Can't find that on the dictionary

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In summary, Dirac's equation is a covariant relativistic equation, meaning it is a differential equation that describes fields transforming in a known way under the symmetry group of the theory. This is in contrast to noncovariant equations, which do not exhibit this behavior. An example of this is the electromagnetic field equations, which are noncovariant when written in terms of E and B, but become covariant when written in terms of the 2-form field. Additionally, solutions to Dirac's equation must also follow certain transformation properties, depending on the spin angular momentum of the field. An example of this is the matrix elements of a finite dimensional representation of the global symmetry group of the space-time manifold. This is relevant to studying Dirac
  • #1
go quantum!
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Why do people mean when they say that Dirac's equation is a covariant relativistic equation?
Thank you for dedicating your time helping me and anyone who finds this question useful for his life.
 
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  • #2
Well, <covariant> usually means 'Minkowski space related', <relativistic equation> means a differential equation with variables (unknown) transforming in a known way under the relativistic symmetry group of the theory.

For example, the field equations of the electromagnetic field written in terms of E and B are said to be noncovariant, while written in terms of the e-m 2-form field, they are covariant, as the 2-form field can be somehow (in a very complicated manner though) related to the geometry of the flat Minkowski space-time.
 
  • #3
In other words, you mean we are at the presence of a differential equation which as the following property. If we find a solution f(x), then f(Tx) is also a solution where by T I mean a lorentz transformation.

Did I get it right?
 
  • #4
It's actually more. f(Tx) is a solution, only if f is a scalar function. if 'f' describes fields with nonzero spin angular momentum, Pf(Tx) is also a solution of the transformed equation, where P describes the matrix elements of some finite dimensional representation of the global symmetry group of the space-time manifold.
 
  • #5
Would you give me an exemple for P?
Maybe I should contextualize what I am working on. I am doing a memoir about Dirac's equation for the course Symmetry Groups in Physics.
 
  • #6
Well, think if the e-m potential. What covariant equation would it obey in one frame of coordinates ? What about another frame ? How would the 2 potentials in the 2 frames be related ?
 

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