Configuration-Space Based Representations in Quantum Theory |

**Schrödinger Picture / Representation**
> s.a. representations in quantum theory; quantum field theory.

* __Idea__: The representation
on L^{2}(\(\cal C\), d*μ*), 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
L^{2}(\(\cal C\), d*μ*) in which
states are time-independent, and observables time-dependent operators.

* __Relationships__: Compared to
the Schrödinger representation,

*ψ*_{H} = exp{i*Ht*/\(\hbar\)}
*ψ*_{S}(*t*), and
*A*_{H}(*t*) = exp{i*Ht*/\(\hbar\)}
*A*_{S} exp{−i*Ht*/\(\hbar\)};

the time evolution of the operators is given by i\(\hbar\) ∂*A*_{H}/∂*t*
= [*A*_{H}, *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*) *ψ*_{I}(*t*)
= i\(\hbar\) ∂*ψ*_{I}(*t*)/∂*t*
, and i\(\hbar\) ∂*A*_{I}(*t*)/∂*t*
= [*A*_{I}(*t*),* H*_{0}]
,

where *H* = *H*_{0} + *H'*,
a free and an interaction part, and the relationship with Schrödinger
representation quantities is

*ψ*_{I}(*t*)
= exp{i*H*_{0}*t*/\(\hbar\)}
*ψ*_{S}(*t*)
, and *H'*_{I}(*t*)
= exp{i*H*_{0}*t*/\(\hbar\)} *H'*_{S}
exp{−i*H*_{0}*t*/\(\hbar\)}
.

* __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].

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