# Fundamental Spacetime Representations of Quantum Antenna Systems
## Appendix D Dirac Interaction Picture  
Consider a Hamiltonian \(H\) that can be written as  
\[H=H_{0}+H^{\prime}(t), \tag{110}\]  
where \(H_{0}\) is the _free_ Hamiltonian of the system (time-independent function), while \(H^{\prime}(t)\) is the _interaction_ (time-dependent) Hamiltonian, which is usually assumed to be turned on at a specific time point. In the Dirac picture, our goal is to _decouple_ the free dynamics of the system, i.e., that governed by a Hamiltonian of the form \(H=H_{0}\) (no interaction or \(H^{\prime}=0\)), from the pure interaction component, i.e., that which is solely due to the term \(H^{\prime}\) in the actual full Hamiltonian (110). Now, let us define the following time-dependent operator  
\[O_{\rm I}(t):=e^{{\rm i}H_{0}t}O_{\rm S}e^{-{\rm i}H_{0}t}, \tag{111}\]  
where \(O_{\rm S}\) is the Schrodinger picture (time-independent) operator. Differentiating (111) with respect to time, we obtain the Heisenberg equation of motion  
\[{\rm i}\frac{{\rm d}}{{\rm d}t}O_{\rm I}(t)=[O,H_{0}]. \tag{112}\]  
Therefore, the interaction picture operator \(O_{\rm I}(t)\) evolves in time according to the exact same law corresponding to the case when the full Hamiltonian is just the free part, i.e., \(H=H_{0}\). What about the evolution of the interaction picture state, which will be denoted by \(\psi_{\rm I}(t)\)? To find out, note that all different pictures, the Schrodinger, Heisenberg, and Dirac, must agree on probability amplitudes. Therefore, for any two pairs of Schrodinger and Dirac states \(\psi_{\rm S}(t)\), \(\varphi_{\rm S}(t)\), and \(\psi_{\rm I}(t)\), \(\varphi_{\rm I}(t)\), respectively, the matrix elements in both pictures must be identical. Formally, we express this by the condition  
\[\langle\psi_{\rm S}(t)|O_{\rm S}\varphi_{\rm S}(t)\rangle=\langle\psi_{\rm I}( t)|O_{\rm I}(t)\varphi_{\rm I}(t)\rangle=\Big{\langle}\psi_{\rm I}(t)e^{-{ \rm i}H_{0}t}\Big{|}O_{\rm S}e^{-{\rm i}H_{0}t}\varphi_{\rm I}(t)\Big{\rangle}. \tag{113}\]  
Clearly, this implies that we should define the Dirac (interaction) state by  
\[|\psi_{\rm I}(t)\rangle:=e^{{\rm i}H_{0}t}|\psi_{\rm S}(t)\rangle. \tag{114}\]  
To find out how this state evolves, we differentiate with respect to time, obtaining  
\[{\rm i}\frac{{\rm d}}{{\rm d}t}|\psi_{\rm I}(t)\rangle=\left(-H_{0}+{\rm i} \frac{{\rm d}}{{\rm d}t}\right)|\psi_{\rm S}(t)\rangle=H^{\prime}|\psi_{\rm S }(t)\rangle \tag{115}\]  
where we have made use of the Schrodinger equation  
\[{\rm i}\frac{{\rm d}}{{\rm d}t}|\psi_{\rm S}(t)\rangle=H|\psi_{\rm S}(t)\rangle \tag{116}\]  
and the form (110). Finally, using the definition (114) of the interacting picture state, the relation (115) becomes  
\[{\rm i}\frac{{\rm d}}{{\rm d}t}|\psi_{\rm I}(t)\rangle=H_{\rm I}(t)|\psi_{\rm I }(t)\rangle, \tag{117}\]  
where the Dirac picture interacting Hamiltonian \(H_{\rm I}\) is given by  
\[H_{\rm I}(t):=e^{{\rm i}H_{0}t}H^{\prime}(t)e^{-{\rm i}H_{0}t}. \tag{118}\]  
Let us summarize the main features of this Dirac picture:  
1. In the Schrodinger picture, the state evolves in time according to the full Hamiltonian while the operators are constant. 2. In the Heisenberg picture, the state is constant (time-independent), but the operator evolves according to the full Hamiltonian. 3. In the Dirac picture, both state and operators evolve with time. However, the time evolution is decoupled into two distinct and independent components. First, all interaction (Dirac) picture operators evolve according to the free Hamiltonian as per the corresponding Heisenberg Equation (A9). Second, the Dirac state \(|\psi_{1}(t)\rangle\) evolves independently according to the dynamic Equation (A14). Therefore, the main advantage of the Dirac picture is that we can concentrate on the essential aspects of interaction as encoded in the Schrodinger-like dynamic Equation (A14). In this equation, Schrodinger or Heisenberg states and operators can be obtained using the usual transformation formula.