Relativistic quantum mechanics

In summary, quantum mechanics and relativistic quantum mechanics have different symmetry groups that govern the behavior of particles. The Schrödinger equation is not covariant under coordinate transformations of special relativity, while the Klein-Gordon and Dirac equations are. This is why in relativistic quantum mechanics, the Schrödinger equation is replaced by the Klein-Gordon and Dirac equations. The rotational symmetries in a relativistic context are dealt with by the Lorentz group, which is a part of the Poincare group. The Klein-Gordon and Dirac equations are different ways of writing the mass-squared Casimir equation, which is used to classify the different types of elementary particles. The SO(3,1
  • #1
Gigi
21
0
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!
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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?
 
  • #5
(special) relativistic theories have a SO(3,1) symmetry group consisting of rotations and boosts, yes.
 
  • #6
Hi...sorry I did not get what you wrote. Is what I said true?

Is the SO(3,1) group the Poincare group? Thanks:)
 
  • #7
Gigi said:
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 [itex]P^2 = P_0^2 - P_i^2[/itex],
and "covariant-spin-squared" operator [itex]W^2[/itex] 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.]
 
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Related to Relativistic quantum mechanics

1. What is the difference between classical mechanics and relativistic quantum mechanics?

Classical mechanics is a branch of physics that describes the behavior of macroscopic objects at non-relativistic speeds, while relativistic quantum mechanics is a theoretical framework that describes the behavior of particles at high speeds and small scales.

2. How does relativistic quantum mechanics explain the behavior of particles?

Relativistic quantum mechanics uses mathematical equations, such as the Schrödinger equation, to describe the wave-like behavior of particles and their interactions with other particles and fields.

3. What is the role of special relativity in relativistic quantum mechanics?

Special relativity is an essential component of relativistic quantum mechanics as it describes how the laws of physics, particularly those governing the behavior of particles, change at high speeds and in different reference frames.

4. Can relativistic quantum mechanics be applied to all particles?

Yes, relativistic quantum mechanics can be applied to all particles, including both matter particles (such as electrons and protons) and force-carrying particles (such as photons and gluons).

5. What are some practical applications of relativistic quantum mechanics?

Relativistic quantum mechanics has numerous practical applications, including the development of technologies such as transistors, lasers, and medical imaging devices. It also plays a crucial role in understanding phenomena such as nuclear reactions, superconductivity, and particle accelerators.

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