Normal ordering and Wick contraction?

[ismaili]

Dear all,

I have a formula at hand but I don't know how to derive it.
$$:e^{ik\cdot X(x)}::e^{ik'\cdot X(x')}: = \exp\{-k_\mu k'_\nu\langle X^\mu(x)X^\nu(x')\rangle\} :\exp\{ik\cdot X(x)+ik'\cdot X(x')\}:$$
where $$::$$ denotes normal ordering, and $$X^\mu(x)$$ is the field. $$\langle X^\mu(x)X^\nu(x')\rangle$$ is the propagator.
I met this problem in the study of vertex operators of string theory. I think this is a field theory stuff but I just can't derive it.
Any help will be appreciated.

 

[eljose79]

Thank you for bringing this formula to our attention. It appears to be related to string theory and involves normal ordering and propagators. To derive this formula, we will need to use some advanced mathematical techniques and concepts from field theory. Here is a brief explanation of how the formula can be derived:

First, we need to understand the concept of normal ordering in quantum field theory. Normal ordering is a way of ordering operators in a specific way to remove any divergences or infinities that may arise in calculations. In this case, we have two exponential operators and we need to normal order them to get rid of any divergences.

Next, we need to use the propagator, which is a mathematical representation of the probability amplitude for a particle to travel from one point to another in space-time. In this case, the propagator is represented by the term <X^\mu(x)X^\nu(x')>, which involves the fields X^\mu(x) and X^\nu(x') at two different points in space-time.

Using these concepts, we can expand the exponential terms and then use the normal ordering operator :: to rearrange the terms in a specific way. This will involve using the commutation relations between the fields and their conjugate momenta, as well as the propagator. After some algebraic manipulations, we will arrive at the final formula.

It is important to note that this derivation requires a strong understanding of quantum field theory and string theory. If you are not familiar with these concepts, it may be helpful to consult with a colleague or a textbook on these topics. Additionally, there may be other factors or assumptions involved in the specific context of your study of string theory that may affect the derivation of this formula.

I hope this explanation helps you in understanding and deriving the formula. If you have any further questions, please do not hesitate to ask. Best of luck with your research.

 

[denian]

Dear colleague,

Thank you for bringing up the topic of normal ordering and Wick contraction. These are important concepts in quantum field theory and are essential in understanding the behavior of fields and particles.

Normal ordering is a mathematical operation that rearranges operators in a quantum field theory expression such that all creation operators are on the left and all annihilation operators are on the right. This is necessary in order to properly calculate expectation values and correlation functions. In the formula you have provided, the normal ordering symbol :: ensures that the operators are arranged in this way.

Wick contraction, on the other hand, is a way to simplify expressions involving multiple fields by using the propagator, which represents the probability amplitude for a particle to propagate from one point to another. In your formula, the Wick contraction is represented by the term \langle X^\mu(x)X^\nu(x')\rangle, which is the propagator between the two points x and x'.

In the context of string theory, these concepts are used to calculate the scattering amplitudes of strings, which are represented by vertex operators. The formula you have provided is a key step in calculating these amplitudes.

I hope this explanation has helped clarify the concepts of normal ordering and Wick contraction. If you need further assistance in deriving the formula or understanding its application in string theory, please do not hesitate to ask for help. Best of luck in your studies.