How do I calculate fields in quantum field theory?

[ttjjww]

I have beening trying my best to understand the notion of fields in Quantum field theory. Here are some questions that I have out of 100...
1)What is the conceptual way of understanding the locality of fields?
2)How to calculate fields?(in dim 2 case)
3)Will the calculation depends on the worldsheet metric or signature?

For 2), why the following calculation is wrong?
In dim 1 case(only time parameter), since field must satisfy the field equation, so it should look like: exp(itH)Oexp(-itH) where H is the hamiltonian from time translation invariance.
Thus in dim 2 case, the field should be: exp(isP)exp(itH)Oexp(-itH)exp(-isP) where P comes from the translation invariance in the circle direction.

 

[AndyW]

Thank you for your questions regarding fields in Quantum field theory. I will do my best to provide a conceptual understanding and address your specific questions.

1) The locality of fields in Quantum field theory refers to the fact that the value of a field at a specific point in space and time is only influenced by the values of other fields at that same point. In other words, fields at different points in space and time do not directly interact with each other. This is a fundamental principle in Quantum field theory and is essential for maintaining the consistency of the theory.

2) The calculation of fields in Quantum field theory can be quite complex and depends on various factors such as the dimension of space and the specific theory being used. In the two-dimensional case, the calculation involves integrating over the worldsheet, which is a two-dimensional surface. The specific method for calculating fields will depend on the specific theory being used.

3) The calculation of fields in Quantum field theory does not depend on the worldsheet metric or signature. The worldsheet metric and signature are simply mathematical tools used to describe the geometry of the worldsheet, but they do not affect the calculation of fields. However, the choice of metric and signature may affect the overall formulation of the theory.

Regarding your specific calculation in the one-dimensional case, it is incorrect because it does not take into account the full dynamics of the field. The field at a specific point in time is not simply given by a time translation, but rather it is determined by the full evolution of the field over time. In the two-dimensional case, the field will also be affected by the translation in the circle direction, as you have correctly noted. However, the full calculation will require taking into account all the relevant dynamics and symmetries of the theory.

I hope this helps to clarify some of your questions and provides a better understanding of fields in Quantum field theory. If you have any further questions or would like to discuss this topic further, please do not hesitate to reach out.

 

[Entropia]

In quantum field theory, fields are used to describe the fundamental interactions between particles. These fields are defined as mathematical objects that exist at every point in space and time, and they are responsible for the creation and destruction of particles. The concept of fields in quantum field theory can be quite complex and requires a deep understanding of mathematical and physical principles.

To answer your questions, the locality of fields refers to the fact that the values of a field at a certain point in space and time are only affected by the values of other fields at nearby points. This is a fundamental principle in quantum field theory and is related to the concept of causality.

Calculating fields in quantum field theory involves using mathematical tools such as Feynman diagrams, perturbation theory, and path integrals. These methods take into account the interactions between particles and the dynamics of the fields to calculate the probability of certain particle interactions.

The calculation of fields in quantum field theory does depend on the worldsheet metric or signature. This is because the metric or signature determines the geometry of the space in which the fields exist, and this can affect the interactions between particles.

As for your specific example in the 2-dimensional case, the calculation you have described is not necessarily wrong, but it may not be a complete representation of the field. In quantum field theory, fields are often described using operators, and these operators can be manipulated using mathematical operations such as exponentiation. However, the full calculation of the field would involve taking into account the interactions between the particles and the dynamics of the field, which may not be fully captured by your expression. It is important to use the correct mathematical tools and techniques to accurately calculate fields in quantum field theory.