Operators fields and classical fields

[quantumfireball]

Whats the intuition behind the concept of current operator in QFT and PP.
For example i know that the charge operaor which correspond to space integral of J-o
when acted upon a fock space of the field of given type gives the total charge in the field
but what about the remaining components ,how to intepret them?


NB: Plz forgive me for being a MORON

 

[blechman]

It is exactly the same as the classical current operator - it represents the flow of charge:

$$J^0\sim\rho$$
$$J^i\sim\rho v^i$$

 

[sir-pinski]

First of all, there is no need to apologize for asking for clarification or for not fully understanding a concept. We all have to start somewhere and learning can be a challenging process.

To answer your question, let's first define what operators and classical fields are in the context of quantum field theory (QFT). Operators in QFT are mathematical objects that act on the state of a quantum system, and their eigenvalues give the possible outcomes of a measurement. In classical field theory, fields are continuous functions that describe the behavior of a physical system.

Now, in QFT, operators can be constructed from classical fields by promoting them to operator-valued fields. This means that instead of just being continuous functions, the fields now have operator components that act on the quantum state. The concept of a current operator arises from this construction, where the classical current field is promoted to an operator-valued field.

The intuition behind the current operator is that it represents the flow of a conserved quantity, such as charge or energy, in a quantum system. In QFT, the total charge or energy in a field is not a fixed value, but rather a probability distribution described by the state of the system. The current operator allows us to calculate the expectation value of this quantity in a given state.

As for the remaining components of the current operator, they represent different aspects of the flow of the conserved quantity. For example, in the case of charge, the remaining components of the current operator represent the flow of charge in different directions or the flow of different types of charge. These components can also be interpreted as the creation and annihilation of particles carrying the conserved quantity.

In summary, the concept of current operator in QFT and PP is intimately connected to the idea of promoting classical fields to operator-valued fields. It allows us to calculate the expectation value of a conserved quantity in a given quantum state and provides insight into the flow and behavior of this quantity in the system. I hope this helps clarify the concept for you. Keep asking questions and never be afraid to seek clarification – that's how we learn.