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The answer is basically the one you've suggested. When we collide particles in e.g. the LHC we are not colliding point particles. We are colliding two wavefunctions that look like semi-localised plane waves. The collision would look something like:

Collision

So classically the two particles would miss each other, but in reality their positions are delocalised so there is some overlap even though the centres (i.e. the average positions) of the two particles miss each other.

I've drawn a green squiggle to vaguely indicate some interaction between the two particles, but you shouldn't take this too literally. What actually happens is that both particles are described as states of a quantum field. When the particles are far from each other they are approximately Fock states i.e. plane waves.

However when the particles approach each other they become entangled and now the state of the quantum field cannot simply be separated into states of two particles. In fact we don't have a precise description of the state of the field when the particles are interacting strongly - we have to approximate the interaction using perturbation theory, which is where those Feynmann diagrams come in.

So to summarise: we should replace the verb collide with interact, and the interaction occurs because the two particles overlap even when their centres are separated. We calculate that interaction using quantum field theory, and the interaction strength will depend on the distance of closest approach.

The OP asks in a comment:

So, that interaction causes two particles to "blow up", and disintegrate into its more elementary particles?

I mentioned above that the particles are a state of the quantum field and that when far apart that state is separable into the two Fock states that describe the two particles. When the particles are close enough to interact strongly the state of the field cannot be separated into separate particle states. Instead we have some complicated state that we cannot describe exactly.

This intermediate state evolves with time, and depending on the energy it can evolve in different ways. It could for example just evolve back into the two original particles and those two particles head off with the same total energy. But if the energy is high enough the intermediate state could evolve into states with different numbers of particles, and this is exactly how particles get create in colliders. We can't say what will happen, but we can calculate the probabilities for all the possible outcomes using quantum field theory.

The key point is that the intermediate state does not simply correspond to a definite number of specific particles. It is a state of the field not a state of particles.

The answer is basically the one you've suggested. When we collide particles in e.g. the LHC we are not colliding point particles. We are colliding two wavefunctions that look like semi-localised plane waves. The collision would look something like:

Collision

So classically the two particles would miss each other, but in reality their positions are delocalised so there is some overlap even though the centres (i.e. the average positions) of the two particles miss each other.

I've drawn a green squiggle to vaguely indicate some interaction between the two particles, but you shouldn't take this too literally. What actually happens is that both particles are described as states of a quantum field. When the particles are far from each other they are approximately Fock states i.e. plane waves.

However when the particles approach each other they become entangled and now the state of the quantum field cannot simply be separated into states of two particles. In fact we don't have a precise description of the state of the field when the particles are interacting strongly - we have to approximate the interaction using perturbation theory, which is where those Feynmann diagrams come in.

So to summarise: we should replace the verb collide with interact, and the interaction occurs because the two particles overlap even when their centres are separated. We calculate that interaction using quantum field theory, and the interaction strength will depend on the distance of closest approach.

The answer is basically the one you've suggested. When we collide particles in e.g. the LHC we are not colliding point particles. We are colliding two wavefunctions that look like semi-localised plane waves. The collision would look something like:

Collision

So classically the two particles would miss each other, but in reality their positions are delocalised so there is some overlap even though the centres (i.e. the average positions) of the two particles miss each other.

I've drawn a green squiggle to vaguely indicate some interaction between the two particles, but you shouldn't take this too literally. What actually happens is that both particles are described as states of a quantum field. When the particles are far from each other they are approximately Fock states i.e. plane waves.

However when the particles approach each other they become entangled and now the state of the quantum field cannot simply be separated into states of two particles. In fact we don't have a precise description of the state of the field when the particles are interacting strongly - we have to approximate the interaction using perturbation theory, which is where those Feynmann diagrams come in.

So to summarise: we should replace the verb collide with interact, and the interaction occurs because the two particles overlap even when their centres are separated. We calculate that interaction using quantum field theory, and the interaction strength will depend on the distance of closest approach.

The OP asks in a comment:

So, that interaction causes two particles to "blow up", and disintegrate into its more elementary particles?

I mentioned above that the particles are a state of the quantum field and that when far apart that state is separable into the two Fock states that describe the two particles. When the particles are close enough to interact strongly the state of the field cannot be separated into separate particle states. Instead we have some complicated state that we cannot describe exactly.

This intermediate state evolves with time, and depending on the energy it can evolve in different ways. It could for example just evolve back into the two original particles and those two particles head off with the same total energy. But if the energy is high enough the intermediate state could evolve into states with different numbers of particles, and this is exactly how particles get create in colliders. We can't say what will happen, but we can calculate the probabilities for all the possible outcomes using quantum field theory.

The key point is that the intermediate state does not simply correspond to a definite number of specific particles. It is a state of the field not a state of particles.

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The answer is basically the one you've suggested. When we collide particles in e.g. the LHC we are not colliding point particles. We are colliding two wavefunctions that look like semi-localised plane waves. The collision would look something like:

Collision

So classically the two particles would miss each other, but in reality their positions are delocalised so there is some overlap even though the centres (i.e. the average positions) of the two particles miss each other.

I've drawn a green squiggle to vaguely indicate some interaction between the two particles, but you shouldn't take this too literally. What actually happens is that both particles are described as states of a quantum field. When the particles are far from each other they are approximately Fock states i.e. plane waves.

However when the particles approach each other they become entangled and now the state of the quantum field cannot simply be separated into states of two particles. In fact we don't have a precise description of the state of the field when the particles are interacting strongly - we have to approximate the interaction using perturbation theory, which is where those Feynmann diagrams come in.

So to summarise: we should replace the verb collide with interact, and the interaction occurs because the two particles overlap even when their centres are separated. We calculate that interaction using quantum field theory, and the interaction strength will depend on the distance of closest approach.