"On-shell" refers to conditions where fields satisfy their equations of motion. In other words, it describes situations where the fields obey the physical laws governing their dynamics, such as the Klein-Gordon equation for scalar fields or the Dirac equation for fermionic fields.
Not necessarily. If a field vanishes on-shell, it means that under the physical conditions described by the equations of motion, the field does not contribute. However, the field might still play a role off-shell (i.e., not satisfying the equations of motion) or in other contexts such as in gauge fixing or in the structure of the theory.
If a field solution vanishes on-shell, it typically means that the field does not contribute directly to physical observables when the system is in a state that satisfies the equations of motion. However, its presence might still affect the theory’s structure, such as symmetries or interactions, indirectly influencing other fields and observables.
Yes, a field that vanishes on-shell can still influence quantum corrections or virtual processes. In quantum field theory, fields contribute to loop diagrams and virtual particle exchanges even if their classical on-shell value is zero. These contributions can affect renormalization, effective potentials, and other quantum corrections.
In some cases, a vanishing on-shell solution might indicate a symmetry of the theory. For example, in gauge theories, certain fields can be set to zero by a choice of gauge, reflecting a gauge symmetry. Similarly, a vanishing solution might be associated with spontaneous symmetry breaking or other symmetry-related phenomena.