In brief, the basic variables are the quantum field and its canonical momentum.
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The canonical momentum is the conserved quantity.
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The Lagrangian indicates an additional detail : the canonical momentum in Lagrangian mechanics is given by:
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Where is the scalar potential and the vector potential, the components of the canonical momentum four-vector is
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Which constitute the standard formulae for canonical momentum and energy of a closed ( time-independent Lagrangian ) system.
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Where the second equality holds after simplifying with the Euler-Lagrange equations of motion and the definition of canonical momentum.
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The classical field is treated as a dynamical variable called the canonical coordinate, and its time-derivative is the canonical momentum.
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For instance the Heisenberg equations for the coordinate and the canonical momentum " m " "'? "'} } of the oscillator are:
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When describing the motion of a charged particle in an electromagnetic field, the canonical momentum "'P "'( derived from the Lagrangian for this system ) is not gauge invariant.
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For landmarks the superposition is a sum of weight kernels weighted by the canonical momentum which determines the inner product, for surfaces it is a surface integral, and for dense volumes it is a volume integral.