1. Thermal operator representation of thermal field theory
Thank you Mr. Chairman.
I would like to talk about the thermal operator representation of thermal field theory.
My name is Kohyama and I belong to Osaka City University.
Collaborators are Mr. Inui and Mr. Niegawa.
H. Kohyama (Osaka City University)
　Collaborators:　M. Inui,　A.Niegawa

2. Contents 1 Introduction 2 Review of thermal field theory 3 Thermal operator representation
1st, Introduction. I will give belief story of this talk.
2nd, Review of thermal field theory.
To remember what is thermal field theory, I am talking about elementary things about thermal field theory.
3rd, Main story, Thermal operator representation.
This is the original work what we have done. Honestly this is not difficult to understand. So if I can talk well, every body here can understand this work.

3. 1 Introduction Thermal operator representation What is to be? Thermal
Amplitude
Thermal
Operator
Vacuum
Amplitude
Introduction.
This is the story about factorization of thermal amplitude.
We have shown that the thermal amplitudes are factorized like this.
Operating some kind of operator to vacuum amplitudes.
We call this operator as “thermal operator”
But here is the question,
What is the thermal operator to be?
We will seek the proper form of the thermal operator.
What is to be?

4. 2. Review of Thermal Field Theory
Before going to the main topic, we had better remember elementary things about thermal field theory.
So I will give a brief review of thermal field theory.
However, probably everyone here is very familiar with this topic.
So if you get bored, you can sleep. I recommend it.
But remember, I beg you, please wake up before the main story.

5. QGP Hadron usual QCD 2.1 Hot and Dense QCD Many phases
This is the famous QCD phase diagram.
At zero temperature zero chemical potential point, we can use usual QCD.
But at some temperature, we have to use thermal QCD.
So to investigate QGP and phase transition structure, thermal field theory is very important.
These days, we believe there are various phase structure, like super conductivity.
About this story, Mr. Inui will talk after me, so I don’t do deeper about this topic.
Instead, I am rather interested in the elementary topic about thermal field theory here.
So I shall talk about theoretical things on thermal field theory.
usual QCD

6. 2.2 Analogy with Statistical Mechanics
Probability Amplitude
Statistical Mechanics
Analogy with Statistical Mechanics.
From quantum mechanics, the probability amplitude finding particle at (x prime , t prime) from (x , t) is expressed like this.
From statistical mechanics, the partition function can be written like this.
We immediately notice the similarity between these two quantity.
Especially, if the time difference is pure imaginary minus i beta and the q prime equals to q then two quantity becomes exactly the same.
So we can write partition function by using probability amplitudes like this.
After this, as in the usual quantum field theory, we apply the path integral method.
Finally, we get this form.
And this partition function would be the one for Thermal field theory.
This manipulation corresponds to determining the boundary condition.
Because the wave function satisfies Klein-Gordon equation, the knowledge of the boundary condition determines the system completely.
Of course the boundary condition relates environment around the particle.
This boundary condition corresponds to the temperature being one over beta.
Path integral
Generating Functional!

7. 2.3 Imaginary Time Formalism
Time difference
Boundary condition
Im t
Re t
Matsubara contour
Imaginary time formalism.
As I said previous page, the time difference is minus i beta, in this sense the time goes pure imaginary direction.
So this is called the imaginary time formalism.
We force the boundary condition like this.
Time path is represented in this graph.
This time path is called Matsubara contour.

8. 2.4 Imaginary time propagator
Thermal average
Propagator
Imaginary time propagator.
As in the case of statistical mechanics, the thermal average is computed following this definition.
One of the most important quantity is the propagator.
Thermal propagator is defined by this.
Here, T means T-product.
And by evaluating this, we can get the propagator in the momentum representation.
This is called Matsubara propagator.
In fact this form was originally derived by Umezawa first.
However this is called Matsubara propagator I don’t know why.
Anyway, in this talk, we use time-momentum mixed representation so I show the propagator in this representation.
Where n is the number density function.
This is the Boson case, of course we can think the Fermion case.
But the difference is not so important in this talk.
So I use boson propagator.
Where

9. Analytic Continuation
2.5 Real time formalism
Im t
Re t
Analytic Continuation
Real time formalism.
In many cases, the imaginary time formalism is easy to make a calculation. So we tend to use imaginary formalism.
But some cases, real time formalism is superior to the imaginary time formalism.
Especially, in this talk, the real time formalism is easier.
Later, I shall talk about this with a concrete example.
So we had better know both formalism.
To get the real time formalism, we just make an analytical continuation from the imaginary time formalism.
Like this, the path would go along real time axis, and then goes imaginary direction, and goes back parallel to the real time axis, and finally goes parallel to the imaginary axis.
We usually name these contours, C-one, C-two, C-three and C-four.
Im t
Re t
Real time contour

10. 2.6 Generating Functional
Im t
Re t
Real time contour
Generating Functional
Generating functional.
Following this contour, we can evaluate the generating function.
Here, we force the constraint that the generating function must be analytical function.
Under this constraint, we compute the generating functional.
Thanks to the constraint, we have to think only one-two path and three-four path.
The other combinations are no relevant here.
So the generating functional becomes like this.
N is the proportional constant.
And Z one-two is from the path one two.
So is Z three-four.

11. 2.7 Real Time Propagators Propagators Real time propagator.
We can derive Propagators by usual method: functional differentiation.
And the propagators are expressed like these in momentum representation.
Of course n is the number density.
Using these propagator, we can evaluate arbitrary physical amplitudes, by the same way in usual Feynman diagram method.

12. 3.Thermal Operator Representation
After a long digression, we finally came here.
Now, I will give you the main story, Thermal operator representation.

13. Thermal operator representation
Amplitude
Thermal
Operator
Vacuum
Amplitude
As I talked in the Introduction,
what we want to do is to factorize thermal amplitudes by thermal operator and vacuum amplitudes like this.

14. 3.1.1 Thermal Operator (Imaginary time)
We introduce the thermal operator.
Translation operator
Distribution operator
First, we introduce the thermal operator in the Imaginary time formalism.
Here, R is the reflection operator.
By R, E plus minus becomes minus E minus plus.
For later convenience, we have introduced E plus minus.
N is the distribution operator.
By N, exponential plus minus E plus minus tau becomes n E plus minus exponential plus minus E plus minus tau.
S is the translation operator.
By S, E plus minus becomes E plus minus mu.
This is the thermal operator.
By using this operator, we can factorize thermal amplitudes.
From now, I will show more details.
Reflection operator
For calculation convenience, we have introduced

15. 3.1.2 Propagator (Imaginary time)
The action of the thermal operator
Propagator.
Here, I want to check the action of the thermal operator.
Thermal propagator is represented like this.
And vacuum propagator is this.
As I said already, we use time-momentum mixed expression.
By operating the thermal operator, we can get this.
Of course, this corresponds thermal propagator.
Therefore, the factorization is actually done.
Thermal propagator can be expressed like this.
Therefore,

16. 3.1.3 N-Point Amplitudes (Imaginary time)
Next, we want to check the factorization of N-point amplitudes.
An N-pint amplitude is expressed by the products of propagators like this.
These delta functions are the momentum conservation delta functions.
We already saw that each thermal propagator is expressed as factorized form like this.
And each thermal operator doesn’t operate the other vacuum propagator.
So we can change orders of the thermal operators.
Therefore, we get this relation.
The factorization of N-point amplitudes is complete.
Therefore,

17. 3.1.4 Loop diagrams (Imaginary time)
The problem of time interval
: for thermal, : for vacuum
using ,
Loop diagrams.
The factorization of loop diagrams is not so trivial as in the case of N-point amplitudes.
Because there is time integration.
Especially, in the imaginary time formalism, the time interval is not the same between thermal amplitudes and vacuum amplitudes.
So we have to check the factorization property carefully.
For simplicity, we consider one loop diagram.
In this case, there is only one time integration.
Time integration of propagator is expressed like this.
And the thermal operator does not relate time, so the operator can go through the outside of integration.
Here, we face the problem of time interval.
Now we rewrite the integration like this.
Fortunately, thermal operator has this relation.
And thanks to this relation, we finally get this.
Therefore, the factorization of one loop diagram is complete.
Therefore,

18. 3.1.5 General Proof (Imaginary time)
Time extension property
Using ,
General proof.
Finally we will prove the factorization property of an arbitrary diagram.
To do that, we have to prove the time extension property when more than one time integration exists.
For simplicity, I will show in the case of two loop diagram.
Two loop diagram is expressed like this.
One integration can be changed by the same technique in the case of one loop diagram.
And using this relation one integration turns into just a operator.
Then we can use the same technique again.
And the problem of time interval is, in fact, not a problem.
No matter how many time integration exist, we can apply the same technique and there is no problem.
Therefore, the factorization of an arbitrary diagram is complete.
Therefore,

19. 3.2.1 Thermal Operator (Real time)
We introduce the thermal operator.
Translation operator
Distribution operator
Real time thermal operator.
Here, we will go the real time formalism.
The procedure is not so different as in the case of the imaginary time formalism.
First, we introduce the thermal operator.
As in the case of the imaginary time formalism, the action of the thermal operator is the same.
But in the real time formalism, we have to care about the epsilon part.
This, of course, expresses the boundary condition.
Reflection operator

20. 3.2.2 Propagators (Real time)
We can easily show the factorization.
Real time propagators.
A propagator in the time-momentum mixed representation is expressed like this.
Here I choose one-one component of the propagator.
And the vacuum propagator is this.
By straight forward calculation, we can check the factorization property like this.
Of course, the other components are expressed by the same way.
Therefore, we get the factorization property of the propagators.
We can also show below.

21. 3.2.3 N-Point Amplitudes (Real time)
Following the same way as in the imaginary time
calculation, we have the following factorization.
N-point amplitudes.
Following the same way as in the imaginary time calculation, we have the following factorization.

22. 3.2.4 Loop diagrams (Real time)
No problem of time interval
: for thermal
: for vacuum
The same!
Real time loop diagrams.
Because in the real time formalism, time interval is the same for the thermal amplitudes and for the vacuum amplitudes.
So there is no problem of time integration.
In this sense, the real time formalism is simpler than the imaginary time formalism.
Therefore, the factorization property is complete.
No problem of time interval
Therefore,

23. 3.2.5 General Proof (Real time)
For an arbitrary diagram, the proof goes as
it does in the imaginary time formalism.
Real time general proof.
For an arbitrary diagram, the proof goes as it does in the imaginary time formalism.
Therefore, the factorization for an arbitrary diagram is complete.
Here, we have completed the factorization both in the imaginary time formalism and real time formalism.

24. Summary and Future works
We have introduced the thermal operator.
We have proved the factorization property.
Future works
Summary.
We have introduced the thermal operator.
We have proved the factorization property.
This work has already been published.
If you are interested in this work, please read our paper.
That is all.
Thank you for listening to me.
To seek the use of this representation.
Reference
[1] M. Inui, H. Kohyama, A. Niegawa, Phys. Rev. D 73, (2006)