Equivalence Relations and Quotient Spaces

  • Albert S. Schwarz
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 307)


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.


Equivalence Class Equivalence Relation Topological Space Gauge Transformation Inverse Image 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Albert S. Schwarz
    • 1
  1. 1.Department of Mathematics, 565 Kerr HallUniversity of CaliforniaDavisUSA

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