Editing LSZ reduction formula (section)

==In and out fields==
[[S matrix|''S''-matrix elements]] are probability amplitudes of [[Transition of state|transitions]] between ''in'' states and ''out'' states.<ref name="Peskin">{{Cite book|last1=Peskin|url=https://www.taylorfrancis.com/books/mono/10.1201/9780429503559/introduction-quantum-field-theory-michael-peskin|title=An Introduction To Quantum Field Theory|last2=Schroeder|date=2018-05-04|publisher=CRC Press|isbn=978-0-429-50355-9|language=en|doi=10.1201/9780429503559}}</ref><ref name="Srednicki">{{Cite book|last=Srednicki|first=Mark|url=https://doi.org/10.1017/CBO9780511813917|title=Quantum Field Theory|date=2007|publisher=Cambridge University Press|isbn=978-0-511-81391-7|location=Cambridge|language=en|doi=10.1017/cbo9780511813917}}</ref><ref name="Weinberg">{{Cite book|last=Weinberg|first=Steven|url=https://www.cambridge.org/core/books/quantum-theory-of-fields/22986119910BF6A2EFE42684801A3BDF|title=The Quantum Theory of Fields: Volume 1: Foundations|date=1995|publisher=Cambridge University Press|isbn=978-0-521-67053-1|volume=1|location=Cambridge|doi=10.1017/cbo9781139644167}}</ref><ref name="Ticciati">{{Cite book|last=Ticciati|first=Robin|url=https://www.cambridge.org/core/books/quantum-field-theory-for-mathematicians/4783BB717856D3572422F05714CA3D5A|title=Quantum Field Theory for Mathematicians|date=1999|publisher=Cambridge University Press|isbn=9780511526428|location=Cambridge|doi=10.1017/CBO9780511526428}}</ref><ref name="Skaar">{{Cite web|last=Skaar|first=Johannes|url=https://www.uio.no/studier/emner/matnat/fys/FYS4170/h23/lsz.pdf|title=The S matrix and the LSZ reduction formula|date=2023|archive-url=https://web.archive.org/web/20231009052645/https://www.uio.no/studier/emner/matnat/fys/FYS4170/h23/lsz.pdf |archive-date=2023-10-09 }}</ref> An ''in'' state <math>|\{p\}\ \mathrm{in}\rangle</math> describes the state of a system of particles which, in a far away past, before interacting, were moving freely with definite momenta {{math|{''p''},}} and, conversely, an ''out'' state <math>|\{p\}\ \mathrm{out}\rangle</math> describes the state of a system of particles which, long after interaction, will be moving freely with definite momenta {{math|{''p''}.}}

''In'' and ''out'' states are states in [[Heisenberg picture]] so they should not be thought to describe particles at a definite time, but rather to describe the system of particles in its entire evolution, so that the S-matrix element:

:<math>S_{\rm fi}=\langle \{q\}\ \mathrm{out}| \{p\}\ \mathrm{in}\rangle</math>

is the [[probability amplitude]] for a set of particles which were prepared with definite momenta {{math|{''p''} }} to interact and be measured later as a new set of particles with  momenta {{math|{''q''}.}}

The easy way {{refn|group=note|A pedagogical derivation of the LSZ reduction formula can be found in Peskin and Schroeder, Section 7.2, <ref name="Peskin"/> also in Srednicki, Section I.5,<ref name="Srednicki"/> in Weinberg, pp.&nbsp;436–438, <ref name="Weinberg"/> in Ticciati, section 10.5 (using <math>\varphi^\prime(f,t)</math> to denote creation operators),<ref name="Ticciati"/> or in lecture notes by Skaar, University of Oslo.<ref name="Skaar"/>}} to build ''in'' and ''out'' states is to seek appropriate field operators that provide the right [[creation and annihilation operators]]. These fields are called respectively ''in'' and ''out'' fields:

Just to fix ideas, suppose we deal with a [[Scalar field (quantum field theory)|Klein–Gordon field]] that interacts in some way:

:<math>\mathcal L= \frac 1 2 \partial_\mu \varphi\partial^\mu \varphi - \frac 1 2 m_0^2 \varphi^2 +\mathcal L_{\mathrm{int}}</math>

<math>\mathcal L_{\mathrm{int}}</math> may contain a [[Nonlinear scalar field theory|self interaction]] {{math|''gφ''<sup>3</sup>}} or interaction with other fields, like a [[Yukawa interaction]] <math>g\ \varphi\bar\psi\psi</math>. From this [[Lagrangian (field theory)|Lagrangian]], using [[Euler–Lagrange equation]]s, the equation of motion follows:

:<math>\left(\partial^2+m_0^2\right)\varphi(x)=j_0(x)</math>

where, if <math>\mathcal L_{\mathrm{int}}</math> does not contain derivative couplings:

:<math>j_0=\frac{\partial\mathcal L_{\mathrm{int}}}{\partial \varphi}</math>

We may expect the ''in'' field to resemble the asymptotic behaviour of the free field as <math>x^0 \to -\infty</math>, making the assumption that in the far away past interaction described by the current {{math|''j''<sub>0</sub>}} is negligible, as particles are far from each other. This hypothesis is named the [[adiabatic theorem|adiabatic hypothesis]]. However [[self-energy|self interaction]] never fades away and, besides many other effects, it causes a difference between the Lagrangian mass {{math|''m''<sub>0</sub>}} and the physical mass {{mvar|m}} of the {{mvar|φ}} [[boson]]. This fact must be taken into account by rewriting the equation of motion as follows:{{Citation needed|date=August 2011}}

:<math>\left(\partial^2+m^2\right)\varphi(x)=j_0(x)+\left(m^2-m_0^2\right)\varphi(x)=j(x)</math>

This equation can be solved formally using the retarded [[Green's function]] of the Klein–Gordon operator <math>\partial^2+m^2</math>:

:<math>\Delta_{\mathrm{ret}}(x)=i\theta\left(x^0\right)\int \frac{\mathrm{d}^3k}{(2\pi)^3 2\omega_k} \left(e^{-ik\cdot x}-e^{ik\cdot x}\right)_{k^0=\omega_k}\qquad \omega_k=\sqrt{\mathbf{k}^2+m^2}</math>

allowing us to split interaction from asymptotic behaviour. The solution is:

:<math>\varphi(x)=\sqrt Z \varphi_{\mathrm{in}}(x) +\int \mathrm{d}^4y \Delta_{\mathrm{ret}}(x-y)j(y)</math>

The factor <math>\sqrt{Z}</math> is a normalization factor that will come handy later, the field <math>\varphi_{\mathrm{in}}</math> is a solution of the [[Homogeneous differential equation|homogeneous equation]] associated with the equation of motion:

:<math>\left(\partial^2+m^2\right) \varphi_{\mathrm{in}}(x)=0,</math>

and hence is a [[free field]] which describes an incoming unperturbed wave, while the last term of the solution gives the [[Perturbation theory (quantum mechanics)|perturbation]] of the wave due to interaction.

The field <math>\varphi_{\mathrm{in}}</math> is indeed the ''in'' field we were seeking, as it describes the asymptotic behaviour of the interacting field as <math>x^0 \to -\infty</math>, though this statement will be made more precise later. It is a free scalar field so it can be expanded in plane waves:

:<math>\varphi_{\mathrm{in}}(x)=\int \mathrm{d}^3k \left\{f_k(x) a_{\mathrm{in}}(\mathbf{k})+f^*_k(x) a^\dagger_{\mathrm{in}}(\mathbf{k})\right\}</math>

where:

:<math>f_k(x)=\left.\frac{e^{-ik\cdot x}}{(2\pi)^{\frac{3}{2}}(2\omega_k)^{\frac{1}{2}}}\right|_{k^0=\omega_k}</math>

The inverse function for the coefficients in terms of the field can be easily obtained and put in the elegant form:

:<math>a_{\mathrm{in}}(\mathbf{k})=i\int \mathrm{d}^3x f^*_k(x)\overleftrightarrow{\partial_0}\varphi_{\mathrm{in}}(x)</math>

where:

:<math>{\mathrm{g}}\overleftrightarrow{\partial_0} f = \mathrm{g}\partial_0 f -f\partial_0 \mathrm{g}.</math>

The [[Fourier coefficient]]s satisfy the algebra of [[creation and annihilation operators]]:

:<math>[a_{\mathrm{in}}(\mathbf{p}),a_{\mathrm{in}}(\mathbf{q})]=0;\quad [a_{\mathrm{in}}(\mathbf{p}),a^\dagger_{\mathrm{in}}(\mathbf{q})]=\delta^3(\mathbf{p}-\mathbf{q});</math>

and they can be used to build ''in'' states in the usual way:

:<math>\left|k_1,\ldots,k_n\ \mathrm{in}\right\rangle=\sqrt{2\omega_{k_1}}a_{\mathrm{in}}^\dagger(\mathbf{k}_1)\ldots \sqrt{2\omega_{k_n}}a_{\mathrm{in}}^\dagger(\mathbf{k}_n)|0\rangle</math>

The relation between the interacting field and the ''in'' field is not very simple to use, and the presence of the retarded Green's function tempts us to write something like:

:<math>\varphi(x)\sim\sqrt Z\varphi_{\mathrm{in}}(x)\qquad \mathrm{as}\quad x^0\to-\infty</math>

implicitly making the assumption that all interactions become negligible when particles are far away from each other. Yet the current {{math|''j''(''x'')}} contains also self interactions like those producing the mass shift from {{math|''m''<sub>0</sub>}} to {{mvar|m}}. These interactions do not fade away as particles drift apart, so much care must be used in establishing asymptotic relations between the interacting field and the ''in'' field.

The correct prescription, as developed by Lehmann, Symanzik and Zimmermann, requires two normalizable states <math>|\alpha\rangle</math> and <math>|\beta\rangle</math>, and a normalizable solution {{math|&nbsp;''f''&nbsp;(''x'')}} of the Klein–Gordon equation <math>(\partial^2+m^2)f(x)=0</math>. With these pieces one can state a correct and useful but very weak asymptotic relation:

:<math>\lim_{x^0\to-\infty} \int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow{\partial_0}\varphi(x)|\beta\rangle= \sqrt Z \int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow{\partial_0}\varphi_{\mathrm{in}}(x)|\beta\rangle</math>

The second member is indeed independent of time as can be shown by differentiating and remembering that both <math>\varphi_{\mathrm{in}}</math> and {{math|&nbsp;''f''&nbsp;}} satisfy the Klein–Gordon equation.

With appropriate changes the same steps can be followed to construct an ''out'' field that builds ''out'' states. In particular the definition of the ''out'' field is:

:<math>\varphi(x)=\sqrt Z \varphi_{\mathrm{out}}(x) +\int \mathrm{d}^4y \Delta_{\mathrm{adv}}(x-y)j(y)</math>

where <math>\Delta_\mathrm{adv} (x-y)</math> is the advanced Green's function of the Klein–Gordon operator. The weak asymptotic relation between ''out'' field and  interacting field is:

:<math> \lim_{x^0\to \infty} \int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow{\partial_0}\varphi(x)|\beta\rangle= \sqrt Z \int \mathrm{d}^3x
\langle\alpha|f(x)\overleftrightarrow{\partial_0}\varphi_{\mathrm{out}}(x)|\beta\rangle </math>