Quantum Field Theory and Topology pp 233-234 | Cite as

# Equivalence Relations and Quotient Spaces

## Abstract

In many situations in physics and mathematics it is reasonable to consider two different objects as equivalent in some sense. For example, in quantum mechanics the state of a particle or system of particles can be described by a nonzero vector in a complex Hilbert space (the state vector). But two vectors *ψ* and *ψ*′ proportional to each other are physically equivalent, that is, they describe the same state. Likewise, an electromagnetic field can be described by a vector potential, but two potentials *A* _{ μ } ^{′} (*x*) and *A* _{ µ }
(*x*) that differ by a gauge transformation (that is, that satisfy *A* _{ μ } ^{′} (*x*) = *A* _{ μ }(*x*) + *∂* _{ μ } *λ*(*x*)) are physically equivalent.