Appendix: The Interaction Picture The technique of expressing operators and quantum states in the so-called interaction picture is of great importance in treating quantum-mechanical problems in a perturbative fashion. It plays an important role, for example, in the derivation of the Born–Markov master equation for decoherence detailed in Sect. 4.2.2. We shall review the basics of the interaction-picture approach in the following. We begin by considering a Hamiltonian of the form Hˆ = Hˆ0 + V.ˆ (A.1) Here Hˆ0 denotes the part of the Hamiltonian that describes the free (un- perturbed) evolution of a system S, whereas Vˆ is some added external per- turbation. In applications of the interaction-picture formalism, Vˆ is typically assumed to be weak in comparison with Hˆ0, and the idea is then to deter- mine the approximate dynamics of the system given this assumption. In the following, however, we shall proceed with exact calculations only and make no assumption about the relative strengths of the two components Hˆ0 and Vˆ . From standard quantum theory, we know that the expectation value of an operator observable Aˆ(t) is given by the trace rule [see (2.17)], & ' & ' ˆ ˆ Aˆ(t) =Tr Aˆ(t)ˆρ(t) =Tr Aˆ(t)e−iHtρˆ(0)eiHt . (A.2) For reasons that will immediately become obvious, let us rewrite this expres- sion as & ' ˆ ˆ ˆ ˆ ˆ ˆ Aˆ(t) =Tr eiH0tAˆ(t)e−iH0t eiH0te−iHtρˆ(0)eiHte−iH0t . (A.3) ˆ In inserting the time-evolution operators e±iH0t at the beginning and the end of the expression in square brackets, we have made use of the fact that the trace is cyclic under permutations of the arguments, i.e., that Tr AˆBˆCˆ ··· =Tr BˆCˆ ···Aˆ , etc. Thus we could move the first factor ˆ eiH0t to the end of the argument of the trace, canceling out the extra factor ˆ e−iH0t at that position. 380 Appendix: The Interaction Picture Equation (A.3) motivates us to now introduce the interaction-picture form of general operators Aˆ(t) and density matricesρ ˆ(t)as ˆ ˆ Aˆ(I)(t)=eiH0tAˆ(t)e−iH0t, (A.4) ˆ ˆ ρˆ(I)(t)=eiH0tρˆ(t)e−iH0t ˆ ˆ ˆ ˆ =eiH0te−iHtρˆ(0)eiHte−iH0t. (A.5) Here the superscript (I) is used to denote interaction-picture operators. Note that the dynamics of the interaction-picture operators Aˆ(I)(t) are fully de- termined by the unperturbed (free) Hamiltonian Hˆ0 rather than by the total Hamiltonian Hˆ . With the new definitions (A.4) and (A.5), our expectation-value equa- tion (A.3) can then be written in the compact form & ' Aˆ(t) =Tr Aˆ(I)(t)ˆρ(I)(t) . (A.6) Let us now determine an evolution equation for the interaction-picture den- sity matrixρ ˆ(I)(t). Using the standard Liouville–von Neumann equation for the density matrixρ ˆ(t) in the Schr¨odinger picture, d ρˆ(t)=−i H,ˆ ρˆ(t) , (A.7) dt we obtain ˆ ˆ d (I) (I) iH0t d −iH0t ρˆ (t)=iHˆ0, ρˆ (t) +e ρˆ(t) e dt dt (A.7) ˆ ˆ (I) iH0t −iH0t =iHˆ0, ρˆ (t) − ie H,ˆ ρˆ(t) e (A.1) ˆ ˆ (I) iH0t −iH0t =iHˆ0, ρˆ (t) − ie Hˆ0 + V,ˆ ρˆ(t) e . (A.8) From (A.4) and (A.5) we see that the application of the time-evolution opera- ˆ tors e±iH0t to the quantities in the second commutator amounts to a transfor- mation of these quantities to the interaction picture. Equation (A.8) therefore becomes d (I) (I) (I) (I) (I) ρˆ (t)=i Hˆ0, ρˆ (t) − i Hˆ0, ρˆ (t) − i Vˆ (t), ρˆ (t) dt = −i Vˆ (I)(t), ρˆ(I)(t) . (A.9) This establishes our first main result: The time evolution of the interaction- picture density operator is given by an equation of the Liouville–von Neu- mann type, but with the interaction-picture perturbation Vˆ (I)(t) instead of the full Hamiltonian Hˆ . Appropriately, (A.9) is called the interaction-picture Liouville–von Neumann equation. Appendix: The Interaction Picture 381 Up to now, we have considered the case of a single system S subject to some perturbation Vˆ whose origin we have not specified further. Let us now enter the setting relevant to decoherence: We suppose that the perturbation is due to the interaction with some external environment E described by an in- teraction Hamiltonian Hˆint, i.e., Vˆ ≡ Hˆint. If we denote the self-Hamiltonians of the system and the environment by HˆS and HˆE , respectively, we may write the total Hamiltonian of the composite system SE in the form ˆ ˆ ˆ ≡ ˆ ˆ ˆ H = H0 + V /HS 01+ HE2 + /H01int2 . (A.10) ˆ ˆ ≡H0 ≡V Following the usual strategy, we are interested in determining the time evo- lution of the reduced density operatorρ ˆS (t). In our case, we would like to obtain the reduced interaction-picture density operator & ' (I) (I) ρˆS (t) ≡ TrE ρˆ (t) . (A.11) We will now address the following two questions: 1. Given the reduced density operatorρ ˆS (t) in the Schr¨odinger picture, what (I) is the corresponding operatorρ ˆS (t) in the interaction picture? (I) 2. What is the appropriate time-evolution equation forρ ˆS (t)? Let us answer the first question. Evaluating the trace in (A.11) gives & ' & ' (A.5) ˆ ˆ (I) iH0t −iH0t TrE ρˆ (t) =TrE e ρˆ(t)e $ & '% (A.10) ˆ ˆ ˆ ˆ iHS t iHE t −iHE t −iHS t =e TrE e ρˆ(t)e e ˆ ˆ iHS t −iHS t =e {TrE [ˆρ(t)]} e ˆ ˆ iHS t −iHS t =e ρˆS (t)e (A.11) (I) ≡ ρˆS (t), (A.12) where in the second-to-last line we have again used the invariance of the trace operation under cyclic permutations of the arguments (or, put slightly differently, the invariance of the trace under unitary transformations of the argument). Equation (A.12) shows that the reduced density operator in the interac- tion picture is obtained by a unitary transformation of the reduced density operator in the Schr¨odinger picture involving the free system Hamiltonian HˆS only, i.e., ˆ ˆ (I) iHS t −iHS t ρˆS (t)=e ρˆS (t)e . (A.13) The analogy of (A.13) to the corresponding transformation for the total interaction-picture density operatorρ ˆ(I)(t), see (A.5), should now be clear. In 382 Appendix: The Interaction Picture each case, the transformation involves only the free Hamiltonian, namely, the (I) Hamiltonian Hˆ0 forρ ˆ (t) and the system Hamiltonian HˆS for the reduced (I) interaction-picture density operatorρ ˆS (t). We note that the definition (A.13) also ensures, as desired, that expec- tation values of system observables AˆS (t) automatically agree in both the Schr¨odinger and interaction pictures, i.e., that & ' & ' (I) (I) AˆS (t) =TrS ρˆS (t)AˆS (t) =TrS ρˆS (t)AˆS (t) , (A.14) where (A.4) ˆ ˆ (A.10) ˆ ˆ (I) iH0t −iH0t iHS t −iHS t AˆS (t) =e AˆS (t)e =e AˆS (t)e . (A.15) The proof of (A.14) is immediate: & ' & ' ˆ ˆ (I) (I) iHS t −iHS t TrS ρˆS (t)AˆS (t) =TrS e ρˆS (t)AˆS (t)e & ' =TrS ρˆS (t)AˆS (t) . (A.16) Finally, let us now answer the second question posed above, i.e., let us find an equation of motion for the reduced interaction-picture density operator (I) ρˆS (t). Taking the trace over the environment on both sides of our evolution equation (A.9) for the full interaction-picture density operatorρ ˆ(I)(t), with (I) (I) Vˆ (t) ≡ Hˆint (t), yields d (I) (I) (I) ρˆ (t)=−iTrE Hˆ (t), ρˆ (t) . (A.17) dt S int A closer inspection reveals that this equation is not of the standard Liouville– von Neumann form, since the right-hand side depends on the full density (I) (I) operatorρ ˆ (t) rather than on the reduced density operatorρ ˆS (t). This fact should not be too surprising to the reader. After all, the state of the environment will generally influence the evolution of the reduced density operator. Thus, in general, we must consider the full interacting system–environment combination to determine the reduced dynamics. (This observation is, of course, independent of whether we work in the Schr¨odinger or the interac- tion picture.) Only once certain approximations are imposed (for instance, the assumption of weak system–environment coupling), we may obtain an evolu- tion equation for the reduced density operator of the system that does not explicitly depend on the time-dependent total (system–environment) density operator. Such master equations are derived in Chaps. 4 and 5. References 1. M. Schlosshauer, Decoherence, the measurement problem, and interpretations of quantum mechanics, Rev. Mod. Phys. 76, 1267–1305 (2004). 2. E. Schr¨odinger, Die gegenw¨artige Situation in der Quantenmechanik, Natur- wissenschaften 23, 807–812, 823–828, 844–849 (1935). 3. A. Pais, Einstein and the quantum theory, Rev. Mod. Phys. 51, 863–914 (1979). 4. H. D. Zeh, On the interpretation of measurement in quantum theory, Found. Phys. 1, 69–76 (1970). 5. H. D. Zeh, Toward a quantum theory of observation, Found. Phys. 3, 109–116 (1973). 6. O. K¨ubler, H. D. Zeh, Dynamics of quantum correlations, Ann. Phys. (N.Y.) 76, 405–418 (1973). 7. E. Joos, H. D. Zeh, The emergence of classical properties through interaction with the environment, Z. Phys. B: Condens.