S-matrix

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In Quantum Mechanics , Scattering Theory or Quantum Field Theory , the S- Matrix relates the final state in the infinite future ( Out-channels ) and the initial state in the infinite past ( In-channels ). The "S" stands for "scattering" or "Strahlung" (radiation). More mathematically, the S-matrix is defined as the Unitary Matrix connecting asymptotic particle states in the Hilbert Space of physical states ( Scattering Channel s). While the S-matrix may be defined for any background ( Spacetime ) that is asymptotically solvable and has no Horizon s, it has a simple form in the case of the Minkowski Space . In this special case, the Hilbert space is a space of irreducible unitary representations of the Inhomogeneous Lorentz Group ; the S-matrix is the Evolution Operator between time equal to minus infinity, and time equal to plus infinity. It can be shown that if a Quantum Field Theory in Minkowski Space has a mass gap, the state in the asymptotic past and in the asymptotic future are both described by Fock Space s. The S-matrix is closely related to the transition Probability Amplitude in Quantum Mechanics and to Cross Sections of various Interaction s; the elements (individual numerical entries) in the S-matrix are known as scattering amplitudes. Poles of the S-matrix in the Complex-energy Plane are identified with Bound State s, virtual states or Resonance s. Branch Cuts of the S-matrix in the Complex-energy Plane are associated to the opening of a Scattering Channel . In the Hamiltonian approach to Quantum Field Theory , the S-matrix may be calculated as a Time-ordered Exponential of the integrated Hamiltonian in the Dirac Picture ; it may be also expressed using Feynman's Path Integral s. In both cases, the Perturbative calculation of the S-matrix leads to Feynman Diagram s. In Scattering Theory , the S-matrix is an Operator mapping free particle ''in-states'' to free particle ''out-states'' ( Scattering Channel s) in the Heisenberg Picture . This is very useful because we cannot describe exactly the interaction (at least, the most interesting ones).
	:<math>a(k)\left 0 Ight Angle	0</math>
	:<math>\mathcal H \mathrm{IN}	\operatorname{span}\{ \left I, k_1\ldots k_n ight angle = a_i^\dagger (k_1)\cdots a_i^\dagger (k_n)\left I, 0 ight angle\},</math>
	:<math>\mathcal H \mathrm{OUT}	\operatorname{span}\{ \left F, p_1\ldots p_n ight angle = a_f^\dagger (p_1)\cdots a_f^\dagger (p_n)\left F, 0 ight angle\}</math>
	:<math>\left I, K 1\ldots K N Ight Angle	C_0 + \sum_{m=1}^\infty \int{d^4p_1\ldots d^4p_mC_m(p_1\ldots p_m)\left F, p_1\ldots p_n ight angle}</math>
	According To	"http://wwwinformationdelightinfo/information/entry/Wigner's_theorem" class="copylinks">Wigner's Theorem , <math>S</math> must be a Unitary Operator such that <math>\left \langle I,\beta ight S\left I,\alpha ight angle = S_{\alpha\beta} = \left \langle F,\beta I,\alpha ight angle</math> Moreover, <math>S</math> leaves the void invariant and transforms IN-space fields in OUT-space fields:
	:<math>S\left0 Ight Angle	\left0 ight angle</math>
	If The System Is Made Up With A Single Particle In Momentum Eigenstate <math>\left K Ight Angle</math>, Then <math>S\left K Ight Angle	\left k ight angle</math>
	:<math>e^{i\alpha}	\left\langle 0U(\infty)0 ight angle^{-1}</math>
	:<math>S\left0 Ight Angle	\left0 ight angle</math>

  :<math>F N(x 1\dots X N) \left\langle 0\mathcal T\phi(x_1)\dots\phi(x_n)0 ight angle</math>




We will use this in the following calculation:

  : <math> \left \langle F,k_1, k_2 a_i^\dagger(p_2)-a_f^\dagger(p_2)I,p_1 ight angle</math>
  :<math> -i\int{d^3x\, f^(p_2,x)\partial_0^\leftrightarrow \left \langle F,k_1, k_2 \phi_i(x)-\phi_f(x)I,p_1 ight angle}</math>
  :<math> i\left(\lim_{t o \infty} - \lim_{t o \infty} ight)\int{d^3x\, f^(p_2,t)\partial_0^\leftrightarrow \left \langle F,k_1, k_2 \phi(x)I,p_1 ight angle}</math>
  :<math> i\int{d^4x\, \left \langle F,k_1, k_2 f^\ddot \phi - \ddot f^\phiI,p_1 ight angle}
  :<math>S {fi} i\int{d^4x\, f^(p_2,x)\left(\Box_x+m^2 ight )\left \langle F,k_1, k_2 \phi(x)I,p_1 ight angle}</math>
  :<math>S {fi} (i)^4\int{d^4x_1\, d^4x_2\, d^4y_1\, d^4y_2\, f^(p_1,x_1)f^(p_2,x_2)f(k_1,y_1)f(k_2,y_2)\left(\Box_{x_1}+m^2 ight )\left(\Box_{x_2}+m^2 ight )\left(\Box_{y_1}+m^2 ight )\left(\Box_{y_2}+m^2 ight )\left \langle 0\mathcal T\phi(x_1)\phi(x_2)\phi(y_1)\phi(y_2)0 ight angle}</math>


\cdotsrac{e^{-iq_{n+m}x_{n+m}}}{\sqrt{(2\pi)^32\omega_k}}
F_{nm}(x_1\dots x_n,y_1\dots y_m)}.

There is a theorem that states (proof omitted) that the S-matrix elements are the residuals of ''f'' calculated on mass-shell:

:$S\_\{fi\}=(i)^\{n+m\}\backslash lim\_\{q\_i\; o\; m^2\}(m^2-q\_1)\backslash cdots(m^2-q\_\{n+m\})f\_\{nm\}(q\_1\backslash dots\; 1\_\{n+m\}).$

The matter is that we do not have an explicit expression for $\backslash phi(x)$, so we have to make a perturbative expansion with $\backslash phi\_i(x)$.

In the end, we obtain:

  :<math>\mathcal C(x 1, X 2) \left \langle 0 \mathcal T\phi_i(x_1)\phi_i(x_2)0 ight angle=\overline{\phi_i(x_1)\phi_i(x_2)}=i\Delta_F(x_1-x_2)



Which means that $\backslash overline\{AB\}=\backslash mathcal\; TAB-:AB:$

In the end, we approach at Wick's theorem:

T ''Wick's theorem''

The T-product of a time-ordered free fields string can be expressed in the following manner:

:$\backslash mathcal\; T\backslash Pi\_\{k=1\}^m\backslash phi(x\_k)=:\backslash Pi\backslash phi\_i(x\_k):+\backslash sum\_\{\backslash alpha,\backslash beta\}\backslash overline\{\backslash phi(x\_\backslash alpha)\backslash phi(x\_\backslash beta)\}:\backslash Pi\_\{k$
ot=\alpha,\beta}\phi_i(x_k):+


:$\backslash mathcal$
+\sum_{(\alpha,\beta),(\gamma,\delta)}\overline{\phi(x_\alpha)\phi(x_\beta)}\;\overline{\phi(x_\gamma)\phi(x_\delta)}:\Pi_{k
ot=\alpha,\beta,\gamma,\delta}\phi_i(x_k):+\cdots.


Applying this theorem to S-matrix elements, we discover that normal-ordered terms acting on void state give a null contribute to the sum. We conclude that ''m'' is even and only completely contracted terms remain.

  :<math>G P^{(n)} \left \langle 0 \mathcal T:v_i(y_1):\dots:v_i(y_n):\phi_i(x_1)\cdots \phi_i(x_p)0 ight angle</math>