Special Relativity: We have been using the word \"covariant\" throughout the cou

Question

Special Relativity: We have been using the word "covariant" throughout the course but what does it mean? It means the object can only have a "lower index" version and no "upper index" version. It means the object has the same form in every inertial frame. It's components has different values in different inertial frames. It means the object is not subjected to Lorentz transformations. It means the object has the same numerical value in every inertial frame. So this object has to be a Lorentz scalar. Special Relativity: We have been using the word "invariant" throughout the course but what does it mean? It means the object is a conserved quantity under Noether's theorem. It means the object has the same form in every inertial frame. It's components has different values in different inertial frames. It means the object is not subjected to Lorentz transformations. It means the object has the same numerical value in every inertial frame. So this object has to be a Lorentz scalar. Classical Field Theory: For the free relativistic charged particle, the Hamiltonian was obtained to be H = squareroot p^2c^2+m^2c^4 and we could go further and say that equations of motion are Hamilton's equations. Why, do you think, is there less interest in this description in this case? This is because the Hamiltonian is total energy E which is a 0-component of a 4-vector. The Hamilton's equations are going to be quantities that transform in any ugly way. This is because the Hamiltonian description is only useful for Quantum Mechanics. This is because the Hamiltonian description will give less information than the Lagrangian description. This is because the Hamiltonian description cannot be subjected to Lorentz transformations.

Explanation / Answer

for question no 6 the correct option b will be correct while studying special theory of relativity actually we talk about the lorentz covarient which sates that if it can be written in terms of lorentz covarient quantities then if they hold in one inertial frame they will hold in any inertial frame

for questio number 7 we have the correct option as d will be correct one because in such cases lorentz scalar is a scalar which invarient under lorentz transformation so if we talk about the invarient it means that we are talking about lorentz scalar and invarient means means same value in all inertial frame

i think for question 8 the actually hamiltonian is mainly used for the discription in statistical mechanics and quantum mechanics in this if we will use it then our description will not be sufficient and we canot reach at a proper conclusion

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