ftr said:How does raising and lowering -annihilation/ creation-(is that energy or particle number?) translate to transition probabilities of path integral.
Well, I am reading this bookPeterDonis said:This question is too vague. Can you find a specific reference that has a specific example that illustrates the issue?
ftr said: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".
PeterDonis said:Could that be because getting the same results is the only relation between the representations?
ftr said:includes the derivation of the commutation and the equivalency between the Schrodinger and Heisenberg pictures
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.ftr said: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.ftr said: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.
vanhees71 said:Dirac's original work on QT answered this question
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.atyy said:The correspondence for relativistic QFT is given by eg. the Osterwalder-Schrader conditions.
If this is true, then you should be able toftr said:I have read many textbooks
samalkhaiat said:If this is true, then you should be able to
1) use the Path Integral formulation to derive:
(a) the Schrödinger equation [itex]i\partial_{t}|\Psi \rangle = H(\varphi , \pi )|\Psi \rangle [/itex], (b) the commutation relations [itex][\varphi , \pi ] = i\delta[/itex], etc., and (c) the quantum Hamilton’s equations (also called Heisenberg equations) [itex]\dot{\varphi} = [iH , \pi][/itex], etc.
2) write the functional integral in a coherent state basis, and used it to derive the commutation relations [itex][a , a^{\dagger}] = \delta[/itex], etc.
If you cannot derive all of the above, then you should consider reading more texts.
L. S. Schulman, “Techniques and Applications of Path Integration”, John Wiley & Sons, Inc. 1980.ftr said:I appreciate some references that you could recommend. Thanks
samalkhaiat said: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.
The operator formulation of quantum field theory describes the evolution of a system in terms of operators acting on a state vector. On the other hand, the path integral formulation describes the evolution of a system in terms of all possible paths that the system can take. Both formulations are equivalent and can be used to calculate physical quantities in QFT.
The operator formulation is more commonly used in quantum field theory, as it is more intuitive and easier to apply to specific systems. The path integral formulation is often used as a conceptual tool to gain a deeper understanding of the underlying principles of QFT.
One advantage of the path integral formulation is that it allows for a more natural treatment of symmetries and gauge invariance. It also provides a more intuitive way of dealing with divergences in calculations. In some cases, the path integral formulation can also simplify calculations compared to the operator formulation.
The two formulations are mathematically equivalent, meaning they describe the same physical system and produce the same results. This was proven by Richard Feynman in his famous work on the path integral formulation of quantum mechanics.
Yes, both formulations can be used interchangeably, although one may be more convenient than the other depending on the specific problem at hand. Some calculations may be easier to perform using the operator formulation, while others may be better suited for the path integral formulation. It is important to understand both formulations in order to fully grasp the concepts of quantum field theory.