Relativistic Dynamics in Hamiltonian Formulation

[Alex Petrosyan]

hi all!

I’m trying to generalise the Caldeira-Leggett Hamiltonian (heat bath + particle) to the case of high velocities. Naturally, the multi-oscillator Hamiltonian needs to change and I have a gnawing suspicion that the multi-particle Hamiltonian is just the sum of single-particle hamiltonians with $$ H = \sqrt{P^2 + m^2} $$

In natural units ofc.

 

[Ambitwistor]

Hello there,

Thank you for your post. The Caldeira-Leggett Hamiltonian is a useful model for studying the dynamics of a particle interacting with a heat bath. As you mentioned, this model is usually described using a multi-oscillator Hamiltonian, but in the case of high velocities, the Hamiltonian needs to be modified.

Your suspicion about the multi-particle Hamiltonian being the sum of single-particle Hamiltonians is partially correct. In the case of high velocities, the Hamiltonian can indeed be written as the sum of single-particle Hamiltonians, but the form of these Hamiltonians will be different from the one you mentioned.

In fact, in the case of high velocities, the single-particle Hamiltonian should be written as $$H=\sqrt{m^2+P^2}-m,$$ where $m$ is the mass of the particle and $P$ is its momentum. This Hamiltonian is known as the relativistic Hamiltonian and it takes into account the effects of special relativity.

Furthermore, the multi-particle Hamiltonian should also include terms that take into account the interactions between the particles and the heat bath. These interactions can be described using the coupling constant $\alpha$, which is a measure of the strength of the interaction.

Therefore, the modified multi-particle Hamiltonian for high velocities would be $$H=\sum_{i=1}^N\sqrt{m_i^2+P_i^2}-m_i+\alpha\sum_{i=1}^N q_iX_i,$$ where $N$ is the number of particles, $m_i$ and $P_i$ are the mass and momentum of the $i$th particle, and $q_i$ and $X_i$ are the position and momentum operators for the $i$th particle.

I hope this helps in your research. Best of luck!