Configuration-Space Based Representations in Quantum Theory

Schrödinger Picture / Representation > s.a. representations in quantum theory; quantum field theory.
* Idea: The representation on L2($$\cal C$$, dμ), where $$\cal C$$ is the configuration space, in which states are time-dependent, observables time-independent.
* Use: It is not convenient for the relativistic theory, since it treats time differently from the space coordinates.
@ References: Hiley & Dennis a1809 [Dirac-Bohm picture]; Stoica a1906 [multi-layered field representation in 3D space].

Heisenberg Picture / Representation > s.a. quantum field theory.
* Idea: The representation on L2($$\cal C$$, dμ) in which states are time-independent, and observables time-dependent operators.
* Relationships: Compared to the Schrödinger representation,

ψH = exp{iHt/$$\hbar$$} ψS(t),   and    AH(t) = exp{iHt/$$\hbar$$} AS exp{−iHt/$$\hbar$$};

the time evolution of the operators is given by i$$\hbar$$ ∂AH/∂t = [AH, H]; The Hamiltonian operator is the same.
* Advantages: Constant phase shifts in the operators due, e.g., to a constant potential are cancelled; The equations of motion for the operators are formally identical to the classical ones; It is convenient for relativistic theory.
* Disadvantages: It is difficult to solve practical problems with it.
@ Compared to Schrödinger representation: Faria et al PLA(02); Nikolić PLA(04)qp/03; de la Madrid qp/05-conf [for unbounded operators, and rigged Hilbert space]; Solomon a0706 [in quantum field theory]; Aharonov a1303-fs [and non-local quantum phenomena]; Partovi a1305 [and Eulerian vs Lagrangian descriptions of fluid dynamics]; de Gosson a1404 [inequivalence]; Franson & Brewster a1811 [limitations].

Interaction Picture / Representation
* Idea: Both states and operators are time-dependent, with evolutions governed by different parts of the Hamiltonian:

H'I(t) ψI(t) = i$$\hbar$$ ∂ψI(t)/∂t ,   and   i$$\hbar$$ ∂AI(t)/∂t = [AI(t), H0] ,

where H = H0 + H', a free and an interaction part, and the relationship with Schrödinger representation quantities is

ψI(t) = exp{iH0t/$$\hbar$$} ψS(t) ,   and   H'I(t) = exp{iH0t/$$\hbar$$} H'S exp{−iH0t/$$\hbar$$} .

* Advantage: Takes into account only the non-trivial evolution of states; The free H evolves the operators.
* Disadvantage: It does not exist in general for a relativistic theory.
* Remark: This representation is often implicitly used in ordinary quantum mechanics, when ignoring "the rest of the world".
@ In quantum field theory: Biswas a0807 [transformation to free fields not unitary].