fluctuations around the classical path correspond
statistics around the Landau configuration. These fluctuations are
studied by perturbation theory, and it is not surprising to
encounter the same techniques in both types of problems.
Paragraph A deals with a basic example: correspondence
between the dynamics of a quantum spin 1/2 and that of the Ising model at
1 dimension. This example illustrates in a very simple way the passage of a
quantum problem to a statistical problem, and allows to specify a
number of matches

And after that:

To a quantum field theory in a D - 1 dimensional space (i.e. a space-time of dimension D) will correspond to a statistical system in a D-dimensional space

when i saw the Ads/Cft correspondence between a quantum 2 space and a 3 space i remembered his sentences.

 

[Heidi]

Could you read the ADS CFT correspondence article in wikipedia? they also talk abous critical points and statistical mechanics.

 

[Heidi]

As the name of the book is "from critical phenomena to gauge theories" I googled these worlds.
I found this article.
https://arxiv.org/abs/1006.4930
It is far far above my level but i think it shows that Le Bellac's ideas have something to do with the ADS CFT correspondence..

 

[Demystifier]

Maybe I was not sufficiently clear, so I will repeat in a more explicit manner.
Let $(n,k)$ denote a theory with $n$ space dimensions and $k$ time dimensions. Le Bellac relates
- $(n,1)$ theory to $(n+1,0)$ theory
By contrast AdS/CFT relates
- $(n,1)$ theory to $(n+1,1)$ theory
or, in thermal equilibrium,
- $(n,0)$ theory to $(n+1,0)$ theory

 

[Heidi]

Wick rotation replaces it/h by 1/kT . In real life we have both. Is it not too much?

 

[Demystifier]

Wick rotation replaces it/h by 1/kT . In real life we have both. Is it not too much?

Strictly speaking, temperature is well defined only in the thermal equilibrium, and in equilibrium nothing changes with time. In this strict sense, we never have both. But in a less strict sense, we often can talk about time-dependent thermal systems. For such purposes one can introduce a complex time, having both a real and an imaginary part.

 

[Heidi]

Is it what Landau does with his i \Gamma in the hamiltonian of an unstable particle?

 

[Demystifier]

Is it what Landau does with his i \Gamma in the hamiltonian of an unstable particle?

Not quite. Imaginary Hamiltonian is one thing, imaginary time is another.

 

[Heidi]

Could you tell me what is , for you , the most interesting or useful
way to use time with an imaginary part?
Have you other examples in mind?
thanks.

 

[Demystifier]

Could you tell me what is , for you , the most interesting or useful
way to use time with an imaginary part?
Have you other examples in mind?
thanks.

Imaginary time can be used as a computational trick for:
- quantum systems in thermal equilibrium
- special relativity
- analytic continuation for better convergence of path integrals
- analytic continuation as a tool in computation of Feynman loops

 

[Heidi]

I found that with time being completely imaginary (Wick rotation)
i would like to see t + i t' in formulas.

 

[Heidi]

the only one i found is the kms contour
https://www.researchgate.net/figure/Figure-D1-Contour-used-in-proof-of-the-KMS-condition_fig4_227342371
have you others?