Is there a paradox involving the Dirac equation and commutation with time?

[ginda770]

I was hoping someone could help me with a seeming paradox involving the Dirac equation. I have taken a non-relativistic QM course, but am new to relativistic theory.

The Dirac equation is (following Shankar)

[tex]i\frac{\partial}{\partial t}\psi = H\psi[/tex]

where

[tex]H = \vec{\alpha}\cdot \vec{p} + \beta m[/tex]

([tex]\psi[/tex] is a four component wavefunction and the alphas and beta are 4 by 4 matrices with constant entries)

It seems to me that any alpha matrix (or almost any other 4 by 4 matrix made up of constants) commutes with [tex]\partial/\partial t[/tex], but not with the hamiltonian [tex]H[/tex]. How can this be true? If [tex]\left[\vec{\alpha},H\right] \neq 0[/tex] and [tex]H = i \left(\partial/\partial t\right)[/tex] how can [tex]\left[\vec{\alpha},\partial/\partial t\right]=0[/tex] ? What am I missing?

 

[ginda770]

Never mind, I figured it out. Stupid question.

 

[Entropia]

There is indeed a paradox involving the Dirac equation and commutation with time. This is known as the "Zitterbewegung paradox" and it arises from the fact that the Dirac equation is a relativistic equation, meaning it takes into account the effects of special relativity such as time dilation and length contraction.

The paradox arises when we consider the commutation relation between the position operator and the Hamiltonian in the Dirac equation. In non-relativistic quantum mechanics, the position operator commutes with the Hamiltonian, meaning they can be measured simultaneously with no uncertainty. However, in the Dirac equation, the position operator does not commute with the Hamiltonian, leading to a non-zero uncertainty between the two.

This paradox can be resolved by considering the fact that the Dirac equation describes a particle with spin 1/2, which has both positive and negative energy states. When we consider both energy states, we see that the position operator does commute with the Hamiltonian for each individual energy state, but not when we consider the full equation with both energy states. This leads to the observed uncertainty in the position and Hamiltonian.

In summary, the paradox arises from the fact that the Dirac equation describes a relativistic particle with spin, and the resolution lies in understanding the behavior of spin and energy states in relativistic quantum mechanics.

 

[Entropia]

There is indeed a paradox involving the Dirac equation and commutation with time. This paradox arises from the fact that the Dirac equation is a relativistic equation, which means it takes into account the effects of special relativity, including the concept of time dilation. In this context, time is not a fixed parameter but rather a variable that is affected by the speed and energy of the particles involved.

The paradox arises because the Dirac equation is based on the concept of a Hamiltonian, which is a mathematical operator that represents the total energy of a system. In non-relativistic quantum mechanics, the Hamiltonian is a constant operator that commutes with all other operators, including the time operator. However, in the relativistic context of the Dirac equation, the Hamiltonian is no longer a constant operator, but rather a function of both space and time. This means that it no longer commutes with the time operator, leading to a paradox.

To resolve this paradox, we need to understand that in the relativistic context, the momentum operator and the time operator do not commute with each other. This means that the time derivative of the wavefunction is not simply related to the Hamiltonian, as it is in non-relativistic quantum mechanics. Instead, the time derivative is related to a combination of the Hamiltonian and the momentum operator, which explains why the alpha matrices do not commute with the Hamiltonian in the Dirac equation.

In conclusion, the paradox involving the Dirac equation and commutation with time is a result of the relativistic nature of the equation, which requires us to rethink our understanding of time and its relationship with other operators. It is a complex issue that highlights the need for a deeper understanding of the principles of special relativity in the context of quantum mechanics.