Relativistic quantum mechanics

[Gigi]

I read in wikipedia the following:

'For example, the Schrödinger equation does not keep its written form under the coordinate transformations of special relativity; thus one might say that the Schrödinger equation is not covariant. By contrast, the Klein-Gordon equation and the Dirac equation take the same form in any coordinate frame of special relativity: thus, one might say that these equations are covariant. More properly, one should really say that the Klein-Gordon and Dirac equations are invariant, and that the Schrödinger equation is not, but this is not the dominant usage. '


Thus I have a question on the transition from quantum mechanics to relativistic quantum mechanics:

In non-relativistic quantum mechanics, if the Hamiltonian commutes with J, the angular momentum operator that is a generator of the SU(2) group, it will be rotationally invariant and after a rotation the system will still obey Schroedinger’s equation.

In relativistic quantum mechanics Schroedingers equation is replaced by the Klein Gordon and Dirac equations. Why?
Is it because the corresponding relativistic Hamiltonians in relativistic QM only commute with generators of the Lorentz group, while they do not commute with generators of the SU(2) group---leading to a lack of rotational invariance?

Is that why when we want to combine relativity with quantum mechanics, only a relativistic hamiltonian would work?

Please help---I am so confused :)!

Many thanks!

 

[malawi_glenn]

Well at first glance, the energy in SE is p^2/2m + V

This is non relativistic expression for energy, so here is the first "proof" that SE is not appropative in relativistic considerations of quantum physics.

 

[blechman]

The SE is invariant under rotations (assuming that it commutes with angular momentum, as you say) but it is NOT invariant under boosts! That's where the trouble comes in.

 

[Gigi]

OK, right that is why we need to replace SE.

One more point here that I still find unclear.
In non-relativistic QM SE is invariant with respect to rotations and translations, if it commutes with the generators of the relevant transformations. The group is SU(2), regarding the rotational symmetry of the non- relativistic Schroedinger equation.

Such a group cannot underline the rotational symmetries of anything relativistic, right?

The rotational symmetries in a relativistic context can only be dealt with Lorentz transforms, which rotational and translational symmetries together by the Poincare group. Right?

 

[blechman]

(special) relativistic theories have a SO(3,1) symmetry group consisting of rotations and boosts, yes.

 

[Gigi]

Hi...sorry I did not get what you wrote. Is what I said true?

Is the SO(3,1) group the Poincare group? Thanks:)

 

[strangerep]

In non-relativistic QM SE is invariant with respect to rotations and translations, if it commutes with the generators of the relevant transformations. The group is SU(2), regarding the rotational symmetry of the non-relativistic Schroedinger equation.

Such a group cannot underline the rotational symmetries of anything relativistic, right?

The rotational symmetries in a relativistic context can only be dealt with Lorentz transforms, which rotational and translational symmetries together by the Poincare group. Right?

SO(3) or SU(2) arise in a relativistic context when one considers the degrees of freedom
of a massive particle at rest. This is actually part of the standard method by which one
classifies what types of elementary particles can exist under the Poincare group.
(This is called "finding the unitary irreducible representations" of the Poincare group.)
The procedure starts by finding all the operators that commute with the Poincare
group generators. These operators are called "Casimir" operators. For the Poincare
algebra, there are only two: the "mass-squared" operator $P^2 = P_0^2 - P_i^2$,
and "covariant-spin-squared" operator $W^2$ which I won't write out in full here.

The eigenvalues of these operators classify the types of elementary particles
(because a particle with one distinct set of these eigenvalues can never have a
different set under Poincare transformations - because the Casimir operators
commute with all Poincare generators).

The Klein-Gordon and Dirac equations are just different ways of writing the
mass-squared Casimir equation above, when represented in particular Hilbert
spaces. For a massive particle, we can find the possible values of spin by
analyzing things in the rest frame, where the only remaining degrees of
freedom are expressed by ordinary rotations, i.e., SO(3) (or its double-cover SU(2)).

Similarly, the Schroedinger equation expresses a Casimir operator from
a different (non-relativistic, i.e., non-Poincare) group.

Is the SO(3,1) group the Poincare group?

SO(3,1) is the Lorentz group (also called the homogeneous Lorentz
group). The Poincare group is the Lorentz group, together with
translations in space and time (and is sometimes called the
inhomogeneous Lorentz group).[BTW, this thread should probably be in the quantum physics forum.]