text:
Engineers vs. God: Chile's Building Codes Take Edge Off Massive Earthquake
By Jim Wyss
Miami Herald, Sep 17, 2015
(Image: A man walks past a lone Chilean flag where surfing schools once stood, swept away by an earthquake-triggered tsunami in Concon, Chile. Courtesy Miami Herald)
At his engineering firm in Santiago, Chile, on Thursday — as aftershocks continued to roll through the city — Eduardo Santos was feeling pretty proud of his industry.
Wednesday night's 8.3-magnitude earthquake had left 11 dead and a 175 houses damaged. While the toll wasn't negligible, the quake — the world's strongest this year — might have leveled less-prepared countries. Read more…
Filed under Media Scan
Vancouver Design Firm to Study Public Space with $50K Prix de Rome Award
by Kevin Griffin
Vancouver Sun, Sep 16, 2015
(Image: Concept design for the redevelopment of the north plaza of the Vancouver Art Gallery showing the
PLALO ring, a floating circular ring for lighting and sound. Courtesy of Milkovich Architects Inc.)
Vancouver needs to focus on building public space in the centre of the city that are as dynamic as the
beaches and seawall around the edge according to a Vancouver architect.
Brian Wakelin said the kinds of urban spaces he's talking about aren't parks but places where people from all
social and economic backgrounds are welcome that can easily hold everything from market stalls to music
concerts. Read more…
buildingSMART Canada Announces Free Membership
bSC has revised its membership structure and is now offering free membership to individuals wanting to get actively involved in and contribute to the movement towards a better built environment for Canada as laid out in the Roadmap to Lifecycle BIM in the Canadian Architecture, Engineering, Construction, Owners and Operations (AECOO) Community published by bSC in January 2015. The call to action, issued through the IBC/bSC Roadmap to lifecycle BIM, involves a movement that must be undertaken with the full engagement, active involvement and continual support from the Canadian AECOO community. This will be done through the creation of a national community of excellence for the built environment which builds on the beliefs and capabilities of bSC's members and includes bSC's new Affiliates initiative.
By joining buildingSMART Canada, individuals become part of a growing community that is implementing collaborative working best practices in Canada and around the world. Members can link up to a worldwide community of practitioners, owners and software developers that are shaping the future with BIM and participate in the development of standards, protocols and programmes through working groups centered on activities described in the Roadmap to Lifecycle BIM. They can also stay abreast of innovative practices and improved project delivery processes through the creation of a national community of excellence. Members will have access to this growing pan-Canadian support network, whose agenda is transparent and aimed at an ambitious yet achievable goal: promoting and enabling the movement towards a better built environment for Canada through the development and use of open standards for BIM.
To sustain the movement, buildingSMART Canada is relying on financial support from Sponsors and Supporters. Sponsors are integral to the operation and activities of buildingSMART Canada. These companies and firms are leaders in the Canadian BIM community, and showcase their commitment to open BIM standard development in Canada. Supporters help facilitate the operations and activities of buildingSMART Canada and are committed to the buildingSMART mandate to see BIM adopted alongside well established open BIM standards. If you are interested in becoming a supporter or a sponsor, please visit our website.
Filed under Industry News
Connecting Canada's BIM Communities: buildingSMART Canada Launches Its Affiliates Initiative
bSC has launched its Affiliates initiative, which is aimed at building and strengthening the Canadian community of excellence for a better built environment based on open standards for BIM. This will be achieved in part by connecting local and regional BIM interest groups across Canada and opening up membership to individuals at no cost.
bSC understands that regional differences in the Canadian construction industry make it difficult to create a solution that is "one-size fits all". The building industry being of provincial jurisdiction, all the while being strongly influenced at the national and municipal levels, bSC's Affiliates initiative aims to create a dynamic community of excellence to develop and promote the use of open standards for BIM in Canada that are appropriate and fulfill these regional needs.
bSC Affiliates will remain independent and have no financial obligations towards – or any expectations from bSC. The intent is to grow the group of members who share the same goals organically while mutually expanding opportunities for sponsorship and support. The Affiliates initiative will provide a shared channel of communications for grassroots stakeholders to remain current with Canadian and international activity and enhance consistent messaging. bSC Affiliates will have first access to the most current news and information on international and national BIM standards and better project delivery practices in Canada as well as have input into the development of Canadian and international standards. bSC Affiliates will also get support from experts engaged with the national body to grow and develop their local or regional community.
Any local or regional organization supporting better practices and better outcomes for the building industry can join buildingSMART Canada as a bSC Affiliate. The initiative not only aims to build on existing local communities but can assist in setting up and developing communities in locations where none are currently in place. bSC Affiliates in turn will share the vision and goals of inter-operability and open standards for BIM while remaining vendor neutral (i.e. representing technological and process solutions from multiple vendors) and promoting the use of open standards for BIM.
Deadline Reminder: 2015 Annual Firm Updates
Firm owners/administrators are reminded that the deadline to complete annual firm updates is Wednesday, September 30, 2015.
Notification was sent by email to all firm administrators on September 10, 2015 about this annual requirement including the online process and deadlines. The AIBC issues notices annually to ensure that each firm's information is up to date to maintain the integrity of the AIBC register. If you haven't already done so, please access your online firm profile (http://aibc.memberpro.net) and review/revise your firm's information. The annual update is in accordance to AIBC Bylaw 15.4. It is the firm's professional responsibility to comply with this bylaw, even if only to confirm that the information we have is correct.
Should you have any questions, please contact AIBC Licensing Coordinator Kim Underwood at kunderwood@aibc.ca.
Art by Architects (AxA) Silent Auction and Exhibition Opening
On September 17, the Metro Vancouver Chapter of the RAIC held its inaugural Art by Architects/arch-kid-ecture fundraiser, an art exhibition and silent auction held at the AIBC Gallery. With 138 works of original art graciously donated by the architecture and design community, half were auctioned off over the course of the evening, with the remainder for sale during the two month run of the show (bidders were asked to hold off taking their art home until November 5).
With partial proceeds going to Architecture for Kids, an incentive of the Marion and Gordon Smith Gallery in North Vancouver to promote the study of architecture in K-12 school curriculum, the evening's event was attended by 100+ curious members from the local architecture community, many of whom left with original art created by many local architects, designers, and interns, even including one original piece created for the event by RAIC Fellow and award winning architect Richard Henriquez (all artwork had been displayed anonymously for the competition).
For full details and the list of winners, visit the RAIC website.
The AxA exhibit will be on display at the AIBC Gallery until November 5, 2015, Mondays to Fridays from 8:30 a.m. to 5 p.m.
Role Call: Municipal Advisory Design Panels
The AIBC invites applications from architects interested in serving on the following design panels:
District of Chilliwack
District of Maple Ridge
City of North Vancouver
District of North Vancouver
District of Squamish
City of Surrey
District of Pitt Meadows
City of Vancouver – Chinatown
Municipality of Whistler
The role of an advisory design panel member is to give impartial, professional advice directly on any proposal or policy affecting the community's physical environment in the public interest.
To learn more about the procedures for serving on a panel, please refer to the following documents:
AIBC Bulletin 65: Advisory Design Panels – Standards for Procedures and Conduct
ADP Frequently Asked Questions
Please fill out an interactive application form and forward your submissions to the attention of Professional Services Coordinator Alexandra Kokol by email (akokol@aibc.ca).
Filed under AIBC Volunteer Opportunities
George Baird in Dialogue – Book Launch and Celebration
Please join us for an evening celebrating the publication of George Baird's major collection of critical essays: Writings on Architecture and the City (Artifice Books on Architecture, London) at InForm Interiors in Gastown. Entry is free and open to anyone, but please come early, as space is limited. This event is made possible through the generosity of Nancy and Niels Bendtsen at InForm Interiors, and James Cheng at JKMCA Architects, and is co-sponsored by UBC's School of Architecture and Landscape Architecture.
George Baird is founder of Baird Neuert Sampson Architects, former Dean and Emeritus Professor at the University of Toronto's Daniel School of Architecture, and the RAIC's 2010 Gold Medalist for his career-long contributions to Canadian architecture and cities. From his pioneering application of post-structuralist theory to architecture in 1969's Meaning in Architecture (Charles Jencks co-editor) to the reflections on Hannah Arendt in his 1995 Space of Appearance, Baird's writing, editing and teaching has probed the essence of creation and city-dwelling. Similarly, his professional practice from the 1975 urban design study On Building Downtown for the City of Toronto, to Governor-General's award-winning buildings demonstrate a rare ability to transfer abstract ideas to the contingencies of city-building and construction.
Trevor Boddy is an architecture critic and urbanist whose writings on cities and buildings have won the Alberta Book of the Year, Western Magazine, Jack Webster, and CICA-UIA Pierre Vago Prizes. He curated the exhibition "Vancouverism: Architecture Builds the City" for its showings in London, Paris and Vancouver/Woodwards 2008-2010, his "Rethink: Behind San Diego's Skyline" is currently running there, and he co-curated the 2014 "Critical Juncture" at London's Victoria and Albert Museum. His text for the major monograph City-Builder: The Architecture of James K. M. Cheng will be published in early 2016.
Copies of Baird's book "Writings on Architecture and Cities" will be available for purchase at the Vancouver launch, and more information is available at http://www.artificebooksonline.com, including this recent review from Domus Magazine: https://www.domusweb.it/en/reviews/2015/05/18/george_baird_s_visionary_realism.html
Time: 5:30 p.m. reception; 6 p.m. George Baird & Trevor Boddy dialogue
Location: InForm Interiors, 50 Water Street, Vancouver
Cost: Free but limited seating, arrive early
Doors Open Vancouver
Doors Open provides an opportunity to enjoy free, behind-the-scenes access to popular City of Vancouver buildings and to learn about what it takes to run a city.
On October 3, you are invited to visit most venues anytime between 10 a.m. and 5 p.m. A few venues will have slightly different hours.
In addition to the two new venues added this year, notable favourites from last year are returning, including:
VPL Central Library
3-1-1 Call Centre and City Studio
The False Creek Energy Centre
Tours at Carnegie Community Centre and the City of Vancouver Tactical Training Centre require advance registration through the Doors Open Vancouver website.
Doors Open venues will be located in the following four walkable hubs:
City Hall and Olympic Village
False Creek Flats
Launch event: October 2, 2015, 12 pm with activities until 5 pm at Vancouver City Hall
Doors Open Vancouver event: October 3, 2015, 10am-5pm
For more information, visit Doors Open Vancouver website.
Building Smart with Mid-Rise Wood Buildings
Building Smart with Mid-Rise Wood Buildings is a half-day workshop focussing on the design, planning and construction of mid-rise wood buildings in British Columbia. Please join us for an exciting exchange of ideas and learn from the practical experience of our panel of local and international experts.
For more details about the topic, visit the Building Smart Workshop page.
This event is sponsored by the Homeowner Protection Office (HPO), Forestry Innovation Investment (FFI), Canadian Wood Council (CWC) and FPInnovations.
Time: 7 a.m. to 12 p.m. (registration and breakfast start at 7 am)
Location: The Westin Bayshore Hotel, Vancouver
Cost: $85 (includes GST)
To register: https://www.gifttool.com/registrar/ShowEventDetails?ID=1881&EID=20693
Lego Literacy Building Challenge
Next month, Surrey Libraries is holding their annual Literacy Building Challenge where local architecture, real estate, property development, construction and related companies gather a team and create lego-designed structures, all in support of early literacy programs.
The 2015 theme is Adventure. Each sponsored team is challenged to use LEGO to build a structure that reflects the bold and adventurous spirit of the community of Surrey.
The event runs over the weekend of October 17-18, 2015.
For more information, contact Cyndy Hill at 604-598-7307 or cahill@surrey.ca.
2015 AIBC Annual Conference: Plenary Session Focuses on Key Concepts for Designing Practice / Practicing Design
Designing Practice / Practicing Design
October 29, 2015 – 1:30-3 p.m.
In designing our practice, what constitutes the business of architecture? Is it about clients, staff, partners, consultants, authorities, contractors? What can we learn from practices in other professions, or from academics who study the structures of successful business teams? In practicing design, what presentations remind us, teach us and inspire us with the qualities of good design? And what are the traditional elements which make "good architecture"? What tools, current or future, will make achieving "good architecture" more feasible and more common? Join us at this plenary panel as we explore these questions.
This plenary session is sponsored by Tremco Roofing & Building Maintenance.
Register today to attend the Designing Practice / Practicing Design plenary session.
Visit the 2015 AIBC Annual Conference website for full details and to view conference pricing information.
2015 AIBC Annual Conference: Movie Night Reception and Screening of "Sagrada: The Mystery of Creation"
Movie Night Reception – October 29, 5-7 p.m.
Movie Night Screening of Sagrada: The Mystery of Creation – October 29, 7-9 p.m.
Enjoy an evening of drinks and appies while networking with your fellow conference delegates and UBC SALA Alumni prior to the screening on October 29 from 5:00-7:00pm, after a full day of exciting Plenary Sessions.
The Movie Night reception is sponsored by UBC School of Architecture and Landscape Architecture.
About Sagrada: The Mystery of Creation
One of the most iconic and enduring human structures ever built, Barcelona's La Sagrada Familia is a unique and fascinating architectural project conceived by Antoni Gaudi in the late 19th century. More than 125 years later after construction began, the basilica still remains unfinished. SAGRADA celebrates Gaudi's vision and the continuing work of countless laborers, artisans, designers and architects as they strive to complete the colossal project while delving into the mysterious process of artistic creation.
Register to attend the Movie Night Reception. and the Movie Night Screening.
2015 AIBC Annual Conference: Featured Sessional Speakers
The 2015 Annual Conference – What's Now – Insights and Directions will be held from October 28 – 30, 2015 at the Vancouver Convention Centre. The conference features plenary sessions, seminars, workshops, panel discussions and an industry exhibitor show. Meet some of our featured speakers below!
Kirsten Murray AIA
NW Traditions, Global Place-making
Session F14 – (1.5 Core LUs)
Kirsten is a principal and owner of Seattle-based Olson Kundig Architects. For over two decades she has focused on a broad range of project types including mixed-use, private residential, adaptive reuse, workplace and urban design. Her work has been published in a variety of national and international media including The New York Times, Architectural Digest, Interior Design, and Architectural Record. Murray's work has garnered national recognition for such projects as Outpost, Tye River Cabin, 1111 E.
Join Kristen at Session F14 – NW Traditions, Global Place-making.
Otto Condon AICP
District-Scale Sustainability – From Portland to DC and Tokyo
Otto Condon is a senior urban designer and planner in the Washington, DC office of ZGF Architects, LLP, with over 25 years of professional experience. He has been extensively involved in the development of plans and projects for downtowns, neighborhoods, housing, institutions, and transit in several cities, and a wide range of site-specific contexts. His urban design and planning approach focuses on solutions that promote goal-based revitalization, placemaking, environmental stewardship and implementation.
Join Otto at Session F10 – District-Scale Sustainability…
Kira Gerwing, MScP
Partnerships for Developing Social Purpose Real Estate
Session F4 – (1.5 Core LUs)
Kira is a Manager of Community Investment for Vancity Credit Union. Her work focuses on enabling the development social enterprise real estate projects that result in community ownership, affordability, and green buildings. Kira's work at Vancity stems from eight years of urban planning in Vancouver's Downtown Eastside neighbourhoods with the City of Vancouver. She has a passion for finding creative and engaging ways to ensure everyone's fundamental right to flourish.
Join Kira at Session F4 – Partnerships for Developing Social Purpose Real Estate.
Don Buchanan MA Planning
LRT, City Building & Placemaking in Surrey
Don Buchanan, MA Planning, is a Transportation Planner in the Rapid Transit and Strategic Projects Section at the City of Surrey. His main areas of focus are rapid transit corridor design (LRT) and developing the multi-modal transportation policy to support the transformation of Surrey City Centre. Previous gigs before joining Surrey include leading the Downtown Eastside Pedestrian Safety Project, working for the regional transportation authority and the David Suzuki Foundation.
Join Don at Session F6 – LRT, City Building and Placemaking in Surrey.
Visit the 2015 AIBC Annual Conference website to view all of our amazing speakers.
View conference pricing and register. To learn more about the 2015 AIBC Annual Conference, please visit the 2015 AIBC Annual Conference website.
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text:
\section{Introduction}

\bigskip \ Integration transformations are very useful in mathematical physics.
In the preceding paper \cite{r1} we have reported a new integration
transformation in $q-p$ phase space%
\begin{equation}
\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(  p-x\right)  \left(
q-y\right)  }h(p,q)=f\left(  x,y\right)  , \label{a1}%
\end{equation}
which is invertible%
\begin{equation}
\iint_{-\infty}^{\infty}\frac{dxdy}{\pi}e^{-2i(p-x)(q-y)}f(x,y)=h(p,q),
\label{a2}%
\end{equation}
and proved that this transformation obeys Parseval theorem. By virtue of the
formula of operators' Weyl ordering expansion \cite{r2,r3}, we find the
following new two-fold $q$-number integration transformation about the Wigner
operator $\Delta \left(  q^{\prime},p^{\prime}\right)  $ \cite{r4,r5,r6} in
phase space quantum mechanics \cite{r7},
\begin{equation}
\iint_{-\infty}^{\infty}\frac{\mathtt{d}p^{\prime}\mathtt{d}q^{\prime}}{\pi
}\Delta \left(  q^{\prime},p^{\prime}\right)  e^{-2i\left(  p-p^{\prime
}\right)  \left(  q-q^{\prime}\right)  }=\delta \left(  p-P\right)
\delta \left(  q-Q\right)  , \label{a3}%
\end{equation}
and its inverse%
\begin{equation}
\iint_{-\infty}^{\infty}\mathtt{d}q\mathtt{d}p\delta \left(  p-P\right)
\delta \left(  q-Q\right)  e^{2i\left(  p-p^{\prime}\right)  \left(
q-q^{\prime}\right)  }=\Delta \left(  q^{\prime},p^{\prime}\right)  ,
\label{a4}%
\end{equation}
where $Q,$ $P$ are the coordinate and momentum operators, respectively.
$\left[  Q,P\right]  =i\hbar.$ These two equations can be applied to studying
mutual converting formulas among $Q-P$ ordering, $P-Q$ ordering and Weyl
ordering of operators.

In the present paper, we extend Eqs.(\ref{a3}) and (\ref{a4}) to the two-mode
entangled case and construct the two-fold complex integration transformation
about the Wigner operator $\Delta \left(  \mu,\nu \right)  $ (in its entangled
form, $\mu,\nu$ are complex) in phase space quantum mechanics. By mentioning
two-mode entangled case, we naturally think of two mutually conjugate
entangled state representations $\left \vert \xi \right \rangle $ and $\left \vert
\eta \right \rangle $ (see Sect. 2 below). In Sect. 3, after deriving the Weyl
ordering of $\left \vert \eta \right \rangle \left \langle \xi \right \vert $ by
virtue of the formula of operator's Weyl ordering expansion, we give the Weyl
ordering form of

$\delta \left(  \nu-a_{1}+a_{2}^{\dagger}\right)  $ $\delta \left(  \nu^{\ast
}-a_{1}^{\dagger}+a_{2}\right)  $ $\delta \left(  \mu-a_{1}-a_{2}^{\dagger
}\right)  $ $\delta \left(  \mu^{\ast}-a_{1}^{\dagger}-a_{2}\right)  $, where
$a_{i},a_{i}^{\dagger}$ are the bosonic creation and annihilation operators
satisfying $\left[  a_{i},a_{j}^{\dagger}\right]  =\delta_{ij}$. Then in Sect.
4, we find a new two-fold complex integration transformation relating the
Wigner operator $\Delta \left(  \mu,\nu \right)  $ and the operator
$\delta \left(  \nu-a_{1}+a_{2}^{\dagger}\right)  $ $\delta \left(  \nu^{\ast
}-a_{1}^{\dagger}+a_{2}\right)  $ $\delta \left(  \mu-a_{1}-a_{2}^{\dagger
}\right)  $ $\delta \left(  \mu^{\ast}-a_{1}^{\dagger}-a_{2}\right)  .$ In
Sect. 5 we tackle some operator ordering problems regarding to $\left(
a_{1}^{\dagger}-a_{2}\right)  $ and $\left(  a_{1}+a_{2}^{\dagger}\right)  $.
In this way, the contents of phase space quantum mechanics can be further enriched.

\section{Brief review of the two mutually conjugate entangled states}

The conception of quantum entanglement has been paid much attention because of
their wide uses in quantum communication and quantum computation. The original
concept of quantum entanglement pointed out by Einstein, Podolsky and Rosen
(EPR) in their famous paper \cite{r8} arguing the incompleteness of quantum
mechanics is for a bipartite system characteristic of continuous variables.
According to the original idea of EPR that two particles' relative coordinate
operator commutes with their total momentum operator, $\left[  Q_{1}%
-Q_{2},P_{1}+P_{2}\right]  =0$, we have set up the bipartite entangle state
representation of continuous variable in two-mode Fock space \cite{r9}, which
is the common eigenvector $\left \vert \eta \right \rangle $ of $Q_{1}-Q_{2}$ and
$P_{1}+P_{2}$, i.e.
\begin{equation}
\left \vert \eta \right \rangle =\exp \left(  -\frac{1}{2}\left \vert
\eta \right \vert ^{2}+\eta a_{1}^{\dagger}-\eta^{\ast}a_{2}^{^{\dagger}}%
+a_{1}^{^{\dagger}}a_{2}^{^{\dagger}}\right)  \left \vert 00\right \rangle ,
\label{b1}%
\end{equation}
where $\eta \equiv \eta_{1}+i\eta_{2}$ is a complex number, $\left \vert
00\right \rangle $ is the two-mode vacuum state. Using $\left[  a_{i}%
,a_{j}^{\dagger}\right]  =\delta_{ij}$, it can be shown that $\left \vert
\eta \right \rangle $ obeys the eigenvector equations
\begin{equation}
\left(  a_{1}-a_{2}^{^{\dagger}}\right)  \left \vert \eta \right \rangle
=\eta \left \vert \eta \right \rangle ,\text{ }\left(  a_{2}-a_{1}^{^{\dagger}%
}\right)  \left \vert \eta \right \rangle =-\eta^{\ast}\left \vert \eta
\right \rangle .\text{ } \label{b2}%
\end{equation}
$\eta^{\prime}s$ real and imaginary part are respectively the eigenvalue of
$Q_{1}-Q_{2}$ and $P_{1}+P_{2}$,
\begin{equation}
\left(  Q_{1}-Q_{2}\right)  \left \vert \eta \right \rangle =\sqrt{2}\eta
_{1}\left \vert \eta \right \rangle ,\text{ }\left(  P_{1}+P_{2}\right)
\left \vert \eta \right \rangle =\sqrt{2}\eta_{2}\left \vert \eta \right \rangle .
\label{b4}%
\end{equation}
$\left \vert \eta \right \rangle $ is proved to be complete relation and
orthonormal property%
\begin{equation}
\int \frac{d^{2}\eta}{\pi}\left \vert \eta \right \rangle \left \langle
\eta \right \vert =1 \label{b6-1}%
\end{equation}
and%
\begin{equation}
\text{ }\left \langle \eta^{\prime}\right \vert \left.  \eta \right \rangle
=\pi \delta \left(  \eta^{\prime}-\eta \right)  \delta \left(  \eta^{\prime \ast
}-\eta^{\ast}\right)  \equiv \pi \delta^{\left(  2\right)  }\left(  \eta
^{\prime}-\eta \right)  . \label{b6-2}%
\end{equation}
On the other hand, $\left \vert \eta \right \rangle ^{\prime}$s conjugate state,
which is the common eigenvector of two particles' center-of-mass coordinate
and relative momentum, is%
\begin{equation}
\left \vert \xi \right \rangle =\exp \left(  -\frac{|\xi|^{2}}{2}+\xi
a_{1}^{\dagger}+\xi^{\ast}a_{2}^{\dagger}-a_{1}^{\dagger}a_{2}^{\dagger
}\right)  \left \vert 00\right \rangle , \label{b7}%
\end{equation}
with $\xi=\xi_{1}+i\xi_{2}$. $\left \vert \xi \right \rangle $ obeys
\begin{equation}
\left(  a_{1}+a_{2}^{\dagger}\right)  \left \vert \xi \right \rangle
=\xi \left \vert \xi \right \rangle ,\text{ \ }\left(  a_{1}^{\dagger}%
+a_{2}\right)  \left \vert \xi \right \rangle =\xi^{\ast}\left \vert
\xi \right \rangle , \label{b8}%
\end{equation}
and%
\begin{equation}
\left(  Q_{1}+Q_{2}\right)  \left \vert \xi \right \rangle =\sqrt{2}\xi
_{1}\left \vert \xi \right \rangle ,\text{ }\left(  P_{1}-P_{2}\right)
\left \vert \xi \right \rangle =\sqrt{2}\xi_{2}\left \vert \xi \right \rangle .
\label{b9}%
\end{equation}
$\left \vert \xi \right \rangle $ is completeness and orthonormal too,%
\begin{equation}
\int \frac{d^{2}\xi}{\pi}\left \vert \xi \right \rangle \left \langle
\xi \right \vert =1 \label{b10-1}%
\end{equation}
and%
\begin{equation}
\left \langle \xi \right.  \left \vert \xi^{\prime}\right \rangle =\pi
\delta \left(  \xi-\xi^{\prime}\right)  \delta \left(  \xi^{\ast}-\xi
^{\prime \ast}\right)  . \label{b10-2}%
\end{equation}
It is found that $\left \vert \xi \right \rangle $ and $\left \vert \eta
\right \rangle $ are of equal importance because
\begin{equation}
\left \langle \eta \right \vert \left.  \xi \right \rangle =\frac{1}{2}\exp \left[
\left(  \eta^{\ast}\xi-\xi^{\ast}\eta \right)  /2\right]  , \label{b11}%
\end{equation}
which is just a complex form of the Fourier transformation, since $\left(
\eta^{\ast}\xi-\xi^{\ast}\eta \right)  $ is pure imaginary, $\left \langle
\eta \right \vert \left.  \xi \right \rangle $ is a Fourier transform kernel.

\section{The Weyl ordering form of $\delta \left(  \nu-a_{1}+a_{2}^{\dagger
}\right)  $ $\delta \left(  \nu^{\ast}-a_{1}^{\dagger}+a_{2}\right)  $
$\delta \left(  \mu-a_{1}-a_{2}^{\dagger}\right)  $ $\delta \left(  \mu^{\ast
}-a_{1}^{\dagger}-a_{2}\right)  $}

Using the complete relation Eqs.(\ref{b6-1}) and (\ref{b10-1}), as well as
Eqs.(\ref{b2}), (\ref{b8}) and (\ref{b11}), we have%
\begin{align}
&  \delta \left(  \nu-a_{1}+a_{2}^{\dagger}\right)  \delta \left(  \nu^{\ast
}-a_{1}^{\dagger}+a_{2}\right)  \delta \left(  \mu-a_{1}-a_{2}^{\dagger
}\right)  \delta \left(  \mu^{\ast}-a_{1}^{\dagger}-a_{2}\right) \nonumber \\
&  =\int \frac{d^{2}\eta d^{2}\xi}{\pi^{2}}\delta^{\left(  2\right)  }\left(
\nu-a_{1}+a_{2}^{\dagger}\right)  \left \vert \eta \right \rangle \left \langle
\eta \right \vert \left.  \xi \right \rangle \left \langle \xi \right \vert
\delta^{\left(  2\right)  }\left(  \mu-a_{1}-a_{2}^{\dagger}\right)
\nonumber \\
&  =\frac{1}{2}\int \frac{d^{2}\eta d^{2}\xi}{\pi^{2}}\left \vert \eta
\right \rangle \left \langle \xi \right \vert e^{\left(  \eta^{\ast}\xi-\eta
\xi^{\ast}\right)  /2}\delta^{\left(  2\right)  }\left(  \nu-\eta \right)
\delta^{\left(  2\right)  }\left(  \mu-\xi \right) \nonumber \\
&  =\frac{1}{2}\left \vert \eta \right \rangle _{\eta=\nu}\left \langle
\xi \right \vert _{\xi=\mu}e^{\left(  \nu^{\ast}\mu-\nu \mu^{\ast}\right)  /2}.
\label{c1}%
\end{align}
where we have simplified the notation of the product of two Delta functions
\begin{align}
&  \delta^{\left(  2\right)  }\left(  \nu-a_{1}+a_{2}^{\dagger}\right)
\delta^{\left(  2\right)  }\left(  \mu-a_{1}-a_{2}^{\dagger}\right)
\nonumber \\
&  \equiv \delta \left(  \nu-a_{1}+a_{2}^{\dagger}\right)  \delta \left(
\nu^{\ast}-a_{1}^{\dagger}+a_{2}\right)  \delta \left(  \mu-a_{1}%
-a_{2}^{\dagger}\right)  \delta \left(  \mu^{\ast}-a_{1}^{\dagger}%
-a_{2}\right)  . \label{c1-1}%
\end{align}
From Eq.(\ref{c1}) we see that once the Weyl ordering of $\left \vert
\eta \right \rangle \left \langle \xi \right \vert $ is known, the Weyl ordering of
$\delta \left(  \nu-a_{1}+a_{2}^{\dagger}\right)  $ $\delta \left(  \nu^{\ast
}-a_{1}^{\dagger}+a_{2}\right)  $ $\delta \left(  \mu-a_{1}-a_{2}^{\dagger
}\right)  $ $\delta \left(  \mu^{\ast}-a_{1}^{\dagger}-a_{2}\right)  $ is
obtained. For this purpose, we recall the Weyl ordered expansion formula of
two-mode operators\cite{r2,r3}%
\begin{equation}
\rho=4\int \frac{d^{2}\beta_{1}d^{2}\beta_{2}}{\pi^{2}}%
\genfrac{}{}{0pt}{}{:}{:}%
\left \langle -\beta_{1},-\beta_{2}\right \vert \rho \left \vert \beta_{1}%
,\beta_{2}\right \rangle \exp \left[  2\sum \limits_{k=1}^{2}\left(  \beta
_{k}^{\ast}a_{k}-a_{k}^{\dagger}\beta_{k}+a_{k}^{\dagger}a_{k}\right)
\right]
\genfrac{}{}{0pt}{}{:}{:}
\label{c2}%
\end{equation}
where $%
\genfrac{}{}{0pt}{}{:}{:}%
\genfrac{}{}{0pt}{}{:}{:}%
$ denotes the Weyl ordering, within which $a_{i},a_{i}^{\dagger}$ can be
permuted, $\left \vert \beta_{1},\beta_{2}\right \rangle \equiv \left \vert
\beta_{1}\right \rangle \left \vert \beta_{2}\right \rangle $ is a two-mode
coherent state, $\left \vert \beta_{i}\right \rangle =\exp[-\frac{1}{2}%
|\beta_{i}|^{2}+\beta_{i}a_{i}^{\dagger}]\left \vert 0_{i}\right \rangle
$\cite{r10}$.$ For the operator $\left \vert \eta \right \rangle \left \langle
\xi \right \vert ,$ using Eq.(\ref{c2}) and the overlaps%
\begin{equation}
\left \langle \xi \right \vert \left.  \beta_{1},\beta_{2}\right \rangle
=\exp \left(  -\frac{1}{2}|\xi|^{2}+\xi^{\ast}\beta_{1}+\xi \beta_{2}-\beta
_{1}\beta_{2}-\frac{1}{2}|\beta_{1}|^{2}-\frac{1}{2}|\beta_{2}|^{2}\right)  ,
\label{c3}%
\end{equation}
and%
\begin{equation}
\left \langle -\beta_{1},-\beta_{2}\right \vert \left.  \eta \right \rangle
=\exp \left(  -\frac{1}{2}|\eta|^{2}-\eta \beta_{1}^{\ast}+\eta^{\ast}\beta
_{2}^{\ast}+\beta_{1}^{\ast}\beta_{2}^{\ast}-\frac{1}{2}|\beta_{1}|^{2}%
-\frac{1}{2}|\beta_{2}|^{2}\right)  , \label{c4}%
\end{equation}
we can derive the Weyl ordering of $\left \vert \eta \right \rangle \left \langle
\xi \right \vert $ in the way%

\begin{align}
\left \vert \eta \right \rangle \left \langle \xi \right \vert  &  =4\int \frac
{d^{2}\beta_{1}d^{2}\beta_{2}}{\pi^{2}}%
\genfrac{}{}{0pt}{}{:}{:}%
\left \langle -\beta_{1},-\beta_{2}\right \vert \left.  \eta \right \rangle
\left \langle \xi \right \vert \left.  \beta_{1},\beta_{2}\right \rangle
\exp \left[  2\sum \limits_{k=1}^{2}\left(  \beta_{k}^{\ast}a_{k}-a_{k}%
^{\dagger}\beta_{k}+a_{k}^{\dagger}a_{k}\right)  \right]
\genfrac{}{}{0pt}{}{:}{:}%
\nonumber \\
&  =4\int \frac{d^{2}\beta_{1}d^{2}\beta_{2}}{\pi^{2}}%
\genfrac{}{}{0pt}{}{:}{:}%
\exp \left[  -\frac{|\eta|^{2}}{2}-\eta \beta_{1}^{\ast}+\eta^{\ast}\beta
_{2}^{\ast}+\beta_{1}^{\ast}\beta_{2}^{\ast}-|\beta_{1}|^{2}-|\beta_{2}%
|^{2}\right. \nonumber \\
&  \left.  -\frac{|\xi|^{2}}{2}+\xi^{\ast}\beta_{1}+\xi \beta_{2}-\beta
_{1}\beta_{2}+2\sum \limits_{k=1}^{2}\left(  \beta_{k}^{\ast}a_{k}%
-a_{k}^{\dagger}\beta_{k}+a_{k}^{\dagger}a_{k}\right)  \right]
\genfrac{}{}{0pt}{}{:}{:}%
\nonumber \\
&  =4e^{-\frac{|\eta|^{2}}{2}-\frac{|\xi|^{2}}{2}}\int \frac{d^{2}\beta_{2}%
}{\pi}%
\genfrac{}{}{0pt}{}{:}{:}%
\exp \left[  -2|\beta_{2}|^{2}+\beta_{2}\left(  \xi-2a_{2}^{\dagger}%
-2a_{1}+\eta \right)  \right. \nonumber \\
&  \left.  +\beta_{2}^{\ast}\left(  \eta^{\ast}+2a_{2}+\xi^{\ast}%
-2a_{1}^{\dagger}\right)  +\left(  \xi^{\ast}-2a_{1}^{\dagger}\right)  \left(
2a_{1}-\eta \right)  +2\sum_{k=1}^{2}a_{k}^{\dagger}a_{k}\right]
\genfrac{}{}{0pt}{}{:}{:}%
\nonumber \\
&  =2%
\genfrac{}{}{0pt}{}{:}{:}%
\exp \left[  \frac{1}{2}\left(  \xi \eta^{\ast}-\eta \xi^{\ast}\right)
+\xi \left(  a_{2}-a_{1}^{\dagger}\right)  +\eta \left(  a_{2}+a_{1}^{\dagger
}\right)  \right. \nonumber \\
&  \left.  -\eta^{\ast}\left(  a_{1}+a_{2}^{\dagger}\right)  +\xi^{\ast
}\left(  a_{1}-a_{2}^{\dagger}\right)  +2a_{2}^{\dagger}a_{1}^{\dagger}%
-2a_{2}a_{1}\right]
\genfrac{}{}{0pt}{}{:}{:}%
, \label{c5}%
\end{align}
where we have used the integral formula%
\begin{equation}
\int \frac{d^{2}z}{\pi}\exp \left(  \zeta \left \vert z\right \vert ^{2}+\xi z+\eta
z^{\ast}\right)  =-\frac{1}{\zeta}\exp \left[  -\frac{\xi \eta}{\zeta}\right]
,\operatorname{Re}\left(  \xi \right)  <0 \label{c6}%
\end{equation}
It then follows from Eq.(\ref{c5}) that%
\begin{equation}
\frac{1}{2}\left \vert \eta \right \rangle \left \langle \xi \right \vert e^{\left(
\eta^{\ast}\xi-\eta \xi^{\ast}\right)  /2}=%
\genfrac{}{}{0pt}{}{:}{:}%
\exp[\left(  \xi-a_{1}-a_{2}^{\dagger}\right)  \left(  \eta^{\ast}-a_{1}%
^{\dag}+a_{2}\right)  -\left(  \eta-a_{1}+a_{2}^{\dagger}\right)  \left(
\xi^{\ast}-a_{1}^{\dag}-a_{2}\right)  ]%
\genfrac{}{}{0pt}{}{:}{:}%
. \label{c7}%
\end{equation}
Substituting Eq.(\ref{c7}) into Eq.(\ref{c1}) we have the Weyl ordered form
\begin{align}
&  \delta \left(  \nu-a_{1}+a_{2}^{\dagger}\right)  \delta \left(  \nu^{\ast
}-a_{1}^{\dagger}+a_{2}\right)  \delta \left(  \mu-a_{1}-a_{2}^{\dagger
}\right)  \delta \left(  \mu^{\ast}-a_{1}^{\dagger}-a_{2}\right) \nonumber \\
&  =%
\genfrac{}{}{0pt}{}{:}{:}%
\exp[\left(  \mu-a_{1}-a_{2}^{\dagger}\right)  \left(  \nu^{\ast}-a_{1}^{\dag
}+a_{2}\right)  -\left(  \nu-a_{1}+a_{2}^{\dagger}\right)  \left(  \mu^{\ast
}-a_{1}^{\dag}-a_{2}\right)  ]%
\genfrac{}{}{0pt}{}{:}{:}%
, \label{c8}%
\end{align}
\textbf{\ }Noting
\begin{equation}
\left[  a_{1}^{\dagger}-a_{2},a_{1}+a_{2}^{\dagger}\right]  =-2, \label{c9}%
\end{equation}
we should consider another ordering other than Eq.(\ref{c8}), i.e.,
\begin{align}
&  \delta \left(  \mu-a_{1}-a_{2}^{\dagger}\right)  \delta \left(  \mu^{\ast
}-a_{1}^{\dagger}-a_{2}\right)  \delta \left(  \nu-a_{1}+a_{2}^{\dagger
}\right)  \delta \left(  \nu^{\ast}-a_{1}^{\dagger}+a_{2}\right) \nonumber \\
&  =\frac{1}{2}\int \frac{d^{2}\eta d^{2}\xi}{\pi^{2}}\left \vert \xi
\right \rangle \left \langle \eta \right \vert e^{\left(  \eta \xi^{\ast}%
-\eta^{\ast}\xi \right)  /2}\delta^{\left(  2\right)  }\left(  \nu-\eta \right)
\delta^{\left(  2\right)  }\left(  \mu-\xi \right)  . \label{c10}%
\end{align}
Similar in the way of deriving Eq.(\ref{c7}), we deduce%

\begin{equation}
\frac{1}{2}\left \vert \xi \right \rangle \left \langle \eta \right \vert e^{\left(
\eta \xi^{\ast}-\eta^{\ast}\xi \right)  /2}=%
\genfrac{}{}{0pt}{}{:}{:}%
\exp \left[  \left(  \eta-\allowbreak a_{1}+a_{2}^{\dagger}\right)  \left(
\xi^{\ast}-a_{1}^{\dagger}-a_{2}\right)  -\left(  \xi-a_{1}-a_{2}^{\dagger
}\right)  \allowbreak \left(  \eta^{\ast}-a_{1}^{\dagger}+a_{2}\right)
\allowbreak \right]
\genfrac{}{}{0pt}{}{:}{:}%
.\label{c11}%
\end{equation}%
\[%
\genfrac{}{}{0pt}{}{:}{:}%
\exp \left[  \left(  \eta-\allowbreak a_{1}+a_{2}^{\dagger}\right)  \left(
\xi^{\ast}-a_{1}^{\dagger}-a_{2}\right)  -\left(  \allowbreak \eta^{\ast}%
-a_{1}^{\dagger}+a_{2}\right)  \left(  \xi-\allowbreak a_{2}^{\dagger}%
-a_{1}\right)  \right]
\genfrac{}{}{0pt}{}{:}{:}%
\]
It then follows
\begin{align}
&  \delta \left(  \mu-a_{1}-a_{2}^{\dagger}\right)  \delta \left(  \mu^{\ast
}-a_{1}^{\dagger}-a_{2}\right)  \delta \left(  \nu-a_{1}+a_{2}^{\dagger
}\right)  \delta \left(  \nu^{\ast}-a_{1}^{\dagger}+a_{2}\right)  \nonumber \\
&  =%
\genfrac{}{}{0pt}{}{:}{:}%
\exp \left[  \left(  \nu-\allowbreak a_{1}+a_{2}^{\dagger}\right)  \left(
\mu^{\ast}-a_{1}^{\dagger}-a_{2}\right)  -\left(  \mu-a_{1}-a_{2}^{\dagger
}\right)  \allowbreak \left(  \nu^{\ast}-a_{1}^{\dagger}+a_{2}\right)
\allowbreak \right]
\genfrac{}{}{0pt}{}{:}{:}%
.\label{c12}%
\end{align}

\section{The mutual integration transformation between $\delta \left(
\nu-a_{1}+a_{2}^{\dagger}\right)  $ $\delta \left(  \nu^{\ast}-a_{1}^{\dagger
}+a_{2}\right)  $ $\delta \left(  \mu-a_{1}-a_{2}^{\dagger}\right)  $
$\delta \left(  \mu^{\ast}-a_{1}^{\dagger}-a_{2}\right)  $ and the Wigner
operator}

In Ref. \cite{r11} for correlated two-body systems, we have successfully
derived the Wigner operator in entangled form, expressed in the entangled
state $\left \langle \eta \right \vert $ representation as
\begin{equation}
\Delta \left(  \mu,\nu \right)  =\int \frac{d^{2}\eta}{\pi^{3}}\left \vert
\nu-\eta \right \rangle \left \langle \nu+\eta \right \vert e^{\eta \mu^{\ast}%
-\eta^{\ast}\mu}, \label{e1}%
\end{equation}
$\Delta \left(  \mu,\nu \right)  $ plays the role of establishing
the relationship between $\eta-\xi$ phase space function and its
Weyl-Wigner quantum correspondence operator. Using the similar
method to deriving Eq.(\ref{c5}), we find that the Weyl ordering of
operator $\left \vert \nu -\eta \right \rangle \left \langle
\nu+\eta \right \vert $ is
\begin{align}
\left \vert \nu-\eta \right \rangle \left \langle \nu+\eta \right \vert  &  =%
\genfrac{}{}{0pt}{}{:}{:}%
\delta \left(  \nu-a_{1}+a_{2}^{\dagger}\right)  \delta \left(  \nu^{\ast}%
-a_{1}^{\dagger}+a_{2}\right) \nonumber \\
&  \times \exp \left[  \eta \left(  \nu^{\ast}-2a_{1}^{\dagger}\right)
-\eta^{\ast}\left(  \nu-2a_{1}\right)  -2\left(  a_{1}^{\dagger}-\nu^{\ast
}\right)  \left(  a_{1}-\nu \right)  +2a_{2}^{\dagger}a_{2}\right]
\genfrac{}{}{0pt}{}{:}{:}%
. \label{e2}%
\end{align}
Substituting (\ref{e2}) into Eq.(\ref{e1}) and performing the integration over
$d^{2}\eta,$ the result is%
\begin{align}
\Delta \left(  \mu,\nu \right)   &  =%
\genfrac{}{}{0pt}{}{:}{:}%
\delta \left(  \mu-a_{1}-a_{2}^{\dagger}\right)  \delta \left(  \mu^{\ast}%
-a_{1}^{\dagger}-a_{2}\right)  \delta \left(  \nu-a_{1}+a_{2}^{\dagger}\right)
\delta \left(  \nu^{\ast}-a_{1}^{\dagger}+a_{2}\right)
\genfrac{}{}{0pt}{}{:}{:}%
\nonumber \\
&  =%
\genfrac{}{}{0pt}{}{:}{:}%
\delta^{\left(  2\right)  }\left(  \mu-a_{1}-a_{2}^{\dagger}\right)
\delta^{\left(  2\right)  }\left(  \nu-a_{1}+a_{2}^{\dagger}\right)
\genfrac{}{}{0pt}{}{:}{:}%
, \label{e3}%
\end{align}
which is also a neat Dirac's $\delta$-operator function within the Weyl
ordering symbol.

As a result of Eqs.(\ref{c8}) and (\ref{e3}), we have
\begin{align}
&  \delta \left(  \eta-a_{1}+a_{2}^{\dagger}\right)  \delta \left(  \eta^{\ast
}-a_{1}^{\dagger}+a_{2}\right)  \delta \left(  \xi-a_{1}-a_{2}^{\dagger
}\right)  \delta \left(  \xi^{\ast}-a_{1}^{\dagger}-a_{2}\right) \nonumber \\
&  =%
\genfrac{}{}{0pt}{}{:}{:}%
\exp[\left(  \xi-a_{1}-a_{2}^{\dagger}\right)  \left(  \eta^{\ast}-a_{1}%
^{\dag}+a_{2}\right)  -\left(  \eta-a_{1}+a_{2}^{\dagger}\right)  \left(
\xi^{\ast}-a_{1}^{\dag}-a_{2}\right)  ]%
\genfrac{}{}{0pt}{}{:}{:}%
\nonumber \\
&  =\int \frac{d^{2}\mu d^{2}\nu}{\pi^{2}}%
\genfrac{}{}{0pt}{}{:}{:}%
e^{\left(  \xi-\mu \right)  \left(  \eta^{\ast}-\nu^{\ast}\right)  -\left(
\eta-\nu \right)  \left(  \xi^{\ast}-\mu^{\ast}\right)  }\delta^{\left(
2\right)  }\left(  \mu-a_{1}-a_{2}^{\dagger}\right)  \delta^{\left(  2\right)
}\left(  \nu-a_{1}+a_{2}^{\dagger}\right)
\genfrac{}{}{0pt}{}{:}{:}%
\nonumber \\
&  =\int \frac{d^{2}\mu d^{2}\nu}{\pi^{2}}e^{\left(  \xi-\mu \right)  \left(
\eta^{\ast}-\nu^{\ast}\right)  -\left(  \eta-\nu \right)  \left(  \xi^{\ast
}-\mu^{\ast}\right)  }\Delta \left(  \mu,\nu \right)  . \label{e4}%
\end{align}
This is the mutual transformation between the entangled form of Wigner
operator $\Delta \left(  \mu,\nu \right)  $ and $\delta \left(  \mu-a_{1}%
-a_{2}^{\dagger}\right)  $ $\delta \left(  \mu^{\ast}-a_{1}^{\dagger}%
-a_{2}\right)  $ $\delta \left(  \nu-a_{1}+a_{2}^{\dagger}\right)  $
$\delta \left(  \nu^{\ast}-a_{1}^{\dagger}+a_{2}\right)  .$

Moveover, the reciprocal transformation of Eq. (\ref{e4}) is%
\begin{align}
&  \int \frac{d^{2}\xi d^{2}\eta}{\pi^{2}}\delta \left(  \eta-a_{1}%
+a_{2}^{\dagger}\right)  \delta \left(  \eta^{\ast}-a_{1}^{\dagger}%
+a_{2}\right)  \delta \left(  \xi-a_{1}-a_{2}^{\dagger}\right)  \delta \left(
\xi^{\ast}-a_{1}^{\dagger}-a_{2}\right)  e^{-\left(  \xi-\mu \right)  \left(
\eta^{\ast}-\nu^{\ast}\right)  +\left(  \eta-\nu \right)  \left(  \xi^{\ast
}-\mu^{\ast}\right)  }\nonumber \\
&  =\int \frac{d^{2}\xi d^{2}\eta}{\pi^{2}}\int \frac{d^{2}\mu^{\prime}d^{2}%
\nu^{\prime}}{\pi^{2}}\Delta \left(  \mu^{\prime},\nu^{\prime}\right)
e^{\left(  \xi-\mu^{\prime}\right)  \left(  \eta^{\ast}-\nu^{\prime \ast
}\right)  -\left(  \eta-\nu^{\prime}\right)  \left(  \xi^{\ast}-\mu
^{\prime \ast}\right)  }e^{-\left(  \xi-\mu \right)  \left(  \eta^{\ast}%
-\nu^{\ast}\right)  +\left(  \eta-\nu \right)  \left(  \xi^{\ast}-\mu^{\ast
}\right)  }\nonumber \\
&  =\int \frac{d^{2}\mu^{\prime}d^{2}\nu^{\prime}}{\pi^{2}}\Delta \left(
\mu^{\prime},\nu^{\prime}\right)  e^{\left(  -\nu^{\prime}\mu^{\prime \ast}%
+\mu^{\prime}\nu^{\prime \ast}-\mu \nu^{\ast}+\nu \mu^{\ast}\right)  }\int
\frac{d^{2}\xi d^{2}\eta}{\pi^{2}}e^{\xi \left(  \nu^{\ast}-\nu^{\prime \ast
}\right)  +\xi^{\ast}\left(  \nu^{\prime}-\nu \right)  }e^{\eta^{\ast}\left(
\mu-\mu^{\prime}\right)  +\eta \left(  \mu^{\prime \ast}-\mu^{\ast}\right)
}\nonumber \\
&  =\int \frac{d^{2}\mu^{\prime}d^{2}\nu^{\prime}}{\pi^{2}}\Delta \left(
\mu^{\prime},\nu^{\prime}\right)  e^{\left(  -\nu^{\prime}\mu^{\prime \ast}%
+\mu^{\prime}\nu^{\prime \ast}-\mu \nu^{\ast}+\nu \mu^{\ast}\right)  }%
\delta^{\left(  2\right)  }\left(  \nu^{\prime}-\nu \right)  \delta^{\left(
2\right)  }\delta \left(  \mu^{\prime}-\mu \right) \nonumber \\
&  =\Delta \left(  \mu,\nu \right)  . \label{e5}%
\end{align}
So Eqs.(\ref{e4}) and (\ref{e5}) are qualified to be new two-fold complex
integration transformations about the Wigner operator $\Delta \left(  \mu
,\nu \right)  $ (in its entangled form) in phase space quantum mechanics.

Further, multiplying both sides of (\ref{e5}) from the left by $\int
\frac{d^{2}\mu d^{2}\nu}{\pi^{2}}D\left(  \nu,\mu \right)  $ and considering
\begin{equation}
Q_{j}=\frac{a_{j}+a_{j}^{^{\dagger}}}{\sqrt{2}},\text{ }P_{j}=\frac
{a_{j}-a_{j}^{^{\dagger}}}{i\sqrt{2}}, \label{b3}%
\end{equation}
we obtain%
\begin{align}
&  \int \frac{d^{2}\mu d^{2}\nu}{\pi^{2}}D\left(  \nu,\mu \right)  \Delta \left(
\mu,\nu \right) \nonumber \\
&  =\int \frac{d^{2}\xi d^{2}\eta}{\pi^{2}}\delta^{\left(  2\right)  }\left(
\eta-a_{1}+a_{2}^{\dagger}\right)  \delta^{\left(  2\right)  }\left(
\xi-a_{1}-a_{2}^{\dagger}\right)  \int \frac{d^{2}\mu d^{2}\nu}{\pi^{2}%
}D\left(  \nu,\mu \right)  e^{-\left(  \xi-\mu \right)  \left(  \eta^{\ast}%
-\nu^{\ast}\right)  +\left(  \eta-\nu \right)  \left(  \xi^{\ast}-\mu^{\ast
}\right)  }\nonumber \\
&  =\int \frac{d^{2}\xi d^{2}\eta}{\pi^{2}}\delta^{\left(  2\right)  }\left(
\eta-a_{1}+a_{2}^{\dagger}\right)  \delta^{\left(  2\right)  }\left(
\xi-a_{1}-a_{2}^{\dagger}\right)  \mathcal{F}(\eta,\xi)\nonumber \\
&  =\int \frac{d^{2}\xi d^{2}\eta}{\pi^{2}}\delta \left(  \eta_{1}-\frac
{Q_{1}-Q_{2}}{\sqrt{2}}\right)  \delta \left(  \eta_{2}-\frac{P_{1}+P_{2}%
}{\sqrt{2}}\right)  \delta \left(  \xi_{1}-\frac{Q_{1}+Q_{2}}{\sqrt{2}}\right)
\delta \left(  \xi_{2}-\frac{P_{1}-P_{2}}{\sqrt{2}}\right)  \mathcal{F}%
(\eta,\xi)\nonumber \\
&  =\mathcal{F}(\frac{Q_{1}-Q_{2}}{\sqrt{2}},\frac{P_{1}+P_{2}}{\sqrt{2}%
},\frac{Q_{1}+Q_{2}}{\sqrt{2}},\frac{P_{1}-P_{2}}{\sqrt{2}}), \label{e7}%
\end{align}
where we have introduced%
\begin{equation}
\mathcal{F}(\eta,\xi)\equiv \int \frac{d^{2}\mu d^{2}\nu}{\pi^{2}}D\left(
\nu,\mu \right)  e^{\left(  \xi^{\ast}-\mu^{\ast}\right)  \left(  \eta
-\nu \right)  -\left(  \eta^{\ast}-\nu^{\ast}\right)  \left(  \xi-\mu \right)
}, \label{e8}%
\end{equation}
which seems a new interesting transformation. Due to%
\begin{align}
&  \int \frac{d^{2}\xi d^{2}\eta}{\pi^{2}}\exp \left[  \left(  \xi-\mu \right)
\left(  \eta^{\ast}-\nu^{\ast}\right)  -\left(  \eta-\nu \right)  \left(
\xi^{\ast}-\mu^{\ast}\right)  \right] \nonumber \\
&  =\int d^{2}\xi \delta \left(  \xi-\mu \right)  \delta \left(  \xi^{\ast}%
-\mu^{\ast}\right)  e^{\nu \left(  \xi^{\ast}-\mu^{\ast}\right)  -\nu^{\ast
}\left(  \xi-\mu \right)  }=1, \label{e6}%
\end{align}
$e^{\left(  \xi-\mu \right)  \left(  \eta^{\ast}-\nu^{\ast}\right)  -\left(
\eta-\nu \right)  \left(  \xi^{\ast}-\mu^{\ast}\right)  }$ can be considered a
basis function in $\xi-\eta$ phase space, or Eq. (\ref{e4}) can be looked as
an expansion of $D\left(  \nu,\mu \right)  $ in terms of $e^{\left(  \xi
-\mu \right)  \left(  \eta^{\ast}-\nu^{\ast}\right)  -\left(  \eta-\nu \right)
\left(  \xi^{\ast}-\mu^{\ast}\right)  },$ with the expansion coefficient being
$\mathcal{F}(\eta,\xi)$.

We can prove that the inverse transform of (\ref{e8}) is
\begin{equation}
\int \frac{d^{2}\xi d^{2}\eta}{\pi^{2}}e^{\left(  \xi-\mu \right)  \left(
\eta^{\ast}-\nu^{\ast}\right)  -\left(  \eta-\nu \right)  \left(  \xi^{\ast
}-\mu^{\ast}\right)  }\mathcal{F}(\eta,\xi)\equiv D\left(  \nu,\mu \right)
.\label{e9}%
\end{equation}
In fact, substituting Eq. (\ref{e9}) into Eq. (\ref{e8}) yields%
\begin{align}
&  \int \frac{d^{2}\xi^{\prime}d^{2}\eta^{\prime}}{\pi^{2}}\mathcal{F}%
(\eta^{\prime},\xi^{\prime})\int \frac{d^{2}\mu d^{2}\nu}{\pi^{2}}e^{\left(
\xi^{\prime}-\mu \right)  \left(  \eta^{\prime \ast}-\nu^{\ast}\right)  -\left(
\eta^{\prime}-\nu \right)  \left(  \xi^{\prime \ast}-\mu^{\ast}\right)  +\left(
\xi^{\ast}-\mu^{\ast}\right)  \left(  \eta-\nu \right)  -\left(  \eta^{\ast
}-\nu^{\ast}\right)  \left(  \xi-\mu \right)  }\nonumber \\
&  =\int \frac{d^{2}\xi^{\prime}d^{2}\eta^{\prime}}{\pi^{2}}\mathcal{F}%
(\eta^{\prime},\xi^{\prime})e^{\left(  \xi^{\prime}\eta^{\prime \ast}%
-\eta^{\prime}\xi^{\prime \ast}+\xi^{\ast}\eta-\eta^{\ast}\xi \right)  }%
\int \frac{d^{2}\mu d^{2}\nu}{\pi^{2}}e^{\left(  \eta^{\ast}-\eta^{\prime \ast
}\right)  \mu+\left(  \eta^{\prime}-\eta \right)  \mu^{\ast}}e^{\left(
\xi^{\prime \ast}-\xi^{\ast}\right)  \nu+\left(  \xi-\xi^{\prime}\right)
\nu^{\ast}}\nonumber \\
&  =\int d^{2}\xi^{\prime}d^{2}\eta^{\prime}\mathcal{F}(\eta^{\prime}%
,\xi^{\prime})e^{\left(  \xi^{\prime}\eta^{\prime \ast}-\eta^{\prime}%
\xi^{\prime \ast}+\xi^{\ast}\eta-\eta^{\ast}\xi \right)  }\delta^{\left(
2\right)  }\left(  \eta^{\prime}-\eta \right)  \delta^{\left(  2\right)
}\delta \left(  \xi-\xi^{\prime}\right)  \nonumber \\
&  =\mathcal{F}(\eta,\xi).\label{e10}%
\end{align}
Additionally, we can prove
\begin{align}
&  \int \frac{d^{2}\xi d^{2}\eta}{\pi^{2}}|\mathcal{F}(\eta,\xi)|^{2}%
\nonumber \\
&  =\int \frac{d^{2}\mu d^{2}\nu}{\pi^{2}}|D\left(  \nu,\mu \right)  |^{2}%
\int \frac{d^{2}\mu^{\prime}d^{2}\nu^{\prime}}{\pi^{2}}\exp \left[  \left(
\mu^{\ast}\nu-\nu^{\ast}\mu \right)  +\left(  \mu^{\prime}\nu^{\prime \ast}%
-\nu^{\prime}\mu^{\prime \ast}\right)  \right]  \nonumber \\
&  \times \int \frac{d^{2}\xi d^{2}\eta}{\pi^{2}}\exp \left[  \left(  \mu
^{\prime \ast}-\mu^{\ast}\right)  \eta+\left(  \mu-\mu^{\prime}\right)
\eta^{\ast}+(\nu^{\ast}-\nu^{\prime \ast})\xi+(\nu^{\prime}-\nu)\xi^{\ast
}\right]  \nonumber \\
&  =\int \frac{d^{2}\mu d^{2}\nu}{\pi^{2}}|D\left(  \nu,\mu \right)  |^{2}\int
d^{2}\mu^{\prime}d^{2}\nu^{\prime}\exp \left[  \left(  \mu^{\ast}\nu-\nu^{\ast
}\mu \right)  +\left(  \mu^{\prime}\nu^{\prime \ast}-\nu^{\prime}\mu^{\prime
\ast}\right)  \right]  \delta^{\left(  2\right)  }\left(  \mu-\mu^{\prime
}\right)  \delta^{\left(  2\right)  }\delta(\nu^{\prime}-\nu)\nonumber \\
&  =\int \frac{d^{2}\mu d^{2}\nu}{\pi^{2}}|D\left(  \nu,\mu \right)
|^{2},\label{e12}%
\end{align}
which shows that the transformations of Eqs.(\ref{e8}) and (\ref{e9}) obey the
Parseval-like theorem as well.

\section{Application}

By using the above transformation we can solve some operator ordering problems
arising from Eq.(\ref{c9}), for example, we want to put $\left(
a_{1}^{\dagger}-a_{2}\right)  ^{n}\left(  a_{1}+a_{2}^{\dagger}\right)  ^{m}$
into its Weyl ordered form. According to Eqs.(\ref{b6-1}), (\ref{b10-1}) and
(\ref{b11}), we have%
\begin{equation}
\frac{1}{2}\int \frac{d^{2}\eta d^{2}\xi}{\pi^{2}}\left \vert \eta \right \rangle
\left \langle \xi \right \vert e^{\left(  \eta^{\ast}\xi-\eta \xi^{\ast}\right)
/2}=1 \label{g1}%
\end{equation}
and
\begin{equation}
\frac{1}{2}\int \frac{d^{2}\eta d^{2}\xi}{\pi^{2}}\left \vert \xi \right \rangle
\left \langle \eta \right \vert e^{\left(  \eta \xi^{\ast}-\eta^{\ast}\xi \right)
/2}=1. \label{g2}%
\end{equation}
As a result of Eqs.(\ref{g1}) and (\ref{c7}), using the eigenvector equations
Eqs.(\ref{b2}) and (\ref{b8}), we obtain \bigskip%
\begin{align}
&  \left(  a_{1}^{\dagger}-a_{2}\right)  ^{n}\left(  a_{1}+a_{2}^{\dagger
}\right)  ^{m}\nonumber \\
&  =\frac{1}{2}\int \frac{d^{2}\eta d^{2}\xi}{\pi^{2}}\eta^{\ast n}\xi
^{m}\left \vert \eta \right \rangle \left \langle \xi \right \vert e^{\left(
\eta^{\ast}\xi-\eta \xi^{\ast}\right)  /2}\nonumber \\
&  =\int \frac{d^{2}\eta d^{2}\xi}{\pi^{2}}\eta^{\ast n}\xi^{m}%
\genfrac{}{}{0pt}{}{:}{:}%
\exp[\left(  \xi-a_{1}-a_{2}^{\dagger}\right)  \left(  \eta^{\ast}-a_{1}%
^{\dag}+a_{2}\right)  -\left(  \eta-a_{1}+a_{2}^{\dagger}\right)  \left(
\xi^{\ast}-a_{1}^{\dag}-a_{2}\right)  ]%
\genfrac{}{}{0pt}{}{:}{:}%
\nonumber \\
&  =\int \frac{d^{2}\eta d^{2}\xi}{\pi^{2}}\sum_{k=0}^{n}\sum_{l=0}^{m}%
\frac{i^{n+m}m!n!}{\left(  n-k\right)  !k!\left(  m-l\right)  !l!}\eta
_{1}^{n-k}\left(  -\eta_{2}\right)  ^{n}\xi_{1}^{m-l}\xi_{2}^{m}\nonumber \\
&  \times%
\genfrac{}{}{0pt}{}{:}{:}%
\exp \left \{  \left[  -2i\eta_{2}\allowbreak+\left(  a_{1}+a_{2}-a_{2}%
^{\dagger}-a_{1}^{\dag}\right)  \right]  \left[  \xi_{1}-\frac{1}{2}\left(
a_{2}^{\dagger}+a_{1}^{\dag}+a_{1}+a_{2}\right)  \right]  \right. \nonumber \\
&  \left.  +\left[  2i\allowbreak \eta_{1}+i\left(  a_{2}^{\dagger}-\allowbreak
a_{1}^{\dag}-a_{1}+a_{2}\right)  \right]  \left[  \xi_{2}-\frac{1}{2}i\left(
a_{1}^{\dag}-a_{1}+\allowbreak a_{2}-a_{2}^{\dagger}\right)  \right]
\right \}
\genfrac{}{}{0pt}{}{:}{:}%
\nonumber \\
&  =-\sum_{k=0}^{n}\sum_{l=0}^{m}\frac{i^{n+m}m!n!}{\left(  n-k\right)
!k!\left(  m-l\right)  !l!}\nonumber \\
&  \times \int \frac{d\left(  -\eta_{2}\right)  d\xi_{1}}{\pi}\left(  -\eta
_{2}\right)  ^{n}\xi_{1}^{m-l}%
\genfrac{}{}{0pt}{}{:}{:}%
\exp \left \{  2i\left[  -\eta_{2}\allowbreak+\frac{1}{\sqrt{2}}\left(
P_{1}+P_{2}\right)  \right]  \left[  \xi_{1}-\frac{1}{\sqrt{2}}\left(
Q_{1}+Q_{2}\right)  \right]  \right \} \nonumber \\
&  \times \int \frac{d\eta_{1}d\xi_{2}}{\pi}\eta_{1}^{n-k}\xi_{2}^{m}%
\exp \left \{  2i\allowbreak \left[  \eta_{1}-\frac{1}{\sqrt{2}}\left(
Q_{1}-Q_{2}\right)  \right]  \left[  \xi_{2}-\frac{1}{\sqrt{2}}\left(
P_{1}-P_{2}\right)  \right]  \right \}
\genfrac{}{}{0pt}{}{:}{:}%
\nonumber \\
&  =-\left(  \frac{1}{2}\right)  ^{n+m}\sum_{k=0}^{n}\sum_{l=0}^{m}%
\frac{m!n!\left(  \sqrt{2}\right)  ^{k+l}}{\left(  n-k\right)  !k!\left(
m-l\right)  !l!}\nonumber \\
&  \times%
\genfrac{}{}{0pt}{}{:}{:}%
H_{n-k,m}\left[  \left(  Q_{1}-Q_{2}\right)  ,i\left(  P_{1}-P_{2}\right)
\right]  H_{m-l,n}\left[  i\left(  Q_{1}+Q_{2}\right)  ,-\left(  P_{1}%
+P_{2}\right)  \right]
\genfrac{}{}{0pt}{}{:}{:}%
, \label{g3}%
\end{align}
where we have used the following integration formula\cite{r13}%
\begin{equation}
\int \frac{dxdy}{\pi}x^{m}y^{r}\exp[2i\left(  y-s\right)  \left(  x-t\right)
]=\left(  \frac{1}{\sqrt{2}}\right)  ^{m+r}\left(  -i\right)  ^{r}%
H_{m,r}\left(  \sqrt{2}t,i\sqrt{2}s\right)  , \label{g4}%
\end{equation}
with $H_{m,r\text{ }}$is the two-variable Hermite polynomials \cite{r14},%
\begin{equation}
H_{m,r}(t,s)=\sum_{l=0}^{\min(m,r)}\frac{m!r!(-1)^{l}}{l!(m-l)!(r-l)!}%
t^{m-l}s^{r-l}. \label{g5}%
\end{equation}
Eq.(\ref{g3}) is a simple approach to turn $\left(  a_{1}^{\dagger}%
-a_{2}\right)  ^{n}\left(  a_{1}+a_{2}^{\dagger}\right)  ^{m}$ into its Weyl
ordering. Similarly, using Eqs.(\ref{c11}) and (\ref{g2}), we see that the
Weyl ordered form of $\left(  a_{1}+a_{2}^{\dagger}\right)  ^{m}\left(
a_{1}^{\dagger}-a_{2}\right)  ^{n}$ is
\begin{align}
&  \left(  a_{1}+a_{2}^{\dagger}\right)  ^{m}\left(  a_{1}^{\dagger}%
-a_{2}\right)  ^{n}\nonumber \\
&  =-\left(  \frac{1}{2}\right)  ^{m+n}\sum_{k=0}^{n}\sum_{l=0}^{m}%
\frac{\left(  -1\right)  ^{l}m!n!\left(  \sqrt{2}\right)  ^{k+l}}{\left(
n-k\right)  !k!\left(  m-l\right)  !l!}\nonumber \\
&  \times%
\genfrac{}{}{0pt}{}{:}{:}%
H_{n,m}\left[  \left(  Q_{2}-Q_{1}\right)  ,i\left(  P_{1}-P_{2}\right)
\right]  H_{m-k,n-l}\left[  \left(  Q_{1}+Q_{2}\right)  ,i\left(  P_{1}%
+P_{2}\right)  \right]
\genfrac{}{}{0pt}{}{:}{:}%
. \label{g6}%
\end{align}

In summary, as a natural extension of our preceding paper\cite{r7},
through employing operators' Weyl ordering expansion the formula we
have found a new two-fold complex integration transformation about
the Wigner operator $\Delta \left(  \mu,\nu \right)  $ (in its
entangled form) in phase space quantum mechanics, which is
invertible and obeys Parseval theorem. The new complex integration
transformation is compatible to $\eta-\xi$ phase space quantum
mechanics which takes entangled state representations as basis.
Under this transformation the relationship between the Weyl ordering
and $\left( a_{1}^{\dagger}-a_{2}\right)  \Leftrightarrow \left(
a_{1}+a_{2}^{\dagger }\right)  $ ordering or $\left(
a_{1}+a_{2}^{\dagger}\right)  \Leftrightarrow \left(
a_{1}^{\dagger}-a_{2}\right)  $ ordering of operators is revealed.
In this way, the contents of phase space quantum mechanics\cite{r5}
can be further enriched. Studies about the multi-mode case are in
progress and new results will be reported later.

\bigskip

redpajama_set_name: RedPajamaArXiv


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text:
#pragma once

#include "il2cpp-config.h"

#ifndef _MSC_VER
# include <alloca.h>
#else
# include <malloc.h>
#endif

#include <stdint.h>

// System.String
struct String_t;
// Mono.Security.ASN1
struct ASN1_t3752917377;

#include "mscorlib_System_Object4170816371.h"

#ifdef __clang__
#pragma clang diagnostic push
#pragma clang diagnostic ignored "-Winvalid-offsetof"
#pragma clang diagnostic ignored "-Wunused-variable"
#endif

// Mono.Security.PKCS7/ContentInfo
struct  ContentInfo_t3081670593  : public Il2CppObject
{
public:
	// System.String Mono.Security.PKCS7/ContentInfo::contentType
	String_t* ___contentType_0;
	// Mono.Security.ASN1 Mono.Security.PKCS7/ContentInfo::content
	ASN1_t3752917377 * ___content_1;

public:
	inline static int32_t get_offset_of_contentType_0() { return static_cast<int32_t>(offsetof(ContentInfo_t3081670593, ___contentType_0)); }
	inline String_t* get_contentType_0() const { return ___contentType_0; }
	inline String_t** get_address_of_contentType_0() { return &___contentType_0; }
	inline void set_contentType_0(String_t* value)
	{
		___contentType_0 = value;
		Il2CppCodeGenWriteBarrier(&___contentType_0, value);
	}

	inline static int32_t get_offset_of_content_1() { return static_cast<int32_t>(offsetof(ContentInfo_t3081670593, ___content_1)); }
	inline ASN1_t3752917377 * get_content_1() const { return ___content_1; }
	inline ASN1_t3752917377 ** get_address_of_content_1() { return &___content_1; }
	inline void set_content_1(ASN1_t3752917377 * value)
	{
		___content_1 = value;
		Il2CppCodeGenWriteBarrier(&___content_1, value);
	}
};

#ifdef __clang__
#pragma clang diagnostic pop
#endif

redpajama_set_name: RedPajamaGithub


text:
Israeli Troops Mount New Raid
Wednesday, February 28th 2007, 5:40 am
NABLUS, West Bank (AP) _ Israel on Wednesday clamped down on this West Bank city for the second time in a week, confining tens of thousands of people to their homes as troops moved house to house in search of wanted militants.
In nearby Jenin, undercover troops in a black car shot and killed three militants in a parking lot, including the chief spokesman for the Islamic Jihad group, Palestinian officials said.
The army has been operating in Nablus, the West Bank's commercial center, since the weekend. Forces briefly withdrew Tuesday, but at dawn, dozens of jeeps backed by bulldozers moved back in.
Nablus, the largest city on the West Bank, with 160,000 residents, is known as a hotbed of Palestinian militant activity. The Israeli army says most of the suicide bombers involved in last year's attacks came from the area. The army's operation in Nablus is the largest in the West Bank since July.
On Wednesday, Israeli forces blocked all entrances to Nablus' crowded Old City _ the focus of the raid _ as they pursued wanted men. Residents said more than 100 jeeps patrolled city streets, enforcing a curfew that confined about 50,000 Old City residents to their homes.
As troops moved house to house, the army broadcast messages on Palestinian TV and radio stations urging wanted men to surrender.
The army said it has arrested five people and uncovered three explosives labs. One Palestinian civilian has been killed since the Nablus operation began.
The wanted militants remained in hiding, but Palestinian officials said 10 people were wounded as troops clashed with youths throwing stones. Fifty people were detained, although many were released, the officials said, and troops surrounded hospitals to check people going in and out.
``We can't continue to operate like this. Our patients are suffering because the tear gas is coming into the hospital,'' said Adnan Kuzeih, manager at Itihad Hospital.
The army said one soldier was seriously wounded by a bomb.
In the Gaza Strip, Palestinian Prime Minister Ismail Haniyeh of Hamas accused Israel of trying to undermine Palestinian efforts to form a unity government. Hamas and the rival Fatah movement are trying to finalize a power-sharing deal reached in Saudi Arabia earlier this month, designed to halt months of infighting.
``It's clear that this Israeli escalation ... is tied directly to Israel's rejection of the Mecca agreement, which strengthened national unity,'' Haniyeh said.
The international community has been reticent about recognizing the new government. The platform falls short of international demands that Hamas renounce violence and recognize Israel's right to exist.
But Palestinian President Mahmoud Abbas, a moderate from Fatah, has been trying to coax the West into accepting the deal, saying it is the best he can get from the violently anti-Israel Hamas.
Visiting European officials said that divisions are beginning to emerge between the European Union and the United States.
The EU external affairs commissioner, Benita Ferrero-Waldner, held talks with Palestinian officials. ``Nobody yet has a clue what the program of the government will look like,'' she said.
EU diplomats acknowledged the Europeans might be willing to engage ``with some elements'' of the coalition government, adding such a selective approach may not be shared by the U.S.
During a recent trip to the region, Secretary of State Condoleezza Rice said Washington expects the new government to accept the international principles on dealing with Israel.
Israeli Prime Minister Ehud Olmert has said he will not deal with the new government if it doesn't accept the international conditions, although he would still be willing to talk to Abbas.
In Jenin, Israeli undercover troops in a black car fired at a vehicle carrying three Islamic Jihad gunmen, witnesses said. The army confirmed its troops killed the men, saying they had helped to plan an attempted suicide bombing in Tel Aviv that was thwarted last week.
Two of the militants were killed instantly, and the third, Ashraf Saadi, was shot and killed after he scrambled out of the car and tried to escape, witnesses said. Saadi was the spokesman for Islamic Jihad, a militant group backed by Iran and Syria, and responsible for dozens of suicide bombings and rocket attacks against Israel.
The army said the troops tried to arrest Saadi but he opened fire on them. Troops returned fire, killing two militants and injuring Saadi. After Saadi fled, continuing to fire, the troops shot him dead, the army said. One soldier was slightly wounded by gunfire.
In the Gaza Strip, Islamic Jihad spokesman Abu Ahmad vowed retaliation. Hamas also swore revenge.
News On 6 NABLUS, West Bank (AP) _ Israel on Wednesday clamped down on this West Bank city for the second time in a week, confining tens of thousands of people to their homes as troops moved house to house in search

redpajama_set_name: RedPajamaCommonCrawl