On the equivalence of operator vs path integral in QFT

[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.

 

[ftr]

Bumping up an old thread with no answers, may this time.

 

[PeterDonis]

How does raising and lowering -annihilation/ creation-(is that energy or particle number?) translate to transition probabilities of path integral.

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

 

[ftr]

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

Well, I am reading this book
https://www.amazon.com/dp/0201360799/?tag=pfamazon01-20
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.

 

[PeterDonis]

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".

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

 

[ftr]

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

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 Schrödinger and Heisenberg pictures in the free three first chapters.

 

[PeterDonis]

includes the derivation of the commutation and the equivalency between the Schrödinger and Heisenberg pictures

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 Schrödinger and Heisenberg formulations of ordinary QM?

 

[A. Neumaier]

physicists write exuberant amounts of paper piles, yet they don't deal with fundamentals like that.

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...

 

[vanhees71]

Well, I am reading this book
https://www.amazon.com/dp/0201360799/?tag=pfamazon01-20
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.

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).

 

[ftr]

Dirac's original work on QT answered this question

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

 

[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/dp/1107005094/?tag=pfamazon01-20

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

 

[A. Neumaier]

The correspondence for relativistic QFT is given by eg. the Osterwalder-Schrader conditions.

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.

 

[samalkhaiat]

I have read many textbooks

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.

 

[ftr]

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.

I appreciate some references that you could recommend. Thanks

 

[samalkhaiat]

I appreciate some references that you could recommend. Thanks

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.

 

[ftr]

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.

Ok, Thanks. I do have a similar book which I did manage to pull it from the stuffed shelves (in two volumes).

https://www.amazon.com/dp/075030801X/?tag=pfamazon01-20