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The Dirac equation is () = If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames: ′ ′ (′) ′ (′) = The two spinors and ′ should both describe the same physical field, and so should be related by a transformation that does not change any physical observables (charge, current, mass, etc.
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In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos.It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group.
After constructing the fiber bundle, one may then consider differential equations, such as the Dirac equation, or the Weyl equation on the fiber bundle. These equations (Dirac or Weyl) have solutions that are plane waves , having symmetries characteristic of the fibers, i.e. having the symmetries of spinors, as obtained from the (zero ...
An alternative version of the Dirac equation whose Dirac operator remains the square root of the Laplacian is given by the Dirac–Kähler equation; the price to pay is the loss of Lorentz invariance in curved spacetime. Note that here Latin indices denote the "Lorentzian" vierbein labels while Greek indices denote manifold coordinate indices.
In the study of Dirac fields in quantum field theory, Richard Feynman introduced the convenient Feynman slash notation (less commonly known as the Dirac slash notation [1]). If A is a covariant vector (i.e., a 1-form ),
Since the Dirac bracket respects the constraints, one need not be careful about evaluating all brackets before using any weak equations, as is the case with the Poisson bracket. Note that while the Poisson bracket of bosonic (Grassmann even) variables with itself must vanish, the Poisson bracket of fermions represented as a Grassmann variables ...