This chapter introduces the general concept of symmetry in both classical and quantum physics. The part devoted to classical physics stresses the importance of conservation laws and the role of Noether’s theorem for their understanding. A fairly general discussion of Noether’s theorem is offered, including ambiguities in the definition of the Noether current, possible existence of generalized, tensor conservation laws, and the inverse of Noether’s theorem. The Hamiltonian formalism is then used to motivate the transition from classical to quantum physics, and the definition of symmetry in the latter. In anticipation of the discussion of spontaneous symmetry breaking in the following chapter, the text here emphasizes the difference between implementing symmetry in terms of transformations on the algebra of physical observables, and in terms of unitary mappings on the Hilbert space of states.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0 and, if A is linear and time-independent, a conservation law, namely the invariance of expectation values of A. For non-Hermitian Hamiltonians, H† comes into play as a distinct operator that complements H in generalized unitarity relations. The above description of symmetries has to be extended to include also A-pseudohermiticity relations of the form AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of A operators that form Klein’s 4-group: parity, time-reversal, parity&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be richer and subtler for non-Hermitian than for Hermitian Hamiltonians.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation
AHA† that leaves the Hamiltonian
H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity
[H,A]=0, and a conservation law, namely the invariance of expectation values of
A. For non-Hermitian Hamiltonians,
H† comes into play as a distinct operator that complements
H in generalized unitarity relations. The above description of symmetries has to be extended to include also
A-pseudohermiticity relations of the form
AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of
A operators forming Klein's 4-group: parity, time-reversal, parity\&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be more subtle for non-Hermitian than for Hermitian Hamiltonians.
Abstract Ehrenfest theorem is extended to the relativistic quantum theory of charged particles, moving under the influence of a classical electromagnetic field. In order to extend the original Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton’s second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws for the energy and momentum.