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In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to another. However, a discrete symmetry can always be reinterpreted as a subset of some higher-dimensional ...
This theorem only applies to continuous and smooth symmetries of physical space. Noether's theorem is used in theoretical physics and the calculus of variations . It reveals the fundamental relation between the symmetries of a physical system and the conservation laws.
Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group). These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics.
Symmetries are of prime importance in physics and are closely related to the hypothesis that certain physical quantities are only relative and unobservable. [5] Symmetries apply to the equations that govern the physical laws (e.g. to a Hamiltonian or Lagrangian) rather than the initial conditions, values or magnitudes of the equations themselves and state that the laws remain unchanged under a ...
This prescription for a quantum field naturally generalizes to any situation where one can enumerate the continuous symmetries of the system, and define duals in a coherent, consistent fashion. The pairing ties together opposite charges in the fully abstract sense. In physics, a charge is associated with a generator of a continuous symmetry.
Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis , every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from ...
In physics, Lie groups often encode the symmetries of a physical system. The way one makes use of this symmetry to help analyze the system is often through representation theory. Consider, for example, the time-independent Schrödinger equation in quantum mechanics, H ^ ψ = E ψ {\displaystyle {\hat {H}}\psi =E\psi } .
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics.