Can't find that on the dictionary

[go quantum!]

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.

 

[dextercioby]

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.

 

[go quantum!]

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?

 

[dextercioby]

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.

 

[go quantum!]

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.

 

[dextercioby]

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 ?