Quantum Field Theory and Non-Relativistic Quantum Mechanics
Summary
Quantum Field Theory (QFT) extends the principles of quantum mechanics to systems with infinitely many degrees of freedom, treating particles as excitations of underlying fields and unifying special relativity with quantum principles. In contrast, non-relativistic Quantum Mechanics (NRQM) focuses on particles evolving under a Hamiltonian at velocities much below the speed of light, capturing phenomena in atomic, molecular and condensed-matter systems. While NRQM provides a precise description of bound states and scattering at low energies, QFT is indispensable for processes involving particle creation, annihilation and relativistic effects. The two frameworks overlap in models where non-relativistic matter interacts with quantised fields, such as electrons coupled to the electromagnetic field. These hybrid settings require techniques from both domains—spectral analysis of self-adjoint operators, renormalisation to handle ultraviolet divergences and careful control of infrared behaviour. Advances in mathematical rigour have clarified the existence and properties of ground states, the removal of infinities by boundary conditions or counter-terms and the construction of unitary dynamics in Fock space. The interplay between QFT and NRQM underpins modern applications in quantum optics, ultra-cold atoms and materials science, where control over particle–field interactions enables design of novel devices and exploration of fundamental symmetries.
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Quantum Field Theory and Non-Relativistic Quantum Mechanics publication trend
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Technical terms
Hamiltonian: Operator corresponding to the total energy of a quantum system, governing its time evolution.
Ground state: Lowest-energy eigenstate of a Hamiltonian, whose existence often ensures physical stability.
Renormalisation: Procedure by which divergent contributions are absorbed into redefined parameters, yielding finite predictions.
Ultraviolet divergence: Divergent behaviour of integrals at high momenta or short distances, typically requiring regularisation or boundary conditions.
Fock space: Hilbert space accommodating states with variable particle number, essential for describing creations and annihilations.
Resolvent: Operator (H − z)^−1 that encodes spectral information and facilitates the construction of self-adjoint extensions.
References
- Ground states for translationally invariant Pauli-Fierz models at zero momentum. Journal of Functional Analysis (2023).
- On the Resolvent of H+A∗+A. Mathematical Physics, Analysis and Geometry (2024).
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