Interaction, representation of

interaction representation

One of the principal possible (along with the Schrödinger representation and the Heisenberg representation) equivalent representations of the dependence of operators $ A $ and wave functions $ \psi $ in quantum mechanics and quantum field theory on the time $ t $. The quantum system, with allowance for the interaction between its various parts (in quantum mechanics) or between its various constituent fields (in quantum field theory), may be described in the Schrödinger representation with the aid of the Schrödinger equation:

$$ \tag{1 } i \hbar \frac{\partial \psi _ {S} ( t) }{\partial t } = \ H \psi _ {S} ( t) = \ ( H _ {0} + H _ {1} ) \psi _ {S} ( t). $$

It is assumed that the complete Hamiltonian $ H $ consists of the Hamiltonian of free (non-interacting) particles or fields, $ H _ {0} $, and the Hamiltonian of interaction, $ H _ {1} $. The Hamiltonians $ H _ {0} $ and $ H _ {1} $ are non-commuting (otherwise the problem would be trivial, since the decomposition of the Hamiltonian $ H $ into $ H _ {0} $ and $ H _ {1} $ would no longer be meaningful) functions of the operators which correspond to the various free fields and which are independent of the time $ t $ in the Schrödinger representation. The unitary transformation

$$ \tag{2 } \psi _ {S} ( t) = \ e ^ {- i \frac{H _ {0} } \hbar t } \phi _ {I} ( t) $$

realizes the transition to the interaction representation in which the wave function satisfies the equation:

$$ \tag{3 } i \hbar \frac{\partial \phi _ {I} ( t) }{\partial t } = \ H _ {I} ( t) \phi _ {I} ( t), $$

i.e. the dependence of $ \phi _ {I} ( t) $ on $ t $ is determined by the interaction Hamiltonian in the representation

$$ \tag{4 } H _ {1} ( t) = \ e ^ {+ i \frac{H _ {0} } \hbar t } H _ {I} e ^ {- i \frac{H _ {0} } \hbar t } . $$

The average value of the operator $ A $ in the Schrödinger representation,

$$ \tag{5 } \overline{A}\; \equiv \overline{A}\; _ {S} = \ ( \psi _ {S} ^ {*} ( t) ,\ A _ {S} \psi _ {S} ( t) ) = $$

$$ = \ \left ( \phi _ {I} ( t) , e ^ {i \frac{H _ {0} } \hbar t } A _ {S} e ^ {- i \frac{H _ {0} } \hbar t } \phi _ {I} ( t) \right ) , $$

may also be understood as the average value with respect to the wave functions $ \phi _ {I} ( t) $ of the operator $ A _ {I} $ in the interaction representation

$$ \tag{6 } A _ {I} ( t) = \ e ^ {i \frac{H _ {0} } \hbar t } A _ {S} e ^ {- i \frac{H _ {0} } \hbar t } . $$

In the interaction representation, operators corresponding to dynamical physical quantities vary with time according to (6) as operators in the Heisenberg representation of free fields, while the variation of the wave functions with time $ t $ is determined by the interaction effects between the fields. In the description of the behaviour of a quantum system as a function of the time $ t $, the interaction representation makes it possible to separate the dependence on the Hamiltonian of the free field $ H _ {0} $, which is usually easy to determine, and to concentrate on the study of equations (3) and (4), which contain complete information on the interaction between the fields. The interaction representation is particularly convenient in use if $ H _ {1} $ contains some small parameter, and the respective solution can be found, with the aid of the theory of perturbations, as a power series in the small parameter.

The invariance of the average values (5) which must be observed, and thus has a physical meaning, with respect to unitary transformations (2) and (6) means that the interaction representation and the Schrödinger and Heisenberg representations are equivalent.

Comments

The interaction representation or interaction picture is also called the Dirac representation or Dirac picture.

References

[a1]	G. Källén, "Quantum electrodynamics" , Springer (1972)