Question

This is a question I came to while reading Srednicki. My overall goal is to understand quantum/QFT in the end so that doing math research in physics-adjacent topics seems more meaningful. I am a mathematician, not a physics student.

Consider a single, spinless, non-relativistic particle with no forces acting on it. The hamiltonian for the system is

H=p⋅p2m.

Srednicki claims that the obvious way to generalize this to a relativistic system is to take

H=+p2c2+m2c4−−−−−−−−−−√.

Is this true simply because H
describes the sum of kinetic and potential energy of the system, and because this gives the relativistic energy-momentum relation?

Srednicki then expands this in inverse powers of c
:

H=mc2+p⋅p2m+⋯.

The way I am conceptualizing this is as follows: If there is no momentum, then the only energy must come from mass, so we have the rest energy mc2
as a term. The second term is the same as the non-relativistic H
, and the rest of the terms are called higher-order corrections. Can you explain what the higher-order corrections are doing? From a mathematical point of view, if H=+p2c2+m2c4−−−−−−−−−−√
is the correct hamiltonian, then expanding just reveals their presence. But from the point of view of trying to conceptualize what is happening, this doesn't satisfy my curiosity.
Options:
A) The higher-order corrections represent additional relativistic effects that become significant as velocities approach the speed of light.
B) Higher-order terms account for quantum fluctuations and deviations from classical behavior in relativistic systems.
C) The higher-order corrections manifest as corrections to the energy due to the particle's interaction with the curved spacetime background.
D) Higher-order terms in the expansion account for corrections arising from the particle's self-interactions and field fluctuations in a relativistic framework.

Answer 1

The higher-order corrections in the relativistic Hamiltonian represent additional relativistic effects that are necessary for accurately describing the energy of a system as velocities approach the speed of light

Step-by-step explanation:

The higher-order corrections in the expansion of the relativistic Hamiltonian H=\sqrt{p^2c^2+m^2c^4} represent additional relativistic effects that become significant as velocities approach the speed of light. These terms are important for refining our understanding of how the total energy of a particle behaves at high speeds and is more than just the sum of the rest energy and classical kinetic energy. When velocities are much less than the speed of light, these additional terms are negligible, and the Hamiltonian simplifies to the classical form of kinetic energy plus the rest mass energy term. However, as the velocity of an object increases and approaches the speed of light, the higher-order terms become relevant and are necessary to accurately describe the energy of the system.