Difference between s-matrix and interactive field propagator?

[its_blitz!]

In an interactive field theory we can compute the amplitude of a particle propagating from y to x by evaluating perturbatively expressions of the form <GS|o(x)o(y)|GS> where GS stands for ground state and o are the field operators. This can be extended to higher number of operators for more particles.

My question is since we can already create states and compute amplitudes for arriving at other states in the interaction theory using the above method for evaluating green's functions, what is the need for defining in and out states and relating them via a scattering matrix that time evolves the in state? Isn't the green's function where the time y0 tends to -infinity and x0 tends to +infinity already giving you the necessary amplitude for an in state to arrive at an out state?

 

[azntoon]

The scattering matrix is important when we are dealing with quantum field theory. It allows us to relate the in and out states which are defined in terms of creation and annihilation operators that act on the vacuum state. The scattering matrix is used to calculate the transition probabilities of a particle going from an initial state to a final state. This is used to describe the behavior of particles over time. In contrast, the Green's function is used to describe the dynamics of a single particle propagating in spacetime. It can be used to calculate the probability amplitude of a particle going from one point in space to another. It is not used to describe the evolution of systems of particles over time.

 

[Entropia]

The main difference between the S-matrix and interactive field propagator lies in their respective approaches to calculating amplitudes for particle interactions. The interactive field propagator, as described in the content, uses perturbative expressions to compute the amplitude of a particle propagating from one point to another. This method is based on the use of field operators and can be extended to higher numbers of operators for more particles.

On the other hand, the S-matrix approach involves defining in and out states and using a scattering matrix to time evolve the in state to the out state. This method takes into account the entire interaction process, rather than just the propagation of a single particle. It also allows for the calculation of scattering amplitudes for multiple particles, including the possibility of particles being created or annihilated during the interaction.

In summary, while the interactive field propagator is useful for calculating the propagation of individual particles, the S-matrix approach provides a more comprehensive understanding of the overall interaction process and allows for the calculation of scattering amplitudes involving multiple particles. Both methods have their own advantages and are used in different contexts within theoretical physics.