The discussion clarifies the relationship between Quantum Mechanics (QM) and Quantum Field Theory (QFT), establishing that QM is not a subset of QFT. Key distinctions include QM's use of position variables and the Born rule, while QFT operates with fields and requires renormalization due to infinite quantities. The conversation highlights that both theories share common postulates like unitarity and cluster decomposition, but differ fundamentally in their treatment of symmetries—Galilean invariance in QM versus Poincaré invariance in QFT. The conclusion posits that QM can be viewed as a non-relativistic limit of QFT, particularly through the lens of second quantization.
PREREQUISITESPhysicists, particularly those specializing in theoretical physics, quantum mechanics, and quantum field theory, as well as students seeking to deepen their understanding of the foundational concepts connecting QM and QFT.
meopemuk said:Strictly speaking, QFT is about variable number of particles.
No. A quantum field theory is primarily about fields, as the name says. Particles appear in general only as approximate asymptotic concepts.meopemuk said:Strictly speaking, QFT is about variable number of particles.
The particle number operator cannot be renormalized in the standard perturbation scheme. To get finite results one needs to renormalize, and to get approximate lifetimes, one needs to sum infinite number of renormalized diagrams - which is neither fixed order perturbation theory nor doable in Fock space.meopemuk said:As far as I understand, the "perturbative/nonperturbative" refers to the method chosen to calculate the S-matrix (including the lifetime of positronium). If you are lucky, you can do that without using the perturbation series, but in most cases it is easier to use the perturbative approach. The chosen calculation method should not have any effect on the (Fock) structure of the Hilbert space. The Fock space structure simply tells you that numbers of particles are good quantum-mechanical observables that commute with each other. So, each Fock sector is a common eigenspace of all particle number operators in the theory.
A. Neumaier said:The particle number operator cannot be renormalized in the standard perturbation scheme. To get finite results one needs to renormalize, and to get approximate lifetimes, one needs to sum infinite number of renormalized diagrams - which is neither fixed order perturbation theory nor doable in Fock space.
Because unrenormalized operators have no meaning in the renormalized theory - all expectations containing it are ill-defined.meopemuk said:Why would we want to renormalize the particle number operator?
Positronium is an unstable composite particle which has a finite lifetime computable by QED.meopemuk said:Besides, in QED we are dealing with stable particles - photons, electrons, protons. What the "lifetimes" have to do with them?