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S-matrix
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=== Interaction picture === A straightforward way to define the ''S''-matrix begins with considering the [[interaction picture]].<ref>{{harvnb|Greiner|Reinhardt|1996}} Section 8.2.</ref> Let the Hamiltonian {{math|''H''}} be split into the free part {{math|''H''<sub>0</sub>}} and the interaction {{math|''V''}}, {{math|1=''H'' = ''H''<sub>0</sub> + ''V''}}. In this picture, the operators behave as free field operators and the state vectors have dynamics according to the interaction {{math|''V''}}. Let <math display="block">\left|\Psi(t)\right\rangle</math> denote a state that has evolved from a free initial state <math display="block">\left|\Phi_{\rm i}\right\rangle.</math> The ''S''-matrix element is then defined as the projection of this state on the final state <math display="block">\left\langle\Phi_{\rm f}\right|.</math> Thus <math display="block">S_{\rm fi} \equiv \lim_{t \rightarrow +\infty} \left\langle\Phi_{\rm f}|\Psi(t)\right\rangle \equiv \left\langle\Phi_{\rm f}\right|S\left|\Phi_{\rm i}\right\rangle,</math> where {{math|''S''}} is the '''S-operator'''. The great advantage of this definition is that the '''time-evolution operator''' {{mvar|U}} evolving a state in the interaction picture is formally known,<ref>{{harvnb|Greiner|Reinhardt|1996}} Equation 8.44.</ref> <math display="block">U(t, t_0) = Te^{-i\int_{t_0}^t d\tau V(\tau)},</math> where {{mvar|T}} denotes the [[time-ordered product]]. Expressed in this operator, <math display="block">S_{\rm fi} = \lim_{t_2 \rightarrow +\infty}\lim_{t_1 \rightarrow -\infty}\left\langle\Phi_{\rm f}\right|U(t_2, t_1)\left|\Phi_{\rm i}\right\rangle,</math> from which <math display="block">S = U(\infty, -\infty).</math> [[Matrix exponential|Expanding]] using the knowledge about {{math|''U''}} gives a [[Dyson series]], <math display="block">S = \sum_{n=0}^\infty \frac{(-i)^n}{n!}\int_{-\infty}^\infty dt_1\cdots \int_{-\infty}^\infty dt_n T\left[V(t_1)\cdots V(t_n)\right],</math> or, if {{mvar|V}} comes as a Hamiltonian density <math>\mathcal{H}</math>, <math display="block">S = \sum_{n=0}^\infty \frac{(-i)^n}{n!}\int_{-\infty}^\infty dx_1^4\cdots \int_{-\infty}^\infty dx_n^4 T\left[\mathcal{H}(x_1)\cdots \mathcal{H}(x_n)\right].</math> Being a special type of time-evolution operator, {{mvar|S}} is unitary. For any initial state and any final state one finds <math display="block">S_{\rm fi} = \left\langle\Phi_{\rm f}|S|\Phi_{\rm i}\right\rangle = \left\langle\Phi_{\rm f} \left|\sum_{n=0}^\infty \frac{(-i)^n}{n!}\int_{-\infty}^\infty dx_1^4\cdots \int_{-\infty}^\infty dx_n^4 T\left[\mathcal{H}(x_1)\cdots \mathcal{H}(x_n)\right]\right| \Phi_{\rm i}\right\rangle .</math> This approach is somewhat naïve in that potential problems are swept under the carpet.<ref name=Greiner_1>{{harvnb|Greiner|Reinhardt|1996}} Chapter 9.</ref> This is intentional. The approach works in practice and some of the technical issues are addressed in the other sections.
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