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Encyclopedia > 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). An expert is someone widely recognized as a reliable source of knowledge, technique, or skill whose judgment is accorded authority and status by the public or their peers. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Scattering theory is a branch of physics and especially of quantum mechanics whose aim is the study of scattering events. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ... In scattering theory, a scattering channel is a quantum state of the colliding system before or after the collision (). The Hilbert space spanned by the states before collision (in states) is equal to the ones spanned by the states after collision (out states) which are both Fock spaces if there... In scattering theory, a scattering channel is a quantum state of the colliding system before or after the collision (). The Hilbert space spanned by the states before collision (in states) is equal to the ones spanned by the states after collision (out states) which are both Fock spaces if there...


More mathematically, the S-matrix is defined as the unitary matrix connecting asymptotic particle states in the Hilbert space of physical states (scattering channels). While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no horizons, 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 spaces. In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... In scattering theory, a scattering channel is a quantum state of the colliding system before or after the collision (). The Hilbert space spanned by the states before collision (in states) is equal to the ones spanned by the states after collision (out states) which are both Fock spaces if there... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... Horizon The horizon is the line that separates earth from sky. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... Homogeneous is an adjective that has several meanings. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... For a system with internal state, (also called stateful system) time evolution means the change of state brought about by the passage of time. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... The Fock space is an algebraic system (Hilbert space) used in quantum mechanics to describe quantum states with a variable or unknown number of identical particles. ...


The S-matrix is closely related to the transition probability amplitude in quantum mechanics and to cross sections of various interactions; 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 states, virtual states or resonances. Branch cuts of the S-matrix in the complex-energy plane are associated to the opening of a scattering channel. In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Cross section may refer to the following In geometry, Cross section is the intersection of a 3-dimensional body with a plane. ... Interaction is a kind of action which occurs as two or more objects have an effect upon one another. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i represents the imaginary number, i2 = −1. ... In physics, a bound state is a composite of two or more building blocks (particles or bodies) that behaves as a single object. ... The Tacoma Narrows Bridge (shown twisting) in Washington collapsed spectacularly, under moderate wind, in part because of resonance. ... In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i represents the imaginary number, i2 = −1. ... In scattering theory, a scattering channel is a quantum state of the colliding system before or after the collision (). The Hilbert space spanned by the states before collision (in states) is equal to the ones spanned by the states after collision (out states) which are both Fock spaces if there...


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 integrals. In both cases, the perturbative calculation of the S-matrix leads to Feynman diagrams. The Hamiltonian, denoted H, has two distinct but closely related meanings. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In theoretical physics, path-ordering is the procedure (or a meta-operator ) of ordering a product of many operators according to the value of one chosen parameter: Here is a permutation that orders the parameters: Examples If an operator is not simply expressed as a product, but as a function... The exponential function is one of the most important functions in mathematics. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ... In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the Schrödinger picture and the Heisenberg picture. ... This article is about a formulation of quantum mechanics. ... In quantum mechanics, perturbation theory is a set of approximation schemes for describing a complicated quantum system in terms of a simpler one. ... In this Feynman diagram, electrons annihilate and become a quark-antiquark pair. ...


In scattering theory, the S-matrix is an operator mapping free particle in-states to free particle out-states (scattering channels) in the Heisenberg picture. This is very useful because we cannot describe exactly the interaction (at least, the most interesting ones). In Dirac notation, we define left |0rightrangle as the void (or vacuum) quantum state. If a^{dagger}(k) is a creation operator, its hermitian conjugate (destruction or annihilation operator) acts on the void as follows: Scattering theory is a branch of physics and especially of quantum mechanics whose aim is the study of scattering events. ... In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ... In scattering theory, a scattering channel is a quantum state of the colliding system before or after the collision (). The Hilbert space spanned by the states before collision (in states) is equal to the ones spanned by the states after collision (out states) which are both Fock spaces if there... The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... A quantum state is any possible state in which a quantum mechanical system can be. ... In mathematics, the Hermitian adjoint of an linear operator is a matching operator (very similar to the inverse operator in concept) defined over a linear space with inner product. ...

a(k)left |0rightrangle = 0

Now, we define two kinds of creation/destruction operators, acting on different Hilbert spaces (IN space i, OUT space f), a_i^dagger (k) and a_f^dagger (k). In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ...


So now

mathcal H_mathrm{IN} = operatorname{span}{ left| I, k_1ldots k_n rightrangle = a_i^dagger (k_1)cdots a_i^dagger (k_n)left| I, 0rightrangle},
mathcal H_mathrm{OUT} = operatorname{span}{ left| F, p_1ldots p_n rightrangle = a_f^dagger (p_1)cdots a_f^dagger (p_n)left| F, 0rightrangle}.

It is possible to prove that left| I, 0rightrangle and left| F, 0rightrangle are both invariant under translation and that the states left| I, k_1ldots k_n rightrangle and left| F, p_1ldots p_n rightrangle are eigenstates of the momentum operator mathcal P^mu. In the Heisenberg picture the states are time-independent, so we can expand initial states on a basis of final states (or vice versa) as follows: The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ...

left| I, k_1ldots k_n rightrangle = C_0 + sum_{m=1}^infty int{d^4p_1ldots d^4p_mC_m(p_1ldots p_m)left| F, p_1ldots p_n rightrangle}

Where left|C_mright|^2 is the probability that the interaction transforms left| I, k_1ldots k_n rightrangle into left| F, p_1ldots p_n rightrangle


According to Wigner's theorem, S must be a unitary operator such that left langle I,beta right |Sleft | I,alpharightrangle = S_{alphabeta} = left langle F,beta | I,alpharightrangle. Moreover, S leaves the void invariant and transforms IN-space fields in OUT-space fields: There is a natural connection, first discovered by Eugene Wigner, between the properties of particles, the representation theory of Lie groups and Lie algebras, and the symmetries of the universe. ... In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ...

Sleft|0rightrangle = left|0rightrangle
φf = S − 1φfS

If S describes an interaction correctly, these properties must be also true:


If the system is made up with a single particle in momentum eigenstate left| krightrangle, then Sleft| krightrangle=left| krightrangle


The S-Matrix element must be non zero if and only if momentum is conserved.

Contents


S-matrix and evolution operator U

a(k,t) = U − 1(t)ai(k)U(t)
phi_f=U^{-1}(infty)phi_i U(infty)=S^{-1}phi_i S

So we have S=e^{ialpha}U(infty) where

e^{ialpha}=leftlangle 0|U(infty)|0rightrangle

because

Sleft|0rightrangle = left|0rightrangle.

Substituting the explicit expression for U we obtain:

S=frac{1}{leftlangle 0|U(infty)|0rightrangle}mathcal T e^{-iint{dtau V_i(tau)}}

You can see that this formula is not explicitly covariant.


L.S.Z. (Lehman, Symanzik, Zimmermann) reduction formula

F_n(x_1dots x_n)=leftlangle 0|mathcal Tphi(x_1)dotsphi(x_n)|0rightrangle

The task is to find an expression for the S-Matrix element using the reduction formula. Before starting to accomplish this, it is useful to show the following trick:

left(lim_{x_0to infty} - lim_{x_0 to infty}right)f^*partial_0^{leftrightarrow}phi=int_{-infty}^{infty} {dx_0,left( f^*ddotphi-ddot fphi*right)},
lim_{t_1,t_2toinfty}int_{t_1}^{t_2}{dtau, frac{partial}{partial t}int{d^3x,psi(x,t)}}=left(lim_{x_0to infty} - lim_{x_0 to infty}right)int{d^3x,psi(x,t)}.

We will use this in the following calculation:

S_{fi}=left langle F,k_1, k_2 | I,p_1,p_2rightrangle=left langle F,k_1, k_2 | a_i^dagger(p_2)|I,p_1rightrangle

This operation is called particle extraction.

=left langle F,k_1, k_2 | a_i^dagger(p_2)-a_f^dagger(p_2)|I,p_1rightrangle

This is true because p is not equal to k.

=-iint{d^3x, f^*(p_2,x)partial_0^leftrightarrow left langle F,k_1, k_2 | phi_i(x)-phi_f(x)|I,p_1rightrangle}
=ileft(lim_{tto infty} - lim_{t to infty}right)int{d^3x, f^*(p_2,t)partial_0^leftrightarrow left langle F,k_1, k_2 | phi(x)|I,p_1rightrangle}
=iint{d^4x, left langle F,k_1, k_2 | f*ddot phi - ddot f^*phi|I,p_1rightrangle}

Remembering that f functions are solutions of Klein-Gordon equation: The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the Schrödinger equation. ...

left( Box + m^2 right ) f^*=0=ddot f^* - nabla^2 f^* + m^2 f^* Rightarrow ddot f^*=left( nabla^2-m^2right)f^*

where Box stands for the D'Alembertian. Substituting this in previous equation we get (integrating by parts two times): In special relativity, electromagnetism and wave theory, the dAlembert operator, also called dAlembertian, is the Laplace operator of Minkowski space. ...

S_{fi}=iint{d^4x, f^*(p_2,x)left(Box_x+m^2right )left langle F,k_1, k_2 | phi(x)|I,p_1rightrangle}.

Now we repeat these operations for all the particle in the system, and finally we get:

S_{fi}=(i)^4int{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^2right )left(Box_{x_2}+m^2right )left(Box_{y_1}+m^2right )left(Box_{y_2}+m^2right )left langle 0|mathcal Tphi(x_1)phi(x_2)phi(y_1)phi(y_2)|0rightrangle}.

This is, of course, the simplest case with only four interacting particles.


Now we Fourier transform (it is not exactly a Fourier transformation) the reduction formula F and we get: The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

f_{mn}(q_1dots 1_{m+n})=int{d^4x_1cdots d^4x_n, d^4y_1cdots d^4y_m, frac{e^{-iq_1x_1}}{sqrt{(2pi)^32omega_k}} cdotsfrac{e^{-iq_{n+m}x_{n+m}}}{sqrt{(2pi)^32omega_k}} F_{nm}(x_1dots x_n,y_1dots 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}lim_{q_ito m^2}(m^2-q_1)cdots(m^2-q_{n+m})f_{nm}(q_1dots 1_{n+m}).

The matter is that we do not have an explicit expression for φ(x), so we have to make a perturbative expansion with φi(x).


In the end, we obtain:

F_p(x)=left langle 0 |mathcal Tphi(x_1)dotsphi(x_p)| 0 right rangle=frac{left langle 0 |mathcal T e^{-iint{dtau, V_i(tau)phi_i(x_1)dotsphi_i(x_p)}}| 0 right rangle}{left langle 0 |e^{-iint{dtau, V_i(tau)}}| 0 right rangle}.

Wick's theorem

Definition of contraction:

mathcal C(x_1, x_2)=left langle 0 |mathcal Tphi_i(x_1)phi_i(x_2)|0right rangle=overline{phi_i(x_1)phi_i(x_2)}=iDelta_F(x_1-x_2) =iint{frac{d^4k}{(2pi)^4}frac{e^{-ik(x_1-x_2)}}{(k^2-m^2)+iepsilon}}.

Which means that overline{AB}=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:

mathcal TPi_{k=1}^mphi(x_k)=:Piphi_i(x_k):+sum_{alpha,beta}overline{phi(x_alpha)phi(x_beta)}:Pi_{knot=alpha,beta}phi_i(x_k):+sum_{(alpha,beta),(gamma,delta)}overline{phi(x_alpha)phi(x_beta)};overline{phi(x_gamma)phi(x_delta)}:Pi_{knot=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.

F_m^i(x)=left langle 0 |mathcal Tphi_i(x_1)phi_i(x_2)|0right rangle=sum_mathrm{pairs}overline{phi(x_1)phi(x_2)}cdots overline{phi(x_{m-1})phi(x_m})
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)|0right rangle

where p is the number of interaction fields (or, equivalently, the number of interacting particles) and n is the development order (or the number of vertices of interaction). For example, if v=gy^4 Rightarrow :v_i(y_1):=:phi_i(y_1)phi_i(y_1)phi_i(y_1)phi_i(y_1):


This is analogous to the corresponding theorem in statistics for the moments of a Gaussian distribution. This article is in need of attention from an expert on the subject. ... Probability density function of Gaussian distribution (bell curve). ...


See also Feynman diagram. In this Feynman diagram, electrons annihilate and become a quark-antiquark pair. ...


See also

The article on Rayleigh scattering for an example of the application of the S-matrix. Rayleigh scattering causing a reddened sky at sunset Rayleigh scattering (named after Lord Rayleigh) is the scattering of light by particles much smaller than the wavelength of the light. ...


Bibliography

The Theory of the Scattering Matrix (Barut, 1967).


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