2 | $0$ | $-\frac{2928}{1225}$ | $\frac{1584}{175}$ | $\frac{48}{35}$ | $-\frac{32}{49}$ | $\frac{15376}{735}$ | $-\frac{3424}{105}$ | $-\frac{240}{7}$ | $0$ | $\frac{3952}{1029}$ | $\frac{2672}{147}$ | $\frac{48}{7}$
3 | $-\frac{216}{125}$ | $\frac{552}{875}$ | $-\frac{2592}{875}$ | $0$ | $0$ | $-\frac{1728}{175}$ | $-\frac{288}{25}$ | $0$ | $\frac{3904}{2205}$ | $\frac{656}{245}$ | $-\frac{288}{49}$ | $0$
4 | $0$ | $-\frac{41472}{6125}$ | $0$ | $0$ | $-\frac{3456}{1225}$ | $-\frac{2592}{1225}$ | $0$ | $0$ | $0$ | $-\frac{608}{343}$ | $0$ | $0$
5 | $-\frac{1536}{1225}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-\frac{1088}{3087}$ | $0$ | $0$ | $0$

(c) Coefficients $\mathcal{C}_{m,\ell}^{n}$ associated to $\,{}_{1,m}\overline{\chi}_{\ell}[\overline{\mathrm{M}}_{a\langle i}^{(n)},\overline{\mathrm{M}}_{j\rangle a}^{(8-n)}]$ $\ell$ | 2
---|---
$m$ $n$ | 0 | 1 | 2 | 3
0 | $-\frac{64}{245}$ | $-\frac{256}{175}$ | $\frac{64}{25}$ | $0$
1 | $0$ | $\frac{1536}{1225}$ | $-\frac{1536}{175}$ | $\frac{384}{175}$
2 | $\frac{64}{245}$ | $-\frac{64}{245}$ | $\frac{64}{5}$ | $-\frac{192}{35}$
3 | $0$ | $\frac{192}{175}$ | $-\frac{2304}{175}$ | $\frac{576}{175}$
4 | $-\frac{1152}{2695}$ | $-\frac{4608}{1225}$ | $\frac{1152}{175}$ | $0$
5 | $0$ | $\frac{768}{245}$ | $0$ | $0$
6 | $\frac{1152}{2695}$ | $0$ | $0$ | $0$

(d) Coefficients $\mathcal{D}_{m,\ell}^{n}$ associated to $\,{}_{2,m}\overline{\chi}_{\ell}[\overline{\mathrm{M}}_{a\langle i}^{(n)},\overline{\mathrm{M}}_{j\rangle a}^{(7-n)}]$ $\ell$ | 2
---|---
$m$ $n$ | 0 | 1 | 2
0 | $0$ | $\frac{576}{175}$ | $0$
1 | $-\frac{576}{1225}$ | $-\frac{1152}{175}$ | $\frac{1152}{175}$
2 | $0$ | $\frac{1152}{245}$ | $-\frac{576}{35}$
3 | $-\frac{192}{175}$ | $-\frac{1728}{175}$ | $\frac{1728}{175}$
4 | $0$ | $\frac{10368}{1225}$ | $0$
5 | $\frac{384}{245}$ | $0$ | $0$

## References

  * Maggiore [2008] M. Maggiore,  _Gravitational waves: Volume 1: Theory and experiments_ , Vol. 1 (Oxford university press, 2008).
  * Buonanno and Sathyaprakash [2015] A. Buonanno and B. Sathyaprakash, Sources of gravitational waves: Theory and observations, in  _General Relativity and Gravitation: A Centennial Perspective_ , edited by A. Ashtekar, B. Berger, J. Isenberg, and M. MacCallum (2015) p. 513, arXiv:1410.7832 [gr-qc] .
  * Blanchet [2014] L. Blanchet, Gravitational radiation from post-Newtonian sources and inspiralling compact binaries, Living Rev. Relativ. 17, 2 (2014), arXiv:1310.1528 [gr-qc] .
  * Barack  _et al._ [2019] L. Barack _et al._ , Black holes, gravitational waves and fundamental physics: a roadmap, Class. Quant. Grav. 36, 143001 (2019), arXiv:1806.05195 [gr-qc] .
  * Bonnor [1959] W. Bonnor, Spherical gravitational waves, Philos. Trans. R. Soc. London, Ser. A 251, 233 (1959).
  * Thorne and Kovàcs [1975] K. Thorne and S. Kovàcs, Generation of gravitational waves. i. weak-field sources, Astrophys. J. 200, 245 (1975).
  * Thorne [1980] K. Thorne, Multipole expansions of gravitational radiation, Rev. Mod. Phys. 52, 299 (1980).
  * Blanchet and Damour [1988] L. Blanchet and T. Damour, Tail-transported temporal correlations in the dynamics of a gravitating system, Phys. Rev. D 37, 1410 (1988).
  * Abbott  _et al._ [2016] B. Abbott _et al._ (LIGO Scientific Collaboration and VIRGO Collaboration), Tests of general relativity with gw150914, Phys. Rev. Lett. 116, 221101 (2016), arXiv:1602.03841 [gr-qc] .
  * Blanchet and Sathyaprakash [1994] L. Blanchet and B. S. Sathyaprakash, Signal analysis of gravitational wave tails, Class. Quant. Grav. 11, 2807 (1994).
  * Blanchet and Sathyaprakash [1995] L. Blanchet and B. S. Sathyaprakash, Detecting the tail effect in gravitational wave experiments, Phys. Rev. Lett. 74, 1067 (1995).
  * Arun  _et al._ [2006a] K. Arun, B. Iyer, M. Qusailah, and B. Sathyaprakash, Testing post-Newtonian theory with gravitational wave observations, Class. Quant. Gravity 23, L37 (2006a), gr-qc/0604018 .
  * Arun  _et al._ [2006b] K. Arun, B. Iyer, M. Qusailah, and B. Sathyaprakash, Probing the non-linear structure of general relativity with black hole binaries, Phys. Rev. D 74, 024006 (2006b), gr-qc/0604067 .
  * Blanchet and Damour [1992] L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46, 4304 (1992).
  * Poisson [1993] E. Poisson, Gravitational-radiation from a particle in circular orbit around a black-hole. i. analytical results for the nonrotating case, Phys. Rev. D 47, 1497 (1993).
  * Wiseman [1993] A. Wiseman, Coalescing binary-systems of compact objects to (post)5/2-Newtonian order. iv. the gravitational-wave tail, Phys. Rev. D 48, 4757 (1993).
  * Blanchet and Schäfer [1993] L. Blanchet and G. Schäfer, Gravitational wave tails and binary star systems, Class. Quant. Grav. 10, 2699 (1993).
  * Foffa and Sturani [2012] S. Foffa and R. Sturani, Tail terms in gravitational radiation reaction via effective field theory, Phys. Rev. D 87, 044056 (2012), arXiv:1111.5488 [gr-qc] .
  * Galley  _et al._ [2016] C. R. Galley, A. K. Leibovich, R. A. Porto, and A. Ross, Tail effect in gravitational radiation reaction: Time nonlocality and renormalization group evolution, Phys. Rev. D 93, 124010 (2016), arXiv:arXiv:1511.07379 [gr-qc] [gr-qc] .
  * Blanchet [1998a] L. Blanchet, Gravitational-wave tails of tails, Class. Quant. Grav. 15, 113 (1998a), gr-qc/9710038 .
  * Faye  _et al._ [2015] G. Faye, L. Blanchet, and B. R. Iyer, Non-linear multipole interactions and gravitational-wave octupole modes for inspiralling compact binaries to third-and-a-half post-Newtonian order, Class. Quant. Grav. 32, 045016 (2015), arXiv:1409.3546 [gr-qc] .
  * Marchand  _et al._ [2016] T. Marchand, L. Blanchet, and G. Faye, Gravitational-wave tail effects to quartic non-linear order, Class. Quant. Grav. 33, 244003 (2016), arXiv:1607.07601 [gr-qc] .
  * Messina and Nagar [2017] F. Messina and A. Nagar, Parametrized-4.5PN taylorf2 approximant(s) and tail effects to quartic nonlinear order from the effective one body formalism, Phys. Rev. D 96, 049907 (2017), arXiv:1703.08107 [gr-qc] .
  * Blanchet [1990] L. Blanchet,  _Contribution à l’étude du rayonnement gravitationnel émis par un système isolé_ , Ph.D. thesis, Université Paris VI, Paris (1990).
  * Christodoulou [1991] D. Christodoulou, Nonlinear nature of gravitation and gravitational-wave experiments, Phys. Rev. Lett. 67, 1486 (1991).
  * Wiseman and Will [1991] A. Wiseman and C. Will, Christodoulou’s nonlinear gravitational-wave memory: Evaluation in the quadrupole approximation, Phys. Rev. D 44, R2945 (1991).
  * Thorne [1992] K. Thorne, Gravitational-wave bursts with memory: The christodoulou effect, Phys. Rev. D 45, 520 (1992).
  * Blanchet [1998b] L. Blanchet, Quadrupole-quadrupole gravitational waves, Class. Quant. Grav. 15, 89 (1998b), gr-qc/9710037 .
  * Favata [2009a] M. Favata, Post-Newtonian corrections to the gravitational-wave memory for quasicircular, inspiralling compact binaries, Phys. Rev. D 80, 024002 (2009a), arXiv:0812.0069 [gr-qc] .
  * Favata [2011] M. Favata, The gravitational-wave memory from eccentric binaries, Phys. Rev. D 84, 124013 (2011), arXiv:1108.3121 [gr-qc] .
  * Nichols [2017] D. A. Nichols, Spin memory effect for compact binaries in the post-newtonian approximation, Physical Review D 95, 084048 (2017).
  * Favata [2009b] M. Favata, Nonlinear gravitational-wave memory from binary black hole mergers, Astrophys. J. Lett. 696, L159 (2009b), arXiv:0902.3660 [astro-ph.SR] .
  * Lasky  _et al._ [2016] P. D. Lasky, E. Thrane, Y. Levin, J. Blackman, and Y. Chen, Detecting gravitational-wave memory with LIGO: implications of GW150914, Phys. Rev. Lett. 117, 061102 (2016), arXiv:1605.01415 [astro-ph.HE] .
  * McNeill  _et al._ [2017] L. O. McNeill, E. Thrane, and P. D. Lasky, Detecting Gravitational Wave Memory without Parent Signals, Phys. Rev. Lett. 118, 181103 (2017), arXiv:1702.01759 [astro-ph.IM] .
  * Goldberger and Rothstein [2006] W. Goldberger and I. Rothstein, Effective field theory of gravity for extended objects, Phys. Rev. D 73, 104029 (2006), hep-th/0409156 .
  * Foffa and Sturani [2014] S. Foffa and R. Sturani, Effective field theory methods to model compact binaries, Class. Quant. Gravity 31, 043001 (2014), arXiv:1309.3474 [gr-qc] .
  * Porto [2016] R. A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633, 1 (2016), arXiv:1601.04914 [hep-th] .
  * Levi [2020] M. Levi, Effective Field Theories of Post-Newtonian Gravity: A comprehensive review, Rept. Prog. Phys. 83, 075901 (2020), arXiv:1807.01699 [hep-th] .
  * Marchand  _et al._ [2020] T. Marchand, Q. Henry, F. Larrouturou, S. Marsat, G. Faye, and L. Blanchet, The mass quadrupole moment of compact binary systems at the fourth post-Newtonian order, Class. Quant. Grav. 37, 215006 (2020), arXiv:2003.13672 [gr-qc] .
  * Henry  _et al._ [2021] Q. Henry, G. Faye, and L. Blanchet, The current-type quadrupole moment and gravitational-wave mode ($\ell$, m) = (2, 1) of compact binary systems at the third post-Newtonian order, Class. Quant. Grav. 38, 185004 (2021), arXiv:2105.10876 [gr-qc] .
  * Larrouturou  _et al._ [2022a] F. Larrouturou, Q. Henry, L. Blanchet, and G. Faye, The quadrupole moment of compact binaries to the fourth post-Newtonian order: I. Non-locality in time and infra-red divergencies, Class. Quant. Grav. 39, 115007 (2022a), arXiv:2110.02240 [gr-qc] .
  * Larrouturou  _et al._ [2022b] F. Larrouturou, L. Blanchet, Q. Henry, and G. Faye, The quadrupole moment of compact binaries to the fourth post-Newtonian order: II. Dimensional regularization and renormalization, Class. Quant. Grav. 39, 115008 (2022b), arXiv:2110.02243 [gr-qc] .
  * Blanchet  _et al._ [2022] L. Blanchet, G. Faye, and F. Larrouturou, The quadrupole moment of compact binaries to the fourth post-Newtonian order: from source to canonical moment, Class. Quant. Grav. 39, 195003 (2022), arXiv:2204.11293 [gr-qc] .
  * Trestini  _et al._ [2022] D. Trestini, F. Larrouturou, and L. Blanchet, The Quadrupole Moment of Compact Binaries to the Fourth post-Newtonian Order: Relating the Harmonic and Radiative Metrics, (2022), arXiv:2209.02719 [gr-qc] .
  * Damour  _et al._ [2014] T. Damour, P. Jaranowski, and G. Schäfer, Non-local-in-time action for the fourth post-Newtonian conservative dynamics of two-body systems, Phys. Rev. D 89, 064058 (2014), arXiv:1401.4548 [gr-qc] .
  * Bernard  _et al._ [2016] L. Bernard, L. Blanchet, A. Bohé, G. Faye, and S. Marsat, Fokker action of non-spinning compact binaries at the fourth post-Newtonian approximation, Phys. Rev. D 93, 084037 (2016), arXiv:1512.02876 [gr-qc] .
  * Marchand  _et al._ [2018] T. Marchand, L. Bernard, L. Blanchet, and G. Faye, Ambiguity-free completion of the equations of motion of compact binary systems at the fourth post-Newtonian order, Phys. Rev. D 97, 044023 (2018), arXiv:1707.09289 [gr-qc] .
  * Foffa and Sturani [2019] S. Foffa and R. Sturani, Conservative dynamics of binary systems to fourth post-Newtonian order in the EFT approach i: Regularized Lagrangian, Phys. Rev. D 100, 024047 (2019), arXiv:1903.05113 [gr-qc] .
  * Foffa  _et al._ [2019] S. Foffa, R. Porto, I. Rothstein, and R. Sturani, Conservative dynamics of binary systems to fourth post-Newtonian order in the EFT approach ii: Renormalized Lagrangian, Phys. Rev. D 100, 024048 (2019), arXiv:1903.05118 [gr-qc] .
  * Blümlein  _et al._ [2020] J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, Fourth post-Newtonian hamiltonian dynamics oftwo-body systems from an effective field theory approach, 2003.01692 [hep-th] (2020), arXiv preprint.
  * Foffa [2014] S. Foffa, Gravitating binaries at 5PN in the post-minkowskian approximation, Phys. Rev. D 89, 024019 (2014), arXiv:1309.3956 [gr-qc] .
  * Bini  _et al._ [2020a] D. Bini, T. Damour, and A. Geralico, Binary dynamics at the fifth and fifth-and-a-half post-newtonian orders, Physical Review D 102, 024062 (2020a).
  * Blümlein  _et al._ [2022] J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, The fifth-order post-Newtonian hamiltonian dynamics of two-body systems from an effective field theory approach: Potential contributions, Nuclear Physics B 983, 115900 (2022), arXiv:2110.13822 [gr-qc] .
  * Almeida  _et al._ [2021] G. L. Almeida, S. Foffa, and R. Sturani, Tail contributions to gravitational conservative dynamics, Physical Review D 104, 124075 (2021).
  * Bini  _et al._ [2020b] D. Bini, T. Damour, and A. Geralico, Sixth post-Newtonian local-in-time dynamics of binary systems, Physical Review D 102, 024061 (2020b).
  * Blanchet and Damour [1986] L. Blanchet and T. Damour, Radiative gravitational fields in general relativity. i. general structure of the field outside the source, Phil. Trans. Roy. Soc. Lond. A 320, 379 (1986).
  * Blanchet [1987] L. Blanchet, Radiative gravitational fields in general relativity. ii. asymptotic behaviour at future null infinity, Proc. Roy. Soc. Lond. A 409, 383 (1987).
  * Papapetrou [1969] A. Papapetrou, Coordonnées radiatives cartésiennes, Ann. Inst. Henri Poincaré A XI, 251 (1969).
  * Madore [1970a] J. Madore, Gravitational radiation from a bounded source. i, Ann. Inst. Henri Poincaré 12, 285 (1970a).
  * Madore [1970b] J. Madore, Gravitational radiation from a bounded source. ii, Ann. Inst. Henri Poincaré 12, 365 (1970b).
  * Sachs and Bergmann [1958] R. Sachs and P. Bergmann, Structure of particles in linearized gravitational theory, Phys. Rev. 112, 674 (1958).
  * Pirani [1964] F. Pirani, Introduction to gravitational radiation theory, in  _Lectures on General Relativity_ , Brandeis Summer Institute in Theoretical Physics, Vol. 1, edited by A. Trautman, F. Pirani, and H. Bondi (Prentice-Hall, Englewood Cliffs, 1964) pp. 249–373.
  * [63] The ancillary file sources.dat.wl contains the expressions for the cubic sources in the radiative construction associated to the interactions $\mathrm{M}\times\mathrm{M}\times\overline{\mathrm{M}}_{ij}$, $\mathrm{M}\times\mathrm{S}_{i}\times\overline{\mathrm{M}}_{ij}$ and $\mathrm{M}\times\overline{\mathrm{M}}_{ij}\times\overline{\mathrm{M}}_{ij}$. The ancillary file kernels.dat.wl contains the explicit expressions of the kernels ${}_{k,m}\overline{K}_{\ell}(\rho,\tau)$ and ${}_{k,m}\overline{L}_{\ell}(\rho,\tau)$ needed in (3.). Note that these files are optimally read using _Mathematica_ , but they can be straightforwardly parsed by any text editor.
  * Gradshteyn and Ryzhik [1980] I. Gradshteyn and I. Ryzhik,  _Table of Integrals, Series and Products_ (Academic Press, 1980).
  * Faye  _et al._ [2012] G. Faye, S. Marsat, L. Blanchet, and B. R. Iyer, The third and a half post-Newtonian gravitational wave quadrupole mode for quasi-circular inspiralling compact binaries, Class. Quant. Grav. 29, 175004 (2012), arXiv:1204.1043 [gr-qc] .


# Scattering in algebraic approach to quantum theory. Associative algebras

Albert Schwarz  A. Schwarz
Department of Mathematics
University of California
Davis, CA 95616, USA,
schwarz @math.ucdavis.edu

###### Abstract

The definitions of scattering matrix and inclusive scattering matrix in the
framework of formulation of quantum field theory in terms of associative
algebras with involution are presented. The scattering matrix is expressed in
terms of Green functions on shell (LSZ formula) and the inclusive scattering
matrix is expressed in terms of generalized Green functions on shell. The
expression for inclusive scattering matrix can be used also for quasi-
particles (for elementary excitations of any translation-invariant stationary
state, for example, for elementary excitations of equilibrium state.) An
interesting novelty is the consideration of associative algebras over real
numbers.

##  1 Introduction

The standard algebraic approach to quantum theory is based on consideration of
associative algebra with involution ∗(algebra of observables). This algebra
will be denoted by $\cal A$, we assume that it has a unit. Usually it is
assumed that $\cal A$ is an algebra over complex numbers (then the involution
should be antilinear), but we consider also the case when $\cal A$ is an
algebra over real numbers. Notice, however, that the case of algebras over
real numbers is less natural; the reader can skip everything related to this
case.

The states are defined as positive linear functionals on $\cal A$ (one says
that the functional $\omega$ is positive if $\omega(A^{*}A)\geq 0$ for
$A\in\cal A$). The set of states is a convex cone denoted by $\cal C$.
Proportional states are identified, hence instead of the cone $\cal C$ one can
consider the convex set ${\cal C}_{0}$ of states obeying $\omega(1)=1$ (the
set of normalized states).

Time translation (evolution operator) acts on $\cal A$ as involution
preserving automorphism. To define particles and their scattering we need also
spatial translations; together time and spatial translations span commutative
group $\cal T.$ The time translations are denoted by $T_{\tau}$ and spatial
translations by $T_{{\bf x}}$ where ${{\bf x}}\in\mathbb{R}^{d}$. The induced
transformations of the space of states are denoted by the same symbols. If
$A\in\cal A$ we use the notation $A(\tau,{\bf x})$ for $T_{\tau}T_{{\bf x}}A.$

Let us consider a stationary translation-invariant state $\omega$ and the pre
Hilbert space $\cal H$ of the corresponding GNS (Gelfand-Naimark-Segal)
representation of $\cal A$. Recall that in this representation there exists a
cyclic vector $\Phi$ obeying $\omega(A)=\langle A\Phi,\Phi\rangle$. (One says
that $\Phi$ is a cyclic vector if ${\cal H}={\cal A}\Phi$). The translations
descend to $\cal H$. In complex case the infinitesimal translations (energy
and momentum operators) are defined on a dense subset of Hilbert space
$\bar{\cal H}$ (of the completion of $\cal H$), in real case they act on a
dense subset of the complexification of $\bar{\cal H}.$ (We use the same
notation for translations in $\cal H$ as for automorphisms of $\cal A$. The
element $A$ of the algebra $\cal A$ and the corresponding operator in $\cal H$
are also denoted by the same symbol. Notice that representing $A$ as an
operator we should consider translation of $A$ as a conjugation with
$T_{\tau}T_{{\bf x}}$, i.e. $A(\tau,{{\bf x}})=T_{\tau}T_{{\bf
x}}AT_{-\tau}T_{-{\bf x}}.$ )

Elements of $\cal H$ can be regarded as excitations of $\omega.$

The elementary space $\textgoth{h}$ is defined as a space of smooth fast
decreasing functions on $\mathbb{R}^{d}\times\cal I$ (all of their derivatives
should decrease faster than any power); it is equipped with $L^{2}$ metric.
(Here $\cal I$ denotes a finite set consisting of $m$ elements.) We should
consider real -valued functions if $\cal A$ is an algebra over $\mathbb{R}$
and complex-valued functions if $\cal A$ is an algebra over $\mathbb{C}.$ It
is convenient to consider the elements of $\textgoth{h}$ as columns of $m$
functions on $\mathbb{R}^{d}.$ The spatial translations act naturally on this
space; we assume that the time translations also act on $\textgoth{h}$ and
commute with spatial translations. In momentum representation the spatial
translation $T_{{\bf x}}$ is represented as multiplication by $e^{i{\bf x}{\bf
k}}$ and the time translation $T_{\tau}$ is represented as a multiplication by
a matrix $e^{-i\tau E({\bf k})}.$We assume that $E({\bf k})$ is a non-
degenerate Hermitian matrix.

To guarantee that time translations act in $\textgoth{h}$ we assume that
$E({\bf k})$ is a smooth function of ${\bf k}$ and has at most polynomial
growth. If $\textgoth{h}$ consists of complex-valued functions then
diagonalizing the matrix $E({\bf k})$ we can reduce the general case to the
case when $m=1$; this remark was used in [8]. ( Notice, however, that the
eigenvalues of $E({\bf k})$ are not necessarily smooth functions of ${\bf
k}$.)

An elementary excitation of $\omega$ is defined as an isometric map
$\Phi:\textgoth{h}\to{\cal H}={\cal A}\Phi$ commuting with translations.

If $\cal A$ is an algebra over $\mathbb{R}$ we assume that the elements of
$\textgoth{h}$ in momentum representation obey the reality condition
$f^{*}({{\bf k}})=f(-{\bf k}).$

We show that imposing the condition of asymptotic commutativity we can define
the scattering matrix and inclusive scattering matrix of elementary
excitations of $\omega.$ To analyze the properties of scattering matrix we
assume that $\omega$ has cluster property. Our results are based on ideas of
[8] . ( This paper was published as Section 13.3 of [14], an improved version
of it was published recently in preprint form.) Notice that the results of [8]
generalize Haag-Ruelle theory dealing with scattering matrix in Lorentz-
invariant local quantum field theories. (See [15] for an exposition of Haag-
Ruelle theory closest to our approach and [16] or [14] for generalization of
this theory to the case when Lorentz-invariance is not assumed.)

If $\cal A$ is a $C^{*}$-algebra over complex numbers (as in [8]) one can
identify (quasi-local) observables with self-adjoint elements of this algebra.
For every normalized state $\omega$ one defines a probability distribution of
the observable $a$ corresponding to a self-adjoint element $A$ in such a way
that for every continuous function $f$ the expectation value of $f(a)$ is
equal to $\omega(f(A)).$ One can consider also global observables
corresponding to infinitesimal automorphisms of $\cal A$, in particular,
energy and momentum corresponding to time and spatial infinitesimal
translations. We can talk about joint spectrum of energy and momentum
operators in $\cal H$ (in the space of excitations of translation-invariant
stationary state $\omega$). We say that $\omega$ is a ground state if the
energy operator in $\bar{\cal H}$ is positive definite.

For algebras over real numbers one should consider skew-adjoint elements
($A=-A^{*}$) instead of self-adjoint elements (for algebras over complex
numbers there exists an obvious one-to-one correspondence $A\to iA$ between
skew- adjoint and self-adjoint elements). The definition of the probability
distribution of physical quantity used in the case of complex numbers does not
work, however, one can use the geometric approach to quantum theory to derive
the probability distribution from decoherence [3].

Elementary excitations of ground state are called particles. Elementary
excitations of arbitrary translation-invariant stationary state are called
quasi-particles. Quasi-particles are in general unstable; this means that the
conditions in the definition of elementary excitation are satisfied only
approximately. For quasi-particles the definition of scattering matrix does
not work, however, the definition of inclusive scattering matrix still makes
sense.

Notice that our considerations can be applied to the scattering of elementary
excitations of any translation-invariant state. In particular, they can be
applied to the scattering of elementary excitations of an equilibrium state.
It is important to notice that our considerations can be applied also in non-
equilibrium situation. The generalized Green functions we are using coincide
with functions considered in Keldysh formalism of non-equilibrium statistical
physics. They appear also in the formalism of $L$-functionals that can be used
to give a simple and transparent derivation of diagram technique for
calculation of generalized Green functions in the framework of perturbation
theory [10],[7],[14].

The ground state is not singled out in any way in our considerations.

The main goal of present paper is to give an exposition of scattering theory
in such a way that it can be easily compared with the definition of scattering
in geometric approach to quantum field theory [4] and in the approach based on
Jordan algebras [5]. In the present paper we consider also the case when $\cal
A$ is an algebra over $\mathbb{R}$; this is necessary for comparison with
papers [4], [5] where we consider both real and complex elementary spaces.
There exists an opinion that complex numbers are important in the formulation
of quantum mechanics. It is true that imposing very natural axioms one can
justify this opinion (see, for example,[1]). One of goals of [3] and present
paper is to formulate axioms that allow us to avoid using complex numbers.

##  2 Scattering matrix.

Let us consider an algebra with involution $\cal A$. We assume that spatial
and time translations act as automorphisms of $\cal A.$ We fix a translation-
invariant stationary state $\omega$; excitations of $\omega$ are defined as
elements of pre Hilbert space $\cal H$ obtained by GNS construction. The
algebra $\cal A$ acts in $\cal H$. In what follows we denote the operator in
$\cal H$ corresponding to an element $A\in\cal A$ by the same letter. We
assume that all operators we are dealing with are smooth. (An operator
corresponding to an element $B\in\mathcal{A}$ is smooth if $B=\int\alpha({\bf
x},t)A({\bf x},t)d{\bf x}dt$, where $\alpha({\bf x},t)\in{\cal
S}({\mathbb{R}}^{d+1}),A\in\mathcal{A}.$)

The element of $\cal H$ corresponding to $\omega$ is denoted by $\Phi.$ We
define an elementary excitation as an isometric map of an elementary space
$\textgoth{h}$ into $\cal H$ commuting with spatial and time translations.
(Recall that an elementary space consists of smooth fast decreasing functions
depending on spatial variable ${\bf x}$ or on momentum variable ${\bf k}$ and
on discrete variable taking $m$ values. We consider simultaneously algebras
over complex an real numbers; correspondingly the functions considered below
are complex-valued or real-valued.)

Let us fix $m$ elements $\phi_{1},\cdots,\phi_{m}\in\textgoth{h}$ and $m$
operators $B_{1},\cdots,B_{m}\in\cal A$ obeying $\Phi(\phi_{i})=B_{i}\Phi.$
The elements $\phi_{i}$ are columns of functions $\phi_{i}^{\alpha}$, together
they can be considered as a square matrix. We assume that this matrix is
invertible and commutes with the matrix $E({\bf k})$ governing the time
translation in $\textgoth{h}$. If this condition is satisfied we say that
$B_{1},\cdots,B_{m}$ are good operators.  We require also that
$B^{*}_{i}\Phi=\Phi(\psi_{i})$ for some $\psi_{i}\in\textgoth{h}.$

Notice that

$B_{j}(\tau,{{\bf x}})\Phi=\Phi(T_{\tau}T_{{\bf x}}\phi_{j}).$

In momentum representation

$B_{j}(\tau,{{\bf x}})\Phi=\Phi({(e^{-i\tau E{({\bf
k})}})}^{\alpha}_{\beta}e^{i{\bf k}{\bf x}}\phi_{j}^{\beta}).$

Let us consider a collection of smooth functions $f=(f^{1}({{\bf
k}}),\cdots,f^{m}({{\bf k}}))$ decreasing faster than any power. ( The space
of smooth fast decreasing functions is denoted by $\cal S$; we can say that
$f\in{\cal S}^{m}.$ ) We define an operator $B(f,\tau)$ acting in
$\overline{\cal H}$ (in the completion of $\cal H$) by the formula

$B(f,\tau)=\int d{{\bf x}}\tilde{f}^{j}_{\tau}({{\bf x}})B_{j}(\tau,{\bf x})$
(1)

where $\tilde{f}^{j}_{\tau}({{\bf x}})$ denotes the inverse Fourier transform
with respect to ${\bf k}$ of the function $f^{j}_{\tau}({{\bf k}})=f^{i}{({\bf
k})}(e^{i\tau E{({\bf k})}})^{j}_{i}.$

The operator $B(f,\tau)$ depends linearly of $f$; it specifies a generalized
vector function of ${{\bf k}},\tau$ that can be written in the form

$B({{\bf k}},\tau)=e^{{i\tau E{({\bf k})}})}\hat{B}(\tau,{\bf k})$ (2)

where $\hat{B}(\tau,{\bf k})$ is a Fourier transform with respect to ${\bf x}$
of $B_{j}(\tau,{\bf x})$ considered as a generalized vector function.

Let us prove that $B(f,\tau)\Phi$ does not depend on $\tau.$ Using (1) we
obtain

$B(f,\tau)\Phi=\int d{{\bf x}}\tilde{f}^{i}_{\tau}({{\bf x}})\Phi({(e^{-i\tau
E{({\bf k})}})}^{\alpha}_{\beta}e^{i{\bf k}{\bf x}}\phi_{i}^{\beta})=$
$\Phi(f^{j}{({\bf k})}(e^{i\tau E{({\bf k})}})^{i}_{j}{(e^{-i\tau E{({\bf
k})}})}^{\alpha}_{\beta}\phi_{i}^{\beta})=\Phi(f^{j}\phi_{j}^{\alpha}).$ (3)

(We have used the fact that the matrix $e^{i\tau E{({\bf k})}}$ commutes with
the matrix $\phi$.)

In what follows we denote $f^{j}\phi_{j}^{\alpha}$ as $f\phi$ where $f$ is
considered as column vector and $\phi$ as a square matrix.

Later we will use this statement in the following form:

######  Lemma 1.

$\dot{B}(f,\tau)\Phi=0$

Here dot denotes the derivative with respect to $\tau.$

######  Definition 2.

Let us consider the function $f^{j}_{\tau}({{\bf x}})$ corresponding to the
collection $f=(f^{1},...,f^{m})$ of smooth fast decreasing functions. We say
that a set $\tau K(f)$ is an essential support of the function
$f^{j}_{\tau}({{\bf x}})$ if for all $n$

$f^{j}_{\tau}({{\bf x}})<C_{n}(1+|{{\bf x}}|^{2}+\tau^{2})^{-n}$

where $\frac{{{\bf x}}}{\tau}\notin K.$

In the case when Fourier transforms $f^{i}({\bf k})$ of functions $f^{i}({\bf
x})$ have compact support one can assume that $K(f)$ is compact, but in
general is not clear that one can find a compact set $K$ obeying the
conditions of this definition.

Let us impose the conditions of asymptotic commutativity on the operators
$A_{j}\in\mathcal{A}.$ This means that

$||[A_{j}(\tau,{{\bf x}}),A_{k}]||<\frac{C_{n}(\tau)}{1+||{{\bf x}}||^{n}}.$
(4)

Here $n$ is an arbitrary integer and $C_{n}(\tau)$ is a polynomial. ( The
condition we have imposed can be weakened, see [8].)

Let us consider the vectors

$\Psi(f_{1},\tau_{1},...,f_{n},\tau_{n})=B(f_{1},\tau_{1})...B(f_{n},\tau_{n})\Phi$

where $f_{i}\in{\cal S}^{m}$ is a collection of $m$ smooth fast decreasing
functions on $\mathbb{R}^{d}.$

We say that $f_{1},...,f_{n}$ are not-overlapping if the sets
${K_{j}}={K(f_{j})}$ do not overlap (more precisely we should assume that the
distance between sets $K_{j}$ and $K_{j^{\prime}}$ is positive for $j\neq
j^{\prime}$).

######  Lemma 3.

If $f_{1},...,f_{n}$ do not overlap the vectors

$\Psi(f_{1},\tau_{1},...,f_{n},\tau_{n})=B(f_{1},\tau_{1})...B(f_{n},\tau_{n})\Phi$

have a limit in $\bar{\cal H}$ as $\tau_{j}$ tend to $-\infty$; this limit
will be denoted by

$\Psi(f_{1},...,f_{n}|-\infty)$

The set spanned by such limits will be denoted by $\cal{D}_{-}.$

Let us sketch the proof of this lemma for the case when
$\tau_{1}=...=\tau_{n}=\tau.$ It is sufficient to check that
$\int_{-\infty}^{0}||\dot{\Psi}(\tau)||d\tau$ is finite. (Here $\Psi(\tau)$
stands for $\Psi(f_{1},\tau,...,f_{n},\tau)$ and $\dot{\Psi}$ denotes the
derivative with respect to $\tau$.) The derivative $\dot{\Psi}$ is a sum of
$n$ terms; every term contains $n$ factors and one of this factors is a
derivative. We can interchange the factors because the commutators can be
neglected as $\tau\to\\-\infty$; this follows from the condition that the
functions $f_{j}$ do not overlap and from the fact that for non-overlapping
families essential supports of functions $f^{i}_{\tau}({{\bf x}},j)$ and
$f^{i}_{\tau}({{\bf x}},j^{\prime})$ are far away for large $\tau$.

We use this remark to shift the factor with the derivative to the right. It
remains to apply the lemma 1.

Let us define the $in$-operators $a^{+}_{in}$ by the formula

$a^{+}_{in}(f\phi)=a^{+}_{in}(f^{j}\phi_{j}^{\alpha})=\lim_{\tau\to-\infty}B(f,\tau).$
(5)

Lemma 3 gives conditions on $f$ that guarantee the existence of this limit as
strong limit on the set $\cal{D}_{-}.$

Let us introduce the asymptotic bosonic Fock space $\mathcal{H}_{as}$ as a
Fock representation of canonical commutation relations

$[b(\rho),b(\rho^{\prime})]=[b^{+}(\rho),b^{+}(\rho^{\prime})]=0,[b(\rho),b^{+}(\rho^{\prime})]=\langle\rho,\rho^{\prime}\rangle$

where $\rho,\rho^{\prime}\in\textgoth{h}.$

We define Møller matrix $S_{-}$ as a map $\mathcal{H}_{as}\to\overline{\cal
H}$ obeying

$a_{in}^{+}(\rho)S_{-}=S_{-}b^{+}(\rho),S_{-}|0\rangle=|0\rangle$

Here $|0\rangle$ stands for the Fock vacuum.

Notice that spatial and time translations act naturally in $\mathcal{H}_{as}.$
The Møller matrix commutes with these translations.

Operators $a_{in}(\rho)$ are defined on the image of $S_{-}$ by the formula

$a_{in}(\rho)S_{-}=S_{-}b(\rho).$

They are Hermitian conjugate to $a_{in}^{+}(\rho).$

One can give a direct definition of Møller matrix by the formula

$S_{-}b^{+}(f_{1}^{k}\phi_{k}^{\alpha})\cdots
b^{+}(f_{n}^{k}\phi_{k}^{\alpha})|0\rangle=\Psi(f_{1},...,f_{n}|-\infty)$

or equivalently

$S_{-}(b^{+}(g_{1})\cdots
b^{+}(g_{n})|0\rangle)=\lim_{\tau\to-\infty}B(g_{1}\phi^{-1},\tau)\cdots
B(g_{n}\phi^{-1},\tau)\Phi.$

If $g_{1},\cdots,g_{n}$ do not overlap the vector

$B(g_{1}\phi^{-1},\tau)\cdots B(g_{n}\phi^{-1},\tau)\Phi$ (6)

describes $n$ distant particles as $\tau\to-\infty.$

It is convenient to require strong cluster property ( see Section 2 of [8]) to
analyze the Møller matrix . (This condition can be weakened.)

Let us make the following

> Assumption. The subset of the space ${\cal S}^{mn}$ spanned by non-
> overlapping families $(f_{1},...,f_{n})$ contains an open dense subset of
> ${\cal S}^{mn}.$

This assumption is not restrictive; see the discussion in Section 4.2 of [8].

Using the above assumption and cluster property one can prove the theorem
below.

######  Theorem 4.

The Møller matrix $S_{-}$ is a well defined isometric operator.

Notice first of all that it is not clear from our definitions that the
$in$-operators and Møller matrices are well defined (they can depend on the
choice of operators $B_{j}$). In other words, the operator $S_{-}$ a priori
can be multivalued. However, we can prove that this operator is isometric and
use the fact that an isometric operator is necessarily single-valued.

To prove that the operator $S_{-}$ preserves the inner product we express the
inner product

$\langle
B(f_{1},\tau_{1})...B(f_{n},\tau_{n})\Phi,B^{\prime}(f^{\prime}_{1},\tau^{\prime}_{1})...B^{\prime}(f^{\prime}_{n^{\prime}},\tau^{\prime}_{n^{\prime}})\Phi\rangle$

in terms of truncated correlation functions. We assume that both
$(f_{1},...,f_{n})$ and $(f^{\prime}_{1},...,f^{\prime}_{n^{\prime}})$ do not
overlap, then it follows from Definition 2 and cluster property that only two-
point correlations $\langle
B^{{}^{\prime}*}(f^{\prime},\tau^{\prime})B(f,\tau)\Phi,\Phi\rangle=\langle
B(f,\tau)\Phi,B^{\prime}(f^{\prime},\tau^{\prime})\Phi\rangle$ contribute in
the limit $\tau_{j},\tau^{\prime}_{j^{\prime}}\to-\infty.$ Calculating the
two-point correlation functions by means of (9) we see that $S_{-}$ is an
isometry. We assumed that the vectors corresponding to families of non-
overlapping functions span a dense subset of $\mathcal{H}_{as}$, hence $S_{-}$
can be extended to an isometric embedding of the space
$\overline{\mathcal{H}_{as}}$ into $\overline{\cal H}.$

Taking $\tau\to+\infty$ instead of $\tau\to-\infty$ we obtain the definition
of Møller matrix $S_{+}$ and of $out$-operators
$a^{+}_{out}(\rho),a_{out}(\rho).$ If Møller matrices are surjective operators
we can define the scattering matrix ($S$-matrix) by the formula
$S=S_{+}^{-1}S_{-}$. In this case one says that the theory has particle
interpretation.111 It is possible that the ”elementary space” $\textgoth{h}$
does not describe all particles existing in the theory (for example, we are
missing some composite particles). In this case we have a chance to get a
theory with particle interpretation extending the space $\textgoth{h}.$

In other words we can say that the theory has particle interpretation if for
dense subset of $\cal H$ the time evolution can be represented as linear
combination of vectors

$e^{-iH\tau}\Psi(f_{1},...,f_{n}|-\infty)=\Psi(T_{\tau}f_{1},...,T_{\tau}f_{n}|-\infty)$

for $\tau<0$ and of vectors

$e^{-iH\tau}\Psi(f_{1},...,f_{n}|+\infty)=\Psi(T_{\tau}f_{1},...,T_{\tau}f_{n}|+\infty)$

fot $\tau>0.$

This means that generically both for $\tau\to-\infty$ and for $\tau\to+\infty$
we obtain a collection of distant particles (the wave functions
$T_{\tau}f_{k}$ have distant essential supports if functions $f_{k}$ do not
overlap). Without assumption that the theory has particle interpretation we
can define scattering matrix by the formula $S=S_{+}^{*}S_{-}$, however this
definition gives a unitary operator only in the case when the image of $S_{-}$
coincides with the image of $S_{+}.$

Until now all of our considerations were applicable both to algebras over real
numbers and algebras over complex numbers. In what follows we restrict
ourselves to algebras over complex numbers. Notice, however, that we can apply
the considerations below to algebras over real numbers complexifying the
elementary space $\textgoth{h}$ and the pre Hilbert space $\cal H$.

Let us diagonalize the matrix $E({\bf k})$; corresponding eigenvalues are
denoted by $\epsilon_{j}({\bf k})$ and eigenvectors are denoted
$\rho_{j}^{\alpha}({\bf k}).$ ( We assume that these eigenvectors constitute
an orthonormal system.) In momentum representation generalized eigenvectors
vectors of time and spatial translations are $\rho_{j}^{\alpha}({{\bf
k}})\delta({{\bf k}}-{{\bf k}}_{0}).$ We consider $in$\- and $out$-operators
corresponding to these eigenvectors as generalized functions of ${\bf k}$;
they are denoted

$a_{in}({{\bf k}},j),a_{in}^{+}({{\bf k}},j),a_{out}({{\bf
k}},j),a_{out}^{+}({{\bf k}},j).$

For example, $\int d{{\bf k}}\sum_{i}f^{i}({{\bf k}})a_{out}^{+}({{\bf
k}},i)=a_{out}^{+}(f\rho)$ where $f\rho=f^{i}({{\bf
k}})\rho_{i}^{\alpha}({{\bf k}})).$ (Here $\rho$ is considered as a square
matrix.) These operators can be interpreted as annihilation and creation
$in$\- and $out$\- operators of particles with momentum ${\bf k}.$ Sometimes
we omit discrete indices characterizing the type of particle in these
operators, then the operators $a_{in}({{\bf k}}),a_{in}^{+}({{\bf
k}}),a_{out}({{\bf k}}),a_{out}^{+}({{\bf k}}))$ should be regarded as
$m$-dimensional vectors and the values of corresponding correlation functions
as elements of tensor product of $m$-dimensional spaces.

If we assume that the theory has particle interpretation, the Møller matrices
establish unitary equivalence of $a_{in}({{\bf k}},j),a_{in}^{+}({{\bf
k}},j),a_{out}({{\bf k}},j),a_{out}^{+}({{\bf k}},j)$ with $b({{\bf
k}},j),b^{+}({{\bf k}},j)$ where $b({{\bf k}},j),b^{+}({{\bf k}},j)$ are
operator generalized functions in ${\cal H}_{as}$ corresponding to
$b(f\rho),b^{+}(f\rho).$

The matrix elements of scattering matrix can be expressed in terms of $in$\-
and $out$-operators:

$\langle S(b^{+}(g_{1})\cdots b^{+}(g_{n})|0\rangle),(b^{+}(h_{1})\cdots
b^{+}(h_{m})|0\rangle)\rangle=$ $\langle S_{-}(b^{+}(g_{1})\cdots
b^{+}(g_{n})|0\rangle),S_{+}(b^{+}(h_{1})\cdots
b^{+}(h_{m})|0\rangle)\rangle=$ $\langle a_{in}^{+}(g_{1})\cdots
a_{in}^{+}(g_{n})\Phi,a_{out}^{+}(h_{1})\cdots a_{out}^{+}(h_{m})\Phi\rangle=$
$\langle\lim_{\tau\to-\infty}B(g_{1}\phi^{-1},\tau)\cdots
B(g_{n}\phi^{-1},\tau)\Phi,\lim_{\tau\to+\infty}B(h_{1}\phi^{-1},\tau)\cdots
B(h_{m}\phi^{-1},\tau)\Phi\rangle$

hence

$\langle S(b^{+}(g_{1})\cdots b^{+}(g_{n})|0\rangle),(b^{+}(h_{1})\cdots
b^{+}(h_{m})|0\rangle)\rangle=$
$\lim_{\tau\to-\infty,\tau^{\prime}\to+\infty}\omega(B^{*}(h_{m}\phi^{-1},\tau^{\prime})\cdots
B^{*}(h_{1}\phi^{-1},\tau^{\prime})B(g_{1}\phi^{-1},\tau)\cdots
B(g_{n}\phi^{-1},\tau))$ (7)

and

$\langle a_{in}^{+}({{\bf k}}_{1},i_{1})...a_{in}^{+}({{\bf
k}}_{n},i_{n})\Phi,a_{out}^{+}({{\bf
k}}^{\prime}_{1},j_{1})...a_{out}^{+}({{\bf
k}}^{\prime}_{m},j_{m})\Phi\rangle=$ (8)
$\lim_{\tau\to-\infty,\tau^{\prime}\to+\infty}\omega(B^{\prime}({{\bf
k}}^{\prime}_{m},j_{m},\tau^{\prime})...B^{\prime}({{\bf
k}}^{\prime}_{1},j_{1},\tau^{\prime})B({{\bf k}}_{1},i_{1},\tau)...B({{\bf
k}}_{n},i_{n},\tau))$

where

$\int d{{\bf k}}\sum_{i}f^{i}({{\bf k}})B({{\bf
k}},i,\tau)=B(f\rho\phi^{-1},\tau),$ $\int d{{\bf k}}\sum_{i}f^{i}({{\bf
k}})B^{\prime}({{\bf
k}},i,\tau^{\prime})=B^{*}(f\rho\phi^{-1},\tau^{\prime}).$

Omitting discrete indices and using (2) we can write

$\langle a_{in}^{+}({{\bf k}}_{1})...a_{in}^{+}({{\bf
k}}_{n})\Phi,a_{out}^{+}({{\bf k}}^{\prime}_{1})...a_{out}^{+}({{\bf
k}}^{\prime}_{m})\Phi\rangle=$
$\lim_{\tau\to-\infty,\tau^{\prime}\to+\infty}\omega(D({{\bf
k}}^{\prime}_{m})B^{\prime}({{\bf k}}^{\prime}_{m},\tau^{\prime})...D({{\bf
k}}_{1})B^{\prime}({{\bf k}}^{\prime}_{1},\tau^{\prime})D({{\bf
k}}_{1})B({{\bf k}}_{1},\tau)..D({{\bf k}}_{n})B({{\bf k}}_{n},\tau))=$ (9)
$\lim_{\tau\to-\infty,\tau^{\prime}\to+\infty}\omega(D({{\bf
k}}^{\prime}_{m})e^{i\tau^{\prime}E({{\bf
k}}^{\prime}_{m})}\hat{B}^{*}({\tau^{\prime},{\bf k}}^{\prime}_{m})...D({{\bf
k}}^{\prime}_{1})e^{i\tau^{\prime}E({{\bf
k}}^{\prime}_{1})}\hat{B}^{*}({\tau^{\prime},{\bf k}}^{\prime}_{1})$ $D({{\bf
k}}_{1})e^{i\tau E({{\bf k}}_{1})}\hat{B}(\tau,{{\bf k}}_{1})..D({{\bf
k}}_{n})e^{i\tau E({{\bf k}}_{n})}\hat{B}(\tau,{{\bf k}}_{n}))$

where $D$ stands for the matrix $\rho\phi^{-1}.$

It is easy to describe the joint spectrum of momentum and energy operators in
${\cal H}_{as}$ (of infinitesimal generators of spatial and time
translations). It consists of points $(\epsilon_{j_{1}}({{{\bf
k}}_{1}})+...+\epsilon_{j_{r}}({{{\bf k}}_{r}}),{{\bf k}}_{1}+...+{{\bf
k}}_{r}).$ For $r=0$ we obtain the point $(0,0)$ corresponding to the vacuum
vector. The points with $r=1$ constitute one-particle spectrum, the points
with $r>1$ belong to multi-particle spectrum. If the theory has particle
interpretation the same formulas describe the joint spectrum of momentum and
energy operators in $\bar{\cal H}.$

The decomposition of the spectrum in one-particle spectrum and multi-particle
spectrum induces the decomposition of $\bar{\cal H}$ into direct sum of one-
dimensional space ${\cal H}_{0}$ containing $\Phi$, one-particle space ${\cal
H}_{1}$ (the closure of the image of the map $\textgoth{h}\to\cal H$ ) and
multi-particle space $\cal M$.

##  3 LSZ formula

The scattering matrix can be expressed in terms of Green functions. These
functions can be defined by the formula

$G_{r}(\tau_{1},{{\bf x}}_{1},i_{1},...,\tau_{r},{{\bf
x}}_{r},i_{r})=\omega(T(B_{i_{1}}(\tau_{1},{{\bf
x}}_{1})...B_{i_{r}}(\tau_{r},{{\bf x}}_{r}))$

where $T$ stand for the chronological product. We defined Green functions in
$(\tau,{\bf x})$-representation, taking Fourier transform with respect to
${\bf x}$ we obtain Green functions in $(\tau,{\bf k})$\- representation;
taking inverse Fourier transform with respect to $\tau$ we get Green functions
in $(\varepsilon,{\bf k})$-representation. For simplicity we are assuming that
that $B_{i}$ are self-adjoint (otherwise we should consider not only $B_{i}$,
but also $B^{*}_{i}$ under the sign of $T$-product).

We have defined Green functions using the operators $B_{i}$ (good operators),
however one can modify this definition replacing $B_{i}$ by other operators
$A_{i}\in\cal A.$

It is easy to calculate the two-point Green function $G_{2}.$ We start with
two-point correlation function

$w_{2}(\tau_{1},{{\bf x}}_{1},i,\tau_{2},{{\bf
x}}_{2},j)=\omega(B_{i}(\tau_{1},{{\bf x}}_{1})B_{j}(\tau_{2},{{\bf
x}_{2}}))=$ $\langle B_{j}(\tau_{2},{{\bf x}_{2}})\Phi,B_{i}(\tau_{1},{{\bf
x}}_{1})\Phi\rangle=\int d{{\bf k}}e^{i{{\bf k}}({{\bf x}_{2}-{\bf
x}_{1}})}\langle e^{-i\tau_{2}E({{\bf k}})}\phi_{j}({{\bf
k}}),e^{-i\tau_{1}E({{\bf k}})}\phi_{i}({{\bf k}})\rangle$

Expressing $G_{2}$ in terms of $w_{2}$ we obtain that in $(\varepsilon,{\bf
k})$-representation

$G_{2}(\varepsilon_{1},{{\bf k}}_{1},i,\varepsilon_{2},{{\bf
k}}_{2},j)=G^{i,j}(\varepsilon_{1},{{\bf
k}}_{1})\delta(\varepsilon_{1}+\varepsilon_{2})\delta({{\bf k}}_{1}+{{\bf
k}}_{2})$

where for fixed ${\bf k}$ the function $G^{i,j}(\varepsilon,{\bf k})$ has
poles with respect to $\varepsilon$ at the points $\pm\epsilon_{s}({\bf k})$
(here $\epsilon_{s}({\bf k})$ are eigenvalues of the matrix $E({\bf k})$)).
Namely,

$G^{i,j}(\varepsilon,{{\bf k}})=A^{i,j}(\varepsilon,{{\bf
k}})+A^{j,i}(-\varepsilon,-{\bf k})$

where

$A^{i,j}(\varepsilon,{{\bf
k}})=\sum_{s}\frac{i}{\varepsilon+\epsilon_{s}({{\bf k}})-i0}\langle
a_{s}({{\bf k}})\phi_{j}({{\bf k}}),\phi_{i}({{\bf k}})\rangle$ (10)

We have used the fact that the matrix $e^{iE({{\bf k}})\tau}$ can be expressed
as a linear combination of exponents $e^{i\epsilon_{j}({{\bf k}})\tau}$ with
matrix coefficients depending on ${\bf k}$:

$e^{iE({{\bf k}})\tau}=\sum_{s}a_{s}({{\bf k}})e^{i\epsilon_{s}({{\bf
k}})\tau}.$

Using the same fact and (7) or (9) it is easy to check that the scattering
matrix can be expressed in terms of asymptotic behavior of Green functions in
$(\tau,{\bf k})$ representation. (One should divide the arguments of Green
function in two groups; in one group we should take the times tending to
$-\infty$, in the second group to $+\infty$. The ordering of times in every
group is irrelevant due to asymptotic commutativity of factors in (7). )

Equivalently one can work in $(\varepsilon,{\bf k})$\- representation taking
inverse Fourier transform with respect to $\tau$ in $(\tau,{\bf
k})$-representation. Then the scattering matrix can be expressed in terms of
poles of Green functions with respect to $\varepsilon$ and residues in these
poles. This is the famous LSZ formula. One can derive it from the following
statements:

Let $E$ denote a Hermitian matrix with eigenvalues $\epsilon_{j}$. Then the
matrix $e^{i\tau E}$ can be written in the form
$\sum_{j}a_{j}e^{i\tau\epsilon_{j}}$ where $a_{j}$ are constant matrices.

Let us assume there exist limits $A_{\pm}=\lim_{\tau\to\pm\infty}e^{i\tau
E}\rho(\tau)$ where $\rho$ is a column vector. Then $\rho(\tau)$ has
asymptotic behavior

$\rho(\tau)\sim e^{-i\tau
E}A_{\pm}=\sum_{k}e^{-i\epsilon_{k}\tau}a_{k}A_{\pm}$

as $\tau\to\pm\infty.$

This implies that the (inverse) Fourier transform $\rho(\varepsilon)$ of
$\rho(\tau)$ has poles at the points $\pm(\epsilon_{k}+i0)$ with residues $\mp
2\pi ia_{k}A_{\pm}.$

We can say that

The asymptotic behavior of $\rho(\tau)$ is determined by the polar part of
$\rho(\varepsilon).$

In what follows we use these statements in a little bit different form. We
represent $\rho(\tau)$ as $e^{-i\tau E}A_{-}\Phi(-\tau)+e^{-i\tau
E}A_{+}\Phi(\tau)+\sigma(\tau)$ where $\sigma(\tau)\to 0$ as $\tau\to\infty.$
We assume that $\sigma(\tau)$ is a summable function, then

$\rho(\varepsilon)=(E-\varepsilon+i0)^{-1}C_{-}+(E+\varepsilon+i0)^{-1}C_{+}+\sigma(\varepsilon)$

where $\sigma(\varepsilon)$ is continuous, $C_{-}$ and $C_{+}$ do not depend
on $\varepsilon.$ We say that the first two summands constitute the polar part
of $\rho(\varepsilon).$ We need the following statement

If $f(\varepsilon)$ is a smooth function then

$f(\varepsilon)\rho(\varepsilon)=(E-\varepsilon-i0)^{-1}C^{\prime}_{-}+(E+\varepsilon-i0)^{-1}C^{\prime}_{+}\sigma^{\prime}(\varepsilon)$
(11)

where $C^{\prime}_{-}=f(E)C_{-},C^{\prime}_{+}=f(-E)C_{+}$ do not depend on
$\varepsilon$ and $\sigma^{\prime}$ is a continuous function.

Notice that in the LSZ formula the operators $B_{i}$ transforming the vector
$\Phi$ into an element of $\Phi(\textgoth{h})$ (of one-particle space) can be
replaced by asymptotically commuting smooth operators $A_{i}\in\cal A$ obeying
a weaker condition: the projections of vectors $A_{i}\Phi$ on the one-particle
space are linearly independent and the projection of $A_{i}\Phi$ on $\Phi$
vanishes. This can be proved if the theory has particle interpretation. ( See
Section 4.6 of [8] for the proof of this fact in less general case; this proof
can be generalized to our setting.) Instead of this condition we can require
the existence of smooth fast decreasing functions $\alpha_{i}^{j}(\tau,{\bf
x})$ such that the operators $B_{i}=\int d\tau d{{\bf
x}}\alpha_{i}^{j}(\tau,{{\bf x}})A_{j}(\tau,{\bf x})$ are good operators.) (
This condition is always satisfied if the joint spectrum of $H,{\bf P}$ in
${{\cal H}}_{0}\oplus{{\cal H}}_{1}$ does not overlap with multi-particle
spectrum.)

Using the formula

$B_{i}(\tau,{{\bf x}})=\int d\tau^{\prime}d{{\bf
x}}^{\prime}\alpha_{i}^{j}(\tau^{\prime}-\tau,{{\bf x}}^{\prime}-{{\bf
x}})A_{j}(\tau^{\prime},{{\bf x}}^{\prime})$

we can express the correlation functions for operators $B_{i}$ in terms of
correlation functions for operators $A_{i}$. The expression looks very simple
in $(\varepsilon,{{\bf k}})$-representation: for example if
$\alpha_{i}^{j}=\alpha_{i}\delta_{i}^{j}$ one should multiply the correlation
functions of operators $A_{i}$ by the product of Fourier transforms of
functions $\alpha_{i}$ with respect to $\tau,{\bf x}.$ The corresponding
expression for Green functions is more complicated due to factors
$\theta(\tau_{i}-\tau_{j})$ entering the definition of chronological product.
However, in scattering theory we are interested in asymptotic behavior of
Green functions in $(\tau,{\bf k})$ representation or in the behavior of polar
parts of Green functions in $(\varepsilon,{\bf k})$-representation. For ${\bf
k}$ in dense open set the behavior of the polar parts of Green functions for
operators $A_{i}$ in $(\varepsilon,{\bf k})$ representation can be described
in the same way as for correlation functions. ( To prove this statement we use
asymptotic commutativity of operators $A_{i}$ and the assumption (2). In the
calculation of scattering matrix we decompose the arguments of Green functions
in two groups; we use the fact that due to asymptotic commutativity the time
ordering inside every group is irrelevant.)

Let us give more precise formulations of the above statements.

We are starting with asymptotically commuting operators $A_{1},...,A_{m}$
obeying $\langle A_{i}\Phi,\Phi\rangle=0.$ We introduce the notation
$T^{A}_{i}({{\bf k}})$ for projections of vectors $A_{i}\Phi\in\cal H$ on
$\Phi(\textgoth{h})$ (on one-particle subspace of $\cal H$) considered as
elements of $\textgoth{h}$ in momentum representation. We assume that these
projections are linearly independent (the matrix $T^{A}({{\bf
k}})=(T^{A}({{\bf k}}))_{i}^{\alpha}$ is non-degenerate).

We consider Green function

$G_{r}(\tau_{1},{{\bf x}}_{1},i_{1},...,\tau_{r},{{\bf
x}}_{r},i_{r})=\omega(T(A_{i_{1}}(\tau_{1},{{\bf
x}}_{1})...A_{i_{r}}(\tau_{r},{{\bf x}}_{r}))$

and their Fourier transforms (Green functions in $(\tau,{\bf k})$\- and
$(\varepsilon,{\bf k})$\- representations. Notice that due to translation
invariance the Green function in $(\varepsilon,{\bf k})$\- representation
contains a delta-function $\delta(\sum\varepsilon_{i})\delta(\sum{{\bf
k}}_{i})$; talking about two-point function (r=2) we always omit this delta-
function. (Hence the two-point Green function is a matrix-valued function of
$(\varepsilon,{\bf k})$.) We can write the two-point Green function in
$(\varepsilon,{\bf k})$-representation as a sum of the polar part (having
first order poles with respect to variables $\varepsilon$) and regular part.
The polar part

$(E({{\bf k}})-\varepsilon+i0)^{-1}C_{-}({{\bf k}})+(E({{\bf
k}})+\varepsilon+i0)^{-1}C_{+}({{\bf k}})$

governs the behavior of Green function in $(\tau,{{\bf k}})$-representation as
$\tau\to\infty$; it is a sum of two summands; one of them ($in$-summand) is
responsible for the limit $\tau\to-\infty$, another ($out$-summand) is
responsible for the limit $\tau\to\infty$.

Let us consider the Green function $G_{r}$ in $(\tau,{\bf k})$-representation.
We assume that the arguments of this function are divided in two groups (with
indices $i$ in the interval $1\leq i\leq a$ and with indices in the interval
$a<i\leq r$.) We assume that the times $\tau_{i}$ with the indices from the
first group tend to $-\infty$ and the remaining times tend to $+\infty.$

In $(\varepsilon,{\bf k})$\- representation the Green function $G_{r}$ can be
represented as a product of the amputated Green function and $r$ two-point
Green functions labeled by index $i.$ We change the signs of the variables
$\varepsilon_{i},{{\bf k}}_{i}$ where $i$ is the index from the second group
to interpret these variables as energies and momenta of outgoing particles. We
define the polar part

$P_{r}(\varepsilon_{1},{{\bf k}}_{1},j_{1},...,\varepsilon_{r},{{\bf
k}}_{r},j_{r})$

of Green function replacing every two-point Green function in this
representation by its $in$-summand of its polar part for indices $i\leq a$ and
by $out$-summand of the polar part for $i>a$.

Let us define operators $A^{\prime}_{i}$ by the formula $A^{\prime}_{i}=\int
d\tau d{{\bf x}}\alpha_{i}^{j}(\tau,{{\bf x}})A_{j}(\tau,{\bf x})$ where
$\alpha_{i}^{j}(\tau,{{\bf x}})$ are smooth fast decreasing functions. Polar
parts of corresponding Green functions are denoted by by $P^{\prime}_{r}.$

It is easy to express the projections $T^{A^{\prime}}_{i}({{\bf k}})$ of
$A^{\prime}_{i}\Phi$ on one-particle space in terms of projections
$T^{A}_{j}({{\bf k}}).$ We obtain

$T^{A^{\prime}}_{i}({{\bf k}})=\alpha_{i}^{j}(E({{\bf k}}),{{\bf
k}})T^{A}_{j}({{\bf k}})$ (12)

where $\alpha_{i}^{j}(E({{\bf k}}),{{\bf k}})=\int d\tau d{{\bf x}}e^{i{{\bf
k}{\bf x}}-iE({{\bf k}})\tau}\alpha_{i}^{j}(\tau,{{\bf x}})$

Equivalently

$T^{A^{\prime}}({{\bf k}})=\alpha(E({{\bf k}}),{{\bf k}})T^{A}({{\bf k}}).$
(13)

To prove (12) we represent $A_{j}(\tau,{{\bf x}})\Phi$ as
$\Phi(T^{A(\tau,{{\bf x}}}_{j})+\rho(\tau,{{\bf x}})$ where $\rho$ belongs to
the multiparticle space. We get

$A^{\prime}_{i}\Phi=\int d\tau d{{\bf x}}\alpha_{i}^{j}(\tau,{{\bf
x}})\Phi(T^{A(\tau,{{\bf x}}}_{j})+\int d\tau d{{\bf
x}}\alpha_{i}^{j}(\tau,{{\bf x}})\rho(\tau,{{\bf x}}).$

Noticing that the second summand lies in the multiparticle space and that for
$\phi\in\textgoth{h}$ we have $\Phi(e^{i{{\bf P}{\bf
x}}-H\tau}\phi)=\Phi(e^{i{{\bf k}{\bf x}}-iE({{\bf k}})\tau}\phi)$ we obtain
(12).

As we noticed the polar part governs the asymptotic behavior in $(\tau,{\bf
k})$\- representation. Using this fact one can prove that
$P^{\prime}_{r}(\varepsilon_{1},{{\bf k}}_{1},i_{1},...,\varepsilon,{{\bf
k}}_{r},i_{r})$ is equal to the polar part of

$\alpha_{i_{1}}^{j_{1}}(\varepsilon_{1},{{\bf
k}}_{1})...\alpha_{i_{r}}^{j_{r}}(\varepsilon_{r},{{\bf
k}}_{r})P_{r}(\varepsilon_{1},{{\bf k}}_{1},j_{1},...,\varepsilon_{r},{{\bf
k}}_{r},j_{r}).$ (14)

Using (11) we obtain that

$P^{\prime}_{r}(\varepsilon_{1},{{\bf k}}_{1},i_{1}...,\varepsilon,{{\bf
k}}_{r},i_{r})=\alpha_{i_{1}}^{j_{1}}(E({{\bf k}}_{1}),{{\bf
k}}_{1})...\alpha_{i_{r}}^{j_{r}}(E({{\bf k}}_{r}),{{\bf
k}}_{r})P_{r}(\varepsilon_{1},{{\bf k}}_{1},j_{1},...,\varepsilon_{r},{{\bf
k}}_{r},j_{r})$

or equivalently

$P^{\prime}_{r}(\varepsilon_{1},{{\bf k}}_{1},...,\varepsilon,{{\bf
k}}_{r})=\alpha(E({{\bf k}}_{1}),{{\bf k}}_{1})\otimes...\otimes\alpha(E({{\bf
k}}_{r}),{{\bf k}}_{r})P_{r}(\varepsilon_{1},{{\bf
k}}_{1},...,\varepsilon_{r},{{\bf k}}_{r})$ (15)

In this formula and in what follows we consider $P^{\prime}_{r}$ and $P_{r}$
as functions taking values in $r$-th tensor power of $m$-dimensional space (we
consider discrete variables in $P_{r}$ and in $P^{\prime}_{r}$ as tensor
indices).

Let us define now the normalized polar part of Green function (closely related
to Green function on shell) by the formula

$\tilde{P}_{r}(\varepsilon_{1},{{\bf k}}_{1},...,\varepsilon_{r},{{\bf
k}}_{r})=(T^{A}({{\bf k}}_{1}))^{-1}\otimes...\otimes(T^{A}({{\bf
k}_{r}}))^{-1}P_{r}(\varepsilon_{1},{{\bf k}}_{1},...,\varepsilon_{r},{{\bf
k}}_{r}).$

This function takes values in $r$-th tensor power of $\textgoth{h}$.

It follows immediately from (13), (15) that

$\tilde{P}^{\prime}_{r}(\varepsilon_{1},{{\bf k}}_{1}...,\varepsilon_{r},{{\bf
k}}_{r})=\tilde{P}_{r}(\varepsilon_{1},{{\bf k}}_{1}...,\varepsilon_{r},{{\bf
k}}_{r}).$ (16)

This means that normalized polar parts of Green functions for
$A_{1},...,A_{m}$ and $A^{\prime}_{1},...,A^{\prime}_{m}$ coincide.

Let us define Green functions on shell taking the residues of normalized polar
parts of Green functions.

Now we can formulate the LSZ formula in the following way:

Matrix elements of scattering matrix coincide with Green functions on shell
(up to sign change in outgoing momenta)

To prove this fact it is sufficient to verify it for good operators
$B_{1},...,B_{m}$ using (7) and (9).

Notice that the matrices $T^{A}$ entering the definition of normalized polar
part are closely related to the polar part of two-point Green function.

##  4 Inclusive scattering matrix

An element $B$ of the algebra $\cal A$ specifies two operators on linear
functionals on $\cal A$. First operator is denoted by the same symbol $B$; it
transforms the functional $\sigma$ into the functional
$(B\sigma)(A)=\sigma(AB).$ The second one (denoted $\tilde{B}$) transforms
$\sigma$ into the functional $(\tilde{B}\sigma)(A)=\sigma(B^{*}A).$ The
vectors in the space $\cal H$ correspond to excitations of the state $\omega$:
the vector $B\Phi$ corresponds to the state $\tilde{B}B\omega.$ Let us
introduce notations

$B(f)=B(f,0),L(g)=\tilde{B}(g\phi^{-1})B(g\phi^{-1}),L(g,\tau)=T_{\tau}L(T_{-\tau}g)=\tilde{B}(f,\tau)B(f,\tau)$

where $g=f\phi\in\textgoth{h}.$ Instead of working with vectors

$\Psi(f_{1},\tau_{1},...,f_{n},\tau_{n})=B(f_{1},\tau_{1})...B(f_{n},\tau_{n})\Phi$

we can work with corresponding states

$\Lambda(g_{1},\tau_{1},...,g_{n},\tau_{n})=L(g_{1},\tau_{1}),...L(g_{n},\tau_{n})\omega.$

It follows from Lemma 3 that these states have a limit as $\tau_{j}\to-\infty$
if $f_{j}=g_{j}\phi^{-1}$ do not overlap. These states will be necessary in
geometric approach (see [4]), however they are useful also in algebraic
approach. Namely, we use these states to construct the inclusive scattering
matrix.

Let us consider the state

$L(g^{\prime}_{1},\tau^{\prime}_{1})...L(g^{\prime}_{n^{\prime}},\tau^{\prime}_{n^{\prime}})L(g_{1},\tau_{1})...L(g_{n},\tau_{n})\omega$

considered as a linear functional on $\cal A$ (as an element of the cone $\cal
C$). We assume that $g^{\prime}_{i}$ as well as $g_{j}$ are not overlapping,
then this state has a limit as
$\tau^{\prime}_{i}\to+\infty,\tau_{j}\to-\infty$; we denote this limit by $Q$.
Notice that $Q$ does not change if we permute $g_{1},...,g_{n}$ (in the limit
$\tau_{j}\to-\infty$ the operators $L(g_{j},\tau_{j})$ commute). Similarly $Q$
does not change if we permute $g^{\prime}_{1},...,g^{\prime}_{n^{\prime}}.$

More generally we can consider a linear functional $\tilde{Q}$ on $\cal A$
defined as a limit of

$L(\tilde{g}^{\prime}_{1},g^{\prime}_{1},\tau^{\prime}_{1})...L(\tilde{g}^{\prime}_{n^{\prime}},g^{\prime}_{n^{\prime}},\tau^{\prime}_{n^{\prime}})L(\tilde{g}_{1},g_{1},\tau_{1})...L(\tilde{g}_{n},g_{n},\tau_{n})\omega$

as $\tau^{\prime}_{i}\to+\infty,\tau_{j}\to-\infty$.( We introduced the
notation $L(\tilde{g},g,\tau)=\tilde{B}(\tilde{f},\tau)B(f,\tau)$.)

Then we define

$\sigma(\tilde{g}^{\prime}_{1},g^{\prime}_{1},...,\tilde{g}^{\prime}_{n^{\prime}},g^{\prime}_{n^{\prime}},\tilde{g}_{1},g_{1},...,\tilde{g}_{n},g_{n}).$

as $\tilde{Q}(1).$ This functional is linear with respect to its arguments
$\tilde{g}^{\prime}_{i},g_{i},\tilde{g}_{j},g_{j}.$ It is well defined if each
of four families $\tilde{g}^{\prime}_{i},g_{i},\tilde{g}_{j},g_{j}$ is non-
overlapping. In bra-ket notations

$\sigma(\tilde{g}^{\prime}_{1},g^{\prime}_{1},...,\tilde{g}^{\prime}_{n^{\prime}},g^{\prime}_{n^{\prime}},\tilde{g}_{1},g_{1},...,\tilde{g}_{n},g_{n})=$
$\langle
1|\lim_{\tau^{\prime}_{i}\to+\infty,\tau_{j}\to-\infty}L(\tilde{g}^{\prime}_{1},g^{\prime}_{1},\tau^{\prime}_{1})...L(\tilde{g}^{\prime}_{n^{\prime}},g^{\prime}_{n^{\prime}},\tau^{\prime}_{n^{\prime}})L(\tilde{g}_{1},g_{1},\tau_{1})...L(\tilde{g}_{n},g_{n},\tau_{n})|\omega\rangle$
(17)

By definition the functional $\sigma$ is inclusive scattering matrix.

To justify this definition we notice that

$\sigma(\tilde{g}^{\prime}_{1},g^{\prime}_{1},...,\tilde{g}^{\prime}_{n^{\prime}},g^{\prime}_{n^{\prime}},\tilde{g}_{1},g_{1},...,\tilde{g}_{n},g_{n})=$
$\lim_{\tau_{j}\to-\infty^{\prime},\tau^{\prime}_{i}\to+\infty}(L(\tilde{g}_{1},g_{1},\tau_{1})...L(\tilde{g}_{n},g_{n},\tau_{n})\omega)(B(f^{\prime}_{1},\tau^{\prime}_{1})...B(f^{\prime}_{n^{\prime}},\tau^{\prime}_{n^{\prime}})B(f^{\prime}_{n^{\prime}},\tau^{\prime}_{n^{\prime}})^{*}...B(f^{\prime}_{1},\tau^{\prime}_{1})^{*})=$
$(\lim_{\tau_{j}\to-\infty}(L(\tilde{g}_{1},g_{1},\tau_{1})...L(\tilde{g}_{n},g_{n},\tau_{n})\omega)(a^{+}_{out}(g^{\prime}_{1})...a^{+}_{out}(g^{\prime}_{n^{\prime}})a_{out}(\tilde{g}^{\prime}_{n^{\prime}})...(a_{out}(\tilde{g}^{\prime}_{1}))$

We have used the relation $(\tilde{M}N\rho)(X)=\rho(M^{*}XN)$ in this
derivation.

The inclusive cross-section can be expressed in terms of inclusive $S$-matrix
defined above. To verify this statement we consider the expectation value

$\nu(a^{+}_{out,k_{1}}({{\bf p}}_{1})a_{out,k_{1}}({{\bf p}}_{1})\dots
a^{+}_{out,k_{m}}({{\bf p}}_{m})a_{out,k_{m}}({{\bf p}}_{m}))$ (18)

where $\nu$ is an arbitrary state. This quantity is the probability density in
momentum space for finding $m$ outgoing particles of the types
$k_{1},\dots,k_{n}$ with momenta ${\bf p}_{1},\dots,{\bf p}_{m}$ plus other
unspecified outgoing particles. It gives inclusive cross-section if

$\nu=\lim_{\tau_{i}\to-\infty}L(g_{1},\tau_{1})...L(g_{n},\tau_{n})\omega$

The inclusive scattering matrix can be expressed in terms of generalized Green
functions (GGreen functions). These functions appear naturally in the
formalism of $L$-functionals [9], [10], [14]; their relation to inclusive
cross-sections is analyzed in [7], [8], [14]. They appear also in Keldysh
formalism and in thermo-field dynamics [11],[12],[13]. GGreen functions can be
defined by the formula

$G(\tau_{1},{{\bf x}}_{1},i_{1},...,\tau_{r},{{\bf
x}}_{r},i_{r},\tau^{\prime}_{1},{{\bf
x}}^{\prime}_{1},i^{\prime}_{1},...,\tau^{\prime}_{r^{\prime}},{{\bf
x}}^{\prime}_{r^{\prime}},i^{\prime}_{r^{\prime}})=$
$(T(B_{i_{1}}(\tau_{1},{{\bf x}}_{1})...B_{i_{r}}(\tau_{r},{{\bf
x}}_{r},)\tilde{B}_{i^{\prime}_{1}}(\tau^{\prime}_{1},{{\bf
x}}^{\prime}_{1})...\tilde{B}_{i^{\prime}_{r^{\prime}}}(\tau^{\prime}_{r},{{\bf
x}}^{\prime}_{r}))\omega)(1)$

where $T$ stand for the chronological product. More precisely we defined
GGreen functions in $(\tau,{\bf x})$-representation, taking Fourier transform
with respect to ${\bf x}$ we obtain GGreen functions in $(\tau,{\bf k})$\-
representation. Using the fact that the matrix $e^{iE({{\bf k}})\tau}$ can be
expressed as a linear combination of exponents $e^{i\epsilon_{j}({{\bf
k}})\tau}$ with matrix coefficients depending on ${\bf k}$ it is easy to check
that the inclusive scattering matrix can be expressed in terms of asymptotic
behavior of GGreen functions in $(\tau,{\bf k})$ representation. (One should
take $r=r^{\prime}$ and assume that
$\tau_{i}\to+\infty,\tau^{\prime}_{i}\to+\infty$ for $i\leq m$ and
$\tau_{j}\to-\infty,\tau^{\prime}_{j}\to-\infty$ for $j>m$. The ordering of
times in every group is irrelevant due to asymptotic commutativity of factors.
)

Equivalently one can work in $(\varepsilon,{\bf k})$\- representation taking
inverse Fourier transform with respect to $\tau$ in $(\tau,{\bf
k})$-representation. Then the inclusive scattering matrix can be expressed in
terms of poles of GGreen functions with respect to $\varepsilon$ and residues
in these poles.

As in LSZ formula for scattering matrix we can work with operators
$A_{1},...,A_{m}$ requiring the existence of fast decreasing functions
$\alpha_{i}^{j}$ such that the operators $B_{i}=\int d\tau d{{\bf
x}}\alpha_{i}^{j}(\tau,{{\bf x}})A_{j}(\tau,{\bf x})$ are good operators.
Using K$\rm\dot{a}$ll$\acute{e}$n-Lehmann representation of two-point GGreen
function we define polar part and normalized polar part of GGreen function.
(We represent GGreen functions in terms of amputated GGreen functions and use
K$\rm\dot{a}$ll$\acute{e}$n-Lehmann representation of two-point GGreen
function in the proof.)

##  5 Fermions

We assumed that operators $B_{i}$ asymptotically commute (4). One can replace
this assumption with the assumption of asymptotic anticommutativity (we
replace the commutator in (4) by anticommutator). Then we should modify also
the definition of truncated correlation functions including some signs.

One can repeat all considerations of present paper in this situation. Slight
modifications are necessary. In particular, instead of bosonic Fock space one
should consider fermionic Fock space (Fock representation of canonical
anticommutation relations.) The particles obey Fermi statistics. To define
inclusive scattering matrix we again consider states

$\Lambda(g_{1},\tau_{1},...,g_{n},\tau_{n})=L(g_{1},\tau_{1}),...L(g_{n},\tau_{n})\omega$

and prove that these states have a limit as $\tau_{i}\to-\infty$ under the
same conditions on $g_{1},...,g_{n}.$ It is important to notice that under
these conditions it follows from the asymptotic anticommutativity of operators
$B_{i}$ that the operators $L(g_{i},\tau_{i})$ commute in the limit
$\tau_{i}\to-\infty.$

##  6 BRST formalism

Methods of homological algebra (=BRST formalism) can be applied in scattering
theory. Recall that in homological algebra together with modules (algebras,
etc) one considers differential graded modules (algebras,…). It is sufficient
to have $\mathbb{Z}_{2}$-grading. A module is $\mathbb{Z}_{2}$-graded if it is
represented as a direct sum of even and odd parts. A differential can be
defined as parity reversing homomorphism $Q$ obeying $Q^{2}=0.$ Homology is
defined as $KerQ/ImQ$ (as a quotient of the submodule consisting of $Q$-closed
elements with respect to the submodule consisting of $Q$-exact elements).

The main idea is to replace a module by simpler (for example, free)
differential graded module. (The new module should be quasi-isomorphic to the
original module, considered as a differential module with trivial grading and
trivial differential. Quasi-isomorphism is defined as a homomorphism commuting
with the differential and inducing an isomorphism on homology.)

The above considerations can be applied to differential $\mathbb{Z}_{2}$\-
graded algebras (algebras with parity reversing BRST operator $Q$ obeying
$Q^{2}=0.$) All physical quantities should be BRST-closed (should belong to
the kernel of $Q$); one should neglect the BRST-exact quantities (the elements
of the image of $Q$). The BRST-operator on algebra should satisfy graded
Leibniz rule: $Q(AB)=Q(A)B\pm AQ(B)$ (plus sign if $A$ is even, minus sign if
$A$ is odd. If algebra $\cal A$ is realized by operators in differential $\cal
A$-module with differential $\hat{Q}$ then the differential $Q$ on algebra is
defined as supercommutator with $\hat{Q}$, i.e. $QA=[Q,A]$ if $A$ is even and
$QA=[Q,A]_{+}$ if $A$ is odd.

Instead of Hilbert spaces one can consider differential modules equipped with
a structure of pseudo Hilbert space (space with non-degenerate, but indefinite
scalar product). However, the indefinite scalar product should descend to
definite scalar product on homology.

Gelfand-Naimark-Segal (GNS) construction can be generalized to the case when
an algebra is not equipped with involution. In this generalization we start
with a unital associative algebra $\cal A$ and a linear functional $\omega$ on
$\cal A.$ Then we can introduce a (not necessarily symmetric) scalar product
on $\cal A$ by the formula $\langle x,y\rangle=\omega(xy$). We are saying that
$a\in\cal A$ is a right null vector if $\langle x,a\rangle=0$ for every
$x\in\cal A.$ It is easy to check that right null vectors constitute a left
ideal in $\cal A$. Factorizing $\cal A$ with respect to this ideal we obtain a
right $\cal A$-module denoted by $R.$ Similarly factorizing with respect to
left null vectors we obtain a left $\cal A$-module denoted by $L.$ It is easy
to define a pairing between $L$ and $R$; this paring is non-degenerate. If the
algebra $\cal A$ is equipped with involution we can consider an induced
involution on the space of linear functionals; we assume that the functional
$\omega$ is self-adjoint. Then $R$ is complex conjugate to $L$ and the pairing
between $L$ and $R$ can be interpreted as (in general indefinite) scalar
product in $L$. If $\omega$ is a positive functional we come back to the GNS
construction.

Let us suppose now $\cal A$ is a differential algebra with differential $Q$.
This differential specifies a differential on the space ${\cal A}^{\vee}$ of
linear functionals denoted by the same symbol. We assume that $Q\omega=0$ (the
functional $\omega$ is $Q$-closed). This assumption implies that the ideals we
constructed are $Q$-invariant, hence the differential $Q$ descends to the
$\cal A$-modules $R$ and $L$. The pairing between differential modules $R$ and
$L$ respects the differential $Q.$

We will work with differential algebra $\cal A$ equipped with involution ∗
that agrees with the differential. We assume that time translations and
spatial translations act as automorphisms of $\cal A$ and commute with the
differential $Q.$ We fix a translation-invariant stationary self-adjoint
$Q$-closed linear functional $\omega$ that descends to a positive functional
on homology of $\cal A$. Applying the modification of GNS construction to
$\omega$ we obtain a differential pseudo pre Hilbert space $\cal H$ with the
differential (BRST operator) denoted $\hat{Q}$.

We modify the definition of elementary space saying that a differential vector
space $\tilde{\textgoth{h}}$ is an elementary space in new sense if its
homology can be identified with elementary space in old sense. An elementary
excitation of $\omega$ is defined as an linear map of $\tilde{\textgoth{h}}$
in $\cal H$ commuting with space-time translations and differentials (BRST
operators).

We can repeat with minor modifications the construction of Møller matrices and
scattering matrices in new situation. In particular, a scattering matrix
$\tilde{S}$ can be defined as an operator in Fock space $\tilde{F}$
corresponding to the space $\tilde{\textgoth{h}}$. The operator $\tilde{S}$
commutes with BRST operator, hence it descends to homology giving the
scattering matrix $S$ of physical (quasi)particles. (The operator $S$ acts in
the Fock space $\tilde{F}$ corresponding to the space ${\textgoth{h}}.$)

Acknowledgements I am indebted to A. Kapustin for valuable discussions.

## References

  * [1] Kapustin, A., 2013. Is there life beyond Quantum Mechanics?. arXiv preprint arXiv:1303.6917.
  * [2] Schwarz A. Geometric approach to quantum theory. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications. 2020 Apr 1;16:020.
  * [3] Schwarz A. 2021. Geometric and algebraic approaches to quantum theory. arXiv preprint arXiv:2102.09176. Nuclear Physics B, 973, p.115601.
  * [4] Schwarz, A., 2021. Scattering in geometric approach to quantum theory. arXiv preprint arXiv:2107.08557.
  * [5] Schwarz A. Scattering in algebraic approach to quantum theory. Jordan algebras (in preparation)
  * [6] A.S. Shvarts, New formulation of quantum theory, Dokl. Akad. Nauk SSSR, 173, 793 (1967).
  * [7] Schwarz A. Inclusive scattering matrix and scattering of quasiparticles. Nuclear Physics B. 2020 Jan 1;950:114869.
  * [8] Schwarz, A., 2019. Scattering matrix and inclusive scattering matrix in algebraic quantum field theory, arXiv preprint arXiv:1908.09388
  * [9] Schwarz, A and Tyupkin, Yu. , On adiabatic definition of S-matrix in the formalism of L-functionals, International seminar on Functional methods in Quantum Field Theory and Statistics, 1971, Lebedev Institute, 38-42
  * [10] Tyupkin, Yu, On the adiabatic definition of the S matrix in the formalism of L-functionals, Theoretical and Mathematical Physics, 1973, 16:2, 751-756, https://link.springer.com/content/pdf/10.1007%2FBF01037126.pdf
  * [11] Chou, K.C., Su, Z.B., Hao, B.L. and Yu, L., 1985. Equilibrium and nonequilibrium formalisms made unified. Physics Reports, 118(1-2), pp.1-131.
  * [12] Chu, H., and H. Umezawa. A unified formalism of thermal quantum field theory. International Journal of Modern Physics A 9.14 (1994): 2363-2409.
  * [13] Kamenev, Alex, and Alex Levchenko. Keldysh technique and non-linear sigma-model: basic principles and applications. Advances in Physics (2009).
  * [14] A. Schwarz, Mathematical foundations of quantum field theory, World Scientific
  * [15] Araki H, Haag R. Collision cross sections in terms of local observables. Communications in Mathematical Physics. 1967 Apr;4(2):77-91.
  * [16] Fateev, V.A. and Schwarz, A.S., On axiomatic scattering theory. Teoreticheskaya i Matematicheskaya Fizika, 1973, 14(2), pp.152-169.