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Exact laws

A partial listing of physical conservation equations due to symmetry that are said to be exact laws , or more precisely have never been proven to be violated: Conservation Law Respective Noether symmetry invariance Number of dimensions Conservation of mass-energy Time-translation invariance Lorentz invariance symmetry 1 translation along time axis Conservation of linear momentum Space-translation invariance 3 translation along x , y , z directions Conservation of angular momentum Rotation invariance 3 rotation about x , y , z axes Conservation of CM (center-of-momentum) velocity Lorentz-boost invariance 3 Lorentz-boost along x , y , z directions Conservation of electric charge Gauge invariance 1⊗4 scalar field (1D) in 4D spacetime ( x , y , z  + time evolution) Conservation of color charge SU(3) Gauge invariance 3 r , g , b Conservation of weak isospin SU(2) L Gauge invariance 1 weak charge Conservation of probability Probability invaria...
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Approximate laws

There are also approximate conservation laws. These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions. Conservation of mechanical energy Conservation of rest mass Conservation of baryon number (See chiral anomaly and sphaleron) Conservation of lepton number (In the Standard Model) Conservation of flavor (violated by the weak interaction) Conservation of parity Invariance under charge conjugation Invariance under time reversal CP symmetry, the combination of charge conjugation and parity (equivalent to time reversal if CPT holds)
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Differential forms

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In continuum mechanics, the most general form of an exact conservation law is given by a continuity equation. For example, conservation of electric charge q is ∂ ρ ∂ t = − ∇ ⋅ j {\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {j} \,} where ∇⋅ is the divergence operator, ρ is the density of q (amount per unit volume), j is the flux of q (amount crossing a unit area in unit time), and t is time. If we assume that the motion u of the charge is a continuous function of position and time, then j = ρ u {\displaystyle \mathbf {j} =\rho \mathbf {u} } ∂ ρ ∂ t = − ∇ ⋅ ( ρ u ) . {\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot (\rho \mathbf {u} )\,.} In one space dimension this can be put into the form of a homogeneous first-order quasilinear hyperbolic equation: y t + A ( y ) y x = 0 {\displaystyle y_{t}+A(y)y_{x}=0} where the dependent variable y ...
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