Obtaining Dirac equation from symmetries

In summary, it is possible to derive the non-interacting Dirac equation from representations of the Poincare group and the path to do so is more complex than the derivation of the Schrodinger and Klein-Gordon equations. There have been various derivations of the Dirac equation, some more adhoc than others, but the cleanest ones tend to be algebraic in nature.
  • #1
tomkeus
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If we consider nonrelativistic QM, we will find Galilean group under the hood. Thanks to this, group theory enables us to find equations of motion directly from the symmetry principles. For example, if we take only geometric symmetries, we will get that the state space is broken into irreducible spaces characterized by momentum and angular momentum. For elementary system, angular momentum is zero so, condition that state lies in k-irreducible space (k being the momentum) is [tex]|\vec{P}||\Psi\rangle =|\vec{k}||\Psi\rangle[/tex] or [tex]P^2|\Psi\rangle=k^2|\Psi\rangle[/tex] which is Shrodinger equation for free particle [tex](\nabla^2+k^2)\Psi(\vec{r})=0[/tex].

If we now take the Poencare group as group representing space-time symmetries and apply the same reasoning as above we will come to the relation [tex]P_{\mu}P^{\mu}|\Psi\rangle=m^2|\Psi\rangle[/tex] being the Klein-Gordon equation for free particle [tex](\Box+m^2)\Psi(x)=0[/tex].

Now, what is common to both equations is that spin is excluded, that is to say, we are for some reason prevented to take into consideration both the rotational and translational symmetry so we cannot adequatly describe spin-[tex]\frac{1}{2}[/tex] particles.

My question is, is there a way to get Dirac equation the same way Schrodinger and Klein-Gordnon can be directly obtained from symmetries or we have to follow little more complicated path? If we cannot, what prevents us? Is the reason for this that Dirac equation is first order in derivatives? Is there a second order version of Dirac equation?
 
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  • #2
tomkeus said:
My question is, is there a way to get Dirac equation the same way Schrodinger and Klein-Gordnon can be directly obtained from symmetries or we have to follow little more complicated path?

The answers are "yes" and "yes," that is, it is possible to derive the non-interacting Dirac equation from (infinite-dimensional unitary) representations of the Poincare group (special-relativistic version of the Galilean group), and the path is substantially more complicated.
 
  • #3
Yup.

Dirac's eqn has been derived in many different contexts with different starting assumptions.. Some are a little adhoc, others are not (usually in increasing lvls of complexity).

The cleanest derivations imo tend to be algebraic in nature.
 

1. What is the Dirac equation?

The Dirac equation is a mathematical equation that describes the behavior of fermions, which are particles with half-integer spin, in quantum mechanics. It was developed by physicist Paul Dirac in 1928 and is used to describe the behavior of fundamental particles such as electrons and protons.

2. How is the Dirac equation obtained from symmetries?

The Dirac equation is obtained from symmetries through a mathematical framework called group theory. This theory allows scientists to understand the symmetries of a physical system and use them to derive equations that describe its behavior. In the case of the Dirac equation, group theory is used to describe the symmetries of spacetime and the behavior of fermions within it.

3. What are the symmetries involved in obtaining the Dirac equation?

The symmetries involved in obtaining the Dirac equation are the symmetries of spacetime, specifically Lorentz symmetry. This symmetry describes the invariance of physical laws under transformations of time and space. In addition, the Dirac equation also involves the symmetries of spin, which is a property of particles related to their angular momentum.

4. Why is it important to obtain the Dirac equation from symmetries?

Obtaining the Dirac equation from symmetries is important because it provides a fundamental understanding of the behavior of fermions in quantum mechanics. It also allows scientists to make predictions and calculations about the behavior of these particles, which has practical applications in fields such as particle physics and quantum computing.

5. Are there other equations that can be obtained from symmetries?

Yes, there are other equations that can be obtained from symmetries, such as the Maxwell equations for electromagnetism and the Schrodinger equation for non-relativistic quantum mechanics. Group theory and symmetry principles are fundamental tools in theoretical physics and are used to derive many important equations in different areas of physics.

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