S-matrix elements are amplitudes of transitions between in states and out states.[2][3][4][5][6] An in state describes the state of a system of particles which, in a far away past, before interacting, were moving freely with definite momenta {p}, and, conversely, an out state describes the state of a system of particles which, long after interaction, will be moving freely with definite momenta {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:
is the probability amplitude for a set of particles which were prepared with definite momenta {p} to interact and be measured later as a new set of particles with momenta {q}.
The easy way [note 1] 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 Klein–Gordon field that interacts in some way which doesn't concern us:
We may expect the in field to resemble the asymptotic behaviour of the free field as x0 → −∞, making the assumption that in the far away past interaction described by the current j0 is negligible, as particles are far from each other. This hypothesis is named the adiabatic hypothesis. However self interaction never fades away and, besides many other effects, it causes a difference between the Lagrangian mass m0 and the physical mass m of the φboson. This fact must be taken into account by rewriting the equation of motion as follows:[citation needed]
This equation can be solved formally using the retarded Green's function of the Klein–Gordon operator :
allowing us to split interaction from asymptotic behaviour. The solution is:
The factor √Z is a normalization factor that will come handy later, the field φin is a solution of the homogeneous equation associated with the equation of motion:
and hence is a free field which describes an incoming unperturbed wave, while the last term of the solution gives the perturbation of the wave due to interaction.
The field φin is indeed the in field we were seeking, as it describes the asymptotic behaviour of the interacting field as x0 → −∞, though this statement will be made more precise later. It is a free scalar field so it can be expanded in plane waves:
where:
The inverse function for the coefficients in terms of the field can be easily obtained and put in the elegant form:
and they can be used to build in states in the usual way:
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:
implicitly making the assumption that all interactions become negligible when particles are far away from each other. Yet the current j(x) contains also self interactions like those producing the mass shift from m0 to 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 and , and a normalizable solution f (x) of the Klein–Gordon equation . With these pieces one can state a correct and useful but very weak asymptotic relation:
The second member is indeed independent of time as can be shown by differentiating and remembering that both φin and f 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:
where Δadv(x − y) is the advanced Green's function of the Klein–Gordon operator. The weak asymptotic relation between out field and interacting field is:
The reduction formula for scalars
The asymptotic relations are all that is needed to obtain the LSZ reduction formula. For future convenience we start with the matrix element:
which is slightly more general than an S-matrix element. Indeed, is the expectation value of the time-ordered product of a number of fields between an out state and an in state. The out state can contain anything from the vacuum to an undefined number of particles, whose momenta are summarized by the index β. The in state contains at least a particle of momentum p, and possibly many others, whose momenta are summarized by the index α. If there are no fields in the time-ordered product, then is obviously an S-matrix element. The particle with momentum p can be 'extracted' from the in state by use of a creation operator:
where the prime on denotes that one particle has been taken out. With the assumption that no particle with momentum p is present in the out state, that is, we are ignoring forward scattering, we can write:
because acting on the left gives zero. Expressing the construction operators in terms of in and out fields, we have:
Now we can use the asymptotic condition to write:
Then we notice that the field φ(x) can be brought inside the time-ordered product, since it appears on the right when x0 → −∞ and on the left when x0 → ∞:
In the following, x dependence in the time-ordered product is what matters, so we set:
It can be shown by explicitly carrying out the time integration that:[note 2]
so that, by explicit time derivation, we have:
By its definition we see that fp (x) is a solution of the Klein–Gordon equation, which can be written as:
Substituting into the expression for and integrating by parts, we arrive at:
That is:
Starting from this result, and following the same path another particle can be extracted from the in state, leading to the insertion of another field in the time-ordered product. A very similar routine can extract particles from the out state, and the two can be iterated to get vacuum both on right and on left of the time-ordered product, leading to the general formula:
Which is the LSZ reduction formula for Klein–Gordon scalars. It gains a much better looking aspect if it is written using the Fourier transform of the correlation function:
Using the inverse transform to substitute in the LSZ reduction formula, with some effort, the following result can be obtained:
Leaving aside normalization factors, this formula asserts that S-matrix elements are the residues of the poles that arise in the Fourier transform of the correlation functions as four-momenta are put on-shell.
Reduction formula for fermions
Recall that solutions to the quantized free-field Dirac equation may be written as
where the metric signature is mostly plus, is an annihilation operator for b-type particles of momentum and spin , is a creation operator for d-type particles of spin , and the spinors and satisfy and . The Lorentz-invariant measure is written as , with . Consider now a scattering event consisting of an in state of non-interacting particles approaching an interaction region at the origin, where scattering occurs, followed by an out state of outgoing non-interacting particles. The probability amplitude for this process is given by
where no extra time-ordered product of field operators has been inserted, for simplicity. The situation considered will be the scattering of b-type particles to b-type particles. Suppose that the in state consists of particles with momenta and spins , while the out state contains particles of momenta and spins . The in and out states are then given by
Extracting an in particle from yields a free-field creation operator acting on the state with one less particle. Assuming that no outgoing particle has that same momentum, we then can write
where the prime on denotes that one particle has been taken out. Now recall that in the free theory, the b-type particle operators can be written in terms of the field using the inverse relation
where . Denoting the asymptotic free fields by and , we find
The weak asymptotic condition needed for a Dirac field, analogous to that for scalar fields, reads
and likewise for the out field. The scattering amplitude is then
where now the interacting field appears in the inner product. Rewriting the limits in terms of the integral of a time derivative, we have
where the row vector of matrix elements of the barred Dirac field is written as . Now, recall that is a solution to the Dirac equation:
Solving for , substituting it into the first term in the integral, and performing an integration by parts, yields
Switching to Dirac index notation (with sums over repeated indices) allows for a neater expression, in which the quantity in square brackets is to be regarded as a differential operator:
Consider next the matrix element appearing in the integral. Extracting an out state creation operator and subtracting the corresponding in state operator, with the assumption that no incoming particle has the same momentum, we have
Remembering that , where , we can replace the annihilation operators with in fields using the adjoint of the inverse relation. Applying the asymptotic relation, we find
Note that a time-ordering symbol has appeared, since the first term requires on the left, while the second term requires it on the right. Following the same steps as before, this expression reduces to
The rest of the in and out states can then be extracted and reduced in the same way, ultimately resulting in
The same procedure can be done for the scattering of d-type particles, for which 's are replaced by 's, and 's and 's are swapped.
Field strength normalization
The reason of the normalization factor Z in the definition of in and out fields can be understood by taking that relation between the vacuum and a single particle state with four-moment on-shell:
Remembering that both φ and φin are scalar fields with their Lorentz transform according to:
where Pμ is the four-momentum operator, we can write:
Applying the Klein–Gordon operator ∂2 + m2 on both sides, remembering that the four-moment p is on-shell and that Δret is the Green's function of the operator, we obtain:
So we arrive to the relation:
which accounts for the need of the factor Z. The in field is a free field, so it can only connect one-particle states with the vacuum. That is, its expectation value between the vacuum and a many-particle state is null. On the other hand, the interacting field can also connect many-particle states to the vacuum, thanks to interaction, so the expectation values on the two sides of the last equation are different, and need a normalization factor in between. The right hand side can be computed explicitly, by expanding the in field in creation and annihilation operators:
Using the commutation relation between ain and we obtain:
leading to the relation:
by which the value of Z may be computed, provided that one knows how to compute .
Notes
^A pedagogical derivation of the LSZ reduction formula can be found in Peskin and Schroeder, Section 7.2, [2] also in Srednicki, Section I.5,[3] in Weinberg, pp. 436–438, [4] in Ticciati, section 10.5 (using to denote creation operators),[5] or in lecture notes by Skaar, University of Oslo.[6]
^Pulling the operators from time-ordering is not entirely trivial since neither nor commutes with time-ordering . When we apply both the differential and the integral operators, however, the problems cancel out and the combined operator commutes with time-ordering.[5]
إسماعيل عبد المعين معلومات شخصية الميلاد سنة 1916 الخرطوم بحري تاريخ الوفاة 28 مايو 1984 (67–68 سنة) مواطنة السودان الحياة العملية المهنة موسيقي، وملحن تعديل مصدري - تعديل إسماعيل عبد المعين (1916- 28 مايو 1984)، موسيقي وملحن سوداني، ويسميه البعض الأب المجهول لل…
This article relies excessively on references to primary sources. Please improve this article by adding secondary or tertiary sources. Find sources: No Humans Involved novel – news · newspapers · books · scholar · JSTOR (July 2013) (Learn how and when to remove this template message) This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help to improve …
Cet article est une ébauche concernant une chanson. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. Pour la chanson homonyme de Hank Williams, voir You Win Again. You Win Again Single de Bee Geesextrait de l'album E.S.P. Face B Backtafunk Sortie août 1987 (États-Unis)octobre 1987 (Royaume-Uni) Durée 3:54 (version single)4:01 (version album) Genre Pop Format 45 tours, maxi 45 tours, CD Auteur-compositeur Bar…
Pat Kenny Pat Kenny (eigentl. Patrick Kenny; * 29. Januar 1948 in Dublin) ist ein irischer Hörfunk- und Fernsehmoderator. Kenny studierte Chemieingenieurwesen am University College Dublin. Ab Mitte der 1970er Jahre wurde er bei RTÉ Radio 1 aktiv, erst als Programmsprecher und Radio-DJ, später begann sein eigenes tägliches Vormittags-Magazin Today. Mitte der 1980er Jahre wurde er zusätzlich beim Fernsehprogramm von RTÉ tätig. Er moderierte kurzzeitig das Magazin Today Tonight und seine ers…
Het seizoen 1974/75 van de Beker van België in het voetbal begon in 1974 en eindigde op 1 juni 1975 met de finale in het Heizelstadion in Brussel. De beker ging naar RSC Anderlecht dat Antwerp FC in de finale versloeg. Laatste 16 Dit schema toont 11 van de 16 langst overgebleven clubs en de wedstrijden vanaf de 1/8ste finales. De 1/8ste finales worden net als de kwartfinales en halve finales gespeeld in één match. De finale wordt gespeeld in één wedstrijd op neutraal terrein. 1/8e fi…
This article is about the ruined Elizabethan mansion. For the district of Runcorn, see Runcorn. For the power station, see Rocksavage Power Station. For the 2005 album, see Rock Savage (album). Building in Cheshire, EnglandRocksavageRuins of Rocksavage c. 1818; the octagonal towers flanked the gatewayGeneral informationStatusRuined (limited fragments remain)Architectural styleElizabethan, prodigy houseTown or cityClifton, Runcorn, CheshireCountryEnglandConstruction started1565Completed1568Demoli…
Genre of music-themed action video game Players using a dance pad to play Dance Dance Revolution, one of the most successful rhythm games Part of a series onAction games Subgenres Action-adventure Metroidvania Battle royale Fighting Beat 'em up Hack and slash Platform fighter Platform Metroidvania Rhythm Action RPG Shooter Artillery Arena First-person Hero Light gun Third-person Tactical Shoot 'em up Bullet hell Twin-stick Sports Racing Stealth Survival Vehicle sim Topics Capture the flag Cover …
جغرافيا ماليزيامعلومات عامةالبلد ماليزيا القارة آسيا الحدود تايلاندبرونايإندونيسياسنغافورةالفلبين الأرض والتضاريسالمساحة 330٬803 كم² أعلى نقطة جبل كينابالو أدنى نقطة المحيط الهندي تعديل - تعديل مصدري - تعديل ويكي بيانات خريطة ماليزيا. تقع ماليزيا في منطقة جنوب شرق آسيا، وأ…
Japanese fashion magazine This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Seventeen Japanese magazine – news · newspapers · books · scholar · JSTOR (Febr…
American judge S. Arthur SpiegelSenior Judge of the United States District Court for the Southern District of OhioIn officeJune 5, 1995 – December 31, 2014Judge of the United States District Court for the Southern District of OhioIn officeMay 23, 1980 – June 5, 1995Appointed byJimmy CarterPreceded byDavid Stewart PorterSucceeded bySusan J. Dlott Personal detailsBornS. Arthur Spiegel(1920-10-24)October 24, 1920Cincinnati, OhioDiedDecember 31, 2014(2014-12-31) (aged 94)C…
Coordenadas: 45° N 75° O Ontário OrientalEastern Ontario (inglês)Est de l'Ontario (francês) Localização ██ Área central ██ Área Estendida██ Área central ██ Área Estendida Coordenadas 45° N 75° O País Canadá Província Ontário Características geográficas Área total 35,296 km² População total (2006) 1,603,625 hab. Densidade 0 hab./km² Fuso horário UTC−5 (EST) Horário de verão UTC−4 (EDT) Código postal K Código de área 613…
Confederate fort Not to be confused with Fort Fisher, part of the Union Army works at the Siege of Petersburg. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Fort Fisher – news · newspapers · books · scholar · JSTOR (February 2013) (Learn how and when to remove this template message) Fort FisherNew Hanover Cou…
For the Fred Frith album, see The Happy End Problem. Erdős–Szekeres conjecture redirects here. For their theorem on monotonic subsequences, see Erdős–Szekeres theorem. The happy ending problem: every set of five points in general position contains the vertices of a convex quadrilateral In mathematics, the happy ending problem (so named by Paul Erdős because it led to the marriage of George Szekeres and Esther Klein[1]) is the following statement: Theorem — any set …
PT-91 Twardy merupakan Tank MBT buatan Polandia yang didesain oleh Obrum (Ośrodek Badawczo-Rozwojowy Urządzeń Mechanicznych atau Penelitian dan pengembangan untuk kegunaan mekanis) dan diproduksi oleh Bumar Labedy. PT-91 merupakan pengembangan pesat dari tank Rusia T-72M1 yang sempat diproduksi dan dipakai oleh Polandia. Tidak seperti modifikasi T-72 pada umumnya, tank ini menggunakan komponen yang eksklusif dibuat secara domestik. Saat ini Polandia dan Malaysia adalah pengguna utama PT-91. M…
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Pure 2010 film – news · newspapers · books · scholar · JSTOR (June 2019) (Learn how and when to remove this template message) 2010 Swedish filmPureFilm posterSwedishTill det som är vackert Directed byLisa LangsethWritten byLisa LangsethStarringAlicia Vikande…
University of East SarajevoУниверзитет у Источном Сарајеву Univerzitet u Istočnom SarajevuTypePublicEstablished1992; 31 years ago (1992)[1]Budget€14.00 million (2015)[2]RectorMilan Kulić [3]Academic staff933[4]Students8,049 (2018–19)[5]Address30 Vuk Karadžić Street[4], Lukavica, Republika Srpska, 71123, Bosnia and HerzegovinaCampusUrbanColorsBlue and Red Websitewww.ues.rs.ba Th…