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.

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/\(\hbar\)};

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].

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: 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].


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