# What is the Hamiltonian density for a massive Dirac field?

Your Name]In summary, the conversation discusses the expansion of \gamma^{\mu}\partial_{\mu} and the resulting Hamiltonian density from the given Lagrangian. The confusion arises from the anti-commuting property of \gamma^{\mu} matrices, leading to a minus sign in the first term of the Hamiltonian density. Remembering this detail is important in order to have the correct sign for the Hamiltonian density. The person responding offers clarification and encourages collaboration and seeking help when needed in the pursuit of scientific understanding.
Hey guys,

So here's the deal. Consider the Lagrangian

$\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi$

where $\bar{\psi}=\psi^{\dagger}\gamma^{0}$.

I need to find the Hamiltonian density from this, using

$\mathcal{H}=\pi_{i}(\partial_{0}\psi_{i})-\mathcal{L}$

So I get the following:

$\mathcal{H}=i\bar{\psi}\gamma^{i}\nabla\psi+\bar{\psi}\psi m$

But my teacher writes

$\mathcal{H}=-i\bar{\psi}\gamma^{i}\nabla\psi+\bar{\psi}\psi m$

And I don't know how he gets that minus factor. The only part where I could be going wrong is when I expand $\gamma^{\mu}\partial_{\mu}$...I'm using $\gamma^{\mu}\partial_{\mu}=\gamma^{0}\partial_{0}-\gamma^{i}\partial_{i}$...because the metric signature is (+,---). But I guess this is wrong?

Hi there,

Thank you for sharing your question about the Lagrangian and Hamiltonian densities. It looks like you have a good understanding of the equations and have made some progress in finding the Hamiltonian density. However, I see where the confusion may be coming from.

When expanding \gamma^{\mu}\partial_{\mu}, it is important to remember that the \gamma^{\mu} matrices are anti-commuting. This means that when you switch the order of the matrices, you will get a minus sign. So when you expand \gamma^{\mu}\partial_{\mu}=\gamma^{0}\partial_{0}-\gamma^{i}\partial_{i}, the second term should actually be -\gamma^{i}\partial_{i} due to the anti-commuting property.

This is why your teacher's Hamiltonian density has a minus sign in front of the first term, because they correctly expanded \gamma^{\mu}\partial_{\mu}=\gamma^{0}\partial_{0}+\gamma^{i}\partial_{i}. This may seem like a small detail, but it is important in order to have the correct sign for the Hamiltonian density.

I hope this helps clarify things for you. Keep up the good work and don't hesitate to ask for help when needed. Science is all about collaboration and learning from each other. Best of luck with your studies!

## 1. What is a Hamiltonian density?

A Hamiltonian density is a mathematical quantity used in the field of theoretical physics to describe the total energy of a system. It is derived from the Hamiltonian, which is a function that represents the total energy of a system in terms of its position and momentum variables.

## 2. What is a massive Dirac field?

A massive Dirac field is a theoretical quantum field that describes the behavior of particles with half-integer spin, such as electrons. It is characterized by its mass and spin, and is governed by the Dirac equation, which describes the evolution of the field over time.

## 3. How is the Hamiltonian density for a massive Dirac field derived?

The Hamiltonian density for a massive Dirac field is derived from the Lagrangian density of the field. This involves applying the Euler-Lagrange equations, which relate the Lagrangian to the Hamiltonian, and incorporating the mass term into the equation.

## 4. What is the physical significance of the Hamiltonian density for a massive Dirac field?

The Hamiltonian density for a massive Dirac field represents the total energy of the field at a given point in space and time. It is used to calculate various physical properties of the field, such as its energy density and momentum density, which have important implications in understanding the behavior of particles.

## 5. How is the Hamiltonian density for a massive Dirac field used in practical applications?

The Hamiltonian density for a massive Dirac field is used in various applications in theoretical physics, such as in quantum field theory and particle physics. It is also used in calculations for the behavior of particles in high-energy accelerators, and in the development of new theories and models in physics.

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