Equivalence Relations and Quotient Spaces
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.