# A On the equivalence of operator vs path integral in QFT

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1. Feb 17, 2017

### ftr

I have read many textbooks and googled google times for a clear explanation, but I could not find one. How does raising and lowering -annihilation/ creation-(is that energy or particle number?) translate to transition probabilities of path integral.

2. Apr 28, 2017

### ftr

3. Apr 28, 2017

### Staff: Mentor

This question is too vague. Can you find a specific reference that has a specific example that illustrates the issue?

4. Apr 28, 2017

### ftr

Well, I am reading this book
https://www.amazon.com/Quantum-Particles-Strings-Frontiers-Physics/dp/0201360799
Quote from the discription
"The book is unique in that it develops all three representations of quantum field theory (operator, functional Schrödinger, and path integral) for point particles and strings. In many cases, identical results are worked out in each representation to emphasize the representation-independent structures of quantum field theory"

Yet I don't see in this book or any other place deriving(showing) the relation between these representations, they only tell you "the results come out to be the same".
It sound very strange for me that the physicists write exuberant amounts of paper piles, yet they don't deal with fundamentals like that.

Last edited by a moderator: May 8, 2017
5. Apr 28, 2017

### Staff: Mentor

Could that be because getting the same results is the only relation between the representations?

6. Apr 28, 2017

### ftr

I expect that the relation should be a form similar to a post I made

For anybody who is interested in the subject of "Matrix Mechanics" I recommend this book "heisenberg's quantum mechanics "

http://www.worldscientific.com/worldscibooks/10.1142/7702

which includes the derivation of the commutation and the equivalency between the Schrodinger and Heisenberg pictures in the free three first chapters.

7. Apr 28, 2017

### Staff: Mentor

So in other words, you're looking for a proof of the equivalence of the canonical and path integral formulations of QFT, similar to the proof of the equivalence between the Schrodinger and Heisenberg formulations of ordinary QM?

8. Apr 29, 2017

### A. Neumaier

The main reason is that the precise relations are not known, or known only in toy cases, being generalized intuitively from the case of ordinary quantum mechanics (1+0-dimensional field theory). In the latter case, you can read about the precise relation with mathematical rigor for example in the first chapter of the book by Glimm and Jaffe.

The second reason is that practical theoretical physicists judge techniques by their results and don't need the fundamentals if they get more complicated than the calculations they need to do. Publication pressure then sweeps all these fundamentally desirable but intellectually frustrating under the carpet....

9. Apr 29, 2017

### vanhees71

Then you've simply not looked at the right place. Dirac's original work on QT answered this question, and it's of course marvelously worked out in his famous textbook. I've also trouble to find any modern textbook that doesn't give these connections.

Take the representation-free formulation (Dirac's bra-ket formalism) and use either the position representation to get wave mechanics or the harmonic-oscillator basis to get matrix mechanics. That's it. It's easier than rocket science ;-).

There are some older textbooks overemphasizing wave mechanics, which I don't like, although these books are partially very good otherwise (like Sommerfeld's classic "Atombau und Spektrallinien" (which I'd translate freely with "Atomic structure and spectral lines"), which has the clear advantage of not getting lost in philosophical speculations but are written in the good Sommerfeld style, working out clearly and comprehensively the mathematical methods and applying them to real-world problems).

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10. Apr 29, 2017

### ftr

Yes, I have seen the QT, but I am talking about QFT, at Least it is very hard for me to see the generalization.

11. Apr 29, 2017

### atyy

The correspondence for relativistic QFT is given by eg. the Osterwalder-Schrader conditions. If one constructs a path integral theory satisfying such axioms, then one knows the corresponding quantum theory satisfying the Wightmann axioms exists. Rigourous relativistic QFTs in 2 and 3 spacetime dimensions have been constructed by such methods.

http://www.arthurjaffe.com/Assets/pdf/CQFT.pdf

https://www.amazon.com/Quantum-Mechanics-Field-Theory-Mathematical/dp/1107005094

http://www.rivasseau.com/resources/book.pdf

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12. Apr 29, 2017

### A. Neumaier

No. The latter gives the connection between the Minkowski theory and the Euclidean theory in the Wightman formulation only. Both have also a canonical and a path integral version, and the relation between these versions is logically precise only in the Euclidean case. The relation between Euclidean path integrals and Minkowski path integrals is not rigorous at all.

13. Apr 29, 2017

### samalkhaiat

If this is true, then you should be able to

1) use the Path Integral formulation to derive:

(a) the Schrödinger equation $i\partial_{t}|\Psi \rangle = H(\varphi , \pi )|\Psi \rangle$, (b) the commutation relations $[\varphi , \pi ] = i\delta$, etc., and (c) the quantum Hamilton’s equations (also called Heisenberg equations) $\dot{\varphi} = [iH , \pi]$, etc.

2) write the functional integral in a coherent state basis, and used it to derive the commutation relations $[a , a^{\dagger}] = \delta$, etc.

If you cannot derive all of the above, then you should consider reading more texts.

14. Apr 29, 2017

### ftr

I appreciate some references that you could recommend. Thanks

15. Apr 29, 2017

### samalkhaiat

L. S. Schulman, “Techniques and Applications of Path Integration”, John Wiley & Sons, Inc. 1980.

It is (in my opinion) the best book ever written on path integral for both mathematicians and physicists.

16. Apr 29, 2017

### ftr

Last edited by a moderator: May 8, 2017