Discussion Overview
The discussion revolves around the definition of Hamiltonian density and the derivation of energy current in the context of quantum mechanics and field theory. Participants explore the continuity equation for energy and its relationship to the energy-momentum tensor, while seeking expressions for energy current analogous to particle current.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants discuss the continuity equation for energy, suggesting it can be expressed using the energy-momentum tensor.
- There is uncertainty about the existence and definition of Hamiltonian density, with some proposing it relates to the expectation value of the Hamiltonian operator.
- Participants express difficulty in finding a straightforward expression for energy current, contrasting it with the more accessible derivation of electron current density.
- One participant mentions that the energy current can be derived from the energy-momentum tensor, specifically identifying the 00-component as related to Hamiltonian density.
- Another participant questions whether an energy current exists that can be expressed simply, similar to the particle current.
- Some participants suggest that the energy current is a general result and can be related to the energy-momentum tensor's components.
- There are references to literature, including Weinberg's work, but participants express that the derivations may not apply to non-relativistic systems.
- One participant indicates they are looking for an expression for energy current that does not involve integrals, as they need it for calculating thermal conductivity.
- Concerns are raised about the correctness of provided formulas and their dependence on the context of the system being analyzed.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the definition of Hamiltonian density or the existence of a simple expression for energy current. Multiple competing views and uncertainties remain regarding the derivation and applicability of energy current in different contexts.
Contextual Notes
Limitations include the potential dependence on the definitions of terms used, the unresolved nature of mathematical steps in deriving energy current, and the distinction between relativistic and non-relativistic frameworks. Some participants express confusion regarding the transition from particle current to energy current.